CIRCULAR HARMONIC SYSTEM - CHAS

Posted by: alfredo capurso

CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 04:17 AM

I would like to make known a discovery that may be of interest to you.

My work as a piano tuner technician over the last 30 years has enabled me to carry out research on sound and beats, leading to the construction of a new model for temperament of the musical scale. The model, which I have called CHAS (Circular HArmonic System), overcomes the flaws inherent in equal temperament, and describes a powerful attractor, as well as a new concept of "purity".

The main results of the research are:

•a scale combining the prime numbers 2, 3 and 5, solving an age-old problem
•a self-contained physical and geometric entity, a perfectly symmetrical and synchronic attractor, determined by flows of beats
•an algorithm enabling construction of infinite microtonal scales and the rewriting of today’s musical scale
•an “s” variable that can produce infinite beats curves, calculate infinitesimal degrees of inharmonicity and predict developments within the system
•a new geometric average deriving from two proportional ratios: one linear and one exponential
•a resonance constant that may be adopted as a new reference standard in calculating the frequencies of partials
•two indications regarding the torsion of a plane and the helix

The system was presented in March 2007 at a conference in Messina, Italy: “Mandelbrot and Fractal Geometry Forty Years Later”. This February an article on CHAS was published by the Research Group for the Teaching and Learning of Mathematics of the University of Palermo (Italy), and can be accessed at: http://dipmat.math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

The CHAS model constitutes a key advance in music as well as in the physics of sound and of related phenomena.

In music, the model provides an algorithm enabling the construction of infinite microtonal scales. CHAS refines the equal semitone scale and solves the age-old problem of combining the prime numbers 2, 3 and 5, to deliver an extraordinarily euphonic and resonant set of sounds. All musical instruments may be tuned to this optimum scale. The system is accurate and natural because it draws on phenomena that are intrinsic to vibrating strings.

In physics and geometry, the CHAS model proves for the first time that flows of beats can determine a perfectly symmetrical and synchronic attractor in a dynamic system. Knowledge of the existence of this attractor may lead to further research into the nature of relationships between the energy of vibrating matter and the beats involved in resonance and interference phenomena. Inharmonicity, which has always been calculated in an approximate way, can now be calculated with infinitesimal accuracy.

I am seeking to promote wider understanding and application of the CHAS model. I would welcome any form of collaboration or support that you may be able to offer.

I hope you enjoy it. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 08:30 AM

Alfredo:

I will study the paper with great interest. I am particularly interested in how anything can be a "dynamic, stable and perfectly resonant system." as mentioned in the abstract. Also, I am wondering what a "synchronic attractor" might be.

The theory of piano tuning fascinates me, but lately I am realizing its usefulness is limited in aural tuning. Aural tuning is all about compromises, compromises that can be heard. They don’t need to be theorized to be heard, just listened to and accepted. I am thinking that the theory is really only necessary for designing a mathematical model so that ETDs can make the compromises without actually “hearing” them. And then there is the final limit on accuracy imposed by the pinblock and rendering points. Not to mention what the next passing thunderstorm may do to a tuning!

Thanks for posting!
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 10:27 AM

Alfredo:

I have not finished reading your paper, but so that you have something to consider, check out This Topic . It is about what is called "Mindless Octaves" and may be what your algorithm "(3 − Δ)^ (1/19) = (4 + Δ)^ (1/ 24)" describes.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 12:25 PM

Alfredo:

I still have not finished reading your paper, and am having a harder and harder time understanding it. Perhaps there is a translation problem. But there is also a math error.

In paragraph 3.3 you state:

(3 − Δ)^ (1/19) = (4 + (Δ * s / s1))^ (1/ 24)
equals:
(3 − (Δ * s1))^ (1/19) = (4 + (Δ * s))^ (1/ 24)

This is no more true than stating:

4^ (1/2) = (4 * (4/2))^ (1/3)
equals:
(4*2)^ (1/2) = (4 * 4)^ (1/3)

Also, you make a big point of the numbers of 2, 3 and 5 being prime numbers, which is true, but do not explain why that is a problem. It is a moot point anyway, because 2, 3 and 5 are the numbers of partials, but are not frequency multipliers due to inharmonicity.

I’ll try to finish the paper and get what I can out of it.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 12:45 PM

Well observed Jeff.
Maybe a kind of scientific hoax of the category "cello scrotum":
http://www.timesonline.co.uk/tol/life_and_style/health/article5601050.ece
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 01:25 PM

Bernhard:

Still waiting for your paper.... smile
Posted by: bobrunyan

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 04:23 PM

Jeff,

I'm having a hard time understanding the document as well, but the math error that you point out is not necessarily an error. He doesn't say that the two equations are always equal but only "if s is a fraction (s/s1)" and that "the denominator multiplies delta in the left-hand expression so that:" the two equations are equal.

Bob
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/08/09 07:08 AM

Bob:

Not sure if I follow what you are saying, but I wondered also how "s" could be a fraction "s/s1" unless "s1" equals 1, in which case the equations would be equal, but why bother? I don't know if Alfredo will respond back or not. But, I don't understand what practical use his paper may have, let alone what he is saying.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/08/09 07:30 AM

Thanks for your interest. No need to rush, better you finish reading the paper, then we'll talk about anything you like. a.c.
Posted by: David Jenson

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/08/09 07:34 AM

Originally Posted By: UnrightTooner
Alfredo:

Aural tuning is all about compromises, compromises that can be heard. They don’t need to be theorized to be heard, just listened to and accepted.

I agree. Keep your ears in the mix and listen. In the day-to-day work of tuning and repairing everything from clunkers to concert instruments I find math theory unnecessary and distracting.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/08/09 07:50 AM

Alfredo:

I cannot understand your paper. Can you explain the basic concepts?
Posted by: RPD

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/08/09 10:13 AM

I'm always grateful for those who are willing to dive deeply into the math and try new things...its interesting to me, but not terribly relevant immediately to my work as a tuner. Those who are willing to take the time to try new processes and write on the results are, I think, bettering the profession and I thank them/you all!

For me though, I tend to stay with simple, tried methods that don't require my time in research...something there is precious little of in the crazy schedule.

RPD
Posted by: Jeff A. Smith, RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/10/09 09:03 PM

Hi Alfredo,

I think most of us would be interested in knowing if your theories have changed the way you actually tune a piano, and if what you're trying to achieve in the process has changed at all.

Are there any problems in piano tuning and its practical musical application that your theories seek to remedy?

Jeff S.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/11/09 07:35 AM

Alfredo hasn’t elaborated on his paper. This Topic may die soon. Let me toss out something from the paper to see if anything productive comes from it.

The use of a Chas ratio of 2.0005312… is mentioned rather than the theoretic 2:1 octave ration. But should a fixed ratio be used in tuning at all? And if a fixed octave ratio, and therefore a fixed ratio for each interval, is used what are the results?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/11/09 01:28 PM

Tooner:

About theory and tuning - you may ignore theory and tune aurally in a casual or personal way, or use an ETD without careing what’s behind it. Maybe a simple question of knowledge and consciousness that is up to you.

About tuning and compromise - untill today we (aural tuners) could only think in terms of compromise becouse we had to get by with Equal Temperament and its unjustified premises, two unjustified assumptions that Chas model discards (section 3.0). Chas demontrates that the ratio 12th root of 2 is unsuitable, not only becouse of inharmonicity, but becouse it produces intervals incresingly narrow (12ths, 19ths and so on) together with intervals incresingly wide (10ths, 17ths ecc. - section 4.3 - graph 5). E.T. premises come out to be missleading.

Also the idea of “pure interval” is missleading. For example, the theoretical ratio 19th root of 3 (i.e. pure 12ths) is unprofitable, being an extreme case. Infact 19th root of 3 widens 3ths, octaves, 10ths, 17ths ecc. more than necessary and spoiles the symmetry of beats (sections 3.4 – 3.5).

About mindless octaves . I’ve had a look at the topic, then I visited Bill Bremmen’s site. I can not say whether he tunes Chas or not. Maybe he can tell you/us.

About Chas model’s relevance - Chas describes the precise form of a dynamic set. In section 2.0 you can read:
“Purity no longer derives from a single combination (refering to pure octaves) or from a pure ratio (refering to 2:1, 3:1, 5:1), but from a new set which is pure because it is perfectly congruent and coherent”.

In section 3.0:
“In this set the ratio must be identifiable both in the single elements (frequencies) forming the scale foreground, as well as in the differences (beats) arising from the infinite combinations of its elements, and forming the background. Each frequency or element in the scale must contain and bear witness to this bi-frontal ratio, which is pure in that it is natural, exactly proportional and perfectly synchronic”.

Chas ratio (1.0594865443501…) does not produce a pure interval nor a compromise, it leads to a pure set, pure in terms of beats and frequencies proportions. Tuning Chas you do not go for a compromise, you go for an optimum scale.

Stopper:

I’m sorry, I was not interested in your “cello scrotum”.
I understand you are commercializing an ETD device, at a cost of $ 600. Could I know on wich basis? Does reading about scrotum help?

Bob:

Thanks for your post.

RPD:

Thanks for your encouragement.

Jeff S.:

About problems in piano tuning - mainly, wrong teachings. For example, still today you learn 5ths must be narrow, Chas proves that this is not correct. In section 4.8 – graph 10, Chas model shows how and why the difference curve for ratio 3:2 (as for ratio 9:8) inverts its progression. From the middle-high register upwards, 5ths go purer and purer, to become wider and wider. The inversion of 5ths allow you to widen octaves in the correct mesure. You can immagine how Chas has changed the way I tune pianos.

Tooner:

what do you mean when you say: Alfredo hasn’t elaborated on his paper. This Topic may die soon.?

A fixed and correct ratio, as a standard reference, is more than productive, is the most precious figure that you could ever wish to find. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/11/09 02:27 PM

Alfredo:

I am glad you have now elaborated, and the Topic has not died.

You stated: “Chas ratio (1.0594865443501…) does not produce a pure interval nor a compromise, it leads to a pure set, pure in terms of beats and frequencies proportions. Tuning Chas you do not go for a compromise, you go for an optimum scale.”

I am gong to play the devil’s advocate to try to get to your basic concept.

The Chas ratio for an octave is only about 5/10,000 larger than 2. The difference between A5 being tuned to a theoretical 2:1 octave above A4 (A440) and a Chas octave is about ¼ Hz or about 1/2 cent. This is less stretch than you would have when tuning a 2:1 octave and taking into account inharmonicity. And speaking of inharmonicity, I have looked for how Chas applies a piano’s iH and have not found it. So, I have to assume that the Chas ratio is used only on theoretical tones and not tones with iH. And yet, one of your criticisms of ET is that it does not account for iH, but neither does Chas, so how is Chas superior? (Please do not take offense, as I said I am playing the devil’s advocate.)

So, how do you actually use Chas to tune?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/12/09 12:09 PM

Originally Posted By: alfredo capurso
…..

Jeff S.:

About problems in piano tuning - mainly, wrong teachings. For example, still today you learn 5ths must be narrow, Chas proves that this is not correct. In section 4.8 – graph 10, Chas model shows how and why the difference curve for ratio 3:2 (as for ratio 9:8) inverts its progression. From the middle-high register upwards, 5ths go purer and purer, to become wider and wider. The inversion of 5ths allow you to widen octaves in the correct mesure. You can immagine how Chas has changed the way I tune pianos.

…..


Alfredo:

I know that you wrote the above in response to Jeff’s post, but since you made it public I hope you don’t mind me commenting on it.

The idea of fifths becoming wide fascinates me and so I worked out the math to see what the result would be with the Chas ratio.

F7 (note 81) is 32 semi-tones above A4/A440 (note 49) and C8 is 39 semi-tones above A4/A440 (note 49). So if we take 440Hz and multiply it by the Chas ratio to the power of the number of semi-tones and multiply that times the partial number we will arrive at the frequency of the partials in question.

440 * 1.0594865443501^32 * 3 = 8387.4
440 * 1.0594865443501^39 * 2 = 8379.2

Since the third partial of F7 is higher than the second partial of C8, even the highest fifth in the Chas system is still a narrow interval.
Posted by: Jeff A. Smith, RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/12/09 01:52 PM

I'll let Jeff D. discuss the math with you, Alfredo.

For my part, I can say that the old teachings stated the beat rates of fifths should increase as one tuned up the scale. That model was based largely on a decades-old mathematical model, worked out before inharmonicity was understood.

Current understanding leans more toward the idea that the beat rates of fifths should either stay roughly the same as one moves up the scale or decrease, until the fifths become pure and possibly even wide (like you yourself seem to be saying).

My point is that, while your mathematic approach to this issue may be unique, the practical way fifths are understood and tuned -- at least here in America, among those in touch with current thought -- has already changed away from the old model. (My opinion and Jeff D.'s may not be quite the same on all aspects of this issue.)

Anyway, Alfredo, thanks for sharing and good luck.

Jeff S.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/12/09 02:42 PM

J. Smith:

Actually, I think I now understand how fifths could become wide and could probably come up with some math based scenarios incorporating iH to prove it. But I am not so sure that a fixed frequency ratio will cause this to happen, or even sticking to a specific interval type (which I don’t believe will produce a fixed ratio). For instance, if pure fifths are tuned, then of course the fifths never will become wide, even though this is a great deal of stretch and requires about an 8:4 octave in the temperament.

But what I would really like is for someone that says that fifths become wide to explain why and give an example. But something better than the usual ketchup-on-everything, because-of-inharmonicity platitude.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/12/09 04:38 PM

Thanks for bringing up my name. I was ill for 3 weeks in April and my computer also died. I finally got things going again.

I must first say that even though I have a university degree and some grad school, I studied music and foreign languages. I never had any math beyond 9th grade Algebra and 10th grade Geometry. However, I still do retain the basic concepts that I learned. Since I was a liberal arts student at the university, the entrance tests I took revealed that I needed no more math or science, so I never studied any.

As a piano technician, I have viewed the work I do as more of a mechanical skill than anything else. Yes, my musical training was helpful but there are excellent piano technicians who are not musicians at all and who also do not know any higher math.

I would say that I have often observed the very finest piano technicians who could not put into words or describe in writing just how they do what they do. They will tell you that tuning is an art and that excellence is achieved by practice and perseverance. Few of us would argue that.

What I know how to do has been the result of reading, watching and listening to many diverse sources. I have taken ideas from here and there and put them together to formulate the set of knowledge and skills I possess today.

The idea which I called "mindless octaves" is a very simple one indeed. At one PTG Convention, I observed Steve Fairchild RPT demonstrate it but I really didn't understand what he was doing at the time. Only later did I realize that the concept I had hit on purely by trial and error was what Steve had been doing.

I started tuning aurally in 1968 using a C Fork and a Braide White type of 4ths and 5ths sequence. The manual I had said that octaves were to be tuned "pure". The exception was that from on or about C6 to C8, about 2 beats per second should be put in each octave and that was what was called "stretching the octaves.

Obviously, the information I had was crude but from my perspective today, there have been countless others who have basically learned the same concepts. So, I too am interested when anyone can explore and discover any more about tuning than is generally known by most piano technicians. ETDs have been a great help in providing a tool for technicians to tune better than they could using aural skills but more and more people start out using them and know virtually nothing about how they work. They may know of inharmonicity but they rely on the device to solve the problem for them and don't have any idea whether the solution provided could be made better or not.

In 1985, I first heard of the concept of unequal temperament. I attended a demonstration by Professor Owen Jorgensen RPT. I was not at all influenced or persuaded to try any of that at the time. However, about 4 years later, I heard a pianist playing Brahms on a piano at a dealer while I was working on a piano in the workshop. Again and again, I heard beauty I had never experienced before. The experienced convinced me of the fact that there was something else to be tried. Once I did, I was hooked.

The "mindless octaves" idea is nothing more than making an exact compromise between the double octave and the octave and 5th (12th). I use the sostenuto pedal to do it. When I used the idea, I heard clarity and beauty that I had never heard before and the feedback from customers kept me at it. To me, it was such a simple technique that it became habitual, within my muscle memory to perform, I didn't really have to concentrate when doing it, I could be thinking about something else, it was just mindless yet the technique yielded such consistent and perfected results each and every time. I later discovered that the aural technique could be as accurate and consistent as any ETD would provide. I use it today when I set up a custom ETD program and when I conduct a PTG Exam Master Tuning.

The EBVT and its variants developed also entirely by ear. It was difficult for me to find a way to describe it in writing but I knew what I wanted to hear. I had rejected the idea of Equal Temperament (ET) 9 years before I started working on the EBVT. Basically the idea is that yes, ET provides for an absolute compromise that divides and distributes the comma equally and in small increments between all 24 Major and minor keys.

The problem with it is that as a compromise, it actually goes too far because it eliminates any distinction from one key or tonality to another. It ignores the idea that as musicians, we expect each key signature that we use to have its own purpose or distinct quality.

The modern piano is meant to play all kinds of music from all periods in all possible keys signatures. ET certainly allows for that but erases the color. The goal then is to find a compromise that retains a distinct character for each key but does not produce harshness that the contemporary sensibility (ear) cannot accept.

I became well aware that even though most piano technicians thing in terms of and firmly believe in ET and most often know of nothing other that ET, most aural tuners tune it imperfectly. That means therefore that everyone has a tolerance for deviation from ET that is acceptable.

The goal then was to work within that range of tolerance to create a mild version of a Well-Temperament, most often called Victorian style. Create a sequence from A within the F3 to F4 octave that most technicians today find familiar. Make it simple and easy to remember. I found that the Equal Beating concept helped with that: simplicity and the ability to replicate the idea accurately time and again. I also found a serendipitous bonus to that: Equal Beating M3s and M6s as well as other intervals have an uncanny ability to cancel themselves out. A triad with equal beating intervals can sound much "purer" than it really is. Intervals which might be considered too harsh out of context, get "swallowed" in the sound as a whole.

Thus the EBVT and the "mindless octaves" (which is also an equal beating concept) provide for an overall sound that is far more appealing than the most perfected ET can ever hope to be. It has this "crystal clear" sound to it. The "pipe organ" effect is another manifestation of it.

Of course, how good it sounds is my opinion. But it is the only way I have tuned most pianos (except for an occasional other kind of non ET) since I first began working with it in 1992. In my community, there are many fine technicians. Our PTG chapter has 21 members and 18 of them are RPTs. Several RPTs live within a 10 mile radius of me. So, people have their choice of whom to call and the customers of my local colleagues have their reasons for their loyalty. Having said that, I find time and again, year after year, the customers whose pianos I tune tell me that I have made the piano sound so much better than anyone else ever did. They offer descriptions such as "more musical", clarity, brilliance, pipe organ, and on and on.

It is the result of not accepting what I first leaned 40 years ago as the truth and whole truth and never seeking anything beyond it. It is the result of constantly looking for something better, a refinement of technique, applying other ideas, seeking limits, making something new and different from a combination of ideas and techniques.

So, while I have absolutely no idea of what Alfredo is talking about at this point, I say, go for it, I may find something I like after all and it could end up being a way to quantify and describe what could only have been called "art" or "instinct" in the past.

I must get back to work now. Cheers.
Posted by: Robert Scott

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/12/09 04:44 PM

Jeff,

This example may be able to help. I ran the following experiment using TuneLab with some artificially entered (but still possible) inharmonicity readings. The made-up IH numbers are: C1: 0.2 C3: 0.05 C4: 0.2 C6: 2.5 Four readings is the absolute minimum number to define an IH model in TuneLab. I then let TuneLab auto-adjust a tuning curve using 6:3 octaves in the low bass and 2:1 octaves in the high treble. This resulted in a stretch of +37 cents at C8. High, but still within reason. Then I temporarily switched the treble interval to the 3:2 fifth to see what the fifths would be like. It turns out that with these settings, the 3:2 fifths transition between narrow and wide at about G7, ending up at about 1.2 cents wide at C8 (F7-C8).

This behavior is not typical. I deliberately chose the IH numbers to make the model think that the IH was increasing rapidly as you move up the scale. This rate of increase and not the absolute IH, it turns out, is the critical factor in determining how octaves and fifths compare. Normally 3:2 fifths are narrower than 2:1 octaves. But with a sufficiently high rate of increase of IH, striving to make 2:1 octaves beatless can make fifths wide.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/13/09 08:22 AM

Mr. Scott:

Thank you so very much for giving an example and explanation. I guess I had worried about this because I was thinking that if my fifths didn’t become wide, I wasn’t tuning “correctly”. But since this happens only in the very high treble, due to a greater slope of the iH curve, then fifths becoming wide is an inherent anomaly of some pianos, not the result of a tuning style.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/13/09 09:29 AM

Bill:

Welcome back and am glad to hear that both you and your computer are doing better. I hope neither of you had the Swine Flu Virus. smile
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/15/09 06:27 AM

Tooner:

In my opinion, it is very important to distinguish theory from practice. If you do not, thinks will not work. For example, this is way the numbers you have used to find pure fifths in Chas could not work.

Jeff S. wrote "I think most of us would be interested in knowing if your theories have changed the way you actually tune a piano...". So I answered on the practical side of the matter.

Theory can only be singular, ways to get to the one theory could be many. So, generally speacking, the practical way to tune Chas requires that fifths go the way I said, but only to counterbalance string lenthening, and the bridge and the harmonic board's adjustement, and only if the piano you are tuning were flat.

Going back to theory, I'll add more this afternoon.

Bill Bremmen:

Thanks for your post. When you say "The "mindless octaves" idea is nothing more than making an exact compromise between the double octave and the octave and 5th (12th).", it makes me think we have had the same experience and we are supporting the very same euphonic set of sounds. Have you made any progress from the practical ground to theory? We could compare figures and get a more precise idea. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/15/09 10:07 AM

Originally Posted By: alfredo capurso
Tooner:

In my opinion, it is very important to distinguish theory from practice. If you do not, thinks will not work. For example, this is way the numbers you have used to find pure fifths in Chas could not work.

.....


Yes I know it could not work. That is why I pointed it out, as the "Devil's Advocate", so that you now have the opportunity to present a more complete theory. I am not your enemy.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/15/09 03:31 PM


Tooner:

Do play the devil’s advocate and I’ll thank you for that, only I’d kindly ask you to read carefully Chas article, so that we don’t go in circles. Thank you very much.

You asked for Chas basic concepts. In the article you’ll find Chas basic concepts but, little by little, we can look at them together – In the abstract you read about the goals: Is E.T. improvable? Can we find a rule to manage string inharmonicity? Can we theorize stretched octaves?

Chas being a theory refers to E.T., i.e. our current international theoretical system. In section 2.0 you can read: “Thus two questions arise. The first: is it correct to theorise that the octave interval must have a 2:1 ratio? The second: which temperament model today is reliable in theoretical terms and is commonly applied in the practice of tuning?”

Answering these questions, Chas asserts that E.T. 2:1 theoretical ratio for the octave is a cultural/historical teaching, maybe deriving from the debatable idea that pure intervals sound better. A reliable model should be free of cultural or historical heritages.

For centuries we have calculated scale frequencies values giving the 2:1 ratio for granted. Today we stretch octaves, so “which temperament is commonly applied in the practice of tuning?”. We are not applying E.T., since it theorizes pure octaves.

Chas model theorizes stretched octaves and combines theoretical harmonic partials. We could look at it the other way around: Chas model combines the scale effects of theoretical harmonic partials 2, 3 and 5 in a new set. The combination of prime numbers in a scale of sounds has been an age-old theoretical problem, today its solution stretches octaves and finds a new beats function.

Chas is a time-rhythm based temperament model that finds the biunivocal relationship between frequencies and beat frequencies. Chas describes a set where beats play the fundamental role (section 2.0).

So, Chas scale frequencies values come out as the result of synchronic beats, i.e. today a theoretical system based on proportional beats can order a scale of proportional frequencies, the opposite of what has been done so far.

Is it because of inharmonicity that we can not apply E.T.? Maybe not only. In fact E.T. calculates 13 frequencies and enlarges the scale by cloning this 13 sounds module. In section 4.3 – graph 5 we see the effects on the beats.

Chas model, referring to the traditional semitonal scale, adopts a two-octave module. From section 3.0: “A two-octave module gives the scale set an intermodular quality. From the minor second degree to the nth degree, all intervals will now find their exclusive identity”.

What does “all intervals will now find their exclusive identity” mean? It means that intervals greater than an octave, in terms of beats, can all play and support a ratio for the entire set.

Any questions so far? a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/16/09 06:05 AM

Alfredo:

I am trying to think of the best way for us to communicate on this subject. I think a broader discussion, rather than a paragraph-by-paragraph study of your paper is worth a try.

There are some terms that we each may use but we each may use differently. ET is one of them. I understand that by ET, you mean the frequencies based on the twelfth root of two. To me, it means a scale where all keys have the same color and the feature of this type of scale is that M3s and M6 beat progressively faster. Since we are talking about your paper, we will use your definition.

I would say that since all pianos have iH, that any piano that has ever been tuned aurally has not been tuned to ET. And any piano that was tuned aurally by using temperament tests outside the initial octave (and there are many sequences that use more than one octave to set the temperament) will produce an “intermodular quality”. As you mentioned “A two-octave module gives the scale set an intermodular quality. From the minor second degree to the nth degree, all intervals will now find their exclusive identity”.

I really want to understand how you are tuning. Can you explain your tuning sequence? This may help me understand how you are using your discovery and thereby understand your paper.

By the way, I’ve been to Sicily. Augusta Bay is one of my favorite ports, very relaxing.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/16/09 02:06 PM

Tooner:

As you like, we can discuss broaderly, as long as we do not go to far from the main goal of this Topic.

You say: "To me, it (ET) means a scale where all keys have the same color and the feature of this type of scale is that M3s and M6 beat progressively faster".

To me it means the same, Chas theory derives from an ET approach and produces ET frequencies values, but talking about the original theory it will mean octave ratio = 2:1.

Then you say: "sequences that use more than one octave to set the temperament will produce an “intermodular quality”.

I agree but, in my opinion, a sequence is a tuning routine, not to be confused with a theory. Anyway, i'll write down the sequence I use and post it.

I'm very glad you've had a good time in Sicily.

Bremmer:

May I ask you what is the difference betwin "Maindless" and EBVT? Are frequencies values in Hz available?

Robert Scott, thanks for your post. a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/17/09 07:27 AM

Jeff S.:

Thanks for contributing, please stay in touch.

You say: “…old teachings stated the beat rates of fifths should increase as one tuned up the scale. That model was based largely on a decades-old mathematical model, worked out before inharmonicity was understood. Current understanding leans more toward the idea that the beat rates of fifths should either stay roughly the same as one moves up the scale or decrease…”.

What you are saying gives me the opportunity to underline, for some of our colleagues, the importance of a reliable theory. In my opinion, even when we can “lean toward an idea”, we are left doubtful. An idea can be interesting, fascinating, even brilliant but it is not quite like having a precise and correct mathematical model deriving from a reliable theory. Should fifths stay roughly the same as one moves up the scale? How roughly? Should fifths decrease? About IH, you say it has been understood, i'm not that sure.

You end up saying: “My point is that… the practical way fifths are understood and tuned… has already changed away from the old model”.

I simply agree, we have left E.T. original theory behind and we are now lacking a comprehensive model, well described by a solid theory that takes inharmonicity in account, what we may call a “inharmonic theoretical model”. This is what Chas is meant to be. a.c.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/17/09 02:25 PM

Alfredo, the Equal Beating Victorian Temperament (EBVT) is a non-equal temperament. The "mindless octaves" concept is an octave stretching technique and therefore it has nothing to do with the initial temperament octave, it is only a way of expanding the temperament over the rest of the piano.

Contrary to what many technicians seem to believe, causing double octaves and 12ths to beat equally (which is the mindless octave concept) does not require the temperament to be equal. Obviously, I use it with the EBVT and any other non-equal temperament, 18th Century style to the present. It would not work with the far more unequal temperaments of the 17th Century and earlier. It does work for 1/7 Comma Meantone and any other mild meantone temperament but an exception must be made when tuning the double octave G#-G# and comparing it to the D#-G# 12th since in any meantone, the G#-D# 5th is a wide interval.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/18/09 12:58 PM

Bill, thanks for replying. So, with EBVT M3s and M6 do not beat progressively faster. Does one have to use an EDT to tune the EBVT? What could aural tuners use? Are there any figures?

Reading in your previous post about cristal sound, pipe organ effect, customer satisfaction, I was really thinking you could be tuning Chas.

Actually, Chas is an inharmonic ET, adopts 12ths and 15ths as the scale constants wich determine model's scale incremental ratio. Nothing to do with a non-equal temperament. I'd really like to try your temperament. a.c.
Posted by: Erus

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/18/09 01:15 PM

Check Bill's website, you can find the cents offset and aural sequences there:

http://billbremmer.com/ebvt/
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/19/09 09:26 AM

Thanks Erus, the EBVT was developed aurally and is the preferred way to tune it. However, so many technicians wanted the numeric data that I asked Professor Owen Jorgensen RPT to calculate them for me. The principal reason for aural tuning preference is that an ETD does not stretch the octaves the way I do aurally and it does make a significant difference in the final results. If you are an aural tuner, you will find the instructions very easy to follow.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/20/09 06:26 AM

Originally Posted By: alfredo capurso

Stopper:

I’m sorry, I was not interested in your “cello scrotum”.
I understand you are commercializing an ETD device, at a cost of $ 600. Could I know on wich basis? Does reading about scrotum help?


Alfredo,
I have noticed that you are not interested in the cello scrotum article, unfortunately you can´t understand my intention then. This article is of great value, as we can learn from it, that even if something has been published in a serious scientific medium, we have to be very careful about the content.

The tuning software i am marketing is based on the tuning method i published in 1988 in euro-piano (based on the 19th root of three in case of abscence of inharmonicity, i.e. the theoretical case) and my own discovery of perfect beat symmetry in the 19th root of three temperament, dating from 2004.

I am presenting the software and some theory at the italian piano technicians convention (7-11 July 2009) in Cavalese, Italy. It is planned that i am tuning a Fazioli grand piano with my tuning software. There is a second Fazioli grand piano present to be tuned from someone else in a different (standard or whatever) tuning. You are welcome to participate and to tune it with your chas method!

Bernhard Stopper
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/20/09 07:31 AM

Bernhard,

you say: "even if something has been published in a serious scientific medium, we have to be very careful about the content."

I'm sorry, my english is poor. I thought you were trivially insinuating that the content of the article about Chas is probably more rubbish. Yeah, how could you immagine readers being that naive or stupid and how could you have got to any conclusion without reading the article. Instead you were suggesting to read the article with care! So I have to thank you.

"The tuning software i am marketing is based on the tuning method i published in 1988 in euro-piano (based on the 19th root of three in case of abscence of inharmonicity, i.e. the theoretical case)...".

I could more or less read about this in an other Topic, but, like some others, I could not really understand your discovery. Could you please tell me more or do I need to necessarly come to Cavalese? By the way, thank you for inviting me but, having listened to your two recorded tracks, I think that sending you a sample of Chas will be enough.

Is there any graph or official document of yours available? a,c,
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/20/09 07:54 AM

Alfredo:

Could you put an audio example of your Chas tuning on the internet and provide us with a link?

I hope you will still post your tuning sequence. The numbers for any tuning scheme are fascinating to me, but to understand what is really happening, when used on different pianos, the sequence means more. For example, the width of the mindless octave’s twelfth and double octave are determined by the piano’s iH and the width of the fourth formed by the lower notes of the intervals. This is easier to understand when looking at the sequence of tuning the intervals to be equal beating. It would be less easy to understand with an equation, because if iH is not included the theory is incomplete, and if iH is included then the theory must show how it is affected by different values and slopes of iH.

Oh, and I think Mr. Stopper’s cello comment had many meanings. Take your pick.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/20/09 05:58 PM

Tooner, thanks for your feedback.

Your idea of a Chas tuning example will probably work much faster than words and numbers. Last year, with the help of a brilliant sound expert, we digitally compared ET and Chas frequencies, in answer to an italian collegue who thoght Chas tuning could not work for pipe organs. I'll start puting these evidencies on internet and I'll proceed with a recording from a real piano.

I've almost finished with my sequence. Meanwhile I can propose some reasoning about Chas tuning in practice.

Chas model, as you may have red in section 2.0, theorizes an ET sound set that, to deal with inharmonicity, is not based on a theoretical pure interval. So, at the end of our tuning we should’nt look for any pure interval. We’ll also see that, to translate Chas inharmonic theory into practice, we’ll need to temporarly raise all frequencies above average Chas inharmonicity theoretical values. Nothing to worry about because anyway this is more or less what we have empirically done so far, although only on the bases of an approximate calculation of inharmonicity.

So, in addition to precise theoretical inharmonicity’s Chas standard values, we’ll also consider the sound-board and the strings while-tuning settling. In fact, once we have tuned and stabilized middle strings, tuning the right and left string of each flat note, from middle-high register upwords, will cause an overall lowering of frequencies. No ETD can foresee or evaluate the fall in frequencies, consequent piano settling (by measuring crhomatic 12ths after your ETD tuning, you may confirm this statement).

So again, considering inharmonicity and depending on how flat our piano was, while tuning the middle strings we will temporarly have to go for a more accentuated stretch. Anyway, the final evidences I can find after tuning are the ones that Chas theory describes:

1)the well known ET progression of M3’s, M6’s, M10’s, M17’s
2)the Chas inharmonic progression of 4ths and 5ths including inversion of the latter
3)the Chas inharmonic “S-shaped progression” for the octaves
4)the constant, equal beating of Chas delta-wide 15ths and delta-narrow 12ths

Let’s have a look at these evidences. In point 1 we find nothing new: it is a well known fact that ET progression of RBI comes from the geometrical esponential increase of frequencies. So, to get stretched octaves in that kind of geometrical progression we only need to use any ratio higher than 2^1/12. Thus one question arise: how much higher does the most correct theoretical incremental ratio need to be?

In point 4, I mention equal beating. Well, I do not seem to be the first, having red Bremmer’s posts, and that makes me very happy. Bremmer well describes the extraordinary effects of 12ths and 15ths equal beating, something that he him self, with other collegues, are still experiencing. Thus a second question arises: how do you get to the most correct equal beating value and still enjoy an ET progression of RBI?

In point 3, I am talking about an S-shaped progression for octaves, a shape that should be familiar to us, since Railsback's measurements. Two more questions arise: is there a chance to find the most theoretically correct standard curve deviation from the 2:1 ratio? Will it ever be possible to adopt a natural and reliable standard curve of reference that deals with inharmonicity?

In point 2 you may find a fresh piece of news: the precise beats progression for 4ths and 5ths and the observation that the latter’s beat curve invert. Then one last question arises: when should 5ths ideally invert?

Simply answering to all these questions would take you straight to Chas model and there you may also find the relevance of any comprehensive theory.

Please, tell me about the many other cello meanings...I might like them more. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/21/09 08:13 AM

Alfredo:

You wrote –

”1)the well known ET progression of M3’s, M6’s, M10’s, M17’s
2)the Chas inharmonic progression of 4ths and 5ths including inversion of the latter
3)the Chas inharmonic “S-shaped progression” for the octaves
4)the constant, equal beating of Chas delta-wide 15ths and delta-narrow 12ths”


I think I now comprehend what you believe the Chas system to be and we can communicate better now. These four points are a great way to focus the discussion.

”1)the well known ET progression of M3’s, M6’s, M10’s, M17’s”

The progressive beating of intervals made from non-iH tones will occur as long as each semi-tone interval is the same. The semi-tone could be 1 cent or 1000 cents. The 2:1 octave ratio does not affect this. Of course, the cent (being 2^1/1200) is an entity of a theoretical 2:1 octave scale. But the unit of measurement could be something else. As long as the semi-tone ratio is the same for all semi-tones and greater than 1, all intervals made from non-iH tones will beat progressively faster. And for iH tones an important question is: what is the ratio of? Is it the ratio of the theoretical fundamentals or the first partials? This is important to understand whether the ratio is 2^1/12, 3^1/19, or the Chas ratio. It may be better to go into fixed tuning ratios deeper in another post.

”2)the Chas inharmonic progression of 4ths and 5ths including inversion of the latter”

As Mr. Scott showed in his post, it is the value and slope of the iH curve that produces the wide beating of the fifths. It is inherent in the scaling of some pianos, although a wider octave in the high treble can make this happen at a lower note. But is this a characteristic that has value in itself? Do listeners prefer a high treble with wide fifths? Or is this a feature of something else that is important, but is not a goal in itself. But then I have also showed that the Chas ratio does not produce wide fifths without iH. But here we get into fixed tuning ratios again.

”3)the Chas inharmonic “S-shaped progression” for the octaves”

I almost always see the same Railsback curve. I wonder if the piano was not tuned very well, or if it had scaling problems, or if the frequencies were not measured accurately. I am sure various pianos and various tuning preferences would show an S curve also, but with differing values. A piano tuned with 2:1 octaves (which is greater than a 2:1 frequency ratio because of iH) will show an S curve. As will also a graph of frequencies of non-iH tones with a semi-tone ratio greater than 2^1/12. Since the Chas ratio is greater than 2^1/12 it will produce an S curve. But that does not mean it accounts for iH, just that the ratio is greater than 2^1/12.

”4)the constant, equal beating of Chas delta-wide 15ths and delta-narrow 12ths”

OK, I was correct in thinking that the mindless octave is the basis of Chas. I have to be a bit intuitive on what I am going to say here because I have not actually worked out the math. Since the beat speed of the 12th and 15th is dependant on the width of the fourth that is formed from the lower notes of the intervals, then neither mindless octaves nor Chas prescribe the overall stretch of a tuning, but only a final outcome from an initial stretch. Also, again being intuitive, unless 12ths become wide first, I don’t think fifths can become wide. After all, a 12th is an octave and a fifth. If these together are not wide (and in the mindless octave and Chas they are not) how could the fifth be wide separately?

Maybe we can get into fixed and variable semi-tone ratios another time. I don’t have them really figured out yet.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/23/09 02:54 PM

To all aural tuning collegues :

I decided to go further my tuning sequence mainly for two reasons: firstly, because hundreds of interesting pages have and are been written about the most original and convenient sequencies, none of them leading to a solid, reliable theory that could deal with inharmonicity, so leaving tuners in a state of uncertainty. A sequence will be debatable, a mathematical evidence won’t. Secondly, because I do not think the sequence I use is any special, nor time saving or more confortable for listening to or comparing beats. In my opinion, any sequence can eventually work, as long as you clearly know what you can be aiming at and why, how and where you’ll get it.

The only novelty may regard the overall approach and the interweaving of SBI, i.e. 8ves, 4ths and 5ths beat curves, the results of research that opened to Chas algorithm. Chromatic 4ths are not only similar, going up the scale they get tiny little wider, chromatic 5ths are not only similar, from low notes they first stretch down and get tiny little narrower, in between C3 and C4 they invert and stretch up toward there pure ratio, going tiny less and less narrow.

An italian collegue pointed out that SBI are much harder to evaluate than RBI. True, I would also agree in saying RBI give you the general idea of what you are doing in the shortest lapse of time. Nevertheless in my opinion, if one truly wanted to achieve excellence in aural tuning, would have to master a maximum control of any interval’s beat. A matter of wrist, both in the figurative and the anathomic sense, and a matter of rhythmics. In my case, SBI control took me to the 7th decimal point (section 4.5).

So what happened was, first I empirically found the univocal SBI and RBI chromatic proportional order, finding an astonishing euphonic set that would prove how inharmonicity can be made tractable. Then I simply elaborated its essence, to finally construct an updated and comprehensive ET IH EB temperament model (lucky us with all those abbs.), reliable in both theoretical and practical terms. Since I know all this comes from practice, simplicity and utmost exactitude, I’m disclosing Chas model with a serene soul.

In tuning, as I have learned, each sound is only temporarly tuned, since every single added sound may indicate the need to correct previously tuned notes. At the end, it is the Chas form that releases me from all doubts and only then I am absolutely certain to have done my best. Anyway, here are a few suggestions introducing and commenting the sequence.

A - do not take this tuning sequence as a must -
B - octaves, 4ths and 5ths shape the skelethon of the entire set -
C - start tuning only middle string, mute from C6 down to strings crossing, dampers up -
D - tuning single strings and unisons, get always the same moderate sound intensity -
E - octaves have a low beat-threshold and a high beat-threshold, this helps me when tuning octaves in middle register -
F - possibly, stabilize middle string frequencies by playing a Forte sound -
G – do not tire your ears, by playing louder you will not hear better nor more -

wide or narrow is referred to the note we are ment to tune

Step 1 – A4 – (Hz) from 440.0 to 442.0 (concert or studio) - from 441.5 to 443.0 (for flat pianos)
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Step 2 – A4-A3 - tiny little narrow, just on the beating threshold
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Step 3 – A3-D4-(A4) - wide, close to 1 beat/sec. – D4-(A4) faintly beating
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Step 4 – A3-E4-(A4)
check overlaping 5ths and adjacent 4ths to set up Chas ET EB inharmonic octave:
A3-E4 about 1,5 beat/3s - sensibly faster than D4-(A4)
E4-(A4) about 2 beats/1s - sensibly faster than A3-D4
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Step 5 – E4-B3 – narrow - tiny little faster beat than A3-D4, sensibly slower beat than E4-(A4)
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Step 6 – B3-F#4 - narrow - little slower beat than A3-E4 since 5ths have already inverted
faster beat than D4-(A4) evaluate M6 A3-F#4
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Step 7 – F#4-C#4 – narrow - faster beat than E4-B3, sensibly slower beat than E4-(A4)
evaluate two M3’s progression + one M6
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Step 8 – C#4-G#4 – narrow - slower beat than B3-F#4, tiny little faster than D4-(A4)
evaluate three M3’s progression + two M6’s progression
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Step 9 – G#4-D#4 – narrow - tiny little slower beat than E4-(A4), faster than F#4-C#4
evaluate four M3’s progression
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Step 10 – D#4-A#3 – narrow - tiny little faster beat than A3-D4, tiny little slower than E4-B3
evaluate five M3’s progression
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Step 11 – A#3-F4 – narrow - tiny little slower beat than A3-E4,
tiny little faster beat than B3-F#4
evaluate seven M3’s progression
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So far, apart from A3-D4, we have stretched narrow - now we’ll stretch wide
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Step 12 – D4-G4 – widw - tiny little slower beat than G#4-D#4, faster beat than F#4-C#4
evaluate eight M3’s progression + three M6’s progression
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Step 13 – G4-C4 - wide - tiny little slower beat than B3-F#4,
tiny little faster beat than C#4-G#4 evaluate nine M3’s progression + four M6’s progression
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Beats curves are meant to be tuned temporarly. While you are tuning, bear all (few) doubts in mind.
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Step 14 – A#3-A#4 – wide - increase octaves beat’s speed very slowly – 5ths go very, very slowly towards pure – F4-A#4 tiny little faster beat than D4-(A4), as for the next 4ths
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From the octave beat threshold, first signs of beating come to us in a shorter and shorter lapse of time, this helps to S-shape octaves stretch
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Step 15 – B3-B4 - wide - increase octaves beats speed very, very slowly - 5ths towards pure
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Step 16 – C4-C5 - wide - increase octaves beats speed very, slowly - 5ths towards pure
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Step 17 – C#4-C#5 - wide - increase octaves beats speed very slowly – 5ths start transiting pure - evaluate one M10
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Step 18 – D4-D5 - wide - increase octaves beats speed very slowly – 5ths are transiting pure - evaluate M10’s progression

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Step 19 – D#4-D#5 - wide - increase octaves beats speed very slowly – 5ths are transiting pure - evaluate M10’s progression

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Step 20 – E4-E5 - wide - increase octaves beats speed very slowly –
5ths have transit pure, evaluate M10’s progression –
chromatic M12s, like A3-E5 must be constant and temporarly tuned pure (on normally out of tune pianos) -
Step 21 – F4-F5 – wide
Step 22 – F#4-F#5 – wide
Step 23 – G4-G5 – wide
Step 24 – G#4-G#5 – wide
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Step 25 – A4-A5 – double octaves like A3-A5 must be constant and temporarly beat with a rate of almost 1b/s
increase octaves beats speed very slowly –
5ths are very slowly widening, evaluate M10’s progression –
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Step 26 – A#4-A#5 – wide - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
Step 27 – B4-B5 – wide - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
Step 28 – C5-C6 – wide - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
---------------------------------------------------------------------------------------------
Go back down for G#3 to lower notes using SBI, RBI and EB, never lose control of beats proportions. 5ths will get slower, so will 4ths. Unison all these registers from your left hand moving right, except last muted string on C6, then go up to higher notes. Chas delta-wide 15ths and delta-narrow 12ths beat’s rate is about1b/3s.
Tune as you know, middle string first, then unison previous note’s right string (C6), next left (C#6), next middle (D6), previous right (C#6), next left, next middle and so on, checking also M17ths progression. While tuning, do not stop evaluating strings and sound table rigidity/elasticity, so you’ll be able to conveniently set up middle strings. In fact, on pianos you have recently tuned, more often grand’s, to get to final Chas delta-wide 15ths and delta-narrow 12ths, could be enough to temporarly set up a milder 12thsV15ths proportion. The exact opposite, in case of badly flat pianos. This may produce a difference.

Tooner:

Thanks. You say:

“And for iH tones an important question is: what is the ratio of? Is it the ratio of the theoretical fundamentals or the first partials? This is important to understand whether the ratio is 2^1/12, 3^1/19, or the Chas ratio.”

Well, you judge. The intermodular combining of partials 3 and 4 in the way Chas algorithm does, includes partial 5, which in the scale is semi-tone 28 (adjacent 3th number 7). In fact partial 4, resulting from 6 adjacent M3 and 8 adjacent m3, can intermodularly mediate, together with partial 3, all partials. Pure theoretical ratio 3^1/19 = 1.0595260647382…, like any other ratio higher than Chas 1.0594865443501, increases differences, and therefor beats, on sounds relative to partial 5 and 10.

After all, I’m not talking about personal taste, one may prefere pure 12ths or pure 19ths (6^1/31), some others pure 5ths or pure 3ths, as we have seen. Chas model explains the reasons for aiming at a purly proportional and synchronic frequenciesVbeats set ratio. Fairly proportioned beats open to a proportional set of sounds, a pure set.

As you will have red, I use higher ratios than Chas only to compensate while-tuning strings and sound-board settling, so to finally get to the ET EB Chas form. Chas theory is meant to describe a new way to interpret beats, why and how profiting from beats, and to show the beauty of Chas form its self. Frequencies, throgh beats, can stir up (or awaken) all scale sounds partials and so lead to an extraordinary resonant set.

Then you ask:

“Do listeners prefer a high treble with wide fifths?”

We have no reason to talk about preferences.

And then:

“But then I have also showed that the Chas ratio does not produce wide fifths without iH.”

I do apply Chas in iH cases, so I do not get your point.

“Since the Chas ratio is greater than 2^1/12 it will produce an S curve. But that does not mean it accounts for iH, just that the ratio is greater than 2^1/12.”

To me Chas S-shaped octave curve meant that we can deal with iH. So far we have related the necessity to stretch octaves only to iH. Chas model suggests we have to recalculate iH’s effect, since up to now we’ve calculated iH giving for granted two unjustiefied ET assumptions (section 3.0). Moreover, Chas octave quotients are closest to pure n/n+1 quotients (section 4.5).

You say:

“OK, I was correct in thinking that the mindless octave is the basis of Chas”.

Well, I’d rather say that Chas theory is the ET height of a base EB idea, call it mindless or whatever. Bill Bremmer says: "The "mindless octaves" concept is an octave stretching technique and therefore it has nothing to do with the initial temperament octave, it is only a way of expanding the temperament over the rest of the piano."

Telling you about my self, I first established in practice a congruent and coherent assumptions-free ET, then I elaborated the observable constants 12thsV15ths EB producing a comprehensive theory that could correct and apdate the approach to ET and iH.

“Since the beat speed of the 12th and 15th is dependant on the width of the fourth that is formed from the lower notes of the intervals, then neither mindless octaves nor Chas prescribe the overall stretch of a tuning, but only a final outcome from an initial stretch.”

This is not exactly correct. If mindless octave idea was intented for an ET scale, then mindless octave idea would be aiming at Chas model. But then he him self says that EBVT is a non-equal temperament. So, without “mindless” formula and with no precise frequencies values I can only say that mindless EB idea on his own, using your words, does not prescribe the overall stretch of a tuning, nor an initial 4th strecth in its sequence (since 4ths are meant to be similar). Chas model prescribes both. In fact, Chas tuning overall strecth and 4ths wideness are determined by Chas ET algorithm and its resulting incremental ratio. In the sequence I use, 4ths, 5ths + A3-A4 octave are the foundation of the whole. Nevertheless Chas is not featuring a strict form: with Chas algorithm, you could actually figure out other kind of EB, explore new ET’s and finally deal with iH, aware of what you are doing. For example:

(9/8 – Δ)^(1/2) = (4 + Δ*s)^(1/24)
s = 1
Δ = 0,00247997487864…
Semi-tone scale incremental ratio = 1,059490455417770…
1st partial’s ratio = 2,00061989765139…

Am I making any progress?

Bill Bremmer:

Do you think Professor Jorgensen could get to know about Chas?

Thank you. a.c.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/23/09 07:30 PM

Alfredo,

I am sure that Professor Jorgensen would be interested in your concepts. However, he is quite elderly and in frail health. He never has communicated via e-mail. I would suggest that you think about what kind of package of information you would like to present to him, print it out and send it to him by conventional mail in a large envelope.

He will reply to you in a hand written letter. Make sure you supply him with the properly written postal address as it is in your country. Professor Jorgensen understands the mathematics of tuning theory.

I have a very difficult time deciphering the math on these posts but I believe it is mostly because of the substitute symbols which are used. Since I am not familiar with the symbols you and other people may use, the mathematics I often see are beyond anything I can work with.

However, I can see from a cursory look at your sequence that what you describe is Equal Temperament (ET). Two points which confuse me are that I saw in earlier posts that you seemed to be denouncing ET and that attracted my attention. Yet, when I see your sequence, it looks like a very typical method of constructing ET using 4ths and 5ths. But that which has me most confused is that you apparently describe the first octave to be tuned as slightly narrow rather than slightly wide. Unless this was an error in transcription, it has me completely confused as to just what you are attempting to accomplish.

You mention lengthy scientific papers, none of which I have read and I am afraid they may prove to be unreadable by me. I have no education in higher mathematics. To me, aural tuning is a mechanical procedure which does have some foundation in mathematics but in the end is a physical job performed by a technician who listens and makes adjustments according to what is heard. Many technicians know nothing at all about tuning theory yet they manage to tune excellently. I have always said, "The essence of aural tuning is the perception and control of beats".

In my understanding, ET can exist with any conceivable amount of stretch or even within an octave which is deliberately narrowed. Stretching or narrowing an octave does not change any temperament, either ET or non-ET, it merely changes how the octaves sound but there is, of course an effect to be heard from even the smallest change to the size of the initial octave. However, that kind of effect is relatively small compared to the kind of effect which can be heard by deliberately tuning a non-ET. Any non-ET will also be affected by octave stretching or narrowing decisions.

If you would like Professor Jorgensen's mailing address, please send me a private message either on this forum or to my e-mail: billbrpt@charter.net. I do not believe it would be proper to post that information for all to see even though it can be obtained easily.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/25/09 05:54 AM

Alfredo:

I am a bit confused. Parts of your sequence seem to indicate non-ET, like when saying a fourth should be narrow. And other parts seem to indicate ET, like when checking M3s and M6s for progression. It may be the use of the terms wide and narrow, and hopefully we can clear this up.

You wrote: “wide or narrow is referred to the note we are ment to tune” When I think of an interval being wide or narrow I think of it being wider or narrower than just intonation (beatless), so it would not matter which note is meant. But perhaps you mean wide and narrow to mean faster beating or slower beating, like a doorway being wide or narrow? Or perhaps by wide or narrow you meant sharp or flat (or even flat or sharp)?

Also, when you submitted your paper to the University for publishing, what was the process for acceptance? Was it checked by the math department? Did you have to defend the paper to a board of professors? Did anyone at the University understand it? Did they agree with it?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/25/09 01:09 PM

Bill, Tooner, all collegues,

I’m so sorry, I used “narrow” and “wide” instead of “flat” and “sharp”. I had highlighted in red those notes to be tuned flat, and blue notes to be tuned wide, but when copying the text into the topic colours desappeard. So I wrote “wide or narrow is referred to the note we are ment to tune” forgeting that you would have understood that as been referred to the interval. When you see Step 2 - A4-A3 – narrow, means that A3 is flat, wich makes this octave wide (like any Chas octave); when you see Step 7 - F#4-C#4 – narrow, means that C#4 is flat, wich makes this 4th wide.

Bill,

Chas is an ET model’s theory, maybe the first ET model’s theory since traditional ET formula. Actually traditional ET, differently than what you could call a well described theory, looks more like an algebraic technique to maintain the pure octave and to distribute the so-called commas equally across 12 semitones (section 1.5).

To me, ET’s algebraic instrument can result been perfect, since nature seems to speak an algebraic language, nevertheless I’m denouncing traditional ET’s assumptions regarding the one octave module and the 2:1 octave ratio. So you were not wrong.

When iH was discovered, we runned to the conclusion that we could never put traditional ET into practice, because of iH, so ET could only be thought as an abstract “theory”. Probably then we also decided that no temperament theory can help in tuning.

I’m trying to correct this thinking, when I say: we could not put traditional ET into practice not really because of iH, but because traditional ET is a lame theory from birth.

In fact, traditional ET theory was spoilt by the “theoretical one octave module” and by “mathematical ratio 2:1”. These theoretical and mathematical assumptions, both wrong, lamed traditional ET and made it unrealizable and consequently unpleasent.

Since traditional ET could never be put into practice, we do not really know what tuners and musicians have been talking about in the past, when referring to traditional ET. Today two things are clear: RBI, like 3ths, 6ths, 10ths, 17ths and so on should have a smooth progression, octaves should be stretched.

Now, stretched-octaves do not come from traditional ET. So I ask: do we know of a reliable ET stretched-octaves theory?

You say you enjoy equal beating tuning in your dayly work, so I ask: do we know of a reliable EB ET stretched-octaves theory?

I'm sure it could be of great meaning for you to read Chas article, even only the two sections about approach and description of Chas model (2 pages). Meanwhile I'll prepare a mathematical description symbol-free.

Thank you also for your indications regarding Professor Jorgensen. You have been so kind, I'll follow your advice.

Tooner,

You ask: "Was it checked by the math department?"

Well, what do you think?

"Did you have to defend the paper to a board of professors?"

I had to rewrite the article 3 times, to explain things that on the way had resulted obscure. It took me almost 2 years.

"Did anyone at the University understand it?"

Yes, Chas maths is not that difficult and I'll demonstrate that.

"Did they agree with it?"

They checked Chas maths without playing any other role. We'd better talk about how could anyone disagree, don't you think?

Thanks, a.c.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/25/09 02:14 PM

This discussion about how you cannot tune equal or whatever temperament because of this, that or the other reminds me of the old saying: There is no problem so difficult that you cannot look at it in such a way to make it much more difficult!
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/26/09 05:25 PM

Bill,

You say:

"In my understanding, ET can exist with any conceivable amount of stretch or even within an octave which is deliberately narrowed."

So far, if you are talking about progressive RBI, I'm with you. Then I start having difficulties:

"Stretching or narrowing an octave does not change any temperament, either ET or non-ET,..."

What do you mean, saying: does not change any temperament?

"it merely changes how the octaves sound but there is, of course an effect to be heard from even the smallest change to the size of the initial octave."

I ask: what effect will the smallest change to the size of the initial octave have?

You end up saying:

"Any non-ET will also be affected by octave stretching or narrowing decisions."

When you started saying:

"Stretching or narrowing an octave does not change any temperament, either ET or non-ET,...".

So, may I ask you for a wider explaination? a.c.
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/26/09 07:27 PM

I finally had a chance to stick my nose into Mr. Capurso's paper, and I must say that it would be difficult to write a less understandable explanation of his thesis than he has done--perhaps it suffered in the translation. I went only so far as to find a math error, and perhaps Mr. Capurso would be willing to address my confusion.

Equation 4 is fine, Equation 5 is fine, but Equation 6 does not follow from Equation 5. To demonstrate this, I did a simple example in MathCAD. I arbitrarily selected a value of .1 for delta,and a value of 2 for S1. MathCAD solved for S, whose value is -0.3244117.... Now, if Equation 6 is valid, then we should be able to state that (3-.1*2)^(1/19) = (4+(-.3244117))^(1/24). However, this equality is invalid, and would only be correct if S1 = 1, which would cause Equation 5 to degenerate back into Equation 4. The error in the equality did not change much even for tiny values of delta. Perhaps, Mr. Capurso meant to suggest an approximate equality in Equation 6, or perhaps I made a mistake in my analysis.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 07:38 AM

Roy:

I noticed the same error and posted about it in the beginning of this Topic. I am trying to go beyond the mathematical explanation and pursue the concept by looking at the tuning sequence.
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 08:38 AM

So you did--it was so long ago in the thread that I missed it. It seems to me that the thesis in question is based in, or at least presented as based in, mathematics, and therefore must be judged on that basis. Mr. Capurso makes many hyperbolic statements and claims throughout his article, and if they are not supported by the analysis, what is his basis for making them?

I hope that Mr. Capurso will address the issue we have both raised. If not, I will be forced to judge his words as hollow.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 08:47 AM

Roy:

That is you prerogative, of course. I am looking for a gem in the rubble. And even if there is not one, there may be something else to discover. If not for me, perhaps for Alfredo. He surely spent a great deal of effort. I think he is in earnest.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 11:01 AM

Alfredo:

Thanks for the clarification on wide meaning sharp and narrow meaning flat.

I now understand your tuning sequence. Fourths beat progressively faster, while fifths beat progressively slower, become beatless, and then beat progressively faster but on the wide side of just intonation. This causes octaves to beat progressively faster also.

The fixed Chas ratio cannot do this when applied to either the note’s frequencies, nor to the beat speeds of the intervals. However, perhaps it describes the “change in the rate of change” of the beat rate curve or perhaps the frequency curve, which is a real ski slope.

Does this sound like what you are trying to say in your paper?
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 11:57 AM

Originally Posted By: UnrightTooner
Roy:

That is you prerogative, of course. I am looking for a gem in the rubble. And even if there is not one, there may be something else to discover. If not for me, perhaps for Alfredo. He surely spent a great deal of effort. I think he is in earnest.


Well, I hope you succeed, but I have my doubts. Even as I just read beyond Equation 6, Mr. Capurso starts talking about different values of s, without saying what the value of s1 would be. This paper should not have been published in its present condition. The figures are not properly annotated or explained, the claims he makes in the text are not backed up in the math, the math has at least some errors, and the whole presentation is loose, rambling, with extraneous information included, and essential explanations left out.

As you say, there may be a gem lurking in there, but one would have to start from the very basic premise, and then attempt to derive the analysis on one's own, I think.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 01:34 PM

Roy123,

Thanks for joining in. You say:

..."I arbitrarily selected a value of .1 for delta,and a value of 2 for S1."...

"The error in the equality did not change much even for tiny values of delta. Perhaps, Mr. Capurso meant to suggest an approximate equality in Equation 6, or perhaps I made a mistake in my analysis."

In this case we do not find any approximation. In section 3.3 you read: ..."When we add in the s variable, a rational number...". So, you can add an s value and you are not supposed to tuch delta, with or without s. Delta is not discretional. I'm sorry if my explaination was not as good you could have done.

Tooner,

I'll be back tomorrow and I'll answer you. thanks for your words and your attention, a.c.
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/27/09 02:22 PM

Thanks for your reply, Mr. Capurso, but your explanation does not make sense. You calculate delta from Equation 1 in your report. If we now use that value of delta in equation 5, we can see by inspection that the only possible value of s/s1 is 1. If we set s = s1 = 1, then all is fine, but Equations 5 and 6 are the same as Equation 1. If we take ANY other values for s and s1, such as s = s1 = 2, then Equation 5 still works, but Equation 6 doesn't, which is what I originally said. Basically, Equation 6 does not follow from Equation 5. The math is not correct.

Mr. Capurso, you have always been polite with your responses, and therefore it behooves me to behave similarly. However, both Tooner and I have addressed a serious question to you about your thesis--namely that Equation 6 is not a mathematically correct form of Equation 5. Therefore, unless you are willing to explain the veracity of your derivation, or to declare your mistake and correct your paper, it becomes difficult for me to take you seriously. Sorry to be blunt, but as the author of a paper that you present publically, you have an obligation to address any mistakes that may be in it, or withdraw it from the public until it can be corrected.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/28/09 06:00 PM

Roy123,

You say: "Mr. Capurso, you have always been polite with your responses, and therefore it behooves me to behave similarly."

I'm glad and I take that as a promise. I must admit that, reading about your conclusions in such final terms did not help my chemistry:

"The figures are not properly annotated or explained, the claims he makes in the text are not backed up in the math, the math has at least some errors, and the whole presentation is loose, rambling, with extraneous information included, and essential explanations left out."

I'm sure the text can be improved, but when you find extraneous information you can skip it, like we would do on any text, and when you are missing explanations you can ask me. First you claimed for maths basis, here you also talk about style, and you also say "the maths has at least some errors", when you should not be that sure.

You say: "Therefore, unless you are willing to explain the veracity of your derivation, or to declare your mistake and correct your paper, it becomes difficult for me to take you seriously."

Ok, let's declare our mistakes, by the way, is it clear why you cannot modify delta?

Then you say: ..."If we now use that value of delta in equation 5, we can see by inspection that the only possible value..."

You are not supposed to use delta value deriving from Equation 1. The value of delta will continuously change, depending on s.

What Equation 6 shows is that, if s is a fraction, the denominator will effect delta (i.e. differencies, i.e. beats) in the left expression (i.e. on partial 3). To check this, after having chosen a fractional s value, calculate the incremental factor (i.e. scale ratio), build up your scale values and you will be able to ascertain that the differencies on partial 3 and 4 will have the same proportions of your s fractional value.

"Sorry to be blunt, but as the author of a paper that you present publically, you have an obligation..."

I do not know what you are worried about, I think I'm aware of my obbligations, why would I be here?

Please, let me know if now Chas algorithm works better.

Tooner,

I have to postpone your question, hope you do not mind. a.c.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/28/09 07:17 PM

Alfredo, while I can make no comment or judgment on the math, I can make a comment on the way the written sequence is described. I will have to take into account that you may not be familiar with the way temperament sequences are described in American (and probably any other variety of) English.

No interval, octave 3rd, 4th, 5th, etc. can be "sharp" or "flat" even though many people will describe them that way. An interval can only be beatless (also called "pure" or "just intonation"), wide or narrow (from the point where it does not beat).

Now, having said that, in order to widen a beatless interval, one may flatten the bottom note or sharpen the top note. To narrow an interval, one may sharpen the bottom note or flatten the top note.

In ET, 5ths are always slightly narrow and therefore some people say that they are flattened and we know what they mean but it is not the correct way to describe a tempered 5th. This is the most common example of misuse of the terms, "sharp" and "flat" when describing the tempering of intervals but it applies to all intervals.

So, I believe you need to review your written instructions for construction a temperament. The way you have described it is quite confusing. You have said that an octave should be slightly "narrow" when you really meant that the octave should be slightly wide. I believe there are some other examples of that where you say a 4th should be narrow when you meant it should be wide and the same possibly with other intervals where you have effectively said the opposite of what you mean.

I am sure that if you sent that material to Owen Jorgensen, he would write back the same as I have said and would provide corrections in red ink.

Writing temperament sequence instructions is very difficult and it is easy to make very bad errors and for the writer to not see them. I know this from experience and I am grateful for those who have helped me correct those kind of errors on many occasions.

Regards,
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 07:22 AM

Bill:

Yes, there is a language error in how the sequence is written. I was able to decipher the sequence when I understood the error.

I think an Equal Temperament can be constructed with wide fifths. It certainly can be constructed with just fifths, so why not wide?

What Alfredo seems to be doing is having ever increasing octave widths. Looking at it with non-iH tones, I would say that the temperament octave would be about 1202 cents wide, and each octave higher being an additional 2 cents wider in order for the fifths to become wide. I normally think of ET as having beat rates that increase for all intervals. Seems very odd to think of one as having an interval that beats slower and then faster on the wide side, but such a beat rate can still be considered to be progressive. I think sometimes my twelfths do this, so probably my nineteenths actually do. Of course, having the fifths do this so low in the piano’s range will make very busy octaves!
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 07:36 AM

Originally Posted By: alfredo capurso
Roy123,

You say: "Mr. Capurso, you have always been polite with your responses, and therefore it behooves me to behave similarly."

I'm glad and I take that as a promise. I must admit that, reading about your conclusions in such final terms did not help my chemistry:

"The figures are not properly annotated or explained, the claims he makes in the text are not backed up in the math, the math has at least some errors, and the whole presentation is loose, rambling, with extraneous information included, and essential explanations left out."

I'm sure the text can be improved, but when you find extraneous information you can skip it, like we would do on any text, and when you are missing explanations you can ask me. First you claimed for maths basis, here you also talk about style, and you also say "the maths has at least some errors", when you should not be that sure.

You say: "Therefore, unless you are willing to explain the veracity of your derivation, or to declare your mistake and correct your paper, it becomes difficult for me to take you seriously."

Ok, let's declare our mistakes, by the way, is it clear why you cannot modify delta?

Then you say: ..."If we now use that value of delta in equation 5, we can see by inspection that the only possible value..."

You are not supposed to use delta value deriving from Equation 1. The value of delta will continuously change, depending on s.

What Equation 6 shows is that, if s is a fraction, the denominator will effect delta (i.e. differencies, i.e. beats) in the left expression (i.e. on partial 3). To check this, after having chosen a fractional s value, calculate the incremental factor (i.e. scale ratio), build up your scale values and you will be able to ascertain that the differencies on partial 3 and 4 will have the same proportions of your s fractional value.

"Sorry to be blunt, but as the author of a paper that you present publically, you have an obligation..."

I do not know what you are worried about, I think I'm aware of my obbligations, why would I be here?

Please, let me know if now Chas algorithm works better.


Mr. Capurso, you continue to miss or evade the point. You claim that Equation 6 can be derived from Equation 5. It can't. There is no reason for me to expound further, the math speaks for itself.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 07:48 AM

Alfredo:

Take your time. Remember what I said before, I am not your enemy.

Something that this discussion is doing for me is making me think about how I think about tuning theory.

I can think of it purely mathematical, with or without iH. And I can think about it purely harmonically (beats) with or without iH. Or I can think about it musically, in how to provide the listener with what they want to hear, or fool them into accepting what they hear as “correct.”

I am guessing that you found a way to tune, and you also discovered some mathematical phenomena and think they are related. I don’t know the evolution of your thinking, so I am only guessing. I am realizing that connecting harmonic tuning theory to mathematical tuning theory is quite a challenge.

Not too long ago I realized how the effects of iH are largely self-correcting on the theoretical beat rates of intervals. But unless there is some reason to express a harmonic tuning style mathematically, why bother? I suppose it is necessary to construct an ETD program. Or in my case, out of the desire to understand what others have said on the subject. What is your motivation to express your tuning style mathematically?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 12:50 PM

Roy:

I agree that how the equations are presented are in error. But I think I now understand what Alfredo was trying to show with these equations.

Although he gives the correct solution for equation 1, and it seems that delta is really a constant and not a variable, I think he means to show that delta can have other solutions dependant on including an “s” factor. But the important thing is that after the “s” factor is applied, that the 19th root of the one term equals the 24th root of the other term. There may be an “s” and also an “s1” because iH affects the third partial differently than the 4th partial. "s1" is shown breifly but not explained.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 01:19 PM

Dear colleagues,

I’m obliged to treat a mathematical matter regarding Chas algorithm. I know most of you will not be interested in this but Ive got no choise.

If anything, Chas model describes the beautiful set I have found in my tuning practice, this makes me hope I can share it with all tuners, despite the following necessary-in-necessity figures.

Bill,

You said: “So, while I have absolutely no idea of what Alfredo is talking about at this point, I say, go for it, I may find something I like after all…”. If you can, please have a go, you may discover that Chas maths is not difficult.

Roy123, Tooner,

I’m going to address the issue you have raised.

In section 3.2 you read: “In the chas algorithm, the Δ variable proportions the differences of two intervals, 8th+5th (12th degree) and 8th+8th (15th degree)…”. So Δ is an unknown quantity.

In section 3.3 you read: “…infinite exponential curves related to oscillations of partial values, and identifiable through a second variable, expressing an “elastic” potential and enabling the system to evolve. When we add in the s variable, a rational number, (s from the concepts of stretching, swinging and spinning), equation (1) becomes:….”, so, Equation (1) becomes Equation (4). This is to say: to our Equation (1) we can add in a rational number, the so called s variable that will change delta value, enabling the system to evolve.

Then you read about the scale effects of s variable:

“The s variable can swing the logarithmic scale… The variable affects the distances and proportion of scale values…”.

Then you read: “If s is a fraction (s/s1)…”, Equation (4) becomes Equation (5). Then you are told about the effect of s/s1 fractional value on the equality: “…the denominator multiples delta in the left-hand expression so that...”, so that Equation (5) equals Equation (6). Let’s check this together:

We choose a fractional value for s/s1:
s = -9
s1 = 8 and use the Equation (5) type, so we have:

Equation (5) type:

(3–Δ)^(1/19) = (4 + (Δ*-9/8))^(1/24)

true for Δ = 0.01018036614 = first found delta from Equation (5) type

Substituting this Δ value:

(3–0.01018036614)^(1/19)=
=(4+(0.01018036614*-9/8))^(1/24)= 1.05933652544275 this is our scale incremental ratio.

You were told that Equation (5) equals Equation (6), so that:

(3–Δ)^(1/19) = (4+(Δ*-9/8))^(1/24) equals
(3–(Δ*8))^(1/19) = (4+(Δ*-9))^(1/24)

It should be that there exists a value of delta (the unknown quantity that s can alter) so that our latter equality produces our previously found scale incremental ratio. Can it be true? Can we find this delta value?

Δ = 0.0012725457675, second found delta from Equation (6) type, so that

(3–(0.0012725457675*8))^(1/19)=
=(4+(0.0012725457675*-9))^(1/24) = 1.05933652544275

The first 24 scale values deriving from our s/s1 fractional value and consequent delta values will be:

Scale values
1,0
Scale ratio 1,059336525442750
1,122193874137110
1,188780959501540
1,259319091150850
1,334042710443460
1,413200169673400
1,497054557496920
1,585884573337010
1,679985453672080
1,779669953287340
1,885269384750260
1,997134719564920
2,115637754664980
2,241172348122290
2,374155728178230
2,515029879948310
2,664263014409130
2,822351124549780
(3-(delta*s1)) 2,989819633859990
3,167225142633750
3,355157277892540
3,554240653076620
3,765136944017540
(4+(delta*s)) 3,988547088091660


Difference on partial 4 (element 24) = 3.98854708809166 - 4 =
= -0.0114529119 wich is our first found (delta*-9/8) and our second found (delta*-9), in fact:

First delta from Equation (5) type = 0.01018036614
0.01018036614*(-9/8) = -0.0114529119

Second delta from Equation (6) type = 0.0012725457675
0.0012725457675*-9 = -0.0114529119

Difference on partial 3 (element 19) = 3 – 2.98981963385999 = 0.01018036614 this is our first found delta from Equation (5) type and our second found (delta*8) from Equation (6) type, in fact:

0.0012725457675*8 = 0.01018036614

Last ceck: divide the difference value on partial 4 by the difference value on partial 3:

-0.0114529119 : 0.01018036614 = -1.125 = -9/8 wich is our discretional s fractional value.

Roy123,

I would not like missing or evading any point.

Bill, Tooner,

I'll be with you asap, thank you. a.c.
Posted by: ROMagister

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 04:09 PM

Good fundamental idea, quite confusing presentation.
If I understood this well, it IS Equal Temperament, but with another ratio: not the classic one where 12 semitones = 1 octave of exactly 2:1 (Pythagorean octave still accepted as axiom in classical ET).
The basic version (s=1) makes an equal compromise between the 'justness' of 3rd and 4th harmonics (octave+fifth vs 2 octaves). "s" is just the compromise parameter which says how important is the error in the 3rd harmonic compared to the error in the 4th harmonic. It can be set "politically" as we want, and the Delta results as a solution of the (implied) equation, also the practical frequency ratio that results.

The 'tweaking knob' of s/s1 may result in different deltas and frequency ratios.

Equation 6 is equivalent to eq.5 only if the Delta in eq.6 is a different Delta from the one in eq.5 (say, notate it Delta').

I just don't see where's the "circular" part of CHAS. The octave being wider than 2:1 they deviate more and more.

The "attractor of size 19*24" is pompously written, since 456 semitones way exceed the audible range (the most used in MIDI is 128 semitones).

I don't understand how this method incorporates the prime number 5. Of course, one can use "politically" the 5th harmonic as the 19th (2 octaves+M3), like it's used in organs with the 1 3/5' Tierce stop. But there it's no inharmonicity, and that stop is meant only to be used together with a fundamental (8') stop. But if used across the whole instrument it deviates way too much from the consonance of 2:1 octaves and 3:2 fifths.

One may use a similarly designed CHAS-like algorithm of equal (or stated-weight) compromise between 3:2 fifths and 5:4 thirds etc.

The difference from classical 2^(1/12) is smaller than the unknown inharmonicity of piano anyway - and that is an unknown depending on many practical details of building.

The suggestion to tune 'narrow' not wide in the central zone I understand it so: the intrinsic piano's C5 (that sounds consonant to C4 on that piano) is in unknown ratio to the C4, but > 2.00. One tunes C5 lower than what sounds consonant to C4, so that the result is closer to 'true' ET (or even Chas) than that piano's inharmonicity may suggest.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 04:31 PM

My own feeling is that my tolerance for narrow octaves is not as great as this calls for. When an octave is 2 Hz narrow in the center of the piano, the piano needs tuning.
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/29/09 04:47 PM

Mr. Capurso, thanks for the additional explanation. As written, your paper is incorrect, or at least highly misleading, because Equations 5 and 6 show the same symbols for delta, S, and S1, and without some explanation, one would make the inevitable assumption that therefore the values of these three variables in both equations would be the same.

In order to make your paper read correctly, I suggest that you add some words to make your intent clear. You could say, for example. "In equation five, we will select values for S and S1, and calculate a new value for delta that makes the equality true. In Equation 6, we keep the same values of S and S1 and compute yet another value of delta that makes the equality true."

With such an explanation, I think your readers would have correctly interpreted your math--I certainly would have.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/30/09 06:04 PM

Roy 123,

thank you for having discounted your inflictions and above all for your suggestion that, in my opiniln also, can help our readers. Now I look forward to knowing about if and how you like Chas model.

ROMagister, BDB,

Chas octaves are not narrow. I'm sorry to have written "narrow" in stead of flat when referring to the note to be tuned. For istance, when in the sequence you read A4-A3 - narrow, I meant to say A3-flat, so A3-A4 is a wide interval.

ROMagister,

I'll have more time tomorrow to replay to your attentive post. Thank you.

Tooner,

you had already understood about Chas octaves, fifths, deltas and s, you devil. One day I'll tell you why you are not my enemy!

Bill,

I'm goin to work on the sequence and submit it to you. How do you like Chas inverting fifths? Did you find all those figures disgusting? a.c.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/30/09 06:19 PM

You say that if you start with a scale value of 1, the octave ratio will be 1,997134719564920 instead of 2, and two octaves will be 3,988547088091660 instead of 4. That is narrow.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/31/09 06:07 AM

Alfredo:

I am going to use analogies to try to describe the problems I see with your paper.

Lets say you get a call to tune a piano, and the customer says that their piano is a motorcycle. So you say your piano could not be a motorcycle, it is a piano. Motorcycles and pianos are not the same thing. So, the customer says that their piano says “Yamaha” on the fallboard and since Yamaha makes motorcycles then their piano is a motorcycle!

This is the problem with saying that equation 5 equals equation 6. Given certain values for the variables the terms can be equal to each other, but that does not make the equations equal to each other.

So you ask where the piano is so that you can go there and tune it and are told that the piano is in the front room, the room with the lovely drapes. From the customer’s point of view this is a perfectly good answer, but does not help you get from where you are to where the piano is.

Your paper seems to be written from your point of view and assuming that what you find to be desirable, everyone else will. By presenting your equations with variables on both sides it is very confusing as to what is being solved. For instance, if I was talking about the Pythagorean Theorem and said that the equation a^2 + b^2 = c^2 will give the length of the hypotenuse, it would be difficult to someone that did not already understand the equation to know that I mean that the Hypotenuse = (a^2 + b^2)^1/2. Your emphasis on delta is confusing. It is of no use in itself, but only as interim step in determining a ratio. It is proper to show and explain delta when showing how your equations are derived, but in the end the equation should be in the form of “ChasRatio = …..”

Then after talking to the customer more about where they live you find out that they live in Haiti. Ok, Haiti can be a nice place (I’ve been there), and it is interesting to think of different ways to get there. But besides not planning on going there, the directions from the customer are just too hard to follow because there are given from their point of view and not yours.

This is how I feel about your tuning from reading your sequence. I have tuned every widening octaves, but probably not to the point of wide fifths so low in the keyboard. It can be OK, but I don’t plan on tuning that way. But besides that, I can find nothing in your paper that goes from a fixed ratio to ever widening octaves. And as I continue to try to understand your paper I read the statement of “s=s/s1” (which can only be true if s1=1, but then what is the point?) and there is no explanation of what units s and s1 are in, nor how s and s1 are determined, nor why s/s1 must be a rational number. Since I am not interested in tuning as you do, the effort becomes too difficult to try to understand how you “get there” from your Chas ratio.

I will probably continue to read the posts to this Topic, and may or may post to it myself, but I doubt if I will put in the effort to really understand your paper. (I never cared for flowered drapes in a front room. I prefer lace.)
Posted by: Roy123

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/31/09 12:07 PM

Tooner, if you read the explanation in my last post, you will see that with the use of different deltas, but the same values for S and S1, in Equations 5 and 6, that both equalities can be obtained, and, in fact, that the semitone ratio calculated for both is the same. I think much of the problem with Afredo's paper is the rather wordy, hyperbolic, and unclear (sorry, Alfredo) presentation. It is simply not written in a way that would be accepted by the scientific or mathematical community. The paper could have been much shorter, crisper, and more lucid.

Having said that, I've not ventured further to see if something is to be gained by using Alfredo's method.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/31/09 02:03 PM

Bill, you kindly say:

“Alfredo, while I can make no comment or judgment on the math, I can make a comment on the way the written sequence is described.”…

I take your's and all our colleagues math understanding to heart, so that we’ll be able to share Chas model in all its aspects.

Think about traditional ET ratio 12th root of 2.
Say you want all intervals to be ET progressive.
Say you want an equal beating on 12ths and 15ths.

You already know that theoretical 12th root of 2 would not satisfy your needs, since those 15ths are theoretically beatless. In fact, you know that the only way to have all intervals being progressive and equal beating 12ths and 15ths is to stretch your 12th root of 2 ratio. So now you are thinking in terms of (12th root of (2 + wide-stretch)).

Your experience tells you that P12’s (pure 12ths, ratio 19th root of3) would give you to wide octaves and 3ths, 10ths and so on, this is why you want 12ths a litle narrow, so you think at 19th root of (3 – stretch), while you want P15’s (pure double-octave, ratio 24th root of 4) to beat equally, say 24th root of (4 + stretch). So you conclude that, in order to have an equal beating on 12ths and 15ths, you can write:

19th root of (3 – stretch) must equal 24th root of (4 + the same stretch). This is Chas algorithm.

ROMagister,

I think you got the point in writing:

"Good fundamental idea, quite confusing presentation.
If I understood this well, it IS Equal Temperament, but with another ratio: not the classic one where 12 semitones = 1 octave of exactly 2:1 (Pythagorean octave still accepted as axiom in classical ET).
The basic version (s=1) makes an equal compromise between the 'justness' of 3rd and 4th harmonics (octave+fifth vs 2 octaves). "s" is just the compromise parameter which says how important is the error in the 3rd harmonic compared to the error in the 4th harmonic. It can be set "politically" as we want, and the Delta results as a solution of the (implied) equation, also the practical frequency ratio that results.

The 'tweaking knob' of s/s1 may result in different deltas and frequency ratios.

Equation 6 is equivalent to eq.5 only if the Delta in eq.6 is a different Delta from the one in eq.5 (say, notate it Delta')."...

I'll answer your questions as soon as I can. Thank you

BDB,

in Chas article you'll find Chas octave ratio: 2.0005312...

The figures in my previous post came from an example, to show the effects of s variable.

Tooner,

I think your contributes are precious because you can think in abstract terms. Sorry for my style.

Roy123,

I think you have already been able to express your opinion about the article style, now if you like, you could help by considering the content. Your point and your suggestion have already solved a question.

Thank you, a.c.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/31/09 03:59 PM

Thanks for this most recent post, quoted below. It is a good explanation of your tuning; I can only wish that your previous writing was this clear.

However, I completely disagree that pure twelfth equal temperament is too wide. At least as tuned by the OnlyPure electronic tuning device, pure twelfth equal temperament yields beautiful, beautiful results.

Quote:
You already know that theoretical 12th root of 2 would not satisfy your needs, since those 15ths are theoretically beatless. In fact, you know that the only way to have all intervals being progressive and equal beating 12ths and 15ths is to stretch your 12th root of 2 ratio. So now you are thinking in terms of (12th root of (2 + wide-stretch)).

Your experience tells you that P12’s (pure 12ths, ratio 19th root of3) would give you to wide octaves and 3ths, 10ths and so on, this is why you want 12ths a litle narrow, so you think at 19th root of (3 – stretch), while you want P15’s (pure double-octave, ratio 24th root of 4) to beat equally, say 24th root of (4 + stretch). So you conclude that, in order to have an equal beating on 12ths and 15ths, you can write:

19th root of (3 – stretch) must equal 24th root of (4 + the same stretch). This is Chas algorithm.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/01/09 01:55 PM

Kent,

thanks for contributing. You tell me that with your ETD you get beautiful results, but I do not know what other ETD you are comparing it with. As you say, I do not even know what tuning curve I would then find in the piano, if 19th root of 3 or something else.

You see, I'm not promoting a precise tuning curve, having this to do with personal/cultural taste, I'm promoting an updated ET theory that is finally applicable in tuning practice.

Chas model discards traditional ET erroneous assumptions. When put into practice, Chas can help aural tuners dealing with iH and can correctly orientate to find the smoothest progression of RBI and SBI.

More precisely, Chas explains why and how 5ths invert, becoming less snd less narrow from the middle-high register goin up. Once you are aware of how your 12ths and 15ths are going, you would be able to fix your favorite tuning curve, while considering both iH and sound-board Vs strings adjustment.

ROMagister, you say:

..."The difference from classical 2^(1/12) is smaller than the unknown inharmonicity of piano anyway - and that is an unknown depending on many practical details of building."...

Today, on well scaled pianos, inharmonicity is made quite even. With Chas correct standard frequency values we will improve the building and scaling of pianos and we'll better control iH. a.c.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/01/09 02:08 PM

Quote:
Today, on well scaled pianos, inharmonicity is made quite even.

I am not certain what that sentence means. In any case, a scale can be designed for other goals than inharmonicity, even now.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/01/09 02:32 PM

This is like trying not to look at a train wreck.

Alfredo:

Can you state what you understand about iH?
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/01/09 09:46 PM

Quote:
Chas model discards traditional ET erroneous assumptions. When put into practice, Chas can help aural tuners dealing with iH and can correctly orientate to find the smoothest progression of RBI and SBI.

More precisely, Chas explains why and how 5ths invert, becoming less snd less narrow from the middle-high register goin up. Once you are aware of how your 12ths and 15ths are going, you would be able to fix your favorite tuning curve, while considering both iH and sound-board Vs strings adjustment.


Of which erroneous assumptions do you speak? Usually, we speak of a mathematical model of equal temperament with no inharmonicity that we know very well doesn't exist on real pianos. Then we try to find the best fit of the model to the inharmonicity-laden piano in front of us. It isn't news that the model of equal temperament doesn't quite fit real pianos. If you have something to contribute, a way of better fitting equal temperament to real pianos, then we are all ears.

These two last posts of yours are providing good descriptions; please keep it up. Now, according to you, why do fifths invert going up the scale?
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/01/09 10:25 PM

Kent, that first line that you quoted confused me. I thought it meant that he was writing about a non-ET and the "erroneous assumptions" that we may have that ET makes the piano sound best. I'm not bringing up that topic or argument here, mind you, it was just what I thought he meant and I was interested.

I would say, however that some of the points which have been raised will apply to a non-ET too, at least the way I prefer to tune a non-ET. It is interesting that theoretically, 5ths will increase in speed but the very last thing anyone wants to hear are "beating 5ths". Since what I normally tune is a mild Well Temperament in which some 5ths are beatless, others tempered a little less than in ET and some a little more than ET, I have long observed how 5ths actually widen when ascending the scale rather than maintain the same width as they do in the temperament/midrange. Regardless of whether they were tempered or not in the midrange, they all eventually become wide.

This will depend, of course on how aggressively or not the octaves are stretched. That kind of choice will make a difference in the part of the scale where 5ths do become wide. My observation has been that it is generally in the 6th octave. I also believe they become wide in the low bass.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/02/09 06:35 AM

I have copyed your posts from an Internet point. In those days I can not go in the web from home.

Today I'll work on my answers, meanwhle I thank you all. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/02/09 08:15 AM

It seems that fifths could not become wide going up the scale, unless 12ths become wide first, which does not happen with mindless octaves, Chas tuning, nor perfect 12ths. Since a 12th is made from a fifth and an octave, the only way to have a wide 5th and have a 12th that is pure or narrow, is to have an octave that is narrow while the fifth is wide. It seems that this could only happen if there is a tuning error.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/02/09 09:51 AM

5ths become wide on PTG Tuning Exam Master Tunings in the 6th octave.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/02/09 10:25 AM

Then I would think that the 12ths become wide, also.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/02/09 09:01 PM

I think they do. Anyway, I have a couple of master tuning records and when I get the time, I will post what I find on that but in a new thread. It doesn't belong here. In any case, the "mindless octaves" idea creates an exact compromise between the double octave and the 12th and I routinely see them invert themselves: the 12th becomes wider than the double octave but I still balance the two.

Then again, does that belong here? Alfredo seems to claim something unique as does Herr Stopper. How do either of them compare to what is considered a standard (a standard to which I freely admit I never adhere except for the purposes of the exam itself). It must be close to 20 years ago that I saw Steve Fairchild demonstrate that 5ths do become wide. He also said that 4ths become narrowed.

Whether they do or not, beating in either 4ths of 5ths cannot be heard beyond a certain point because the coincident partials are too high and too faint. Try it yourself: tune a 5th from G6 to D7. Tune it as wide or narrow as you like and you won't hear any beats. The coincident partials are in the 8th Octave. The beats for 4ths disappear on or about F5. If either is wildly wide or nefariously narrow, does it matter if you can't hear any beats? What does matter are beats that *can* be heard from the wider intervals such as double and triple octaves, 10ths, 12ths and 17ths.

As far as I am concerned, when you get to the top of the 7th octave, none of them matter at all any more, only a sense of pitch does but everything leading up to that must provide a foundation for stretching the top part of Octave 7 and C8 as much as I customarily do. The amount I stretch up there shocks may technicians (when they know the numbers) but I can assure you that many fine aural tuners go beyond that. I can at least justify what I do by making those highest pitches agree with notes in the midrange.

Just for consideration, C4 read at 0.0 on its fundamental will typically read 1.0-2.0 cents when read on C5, 3.0-6.0 when read on C6, 15-20 cents on C7 and 65-80 cents when read on C8. Wouldn't that be a reason to try to stretch the octaves to at least partially conform to the amount of inharmonicity there really is in piano strings?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/03/09 06:57 AM

BDB,

I meant to say that iH is, to a certain extent, predictable.

Tooner,

I would like to thank you again for what you had written:

…“I am looking for a gem in the rubble. And even if there is not one, there may be something else to discover. If not for me, perhaps for Alfredo. He surely spent a great deal of effort. I think he is in earnest.”

I had never told you that I really think Chas is a gem. What a pure chance.

About iH, let me answer with a friend of ours words, from Topic “Tunelab 6:3 octaves”.

“Here’s iH in a nutshell. A piano sting with certain characteristics and at a certain tension should vibrate at a certain pitch, but doesn’t. The difference in cents that the fundamental frequency differs from the theoretical frequency is the string’s inharmonicity constant or inharmonicity coefficient. All of the string’s partials, the fundamental being the first partial, are sharp of their theoretical frequencies. The amount in cents that they are sharp is the square of the partial number times the iH constant.

Example: A string has an iH constant of 0.5. The first partial is 0.5 cents sharp of theoretical, the second 2.0 cents, the third 4.5 cents, the fourth 8.0 cents, and so on.

So you should see that the number of cents sharp that each partial is in relation to its neighbors is not linear, but logarithmic. Matching the 6th partial of one note to the 3rd partial of another will not make the 8th and 4th partials also align.

Now a graph of a piano’s iH approximates a “V”. But since the left hand scale on the graph is logarithmic, it actually is a curve. The iH constant will double around every 8 semi-tones or more. So in the treble, not only do the strings have a higher iH constant, but the iH constant increases more and more. The same thing occurs in the bass with the iH increasing toward the bass.”…

This is what you understand. The italian colleague Giovanni Bettin writes:

“Fino a tempi non lontani la disarmonicità ha rivestito importanza e rilevanza solo per corde poste in stato di tensione e vibranti. Esami più accurati e ricerche effettuate da parte di O. H. Schuck e R. W. Young (1943) e dallo statunitense H. Fletcher, hanno comprovato la sistematicità con la quale si producono gli spostamenti di frequenza dovuti alla disarmonicità, e hanno stabilito le formule in base alle quali calcolarli: diametro delle corda al quadrato (D), diviso; la lunghezza della corda alla quarta potenza (L) moltiplicato per la frequenza (F), il tutto moltiplicato per un fattore costante (K) che deriva dal modulo di elasticità (E) del materiale che costituisce la fonte sonora."

These are good examples of what I understand about how iH is understood. But maybe this was not your question’s target.

At one stage you wrote:

…”I had worried about this because I was thinking that if my fifths didn’t become wide, I wasn’t tuning “correctly”. But since this happens only in the very high treble, due to a greater slope of the iH curve, then fifths becoming wide is an inherent anomaly of some pianos, not the result of a tuning style.”

Would you tell us about your tuning style, especially regarding your 4ths, 5ths, 6ths, octaves, 12ths and 15ths?

You also wrote: “I now understand your tuning sequence. Fourths beat progressively faster, while fifths beat progressively slower, become beatless, and then beat progressively faster but on the wide side of just intonation. This causes octaves to beat progressively faster also.”

Unluckily, I had to prove that Chas maths is errorless, otherwise I would have correct your understanding there and then: 5ths, from low notes, beat progressively faster (narrower), but then 5ths invert and beat as you say “progressively slower, become beatless, and then beat progressively faster but on the wide side of just intonation.” Please remember this as referred only to middle string tuning. Also I would like to understand more about you saying:

“Not too long ago I realized how the effects of iH are largely self-correcting on the theoretical beat rates of intervals.”

Kent, you ask:

“Of which erroneous assumptions do you speak?”

You find your answer in Chas article, section 3.0: “The chas model discards two unjustified assumptions: that the range of the scale module must be 12 semitones, and that the octave, the 12th semitone, must be double the first note.”

Also in Chas Topic you can still read: “About tuning and compromise - untill today we (aural tuners) could only think in terms of compromise becouse we had to get by with Equal Temperament and its unjustified premises, two unjustified assumptions that Chas model discards (section 3.0). Chas demontrates that the ratio 12th root of 2 is unsuitable, not only becouse of inharmonicity, but becouse it produces intervals incresingly narrow (12ths, 19ths and so on) together with intervals incresingly wide (10ths, 17ths ecc. - section 4.3 - graph 5). E.T. premises come out to be missleading.”

You say: “Usually, we speak of a mathematical model of equal temperament with no inharmonicity that we know very well doesn't exist on real pianos.”

Actually, I’m explaining why today we have good reasons to renew our old mathematical model of equal temperament with no inharmonicity, and adopt a mathematical model of equal temperament that can deal with inharmonicity, i.e. Chas theory’s mathematical model.

“If you have something to contribute, a way of better fitting equal temperament to real pianos, then we are all ears.”

Thank you so much for your interest and your encouragement. We know iH requires stretching, so why opposing an updated and reliable stretched-partials ET theory? This is what Chas can prove to be, despite banal prejudices and predictable distrustfulness.

I would like to ask you all:

1 - is it possible to have progressive M6’s (4th+ M3) without a correct ET progression of 4ths, meaning without a correct ET 4ths theoretical and practical progression? I would answer no. Actually, if we had theoretical stretched octaves we could, in fact today with Chas we can.

2 – if we can not get progressive M6’s, what happens to m3’s and how can we get progressive stretched octaves without progressive 6ths (4th+ M3)?

Maybe answering these question explains why we were in need of accuratelly theoretically stretched 4ths.

You ask: “Now, according to you, why do fifths invert going up the scale?”

Because if 5ths were not to invert, goin up the scale 5ths would unconveniently diverge from stretched octaves.

Bill,

You wrote:…“the "erroneous assumptions" that we may have that ET makes the piano sound best.”…

In a way I’m saying what you understood: traditional ET can not make a piano sound best. As I have said, traditional ET is a lame theory because of its two wrong theoretical basic assumptions. As you read in section 3.3, traditional ET is a particular case that we can still find included in Chas mathematical and geometrical entity (s = 0). As shown in section 4.5, traditional ET ratio is the only ratio that, as a tragic matter of fact, perfectly flattens octaves beat-curve.

To conclude, traditional ET manages to theoretically stretch only 3ths, 10ths and 17ths, when we were in need to theoretically and pratically stretch octaves.

You say: “It is interesting that theoretically, 5ths will increase in speed but the very last thing anyone wants to hear are "beating 5ths".”…

An again we would move on a debatable ground. Bill, I’m not talking about preferencies, I’m trying to share an "s" dynamical ET model.

Finally you say:” Regardless of whether they were tempered or not in the midrange, they all eventually become wide. This will depend, of course on how aggressively or not the octaves are stretched. That kind of choice will make a difference in the part of the scale where 5ths do become wide. My observation has been that it is generally in the 6th octave.”.

What you are saying seems to me very close to what I’m saying, talking about middle strings. In my experience, how much you need to stretch C8 will depend on how you have got to F7.

Thanks, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/03/09 08:03 AM

Bill:

I think it would be OK to continue to post about 12ths and 5ths on this Topic. I think “brain-storming” with similar subjects may help clarify what Alfredo is trying to say.

I admit that I cannot hear much more than how the 2:1 octaves beat after a certain point in the scale. But when evaluating the theory behind a tuning system, the only way I can do so objectively is by considering the beating of intervals, including those that cannot be heard. I don’t think the number of cents that any particular note is from a theoretical pitch means much in how a tuning is constructed nor how it sounds.

So thank you for confirming what I thought, but could not know from just listening: that if 5ths go wide, 12ths do also. Now the question this raises is whether a tuning that does not allow wide 12ths will make the high treble sound its best.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/03/09 11:04 AM

Alfredo:

I am sorry. I do not think that the Chas ratio is a “gem”. Like any fixed ratio, it cannot describe how a piano is actually tuned. Theoretical ET has the same problem. However, if we look at the beat rates that a fixed ratio (ignoring iH) predicts, then the beat rates, (or at least the ratios between beat rates) can be used as a model for an aural tuning. In other words, the Chas ratio describes how the beats of 12ths and 15th can be equal, but if the Chas ratio is used to determine the frequencies, this will not be the result. And there still is the problem of predicting 5ths becoming wide and 12ths staying narrow. I don’t believe this is possible. The “gem” that may be lurking is how to produce a variable ratio, that when applied to non-iH tones, will predict beat rates that will result in an aural tuning that works for iH tones.

You quoted what other people wrote about iH, but that is not the same as stating what you understand.

When I posted about the beat rate of your 5ths, I was referring to what they do from the temperament up. I chose not to confuse things at the time with a longer explanation.

OK, about the effects of iH being largely self-correcting on the theoretical beat speeds of intervals. I will try to explain this by using concepts instead of math.

All tuning intervals have nearly coincident partials. The partial of the lower note is higher in the partial series of its note than the partial of the higher note is in its partial series. Since, iH affects higher partials much, much more than lower partials, the first effect of iH is that wide intervals beat slower and narrow intervals beat faster than if the iH affected each partial the same. But there are two other effects of iH that have an opposite effect. Next, iH increases as we go up the scale. So the iH is greater for the upper note of the interval. This effect causes the wide intervals to beat faster and the narrower intervals to beat slower than if the iH was the same for both notes. Finally is the octave stretch. The octave is tuned wider than theoretical due to iH. This means that each interval is also wider than theoretical and wide intervals will beat faster and narrow intervals will beat slower than if the octave was theoretical.

You wrote:

”I would like to ask you all:

1 - is it possible to have progressive M6’s (4th+ M3) without a correct ET progression of 4ths, meaning without a correct ET 4ths theoretical and practical progression? I would answer no. Actually, if we had theoretical stretched octaves we could, in fact today with Chas we can.

2 – if we can not get progressive M6’s, what happens to m3’s and how can we get progressive stretched octaves without progressive 6ths (4th+ M3)?

Maybe answering these question explains why we were in need of accuratelly theoretically stretched 4ths.

You ask: “Now, according to you, why do fifths invert going up the scale?”

Because if 5ths were not to invert, goin up the scale 5ths would unconveniently diverge from stretched octaves.


Answer to #1: You can have 4ths beat slower, remain the same speed, or beat faster and still have M6s beat progressively faster. In fact, since 4ths are 2 cents from just and M6s 16 cents from just, there is enough leeway for the beat speed of 4ths to become faster and slower and faster again while M6s remain progressive. Chromatic M6s need only be within 1 cent of each other for their beat speeds to be progressive while chromatic 4ths must be within 1/8 cent.

Answer to #2: since m3s are inversions of M6s, unless octave widths vary, they will be as progressive or as unprogressive as M6s.

When we only consider non-iH tones, and an octave is stretched to 1203.5 cents wide, 5ths become 702 cents wide and would be just. Any more octave stretch will produce 5ths that beat wide. However when we consider iH tones, unexpected things happen.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/04/09 01:30 PM

Wanting to share Chas theory’s model, I dedicate this paper to Giuseppe Sciurti, to Bill Bremmer and to all colleagues who are not accustomed to using maths symbols.

Think about traditional ET ratio 12th root of 2.
Say you want all intervals to be ET progressive.
Say you want an equal beating (EB) on 12ths (narrow) and 15ths (wide).

You already know that theoretical ratio 12th root of 2 will not satisfy your needs, since those 15ths are theoretically beatless. You also know that the only way to have all intervals going ET progressive, and equal beating 12ths and 15ths is to stretch your 12th root of 2 ratio. So you are thinking in terms of:

(12th root of (2 + widestretch)) (1)

Your experience tells you that P12’s (pure 12ths, ratio 19th root of3) would give you too wide octaves and 3ths, 10ths and so on, this is why you want 12ths a little narrow, so you think at 19th root of (3 – stretch), while you want P15’s (pure double-octave, ratio 24th root of 4) to beat equally, say 24th root of (4 + stretch). So you conclude that, in order to have an equal beating on 12ths and 15ths, you can write:

19th root of (3 – stretch) must equal 24th root of (4 + the same stretch). (2)
This is our Chas ET EB algorithm. In fact saying: stretch = same stretch = Δ you can write:

(3 – Δ)^(1/19) = (4 + Δ)^(1/24) (2.2)

With some calculation you realize that:
For stretch = same stretch = Δ = 0.00212538996469 your conclusion (2) is true:

19th root of (3 – 0.00212538996469) equals 24th root of (4 + 0.00212538996469) =
= 1.0594865443501 = new scale incremental ratio

Now, having achieved your aim, with your new ET EB scale’s incremental ratio you can calculate all your scale’s frequency values. You can also realize what precise numerical terms you were initially thinking in. Back to (1),

(12th root of (2 + wide-stretch)) is now (12th root of (2 + 0.00053127692738))

You know that your scale ratio leads to a unique synchronic event, in fact your 12ths beat (narrow) at the same beat-rate of your 15ths (wide). No other esponential scale ratio will lead to the same 12ths and 15ths scale synchronic event.

What happened was that, instead of making use of only partial 2, say only one string, you used partial 3 and partial 4, i.e. two strings.

Arguing from analogy, for decades you had only flown your single string kite, today you can fly a kite with two strings and have it perfectly still in the wind. Soon you discover your passion for aerial acrobatics, the pleasure of pulling one of your strings, say your right string, and seeing your kite happily swinging for you. So you go back to your previous conclusion (2):

19th root of (3 – stretch) must equal 24th root of (4 + the same stretch)

and include your dynamic desire for swinging:

19th root of (3 – stretch) must equal 24th root of (4 + (the same stretch times swinging)). (3)
This is our Chas ET EB dynamic algorithm. In fact having said: stretch = same stretch = Δ and now saying: right swinging = s you can write:

(3 – Δ)^(1/19) = (4 + Δ*s)^(1/24) (3.3)

Once you decide to make use of your left string, you go back to your last conclusion (3) and improve it:

19th root of (3 – (stretch times left swinging)) must equal 24th root of (4 + (the same stretch times right swinging)).(4)

The latter is our Chas ET EB dynamic algorithm improved. In fact having said: stretch = same stretch = Δ , right swinging = s and now saying: left swinging = s1 you can write:

(3 – (Δ*s1))^(1/19) = (4 + (Δ*s))^(1/24) (4.4)

These are the only symbols we are using:

stretch = same stretch = Δ = ever different unknown value
right swinging = s = discretional variable
left swinging = s1 = discretional variable

Let’s see what happens to some of our theoretical pure scale incremental ratios, for istance those ones more frequently mentioned and deriving from 12th root of 2 (considering pure partial 2), and from 19th root of 3 (considering pure partial 3).

Scale incremental ratio 12th root of 2 = 1.059463094359

Does Chas algorithm include this ratio?

Using equation (4.4) = (3 – (Δ*s1))^(1/19) = (4 + (Δ*s))^(1/24)

s1 = 1
s = 0
Δ = 0.0033858462466

(3 – (0.0033858462466*1))^(1/19) = (4 + (0.0033858462466*0))^(1/24) =
= 1.059463094359 = Scale incremental ratio 12th root of 2

Scale incremental ratio 19th root of 3 = 1.0595260647382

Does Chas algorithm include this ratio?

Using equation (4.4) = (3 – (Δ*s1))^(1/19) = (4 + (Δ*s))^(1/24)

s1 = 0
s = 1
Δ = 0.0057097695742

(3 – (0.0057097695742*0))^(1/19) = (4 + (0.0057097695742*1))^(1/24) =
= 1.0595260647382 = Scale incremental ratio 19th root of 3.

As for theoretical partials 2 and 3, all partials can gush out of Chas algorithm, so we may finally say where all pure ratios find home.

Maybe this proves that the Chas theory describes the first comprehensive harmonic ET model. Now we can play all partials the way we like, no matter what preference, and we can look into practice for our most sought after sound whole.

Please, to quote single lines of this paper use quotation marks.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/04/09 02:00 PM

Tooner,

thanks for your answer.

"I am sorry. I do not think that the Chas ratio is a “gem”."

I well konw that, you only said ...looking for..., only I was surprised for the word you used.

I'm copying your post so that I can read it calmly and answer you.

Meanwhile, would you tell me what you think of symbol-free Chas? Does it help?

There is a fenomenon that I do not really understand, how is it possible to take a lame theory inside and out, one minute referring to it and the minute after negating it. Now theoretical wrong value from traditional ET have a sense, the minute after they do not.

Another very funny thing is how Chas seems not be idoneous for non-iH tones (organs), nor for iH tones. Time.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/04/09 02:02 PM

Alfredo:

That was a much more understandable than your original paper. Very Good!

But once you have the frequencies, what do you do with them? (Your devil's advocate is asking this.)
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/04/09 02:35 PM

Alfredo:

We cross posted each other.

”There is a fenomenon that I do not really understand, how is it possible to take a lame theory inside and out, one minute referring to it and the minute after negating it. Now theoretical wrong value from traditional ET have a sense, the minute after they do not.

Because if we take the beat rates (or at least the ratio between beat rates, including equal beating) that are predicted from a frequency ratio (such as 2^1/12) that does not take into account iH, and then tune a piano with iH using the beat rates we end up with a decent tuning, but a different frequency ratio, one that is non-linear.

So on the one hand, the frequency ratio is wrong, but on the other, the beat rates are correct. And since when tuning aurally, we listen to beat rates, the model works even though it is incorrect. Much like when we think of the sun rising and setting, and there are 365 ¼ days a year, the earth actually spins on its axis 366 ¼ times a year. (Like I said I am a bit of a “fool on the hill”.) And to take the analogy a bit further, in high altitudes the time of moonrise and moonset can change in unexpected ways due to changes in declination, much like the beat rates of some intervals do in the high treble due to iH.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/05/09 07:05 AM

Sorry Folks, made an error:

"And to take the analogy a bit further, in high altitudes the time of moonrise and moonset can change in unexpected ways due to changes in declination, much like the beat rates of some intervals do in the high treble due to iH."

I should have used the word "latitudes" instead of "altitudes". The sentence should be:

"And to take the analogy a bit further, in high latitudes the time of moonrise and moonset can change in unexpected ways due to changes in declination, much like the beat rates of some intervals do in the high treble due to iH."

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/06/09 03:35 PM

Tooner, excuse me if I needed to go back a few posts. Referring to Chas ratio you wrote:

“Like any fixed ratio, it cannot describe how a piano is actually tuned.”…

In my opinion you may be negating, or maybe only undervaluing Chas theory on the basis of your/our xperience with traditional ET and consequent iH calculation.

…”Theoretical ET has the same problem.”…

Sorry if I seem to be fussy but, as I have well explained, traditional ET is a lame pseudo-model, Chas is an impeccable theory that comes from practice.

…”However, if we look at the beat rates that a fixed ratio (ignoring iH) predicts, then the beat rates, (or at least the ratios between beat rates) can be used as a model for an aural tuning.”…

Ok, fair enough, but be aware that the model we have been using so far (pure octaves) was confusing and misleading.

…”In other words, the Chas ratio describes how the beats of 12ths and 15th can be equal, but if the Chas ratio is used to determine the frequencies, this will not be the result.”…

Chas model gives you the chance to control any frequency curve, so that you can get to any interval's beat curve or vice versa. Chas theory, for the first time in 2500 years, gains the frequencies scale on the basis of two constant differences, i.e. a “beat ratio”, actually a bi-frontal ratio since it can perfectly proportionate frequencies too. Does this make a difference?

You say:...”And there still is the problem of predicting 5ths becoming wide and 12ths staying narrow. I don’t believe this is possible.”...”It seems that fifths could not become wide going up the scale, unless 12ths become wide first, which does not happen with mindless octaves, Chas tuning, nor perfect 12ths. Since a 12th is made from a fifth and an octave, the only way to have a wide 5th and have a 12th that is pure or narrow, is to have an octave that is narrow while the fifth is wide.”...

Ok, but you could also think a 12th as being made of 5th + 4th + 5th. What happens if they chromaticaly go: narrower + wider + narrower, narrower + wider + narrower, narrower + wider + narrower, untill 5ths in midrange invert so that 5th + 4th + 5th can go: less narrow + wider + less narrow? Can 12ths remain narrow-constant, and can 15ths remain EB wide-constant in this way? Let me pass you the answer: yes.

...“And as I continue to try to understand your paper I read the statement of “s=s/s1” (which can only be true if s1=1, but then what is the point?) and there is no explanation of what units s and s1 are in, nor how s and s1 are determined, nor why s/s1 must be a rational number. Since I am not interested in tuning as you do, the effort becomes too difficult to try to understand how you “get there” from your Chas ratio.”...

Has the kite analogy solved the hairy question about the use of “s” variable?

...”But besides that, I can find nothing in your paper that goes from a fixed ratio to ever widening octaves.”...

Please, check figures in Chas article, section 4.2. In section 3.2, you can see that the linear ratio regards the 1:1 difference proportion for 12ths and 15ths: …“The 1:1 proportion of the differences related to intervals 0-19 and 0-24 is constant for all degrees 12 and 15. Their ratio, in this exponential scale, expresses a constant of linear proportionality which we find in the chromatic combinations (1-20, 1-25) – (2-21, 2-26) – (3-22, 3-27) etc.” You would be looking for a non linear octave incremental-ratio, maybe a non linear octave difference-ratio will do. These are Chas standard ratio’s effects on 7 octaves:


ET octaves Cents - Chas octaves Cents - Chas-ETdifferences
1200 1200,45982128266 0,45982128266405
2400 2400,91964256533 0,91964256532810
3600 3601,37946384799 1,37946384799216
4800 4801,83928513066 1,83928513065621
6000 6002,29910641332 2,29910641332026
7200 7202,75892769598 2,75892769598431
8400 8403,21874897865 3,21874897864836

Here you would have seen a graph, but this window refuses it.

Chas-ET differences Ratio, i.e. Diff.7 : Diff.6 and so on

1,1666666666667
1,2000000000000
1,2500000000000
1,3333333333333
1,5000000000000
2,0000000000000

One more graph missing.

About iH you kindly wrote:...“The difference in cents that the fundamental frequency differs from the theoretical frequency is the string’s inharmonicity constant or inharmonicity coefficient.”...

Since traditional ET theoretical frequencies derive from two unjustified assumptions, string’s inharmonicity constant or inharmonicity coefficient may need to be corrected, would you agree?

...” The iH constant will double around every 8 semi-tones or more. So in the treble, not only do the strings have a higher iH constant, but the iH constant increases more and more. The same thing occurs in the bass with the iH increasing toward the bass.”...

In Chas article, section 4.3 you read: “In the equal temperament scale, based on a ratio of 2, octave intervals have zero differences. As a direct consequence, the differences for partials other than 2 have ratios which are multiples of 2. The differences, divided by themselves, have a quotient of 2 for combinations 0-12, a quotient of 4 for combinations 0-24, and so on. With the exclusion of partial 2 and its multiples, the difference curves relating to all the other partials move away from each other exponentially in a monotone curve.” If you confront this with what you have stated above you can understand when, in calculating iH, confusion may have taken place.

About iH, thank goodness and thank you, you also said:

“Not too long ago I realized how the effects of iH are largely self-correcting on the theoretical beat rates of intervals.”...”the first effect of iH is that wide intervals beat slower and narrow intervals beat faster than if the iH affected each partial the same. But there are two other effects of iH that have an opposite effect. Next, iH increases as we go up the scale. So the iH is greater for the upper note of the interval. This effect causes the wide intervals to beat faster and the narrower intervals to beat slower than if the iH was the same for both notes.”...

So, this is how iH effects are somehow self-correcting. You end up saying:

“Finally is the octave stretch. The octave is tuned wider than theoretical due to iH.”…

As I’m saying, this is the wrong initial assumption that has taken to wrong iH calculation. Chas model shows how the interweaving of partial 3 and partial 4, gives us the most logical reason for natural octaves stretching. How can we calculate iH moving from wrong premises? Evaluating this + iH self-correcting effects could be as convenient as admiting 2 + 2 = 4.

About 4ths and M6 you wrote:...“You can have 4ths beat slower, remain the same speed, or beat faster and still have M6s beat progressively faster.”...

I quite agree, but my challenge has been finding all intervals precise and univocal beats incremental curves, the only reality that would prove the sound set being perfectly coherent.

...”In fact, since 4ths are 2 cents from just and M6s 16 cents from just, there is enough leeway..."

The game I played did not admit any leeway. This is to say that it was not enough M6s beating progressively faster, actually like for all intervals, also this progression were to be justified by an inner smooth and proportional beats increase. For istance, you think at 12th as the result of a 5th+ octave, which is ok. Yet, as I may suggest, you could try to think at 12ths as three 4ths + M3, set your leeway at zero and see what happens.

No leeway for years and, traslating Chas beat constants and curves into figures, I discovered that Chas maths could manage any number to the infinite decimal point. Do you know how previous temperaments have managed 4th ratio 4/3 = 1.333…? Doing so that 3/2*4/3 = 2, which is simply false.

I would really like to know from Bill, Jeff S., Kent, BDB, Bobranyan, and why not, from our cello expert and our academical style and maths error expert.

Have a nice sunday, regards, a.c.
Posted by: Robert Scott

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/07/09 08:21 AM

alfredo,

In reply to Jeff's:

“The difference in cents that the fundamental frequency differs from the theoretical frequency is the string’s inharmonicity constant or inharmonicity coefficient.”...

You wrote:

"Since traditional ET theoretical frequencies derive from two unjustified assumptions, string’s inharmonicity constant or inharmonicity coefficient may need to be corrected, would you agree?"

The definition of the inharmonicity constant (which Jeff correctly cited) is an intrinsic property of a string, like the "length" or the "thickness". It does not depend on which tuning system is being used (ET or Chas or anything else). It affects the outcome of the tuning, but the tuning does not affect the inharmonicity. So when you say that the "inharmonicity constant or inharmonicity coefficient may need to be corrected", I must disagree.

Robert Scott
Ypsilanti, Michigan
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/07/09 03:54 PM

Mr. Scott:

Thank you for commenting on Alfredo’s error. I am not sure if he knows that you wrote the Tunelab ETD program.

I hope if I write anything in error you will also point it out. The more I learn about the subject, the more I realize there is even more to learn.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/07/09 04:09 PM

Alfredo:

I am having a very nice weekend, thank you, and hope you are too.

I have tried to understand you paper and cannot. So, quoting from it does not help me. However, if you re-word your points like you did in an earlier post I may have a chance. Referencing diagram numbers would be appropriate, though.

I expect to be too busy to make any long posts until later in the week, so please don’t think I am ignoring you.

I do have a challenge for you, though. You wrote:

”Chas model gives you the chance to control any frequency curve, so that you can get to any interval's beat curve or vice versa. Chas theory, for the first time in 2500 years, gains the frequencies scale on the basis of two constant differences, i.e. a “beat ratio”, actually a bi-frontal ratio since it can perfectly proportionate frequencies too. Does this make a difference?“

Very well then, assuming an iH constant of 0.1 for C3 that doubles every 8 semi-tones (and to make it easy, lets continue this down to A0) and I desire a tuning that results in all octaves beating ½ bps wide at the 2:1 partial match, how would the CHAS algorithm be used to determine the fundamental frequencies of the tuning?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/08/09 02:03 PM

Robert,

thanks for what you have written.

You say: "The definition of the inharmonicity constant (which Jeff correctly cited) is an intrinsic property of a string, like the "length" or the "thickness". It does not depend on which tuning system is being used (ET or Chas or anything else). It affects the outcome of the tuning, but the tuning does not affect the inharmonicity."

Can I make three questions?

Say we can tune both non-iH tones and iH tones

Say we tune frequency value A4 = 440.0 Hz

Say that, looking for A4 higher octave interval, we tune frequency A5 = ? Hz

Say that, looking for A4 lower octave interval, we tune frequency A3 = ? Hz

Say we tune, from A3 to A5, two whole equal-tempered octaves i.e. 24 semi-tones

Question n.1: what A3 - A5 frequency values would you expect whith non-iH tones?

Questions n.2 and n.3: what A3 - A5 frequency values increase would you expect with iH tones? can you refer to an avarage increase?

Tooner,

you say: "I expect to be too busy to make any long posts until later in the week, so please don’t think I am ignoring you."

I could never think that, you are helping so much, take your best time.

You ask: ..."how would the CHAS algorithm be used to determine the fundamental frequencies of the tuning?"

To answer you, please let me wait for your answers. Despite dozines of pages anyone can read about iH, I have not made up my mind yet.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/08/09 02:35 PM

Alfredo:

It is wise of you to wait for the answers. You should have a very different perspective when you understand iH more.
Posted by: Robert Scott

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/08/09 03:18 PM

Originally Posted By: alfredo capurso
Robert,
Question n.1: what A3 - A5 frequency values would you expect whith non-iH tones?

I guess by "non-IH tones" you are talking about tones generated by an instrument like a pipe organ that does not have any inharmonicity. And I guess by ET you mean the geometric sequence that exactly doubles every 12 steps (although the term ET is also properly applied to a sequence that more than doubles, as in piano tuning with stretch). If that is what you mean, then A3=220 Hz exactly and A5=880 Hz exactly, assuming the goal was to tune the octaves beatless.

Quote:

Questions n.2 and n.3: what A3 - A5 frequency values increase would you expect with iH tones? can you refer to an avarage increase?

Your question is not well-defined. In the presence of inharmonicity, there is no single agreed-upon implementation of "two whole equal-tempered octaves i.e. 24 semi-tones" as you say. Some say that the octaves should be beatless 4:2 octaves. Some say they should be slightly wide of just. And they are both right. ET encompasses any sequence of pitches where the ratio of consecutive pitches is the same. You can have more stretch or less stretch and still call it ET. So I don't know what false assumptions you are attributing to ET. As far as I know, ET does not make any assumptions at all.

If what you are asking is "what effect will IH have on the tuning of A3-A5", then the answer is generally that more IH will cause any implementation of ET to have more stretch than it would if there were less IH.

Robert Scott
Ypsilanti, Michigan
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 06:26 AM

Robert,

I've copied your answers and I'll try to better define my definitions and questions. Anyhow, I think you got my definitions and I think your answers conferm how iH and ET are generally understood .

Whith my next post I'll also try to be more precise about what you have written:

"I guess by "non-IH tones" you are talking about tones generated by an instrument like a pipe organ that does not have any inharmonicity."

"And I guess by ET you mean the geometric sequence that exactly doubles every 12 steps..."

..."If that is what you mean, then A3=220 Hz exactly and A5=880 Hz exactly, assuming the goal was to tune the octaves beatless."

"So I don't know what false assumptions you are attributing to ET. As far as I know, ET does not make any assumptions at all."...

For the time being we could try to share a well known fact: traditional ET pseudo-model calculates 2:1 octave ratio. This is the first wrong assumption I'm talking about.

As you say, "ET encompasses any sequence of pitches where the ratio of consecutive pitches is the same. You can have more stretch or less stretch and still call it ET."

Here you are talking about ET referring to a geometrical sequence and exactly here you can find the second wrong assumption I'm talking about: traditional ET pseudo-model sequence does not intermodulate octaves. Is there a language distabce in saying that?

Regards, a.c.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 07:17 AM

Alfredo,

It is usually highly recommendable to do research what others have said about the matter, before trumpeting out revolutionary ideas.

Your claim that CHAS ET (equal temperament) is the first ET model that questions the the base of pure octaves for ET is wrong.
Serge Cordier published work about an equal temperament on pure fifths in 1982. (Accord bien temperé et justesse orchestrale, S. Cordier 1982)
So it is his merit of having questioned octave based ET first and replaced it with a different theoretical model.

Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET). The solution you provide as a "discovery" of symmetry is rather an invention for me. There are plenty of other "symmetric" solutions available. For example one could split the pythagorean comma in two equal parts and subtract one part from the twelve fifths and add the other part to the 7 octaves. But such symmetric methods must not end in symmetric beat patterns. The resulting numbers of CHAS in the beat tables don´t show a higher degree of symmtry than let´s say standard ET or Cordier ET to me. (See your table of different ET beat rates)

I also disagree with your claim that ET based on pure duocimes (twelfths) does spoil symmetry. The opposite is true: To verify, generate a beat table for a 19th root of three table (which you have left out in your paper). You WILL find symmetries of highest degree in the resulting beat tables, which are not present in other ET solutions (including yours). And THAT is a discovery (discovered in 2004, as is wrote in an earlier post of this thread).

From your writing about iH, it is obvious that you are not familiar with what an inharmonicity constant is. State of the art of available theoretical tuning models (which are in use for example in advanced electronic tuning devices) CAN handle with inharmonicity. You claims are once more wrong relating this fact, so you may not claim that CHAS is the first tuning model that can handle inharmonicity.

Finally your paper does appear to be an official paper of the university of ..., but i don´t find any names of representatives of the university. That makes your paper very suspect to me.

All your claims appear in those contexts as overblown and whithout reliable fundaments.

Best Regards,

Bernhard Stopper







Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 09:32 AM

Mr. Stopper:

I have to agree with most of what you posted. I cannot, of course, agree with what I do not know. I am still looking forward to your paper and hope that your “Super Symmetry” is more understandable than Alfredo’s “Synchronic Attractor”.

Hopefully, a week from now Alfredo will understand iH much better.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 10:36 AM

Originally Posted By: UnrightTooner
Mr. Stopper:

I have to agree with most of what you posted. I cannot, of course, agree with what I do not know. I am still looking forward to your paper and hope that your “Super Symmetry” is more understandable than Alfredo’s “Synchronic Attractor”.

Hopefully, a week from now Alfredo will understand iH much better.


As i wrote, you can verify it on your own by generating a 19th root of three beating table (Quite easy with todays spreadsheet programs)

I usually do not post material of my own work into a thread of someone others, but a small example of the apparent symmetry in the 19th root of three ET whithout inharmonicity is:

Octave on A4: 0.627
Fifth on D4: -0.627
Fourth on A2: 0.627
Fifth on D3: -0.313
Octave on A3: 0.313
Fourth on A3: 1.255
etc., i.e. beats have (rounded) same values or integer or fractional values of themselves. Such relations are apparent in many ET solutions, but only in the case of 19th root of three this is true for all combinations of octaves fifths and fourths (with distances of octaves, fourths and fifths between them) etc.

Regards,

Bernhard Stopper
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 11:39 AM

Mr. Stopper:

I think it is OK that you piggybacked the Topic because sooner or later the subject of equal beating tests will come up.

Whenever an interval is tuned beatless, there will be intervals with identical beat rates. But when it is decided to keep an interval beatless throughout the keyboard, there is no leeway for different preferences of stretch.

Also, the last beat rate that you mentioned was: “Fourth on A3: 1.255” I suppose the point is that the beatrate is double 0.627. But it would only be exactly double if there was no iH. So, your theory seems to have a similar problem as Alfredo’s.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 12:01 PM

Originally Posted By: UnrightTooner
Mr. Stopper:

So, your theory seems to have a similar problem as Alfredo’s.


This is not the point.
My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not.

The point is, that the acoustic effects caused by the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.

(The fact that fourths can appear to beat at the same rate throughout the keyboard has different reasons. In fact they do not, they only can appear to do so.)

Regards,

Bernhard Stopper
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/09/09 12:18 PM

I look forward to your paper.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/10/09 11:06 AM

Alfredo:

I have some time to reply to your posts now.

You wrote two things that seem to contradict each other:

”Ok, but you could also think a 12th as being made of 5th + 4th + 5th. What happens if they chromaticaly go: narrower + wider + narrower, narrower + wider + narrower, narrower + wider + narrower, untill 5ths in midrange invert so that 5th + 4th + 5th can go: less narrow + wider + less narrow? Can 12ths remain narrow-constant, and can 15ths remain EB wide-constant in this way? Let me pass you the answer: yes.”

”About 4ths and M6 you wrote:...“You can have 4ths beat slower, remain the same speed, or beat faster and still have M6s beat progressively faster.”...

I quite agree, but my challenge has been finding all intervals precise and univocal beats incremental curves, the only reality that would prove the sound set being perfectly coherent.”


You seem to say that the beat rate of all intervals must be progressive, but you also say that 5ths can first be narrower, but then later less narrower (and eventually wide).

There is an answer to such a phenomena, but it requires that all the effects of iH be taken into account.

You asked Mr. Scott for an example, but he did not seem to understand what you wanted to know. I think I understand what you want to know because I have been down this road recently. Much is said about iH, but very little is actually shown. I will work up some figures to give an example and post them within a few days.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/10/09 02:59 PM


Tooner, do not you think it is nice we can read again from Bernhard?

Anyway, you say: “You should have a very different perspective when you understand iH more.”

Konwing you are lovingly saying this, I need to tell you that what gave me a different perspective was a new approach to the scale frequencies values, it was looking for a sound scale’s beat-period’s ratio (or beat rate ratio, if you prefere), something had never been sought after, while I can quietly and friendly tell you that, what has been said so far about iH is common knowledge.

On my part, I prefer not thinking my self as being an iH expert mainly because I think that, the way tuning difficulties have been related to iH has been messed up by a bottom problem that has got nothing to do with iH, i.e. the 2:1 theoretical octave ratio. This is to say: iH exists but tuning difficulties do not derive only from iH.

You said: “…”However, if we look at the beat rates that a fixed ratio (ignoring iH) predicts, then the beat rates, (or at least the ratios between beat rates) can be used as a model for an aural tuning.”

Well, you can then be very happy since the Chas theory’s model is, for the first time, suppling us with a ratio between beat rates (12ths and 15ths), actually a ratio that you could modify by using “s” variable. I’ll be here just waiting for you (all) to realize it. My answers to your latest posts will follow in this topic post's order. Thank you.

Robert, you say: “I guess by "non-IH tones" you are talking about tones generated by an instrument like a pipe organ that does not have any inharmonicity.”

I thank you because your answers to my questions gives us a good chance to check the accuracy of our premises.

“...a pipe organ that does not have any inharmonicity…” Is what you are saying 100% true? Thinking theoretically, what about air pressure? Thinking in practice, have you heard organ’s 12ths and 19ths? Did you like them?

By "non-IH tones" I’m talking about tones generated by a digital instrument, i.e. tones resulting from a single sinusoid.

You say: “...And I guess by ET you mean the geometric sequence that exactly doubles every 12 steps...”

It is so when I mention “traditional ET pseudo-model”.

You say: ...“(although the term ET is also properly applied to a sequence that more than doubles, as in piano tuning with stretch).”

Correct, these are ET sequencies.

You say: “If that is what you mean, then A3=220 Hz exactly and A5=880 Hz exactly, assuming the goal was to tune the octaves beatless.”

So you are saying: ...“assuming the goal was to tune the octaves beatless” you would refer to traditional ET ratio 2:1.

We better stop for some reasoning, I hope you wont mind. For tuners, ET would also mean “progressive 3ths, 6ths, 10ths and so on”, what we can not get from traditional ET pseudo-model, while our-days ET means “progressive 3ths, 6ths, 10ths and so on + octave stretching”.

A few posts ago I highlighted a foundamental detail: unless you correctly and univocally stretch 4ths, you will not be able to obtain the correct progression of all scale’s intervals.

Why? Because 4th interval is the base component for 6ths (4th + M3), octaves (4th + 5th), 10th (4th + 5th + M3) and so on. As Tooner says, we may somehow stretch octaves and, by playing with leeways, obtain progressive M3’s and M6's, M10’s and so on but, those leeways will mess up our 4ths, 5ths and octaves progressions (with us thinking that this happens because of iH).

In other words, even if you can get progressive M3's, M10's and M17's, unless you order progressive 4ths correctly you will not get progressive 5ths nor progressive octaves.

What is normally thoght is that iH does not allow you to put traditional ET pseudo-model in practice, Tooner its self conferms that, saying …“The octave is tuned wider than theoretical due to iH…”.

What I’m stating is: even thinking to non-iH tones, traditional ET pseudo-model can not give you progressive intervals.

Why? Because basic ratio 2:1 is wrong. In fact, basic ratio 2:1 crushes all other intervals ratios, so that you could regularly only find a double ratio on any other clone-copied octave. We still call this a compromise, but really we should call it a ratio crushing. Traditional ET pseudo-model 2:1 ratio reads too much into the scale sound’s ratios.

Let’s go back to you saying: ““assuming the goal was to tune the octaves beatless”.

You may now understand that we have no reasons for tuning the octaves beatless, if not to refer to the ET first sequence, the one coming from a lame ET pseudo-model. You may also understand that, from Chas model’s natural interweaving of ratios 3:1 and 4:1 you get stretched octaves, before what iH could impose, i.e. with or without iH. So, my answer to question 1 is:

A3 = 219.94157505789133667677
A5 = 880.23376184808211425507

You then kindly say: “In the presence of inharmonicity, there is no single agreed-upon implementation of "two whole equal-tempered octaves i.e. 24 semi-tones"...”

I’m not surprised.

“...Some say that the octaves should be beatless 4:2 octaves. Some say they should be slightly wide of just. And they are both right.”

In my opinion, they are both – you choose the word – bewildered? Lost? Puzzled? Disconcerted? Mixed up?

And why can be so? Because they can not rely on a solid stretched octave theory.

“...ET encompasses any sequence of pitches where the ratio of consecutive pitches is the same. You can have more stretch or less stretch and still call it ET.”

True, so true that with Chas algorithm you can finally draw any kind of stretch. Was'nt the kite analogy clear enough?

“...If what you are asking is "what effect will IH have on the tuning of A3-A5", then the answer is generally that more IH will cause any implementation of ET to have more stretch than it would if there were less IH.”

You say: “…more iH…”, “…less iH…”. What I think is that, iH experts should calculate iH on the base of correct premises, something that up to now, on the basis of a lame pseudo-model, could not happen.

All this to say that in stead of unhappily submiting to iH, when we think in terms of euphonicity, what ever preference, we should take the chance to refer to a solid, reliable and comprehensive theory.

Bernhard,

how did you know that I play trumpet?

I'll answer your post in this topic's order, meanwhile I'd only kindly ask you to stop insinuating about the authenticity of Chas article. In case it was your favorite sport, check with your lawyers how far you can go. I understand that something is disturbing you very much, what I can tell you for the time being is that Chas model is meant to give, not to steal. So, please, calm down and stop celloing about.

Regards, a.c.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/10/09 03:46 PM

Originally Posted By: alfredo capurso



Bernhard,

how did you know that I play trumpet?

I'll answer your post in this topic's order, meanwhile I'd only kindly ask you to stop insinuating about the authenticity of Chas article. In case it was your favorite sport, check with your lawyers how far you can go. I understand that something is disturbing you very much, what I can tell you for the time being is that Chas model is meant to give, not to steal. So, please, calm down and stop celloing about.

Regards, a.c.


Alfredo,

You have posted your paper officially and i think you appreciate any comments if they are critical or not.

What is disturbing me, are the pathetic claims in your paper, which seem to be caused by incomplete research about prior art.

And no, i did not insinuate about the authenticity about your paper. I just wondered why nobody of the university staff has co-signed your paper. They usually do no not use such a pathetic language, what may be a reason for this.

Best Regards,

Bernhard Stopper

Posted by: Robert Scott

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/10/09 05:09 PM

Originally Posted By: alfredo capurso

Robert, you say: “I guess by "non-IH tones" you are talking about tones generated by an instrument like a pipe organ that does not have any inharmonicity.”
...

“...a pipe organ that does not have any inharmonicity…” Is what you are saying 100% true? Thinking theoretically, what about air pressure? Thinking in practice, have you heard organ’s 12ths and 19ths? Did you like them?

Clearly you have in mind a definition of the word "inharmonicity" that is different from what the rest of the world means by "inharmonicity". But in terms of the what the rest of the world means by that word, yes, it is 100% true that a pipe organ does not have any inharmonicity. It has harmonics. Those harmonics are true. They are locked to the fundamental. If air pressure changes, then the pitch of the pipe will change, but so will all its harmonics, and they will remain locked. That is the definition of zero inharmonicity. Whether or not 12ths and 19ths sound good does not change this fact.
Quote:

By "non-IH tones" I’m talking about tones generated by a digital instrument, i.e. tones resulting from a single sinusoid.

Well, some digital instruments might generate a single sinusoid, but most of them generate a more complex waveform.
Quote:

...For tuners, ET would also mean “progressive 3ths, 6ths, 10ths and so on”, what we can not get from traditional ET pseudo-model..

You certainly can and do get progressive intervals from the traditional no-stretch ET model, which is the model I was refering to because I was answering your question about non-IH tones. And when non-IH instruments, like pipe organs, are tuned, they most often are tuned this way.

Now when it comes to instruments that have inharmonicity, like the piano, nobody uses the traditional no-stretch ET model. So you are criticizing a model that nobody uses for the piano anyway.
Quote:

...What I’m stating is: even thinking to non-iH tones, traditional ET pseudo-model can not give you progressive intervals...

Nonsense. Of course the intervals are progressive for non-IH tones. Just play any cheap electronic piano (I say "cheap" to make sure it does not simulate inharmonicity, which the expensive ones do). The beat rate of 3rds, 4ths, 5ths, etc. will be prefectly progressive. They will all increase as you go up the scale. (Unless you are inventing a new defintion for the word "progressive" too.)
Quote:

Let’s go back to you saying: ““assuming the goal was to tune the octaves beatless”.

I was not recommending it. I was just trying to guess what you meant by your "question #1".
Quote:

“...Some say that the octaves should be beatless 4:2 octaves. Some say they should be slightly wide of just. And they are both right.”

In my opinion, they are both – you choose the word – bewildered? Lost? Puzzled? Disconcerted? Mixed up?

I would say they are entitled to their own opinion.
Quote:

“...If what you are asking is "what effect will IH have on the tuning of A3-A5", then the answer is generally that more IH will cause any implementation of ET to have more stretch than it would if there were less IH.”

You say: “…more iH…”, “…less iH…”. What I think is that, iH experts should calculate iH on the base of correct premises, something that up to now, on the basis of a lame pseudo-model, could not happen.

As I said before, the IH of a piano string does not rest on any system of tuning, be it lame or otherwise. You say "IH experts" as if IH was some deep notion, accessible only to a few. It is an objective physical measurement that anyone can make with the proper equipment. They do not have to be an expert. By way of analogy, consider "length", which is another objective physical measurement that could be made of a piano string. Would you talk about "length experts"? No, anyone with a tape measure can measure the length. So anyone with an ETD can measure IH.

Robert Scott
Ypsilanti, Michigan
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/11/09 02:18 PM

Alfredo:

I am going to show how to mathematically tune A3, A4 and A5 to 4:2 beatless octaves on a theoretical piano while accounting for iH, show what the octave ratios are, and what the beat speeds of the 2:1 partial matches are.

The model I am going to use is the one mentioned in Young’s paper: iH = 0.1 at C3 and doubles every 8 semi-tones.

The iH of other notes can be determined by multiplying 0.10000 times (2 ^ (1/8)) ^ (the number of semitones above C3). A3 is 9 semitones above C3. So the iH of A3 = 0.1 * (2 ^ (1/8)) ^ 9. Dong likewise with A4 and A5 we have:

iH of A3 = 0.218101
iH of A4 = 0.616884
iH of A5 = 1.74481

We know that the 1st partial of A4 is 440Hz, but the theoretical fundamental is 0.616884 cents lower. We will first determine the theoretical fundamental frequency so that we can then determine the theoretical partial frequencies by multiplying by the partial numbers. Finally, the theoretical partial frequencies are raised a number of cents (determined by multiplying the iH times the square of the partial number) to find the actual partial frequencies.

Using the equation: a = b * (2 ^ (n/1200)) where “a” is the actual frequency, “b” is the theoretical frequency and “n” is the number of cents, we can determine the theoretical frequencies from the actual frequencies and visa versa. So 440.000 / (2 ^ (0.616884/1200) = 439.843. In other words, the theoretical fundamental frequency of a string with a 1st partial of 440.000 Hz that has an iH of 0.616884 cents is 439.843. Multiplying this theoretical fundamental frequency by the partial number “2” gives us a theoretical 2nd partial frequency of 879.686 (2 * 439.843). The iH correction in cents for a partial’s theoretic frequency is the iH times the square of the partial number. So 0.616884 * (2 ^ 2) = 2.46753 cents. Again using the equation: a = b * (2 ^ (n/1200)), but this time to determine the actual frequency, we have the actual frequency of the second partial = 879.686 * (2 ^ (2.46753/1200) or 880.940 Hz. Doing the same for the 4th partial of A4 we have:

1st partial of A4 = 440.000
2nd partial of A4 = 880.940
4th partial of A4 = 1769.43

Next, let’s “tune” A3 for a beatless 4:2 partial match. This means that the frequency of the 4th partial of A3 is at the same frequency as the 2nd partial of A4. In this case the frequency is 880.940. I chose a beatless 4:2 octave because it results in the lower P4 interval in an octave beating at the same speed as the upper P5 interval in the same octave. (If you have questions about why this is so, please ask!!!) Since we know that the 4th partial of A3 must be 880.940 and the correction is the iH of A3 times the square of the partial number, we can say that the theoretical frequency of the 4th partial of A3 = 880.940 / (2 ^ ((0.218101 * (4 ^ 2)) / 1200)) or 879.166. So then the theoretical 4th partial divided by 4 equals the theoretical fundamental, 879.166 / 4 = 219.791 Applying the iH correction we get the actual 1st partial frequency of A3 = 219.791 * (2 ^ ((0.218101 * (1 ^ 2)) / 1200)) or 219.818. Working out the 2nd actual partial the same way gives us:

1st partial of A3 = 219.818
2nd partial of A3 = 439.803
4th partial of A3 = 880.940

And starting with the 2nd actual partial of A5 = 1769.43 we have:

1st partial of A5 = 882.043
2nd partial of A5 = 1769.43

So now we can see what the octave ratios are. The 1st partial of A5 divided by the 1st partial of A4 = 882.043 / 440.000, or 2.00464. But the 1st partial of A4 divided by the 1st partial of A3 = 440.000 / 219.818, or 2.00165. They are not the same. So if we wanted to have beatless 4:2 octaves we would need to have an increasing frequency ratio.

Now let’s look at the beat speeds of the 2:1 partial matches. The 1st partial of A5 minus the 2nd partial of A4 = 882.043 - 880.940, or 1.1 bps. But the 1st partial of A4 minus the 2nd partial of A3 = 440.000 - 439.803, or 0.2 bps. This is more than double per octave, not what would be expected.




Alfredo, I took the time to post this for a number of reasons. First, I think you are ready to look deeper into how iH affects frequencies, beat rates and their ratios. Everything that I showed can be derived from Young’s paper, that you referenced in your paper. That bothers me a little and leads to my second reason.

I have been able to learn a great deal from this Forum. Others have corrected errors when they find them. By doing so, it maintains the integrity of the vast knowledge that can be found in past posts. I feel an obligation to help correct errors, also. Your paper and posts are in error because of your misunderstanding of iH.

And finally, I have never seen anywhere where the math was actually demonstrated to show how to calculate the effects of iH. I thought I ought to do so.

So my point is that fixed frequency ratios and expected beat speeds are mutually exclusive. In fact, you can have only one interval with an expected beat speed progression. All others will vary dependant on the piano’s iH. That is why I said very early in this Topic:

”The theory of piano tuning fascinates me, but lately I am realizing its usefulness is limited in aural tuning. Aural tuning is all about compromises, compromises that can be heard. They don’t need to be theorized to be heard, just listened to and accepted. I am thinking that the theory is really only necessary for designing a mathematical model so that ETDs can make the compromises without actually “hearing” them. And then there is the final limit on accuracy imposed by the pinblock and rendering points. Not to mention what the next passing thunderstorm may do to a tuning!"
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/11/09 03:49 PM

Robert, Tooner, thank you very much for your posts.

Bernhard,

You say: “It is usually highly recommendable to do research what others have said about the matter, before trumpeting out revolutionary ideas.”…

If not your style, I can at least share one non-revolutionary, recommendable and usual idea of yours.

...“Your claim that CHAS ET (equal temperament) is the first ET model that questions the the base of pure octaves for ET is wrong.”

You are missing the point, having developed my arguments I’m now more extensively explaining my logical, mathematical and practical reasons for sharing Chas theory, without really caring whether I’m the first or the last. For me, knowing I’m not alone on this path is only better. Anyway, before 1982, iH’s apparent need for stretched octaves questioned pure octaves first.

On the theoretical ground, what I’m proving is that, with octave’s 2:1 ratio, the sequence of difference quotients n/n+1 (section 4.5) cannot occur.

Why is this difference-quotients sequence important? Firstly, because Chas theorizes a dynamic sound-set and n/n+1 sequence helps to understand why beats have in Chas model the greater function. Secondly, because it proves that, the need for stretching octaves derives from a bottom question.

...“Serge Cordier published work about an equal temperament on pure fifths in 1982. (Accord bien temperé et justesse orchestrale, S. Cordier 1982). So it is his merit of having questioned octave based ET first and replaced it with a different theoretical model.”

Please adjust your throw, may Serge Cordier have recognised all its merits. Let me read his article, only then I’ll be able to value if he banally talked about one more pure ratio, like 19th root of 3, or like (why not?) square root of 9/8, or if he perfected what one would call a reliable (not just a different pure ratio) theoretical model.

If we look at frequencies from a dynamic point of view, the whole theoretical concept of pure ratios is useless, exactly like it would be useless – as you may know - trying to divide any lenth in two halfs without approximation. To zero beats is a theoretical absurd, it would be like zeroing a wave, or considering space as a plane. Instead you can theoretically set an equal beating, even if you may end up having to do with irrational ratios.

...”Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET).”

I agree and I'm glad you recognize that. If you had red the “kite analogy” (in this topic), you would have understood how from Chas model’s algorithm also ratios 4th root of 5/4, 28th root of 5, 31st root of 6 can gush out. This is how, mathematically, Chas model proves to be coming from a comprehensive theory.

...“The solution you provide as a "discovery" of symmetry is rather an invention for me.”

Maybe you meant to say “fantasy”. Anyway, what you can see in Chas article (section 3.5 - Effect of delta on incremental scale ratios), is not fantasies, it is absolutelly correct and unique.

...“There are plenty of other "symmetric" solutions available.”

Good that you say that. As you know though, the partials that Chas model interwaves in its standard version, i.e. partials 3 and 4, just after foundamental tone and partial 2, are more often the most intense.

...”For example one could split the pythagorean comma in two equal parts and subtract one part from the twelve fifths and add the other part to the 7 octaves. But such symmetric methods must not end in symmetric beat patterns.”

You see, I decided to rely on beats. Once I was reasonably sure I could find again and again the one precise beat-set form, having dismissed all cultural influences, I traslated this dynamic set’s form into numbers.

...“The resulting numbers of CHAS in the beat tables don´t show a higher degree of symmtry than let´s say standard ET or Cordier ET to me. (See your table of different ET beat rates).”

Please mention precise numbered sections, otherwise I can not understand what you are referring to. Also remember that Chas has been conceived in dynamic terms, where symmetries of the beating-whole could continuously change.

...“I also disagree with your claim that ET based on pure duocimes (twelfths) does spoil symmetry.”

What I’ve said is: pure 12ths produce to wide M3’s, M6’s, octaves, 10ths and so on. Contextually, I’m stressing on Chas model’s different approach to the concept of “purity”: …“Purity no longer derives from a single combination or from a pure ratio…” (section 2.0).

...”The opposite is true: To verify, generate a beat table for a 19th root of three table (which you have left out in your paper).”

Don’t worry, you can put it in your paper.

...“You WILL find symmetries of highest degree in the resulting beat tables, which are not present in other ET solutions (including yours).”

I'm not competing for the highest degree of symmetry. Anyway, what you could conferm, reading section 4.5, is that Chas model standard version’s quotients values (s = 1) reach the 7th decimal point of the n/n+1 sequence, what no other pure ratio does.

...”And THAT is a discovery (discovered in 2004, as is wrote in an earlier post of this thread).”

What happened to your capital letters? I hope you are not living in a state of constant apprehension, you should know, time adjusts everything, also your claims.

...”From your writing about iH, it is obvious that you are not familiar with what an inharmonicity constant is.”

You are right, I’m more familiar with beats.

...”State of the art of available theoretical tuning models (which are in use for example in advanced electronic tuning devices) CAN handle with inharmonicity.”

For example, only-pure ETD Bernhard Stopper’s device?

...”You claims are once more wrong relating this fact, so you may not claim that CHAS is the first tuning model that can handle inharmonicity.”

It is important that you have understood how Chas model can deal with iH, I don't care being the first or what. Strangely enough though you are confusing Chas theory’s model with a device.

...”Finally your paper does appear to be an official paper of the university of ..., but i don´t find any names of representatives of the university.”

I don't know what you are talking about. Have you suddenly forgotten how to search? Look for: g.r.i.m. university of palermo. You find that heading on every single page of Chas model’s article.

...”That makes your paper very suspect to me.”

You are free to suspect as much as you like, as long as you do not spend words inconsiderately.

...”All your claims appear in those contexts as overblown and whithout reliable fundaments.”

When you state something you should also explain why, otherwise you sound like a defamer. Overblown or underblown, once again this is not the point. As for reliability, Chas model, as you may know, is numerically correct in absolute terms. This, together with its uniqueness have led to Chas model article’s publication.

Tooner, thanks again for your iH calculations.

following Bernhard post you say:

“I have to agree with most of what you posted. I cannot, of course, agree with what I do not know. I am still looking forward to your paper and hope that your “Super Symmetry” is more understandable than Alfredo’s “Synchronic Attractor”.”

I like it when you play the devil’s advocate, more than when you play the echo. Please, personalize all your statements, so that our contributing wont fall into triviality.

To know more about “attractors” you'd better ask a physicist.

Bernhard, you wrote:

...“As i wrote, you can verify it on your own by generating a 19th root of three beating table…”…”… a small example of the apparent symmetry in the 19th root of three ET whithout inharmonicity is...”…”...only in the case of 19th root of three this is true for all combinations of octaves fifths and fourths (with distances of octaves, fourths and fifths between them) etc.”…

As I could tell you, with 19th root of 3 you get a fast beating octave, as your own symmetry exercise’s figures conferm.

You also say:…” My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not.”

If I were you I would start a new claims topic. There you could freely compare Chas theory’s model with your exercises figures. Meanwhile it may happen that you understand how Chas algorithm includes also 19th root of 3 ratio.

...“The point is, that the acoustic effects caused by the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.”

Untill you don’t explain on what basis you make your statements, I can only think your words as an act. Your experience should tell you that, a piano tuned using 19th root of 3 ratio sounds harsh. Anyway tell me please, on wich ratio do you think the piano will, after some playing or after some time, adjust?

Tooner,

You say: …“you also say that 5ths can first be narrower, but then later less narrower (and eventually wide). There is an answer to such a phenomena, but it requires that all the effects of iH be taken into account.”

You’d better figure out what happens.

Bernhard, you say:

..."What is disturbing me, are the pathetic claims in your paper, which seem to be caused by incomplete research about prior art."..."nobody of the university staff has co-signed your paper. They usually do no not use such a pathetic language, what may be a reason for this."

How strange, I've been listening to your claims. As for the rest, as I've said, I'm trying to share a new ET EB model that can deal with iH. Hope you do not mind if, despite my pathetic language, Chas theory's model can also include 19th root of 3.

Thanks a lot, a.c.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/11/09 09:15 PM

Quote:
Your experience should tell you that, a piano tuned using 19th root of 3 ratio sounds harsh.


I have already voiced strong disagreement with this statement. What is your point in repeating it?

Are you suggesting that I and my customers and those who have heard my posted recordings do not know what a well-tuned piano should sound like?

When will you make some recordings available to attempt to back up your statements? I look forward to hearing the consonance in your tunings that will suddenly make mine sound "harsh".

You posted your tuning sequence here with the a number of directions reversed. Wouldn't it be appropriate (and face-saving) to post a corrected sequence in its entirety?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/12/09 07:36 AM

Mr. Swafford:

The harshness of a tuning is subjective. I find pure 12ths create a harsh sound in the tenor and bass, but make a very nice sound in the mid treble. In the high treble I prefer a little more “zing” than pure twelfths will provide. The recordings of Pure Sound and pure 12ths tunings that I have been able to get from the internet agree with what I hear when tuning aurally.

I don’t think this is a well-tuned piano verses a poorly-tuned piano judgment. It is a merely a personal preference.

My primary interest in all this is the mathematics. Unfortunately, there is only a mention of iH in discussions on tuning theory and not an integration of it.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/13/09 11:12 PM

Quote:
The harshness of a tuning is subjective. I find pure 12ths create a harsh sound in the tenor and bass, but make a very nice sound in the mid treble. In the high treble I prefer a little more “zing” than pure twelfths will provide. The recordings of Pure Sound and pure 12ths tunings that I have been able to get from the internet agree with what I hear when tuning aurally.

I don’t think this is a well-tuned piano verses a poorly-tuned piano judgment. It is a merely a personal preference.


Right. It is pointless to have a "debate" that consists of "this tuning is harsh" followed by "no, it isn't."

I had hoped to hear exactly what intervals or chords sound harsh in a pure 12th tuning and what can alleviate that harshness, according to those who hear such harshness. I can't hear such harshness so I haven't a clue where harshness might be coming from.

Actually, my hypothesis is that my execution of Stopper's implementation of a pure 12th tuning would not sound "harsh" to any listener. One shouldn't automatically assume that various tunings claiming to be pure 12th ET are all of the same quality. I have worked hard to learn to execute the Stopper tuning.

Stopper has made some rather bold claims; the difference is, however, that unlike Capurso, he has made it possible for us to duplicate his tuning and verify the claims.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/14/09 11:09 PM

I can't begin to follow all of this discussion or comment on it but I have finally had the time to look up a PTG Tuning Exam Master Tuning to show that 5ths become pure and then wide in an example of ET in which optimum stretch is used in the 5th and 6th octaves and conservative stretch in the 7th. So, for what it is worth:

If any two notes which form a 5th belong to the same octave and are read on the same partial and have the same numeric value, they would be theoretically equal tempered 5ths (2 cents narrow). We don't expect to see that at all and we don't in a master tuning. If the upper note of the 5th is 2 cents larger than the bottom note, the 5th is pure (beatless or just). If the upper note is more than 2 cents higher than the bottom note, the 5th is widened.

On a Yamaha C5 the C5-G5 5th reads: C5: 0.3, G5: 2.2. That is almost pure. The E5-B5 5th reads: E5: 0.6, B5: 3.0. That is 0.4 cents wide. The C6-G6 5th reads: C6: 3.6, G6: 5.1. That is 0.5 cents narrow. The E6-B6 5th reads: E6: 5.2, B6: 12.8. That is 5.6 cents wide. The C7-G7 5th reads: C7: 12.3, G7: 19.6. That is 5.3 cents wide. The E7-B7 reads: E7: 21.8, B7: 25.6. That is 1.8 cents wide.

Regarding 12ths: The Master tuning reads all pitches from C5 to B7 on the 1st partial. Therefore, 12ths can be analyzed similarly. The C5-C6 12th reads: C5: 0.3, G6: 5.1. That is 2.8 cents wide. The E5-G6 12th reads: E5: 0.6, G6: 5.5. That is 2.9 cents wide. Between the 6th and 7th octaves, with the 7th octave being tuned as 2:1 octaves with the 6th, an amount which the majority of technicians feel sounds "flat" and unacceptable, the 12s are still surprisingly wide. The C6-G7 12 reads: C6: 3.6, G7: 19.6. That is a whopping 14 cents wide. The E6-B7 12th reads: E6: 5.2, B7: 25.6. That is a mind blowing 18.1 cents wide!

By the way, I had nothing whatsoever to do with the Master Tuning I just quoted from. It was done late last year by 2 CTE's and a CTE Trainee from Chicago. Both of the CTEs have served more than 10 years as such. One of them is the Chicago Symphony Orchestra Concert Technician and the other is an instructor at the Chicago School of Piano Technology. I personally observed the CTE Trainee's qualifying exam and the results were superior indeed. He will no doubt be certified at the PTG Convention next month. Need I say more about the qualifications of the technicians who crafted this tuning record?

I can quote from other Master Tuning records and I can quote more examples than I have but all would reveal similar findings. Just consider now, that the above quotes are from what is considered to be the most standard form of ET and that the top octave is not stretched nearly as much as most people will do and certainly not as much as many other technicians, including myself do routinely.

So, to me, the issue is certainly not whether 5ths or 12ths become pure, then widened; they certainly do quite naturally. The ideas that there are out there for this to occur lower in the scale and the effects that has are what the issue is. I do what I do and I am pleased with the results and can document exactly how much I stretch octaves either numerically or in clear, readable text. From what I know, so can Bernhard Stopper.

Between what I do and what Bernhard Stopper does, although the two are aurally perceptive as different, there isn't really that much difference numerically or aurally as perceived by both casual and very educated listener alike. The difference amounts to a few cents here and there of manipulation of the temperament octave and midrange. If we take the "ET with pure 5ths" idea which I consider to be too extreme, there still is not all that much difference between a PTG Master Tuning and what would be required to produce those results in the midrange; a couple of cents worth at most.

If there were very much more of a difference in any of these, the results would inevitably be perceived as unacceptable by at least some and that "some" would be far too many to try to convince. There is simply a limit on how far one can go before it is too far and that limit has a very narrow range.

Therefore, it begs the question whenever any one of these new and improved ETs come out which claim to have the ultimate answer to universal beauty, "Just where is that sweet spot?" If we don't hit it this time but get it the next, will anyone really ever notice or care?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/15/09 08:03 AM

Bill:

Thank you very much for posting the figures from the master tunings.

You wrote: ” If any two notes which form a 5th belong to the same octave and are read on the same partial and have the same numeric value, they would be theoretically equal tempered 5ths (2 cents narrow). We don't expect to see that at all and we don't in a master tuning. If the upper note of the 5th is 2 cents larger than the bottom note, the 5th is pure (beatless or just). If the upper note is more than 2 cents higher than the bottom note, the 5th is widened.”

I respectfully disagree. The effects of iH are much, much greater on higher partials than lower partials, and iH is much, much more for higher notes than lower notes. So, the cents differences between theoretical frequencies and lower partials can be very, very different than the cents differences between theoretical frequencies and higher partials. I believe the only way to determine if a fifth is wide is to compare the frequency of the third partial of the lower note with the frequency of the second partial of the upper note. There is no short cut. This is not to say that fifths do not become wide, just that it needs to be demonstrated in another way.

I enjoyed your last paragraph very much:

” Therefore, it begs the question whenever any one of these new and improved ETs come out which claim to have the ultimate answer to universal beauty, "Just where is that sweet spot?" If we don't hit it this time but get it the next, will anyone really ever notice or care?”

Very excellent point! Who really notices or cares? Probably only the person that did the tuning. I sometimes think that many people could hear what I hear, but it is the caring that makes the difference. I have resigned myself to the burden of never being quite satisfied.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/15/09 08:14 AM

Quote:
If any two notes which form a 5th belong to the same octave and are read on the same partial and have the same numeric value, they would be theoretically equal tempered 5ths (2 cents narrow). We don't expect to see that at all and we don't in a master tuning. If the upper note of the 5th is 2 cents larger than the bottom note, the 5th is pure (beatless or just). If the upper note is more than 2 cents higher than the bottom note, the 5th is widened.


No. Re-think your assumptions please. The width of the fifths is determined by the 3:2 partial relationship; the width of fifths is not included in master tuning records. Sorry.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/15/09 09:32 AM

Well, of course I thought about this but consider this: if C3 and G3 both have the same number (which they would not), would that not mean that the 5th is 2 full cents narrow? From the same Master Tuning: C3: -3.7, G3: 0.7. That ;eaves 3 cents difference between the two. Subtract 2 cents that is already between C3 and G3 and it leaves a 5th that is 1 cent narrow. The theoretical 2 cent narrow 5th is widened by fully half the amount we expect theoretically to only 1 cent narrow.

Doesn't this apply to all the rest of it? While I agree with both Jeff and Kent that one would need to compare the actual partials, they are unknown and can't be determined without actually measuring them. Once the partials reach into the 8th octave and higher, that is impossible. They also cannot be heard, so it does not matter that 5ths and 12ths are wide.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/15/09 09:47 AM

Oops, I made a pos/neg error with the above which I can often do. The above 5th is actually 1 cent wide. Now, aurally, of course, it is still narrow but there is still 3 cents width between the two at the fundamental level. Between C6 and C7, C6: 3.6 and C7: 12.3, there is 8.7 cents width in the octave even though the octave sounds pure aurally. If C7 were any higher, there would be a beat, of course and that is the way most technicians would tune it. So, although 5ths and 12ths won't beat crazily sounding wide wherever we can hear them, I still conclude that they do, in fact become wide, even by an amount we can sometimes hear somewhere in the 6th octave on up.

I have often seen this happen when comparing the double octave and 12th electronically. Up to a certain point, the double octave is wider than the 12th but at some point they become approximately equal and then at another point, the 12th becomes wider than the double octave. If the 12th is wide, then the 5th must be wide too.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/15/09 10:15 AM

Originally Posted By: Bill Bremmer RPT
Well, of course I thought about this but consider this: if C3 and G3 both have the same number (which they would not), would that not mean that the 5th is 2 full cents narrow? From the same Master Tuning: C3: -3.7, G3: 0.7. That ;eaves 3 cents difference between the two. Subtract 2 cents that is already between C3 and G3 and it leaves a 5th that is 1 cent narrow. The theoretical 2 cent narrow 5th is widened by fully half the amount we expect theoretically to only 1 cent narrow.

Doesn't this apply to all the rest of it? While I agree with both Jeff and Kent that one would need to compare the actual partials, they are unknown and can't be determined without actually measuring them. Once the partials reach into the 8th octave and higher, that is impossible. They also cannot be heard, so it does not matter that 5ths and 12ths are wide.


Bill:

First, if “C3: -3.7, G3: 0.7” is not a typo, then the algebraic difference is 4.2 cents because C3 is negative and G3 is positive. So, in your line of reasoning, this interval is 2.2 cents wide of just intonation. Would a master tuning have this interval beat wide of just intonation?

But let’s try a different sort of reasoning. Let’s say, just as an example, that C4 is +3.5 cents. This means that the octave is stretched 7.2 cents [3.5 - (-3.7) = 7.2], and again just as an example, this produces a beatless 4:2 octave. So how much is each semitone stretched due to iH? 7.2 / 12 = 0.6 cents. And how much is a fifth stretched due to iH? 0.6 x 7 = 4.2 cents. So we would expect the fifth to beat as if it were 2 cents narrow, because there is no “additional” stretch. The 4.2 cents is due to iH, which affects the 5th the same as it does the octave and all other intervals.

But even this kind of reasoning continues to have problems because the G3 – G4 octave would probably not be stretched 7.2 cents, but more. And there are even other considerations...

EDIT:

Sorry I cross posted. Glad you caught your own error. Still, there are other considerations. But if an interval beats wide, it is wide.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/16/09 06:52 AM


Robert, thank you.

In one previous posts of yours you wrote:

“I don't know what false assumptions you are attributing to ET. As far as I know, ET does not make any assumptions at all.”

Have I succeded to explain you that?

You say:

“Clearly you have in mind a definition of the word "inharmonicity" that is different from what the rest of the world means by "inharmonicity".

Have I talked about iH definition? No, I have and I am talking about the reference model on which basis iH’s effects have been calculated.

About the case of a pipe organ you wrote:

“It has harmonics. Those harmonics are true. They are locked to the fundamental. If air pressure changes, then the pitch of the pipe will change, but so will all its harmonics, and they will remain locked. That is the definition of zero inharmonicity.”

Please tell me, when you say “It has harmonics…..that is the definition of zero inharmonicity”, what are the harmonics values you are/they were referring to, when fixing zero iH?

“Whether or not 12ths and 19ths sound good does not change this fact.”

Whether or not 12ths and 19ths sound good, this may change our reasoning about traditional ET pseudo-model V iH.

“You certainly can and do get progressive intervals from the traditional no-stretch ET model,…”

Do you mean, on a pipe organ? By the way, are you an aural tuner?

…“Now when it comes to instruments that have inharmonicity, like the piano, nobody uses the traditional no-stretch ET model. So you are criticizing a model that nobody uses for the piano anyway.”…

I am criticizing traditional ET pseudo-model because it is a lame reference model, with or without iH.

When I wrote: “...What I’m stating is: even thinking to non-iH tones, traditional ET pseudo-model can not give you progressive intervals...

You answered back:

”Nonsense. Of course the intervals are progressive for non-IH tones. Just play any cheap electronic piano (I say "cheap" to make sure it does not simulate inharmonicity, which the expensive ones do). The beat rate of 3rds, 4ths, 5ths, etc. will be prefectly progressive. They will all increase as you go up the scale. (Unless you are inventing a new defintion for the word "progressive" too.)”

I think you are right, one of us is saying nonsense. In my opinion, if you were familiar with beats you could never say that. Give me a little time, and I’ll help you with precise reference brands and 4ths and 5ths reference intervals. In the meanwhile please, keep your definition of “progressive” but notice that RBI beat-rate progression can also be rough or smooth. A rough-hewn RBI progression will leave disorder amongst SBI, what some aural tuners may well know.

When you said:

“...Some say that the octaves should be beatless 4:2 octaves. Some say they should be slightly wide of just. And they are both right.”

I retourned you my opinion:

“In my opinion, they are both – you choose the word – bewildered? Lost? Puzzled? Disconcerted? Mixed up?”

You answered back saying:

”I would say they are entitled to their own opinion.”

Ok, generally speacking I quite agree, but I’m not writing about an opinion festival. Chas model’s interweaves prime numbers and relates scale’s frequencies on the basis of a new ET theory, a theory deriving from a dynamic approach to beats. In other words, Chas model proves how proportional beats can define an infinite number of scale’s ratios, quite the opposite of what has always been done.

You finally say:

"IH experts" as if IH was some deep notion, accessible only to a few. It is an objective physical measurement that anyone can make with the proper equipment. They do not have to be an expert. By way of analogy,…”

I am referring to those technicians able to make laboratory measurements and calculations with sophisticated technology. Masons still work with a tape (God bless them), land-surveyors don’t anymore. As you may notice, I’m treating a semitone ratio that, compared with traditional ET pseudo-model, differs 0.00002.., and produces an A5 octave difference of 0.2337..

Tooner, thanks for your commitment.

You say:

“I have been able to learn a great deal from this Forum. Others have corrected errors when they find them. By doing so, it maintains the integrity of the vast knowledge that can be found in past posts. I feel an obligation to help correct errors, also. Your paper and posts are in error because of your misunderstanding of iH.”

I feel a similar tipe of obligation, this is way I would ask you to be very responsible and utterly precise when you say “your paper is in error”, as I would also ask you to acknoledge that I do not ignore iH, I precisely think it should be calculated by using a correct reference model.

You kindly wrote:

“The iH of other notes can be determined by multiplying 0.10000 times (2 ^ (1/8)) ^ (the number of semitones above C3). A3 is 9 semitones above C3. So the iH of A3 = 0.1 * (2 ^ (1/8)) ^ 9.”…

This formula is the base of all your calculations. So I ask you: how wide are you calculating the semitone? What partial values would you expect with “zero inharmonicity”?

Certainly you will have red where our cello-expert writes: “…the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.” What do you think he meant, saying “…proper consideration of inharmonicity”?

Bill, Kent,

I will contribute to what you are saying asap, thank you.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/16/09 07:42 AM

”I feel a similar tipe of obligation, this is way I would ask you to be very responsible and utterly precise when you say “your paper is in error”, as I would also ask you to acknoledge that I do not ignore iH, I precisely think it should be calculated by using a correct reference model."

You do not ignore iH in your paper. You mention it a number of times. That does not mean you are applying it correctly. You referenced Young’s paper, but seem to either not understand it, or believe that iH is different than described. Young’s paper describes how iH is calculated and how it affects the frequencies of partials. You have not used those calculations in your paper. I don’t know how I can be more - responsible and utterly precise when I say your paper is in error.

” You kindly wrote:

“The iH of other notes can be determined by multiplying 0.10000 times (2 ^ (1/8)) ^ (the number of semitones above C3). A3 is 9 semitones above C3. So the iH of A3 = 0.1 * (2 ^ (1/8)) ^ 9.”…

This formula is the base of all your calculations. So I ask you: how wide are you calculating the semitone? What partial values would you expect with “zero inharmonicity”?


The width of the semitone has no effect on the iH of a particular note. The frequency of a note does not change a note's iH either, although the frequencies of the partials are affected by the both the fundamental pitch and the iH. The same calculations can be made with an iH of zero. The result will be partials that are whole number multiples of the base frequency.

Maybe you are asking about the width of semitones in regard to how I determined the width of the octaves in the example. If so, I did not calculate any semitones, only octaves. And I can think of a number of ways to calculate the semitones, but I would first have to decide on the width of more octaves so that there would be an accurate nonlinear interpolation.

[EDIT]: Ok, I think I know what you mean, now. Rather than considering the semitone as being an interval that is tuned, it should be considered as a note on the piano. The iH doubles every 8 notes in this example. iH is a function of piano scale design, not tuning.

” Certainly you will have red where our cello-expert writes: “…the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.” What do you think he meant, saying “…proper consideration of inharmonicity”?

I am not certain what Mr. Stopper means. It may be similar to what I call “the largely self-correcting effects of iH on beat rates”. We should just be patient and wait for his paper.

Alfredo, have you tried looking at iH from a fresh perspective? You may feel that this Forum has treated you poorly (It does treat people poorly, sometimes) and are hesitant to accept what is written. Let me suggest that you re-read Young’s paper one paragraph at a time, not going on to the next until it is fully understood. Also, here is a link to the graphs of iH curves for a number of pianos:

http://www.goptools.com/

After opening the link, click on “The Scale Collection - Browse Original Scales of over 40 Pianos “

And by the way, I think very few tuners completely understand all the effects of iH. I am still working on the finer details. But I don't believe it is necessary to understand it to tune well.
Posted by: Robert Scott

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/16/09 08:41 AM

Originally Posted By: alfredo capurso

In one previous posts of yours you wrote:

“I don't know what false assumptions you are attributing to ET. As far as I know, ET does not make any assumptions at all.”

Have I succeded to explain you that?

Unfortunately, no.
Quote:

Have I talked about iH definition? No, I have and I am talking about the reference model on which basis iH’s effects have been calculated.

You may not have explicitly talked about your definition of IH. But one must learn to walk before one learns to run. How can you make accurate statements about IH if you don't understand what it is at a more fundamental level?
Quote:

About the case of a pipe organ ...Please tell me, when you say “It has harmonics…..that is the definition of zero inharmonicity”, what are the harmonics values you are/they were referring to, when fixing zero iH?

A harmonic is a sinusoidal tone whose frequency is an exact whole-number multiple of a fundamental frequency. For example, in the case of a pipe organ, if the fundamental frequency is, say, 511.2 Hz, then the 2nd harmonic is 1022.4 Hz, the 3rd harmonic is 1533.6 Hz, etc. And we don't "fix" zero IH. We observe it. A pipe will behave like this whether we want it to or not.
Quote:

“You certainly can and do get progressive intervals from the traditional no-stretch ET model,…”

Do you mean, on a pipe organ?

Yes, that is one of the places where the traditional no-stretch ET model is actually used. It is not used for pianos.

Robert Scott
Ypsilanti, Michigan
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/16/09 09:32 AM

Quote:
What do you think he meant, saying “…proper consideration of inharmonicity”?”


Equal temperament is defined as that temperament in which the beat speeds of like intervals progress smoothly across the scale.

I believe it is true that pianos tend to be scaled in such a way that beat speeds taken from the mathematical model of equal temperament can be used in the mid-range of pianos with only relatively small modifications due to inharmonicity. The necessary modifications of beat speeds would tend to increase as one moves toward the top and bottom of the piano scales. No?

"Proper consideration of inharmonicity" would be modifying a piano tuning's beat speeds away from that of the mathematical model in order to best preserve the overall smoothness of beat rate progressions (and desired purity of the slow-beating intervals) despite the changing level of inharmonicity through the scale.

The perturbations of inharmonicity upon the mathematical model of equal temperament as applied to piano tuning are significant, and are probably the problem that underlies this discussion. This problem is endlessly fascinating and some brilliant people have devoted a great part of their lives to the subject. A great example of these "perturbations" is the fact that a standard feature in Dr. Al Sanderson's FAC tunings as calculated by the Accu-Tuner is narrow 4:2 octaves in the piano's treble. Dr. Sanderson considered those narrow 4:2 octaves absolutely necessary to provide the desired overall purity of the slow-beating intervals; obviously, many piano techs have agreed over time.

This is but one example.

There probably are still contributions to be made concerning "the proper consideration of inharmonicity", but my opinion is that these contributions should be made with great respect for the coherent effort and considerable intellect that has already been devoted to this subject. After all, many pianos already sound very well tuned.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/16/09 10:47 AM

Mr. Swafford:

I think the problem that is underlying this discussion is bruised egos. I am trying to be more careful.

You also mentioned:

”There probably are still contributions to be made concerning "the proper consideration of inharmonicity", but my opinion is that these contributions should be made with great respect for the coherent effort and considerable intellect that has already been devoted to this subject. After all, many pianos already sound very well tuned.”

Is this information readily available? This also makes me think about how pianos might have been tuned through the centuries, without the use of iH theory. I know the standard text that I learned to tune with is out-of-date, but it still worked. It is interesting that the octave tests that were used were appropriate even though they were not designed in "consideration of inharmonicity." Just musing...
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/16/09 10:51 AM

Originally Posted By: alfredo capurso



Chas model’s interweaves prime numbers and relates scale’s frequencies on the basis of a new ET theory, a theory deriving from a dynamic approach to beats. In other words, Chas model proves how proportional beats can define an infinite number of scale’s ratios, quite the opposite of what has always been done.



You may consider that even more work about different ET from the standard model from other people have been proposed, beside the book of Cordier.

You may read the article of Gary Schulze in the PTG journal from march 1982, where he explicitely describes a theoretical ET model based on the 31th root of six.

An ET based on the theoretical model of the 19th root of three has been proposed by me in euro-piano 3/1988. Inharmonicity consideration is targeted by using inharmonicity affected partials through the use of beats when tuning aurally.

A general mathematical model of tunings has been proposed by Guerino Mazzola 1989 in his book "Geometrie der Töne". In section 2.3.3,

a "convention dependant linear function of the form
Y = uX + v, where u and v are not constants,
with X = ln(f) " is proposed, what finally includes all possible theoretical ET and non ET models.

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/17/09 04:23 PM

Thank you for your feed-backs,

I'm visiting my parents...and I'll soon be with you. a.c.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/21/09 08:49 AM

I don't believe my eyes!

So you guys can not come out with a definition of ET?

In this topic I've seen the most experienced people, I mean guys like Mr. Bremmer, Mr. Swafford, Mr. Scott, Certified Tuning Examinators who had developped new temperaments, new sequences for tuning ET and new techniques on tuning the hole piano or who had designed tuning software for ETDs, etc., and in spite all this knowledge, they get confused by Mr. Carpuso to the point of admiting they don't know what ET is for sure?

Mr. Carpuso doesn't give an answer with all that math stuff. He only plays with some clever equations which relate to nothing in piano tuning's real world.

His sequence is the same sequence up a fifth, down a fourth taught by Randy Potter in his course and he does not explain how to exactly temper fourths and fifths. It will suffice to give a look to his instructions:

"Step 2 – A4-A3 - tiny little narrow, just on the beating threshold"

"Step 3 – A3-D4-(A4) - wide, close to 1 beat/sec. – D4-(A4) faintly beating"


"Step 4 ...
...A3-E4 about 1,5 beat/3s - sensibly faster than D4-(A4)
E4-(A4) about 2 beats/1s - sensibly faster than A3-D4
"
The same "slightly faster than, or 3 beats in 5 seconds or one beat per second" of a ton of SBI's sequences. I feel as I was reading Braid-White book, written a century ago.

It is unbelievable that Mr. Carpuso dares to claim that he has constructed a new model in which there is a variable called 's' that can "calculate infinitesimal degrees of inharmonicity". And says in his first post: "Inharmonicity, which has always been calculated in an approximate way, can now be calculated with infinitesimal accuracy". (Mr. Carpuso confuses iH with stretch).

And it turns out in this thread that he doesn't even know what inharmonicity is, and how it affects beats!

And here we have gentlemen like Mr. Jeff Deutschle and Mr. Robert Scott explaining him what iH is!

Or brave Mr. Bremmer trying to find wide fifths and 12ths in a Master tuning in an effort to understand what Mr. Carpuso says! No way!

Don’t get confused by all this bla, bla, bla. At first I was impressed by this thread when reading Mr. Carpuso's paper, but now I am affraid that it is all about it. It impresses people by using confusing math, which leads nowhere. All that math isn't required to tune an octave slightly wide, a fifth beating 1.5 times in 3 seconds, etc...

Sorry Mr. Carpuso but I think you are an excellent illusionist.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/21/09 12:35 PM

Part of the problem, not just with this paper, is using terminology that varies from moment to moment. "All that math isn't required to tune an octave slightly wide,..." is ambiguous depending on whether one is talking about the interval's pitch numbers being wide or the interval's beats indicating that the interval is wide.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/21/09 02:35 PM

Rafael, the definition of ET is a temperament in which all intervals are tempered equally. It is as simple as that. I learned that from Kent Swafford. I also learned from Kent that it doesn't matter how wide or narrow the octave is. Since he had explained it to me, I heard it from other sources as well. With further work in techniques for constructing an ET, I learned that the scaling of the piano does not matter either.

Please, let us not mention "CTE" too much. No one who has that authority is allowed to "advertise" it. Only one member of any master tuning committee is required to have that title. PTG members who become inactive with tuning exams lose that title. It is only meant to be used for the purposes of conducting an official PTG tuning exam and is not meant to represent a level of skill. It has to do more with the administrative skills than it does with tuning skills. It is well known, however that to qualify to train in that area, the person must possess a higher standard of aural tuning skills than is required to become an RPT.

Therefore, there has been some suggestion within PTG that those who do possess these skills, whether they work with exams or not, be recognized for it. It may come about eventually but presently, there is no title which PTG bestows other than RPT.

So, let's be sensitive as well to those who are not PTG members, especially to those who may, in fact, have superior aural tuning skills. Anyone can use an ETD to test aural tunings by using the exam program which comes with them, if they care to learn how to do it. Anyone can set up a "reference" tuning either by themselves or with assistance from a more highly skilled technician, PTG member or not and regardless of whether that person ever worked as an examiner or not.

I can say, however that when an aural ET is being refined to its ultimate perfection such as during the construction of a Master or reference tuning, the people doing that look for the same amount of tempering in all intervals. There can be no "fudging", no favoring of any one kind of interval over another.
This is true regardless of octave type, size or width, however you may like to think about that. PTG does not have any specifications for that, even for the exams.

If anyone used Bernhard Stopper's model for a PTG exam and did the midrange aurally as required with sufficient accuracy, it would pass the exam. Any other octave width, even a slightly narrow width would work just as well. What matters is consistency, maintaining the same concept throughout. That is not to say that favoring one kind of interval over another may produce pleasing results in some circumstances, that is what I do every day. "To be or not to be ET, that is the question".

Regarding whether 5ths and 12ths become wide or not, they do, I am convinced of that. I learned that very long ago from Steve Fairchild who demonstrated it at a PTG convention. Now, I take what Kent said about my post to heart but the figures as I posted them still suggest as much.

Anyone can do what I do routinely in constructing octaves from F5 to the top and they will see that what I say is true. Beginning on F5 and with the ETD set on F5 (Partial 1), play the F3 and A#3. If you first stop the pattern when F3 is played, then play A#3, the pattern will roll slightly sharp. If you adjust F5 on the ETD so that the pattern rolls equally sharp and flat when F3 and A#3 are played alternately, you will find an ideal spot for F5 to be tuned. The double octave will be slightly wide and the 12th, slightly narrow, each by a very small amount, nearly imperceptible to the ear.

That is the basic "mindless octaves" concept. If by ear, the double octave is made to sound beatless, the 12th will beat noticeably. If the 12th is made beatless, the double octave will have a noticeable beat that may be considered unacceptable. However, when there is an exact compromise between both the double octave and the 12th, both intervals sound apparently in tune, the beat is so slight as to not be really noticeable, especially in a musical context.

That is why I dubbed the concept as "mindless" because if either the double octave or the 12th beats, it sounds "wrong" but when there is that exact compromise between the two, it sounds "right". This proved to be true for me even when tuning an unequal temperament.

Now, if you continue this technique upwards, you will inevitable find a point where both the double octave and the 12th will both stop the pattern and to the ear, both will sound perfectly in tune. When you continue upwards, you will find the exact opposite of what you found at F5. When the 12th stops the pattern, the double octave will be wide, when the double octave stops the pattern, the 12th will be narrow, still each by a very small amount.

This means (at least by my reasoning), that the 12th has become wide and therefore the 5th as well. However, at this point, the coincident partials for the 5th may well be out of hearing range and therefore, however wide they may be won't matter because they cannot be heard. In any case, a slightly wide 5th is not unpleasant to the ear, especially that high up where the sustain is so short. The same applies to single octaves: a slight or even slightly rapid beat in a single octave does not offend the ear, particularly in a true musical context.

I have now long taken to the practice of tuning pure double octaves and 5ths from F6 to the top. Sometimes, I don't start that until C7, it all depends on how wide the single octave sounds. If it is just too crazy, I go back to the double octave and 12th compromise until the single octaves can sound reasonable. So, I may start the pure double octave and 5th idea anywhere between F6 and C7 or maybe even a little higher but eventually, I get to 6:1 octaves in the high treble. I tune the low bass basically the same way. I let the piano tell me what it can take.

To me, this is a far better way to tune the extremes of the piano than to depend on a calculated stretch curve. (The totally advanced features of the Verituner notwithstanding). That curve is based upon assumptions instead of what the piano may really offer. Even if I use a calculated tuning for the middle, I change my partial selection and tune by direct interval at either extreme end. It doesn't take that much more time to do it that way but the results are certainly worth the time and effort.

That is one of the several reasons why I chose to use the SAT over the other choices of ETD which are available. For me, direct interval tuning is my preferred concept because I tell the ETD what I want, it does not tell me. I am in complete control of the results to the very highest and lowest notes and that is the way I want it. I do believe that for those who are used to using the other devices and software, a direct interval method of tuning the high treble and low bass are just as accessible as with an SAT.

I have met and had discussions with Bernhard Stopper and have also heard his tuning. It has a remarkably clear character to it. While I still do not fully understand it, I did gather from what he has said that the 12ths also become wide at some point in his tunings as well.

Others have long ago suggested pure triple octaves including Jim Coleman, Sr. and Virgil Smith. If one considers that 12ths and double octaves converge, then reverse at some point, then so would double octaves and 12th and triple octaves. I would suggest, as my own practice has told me that by the time one reaches the top end of the 7th octave, that is the point where double octaves and 12ths and triple octaves are both pure or at least so close to each other that the difference is insignificant.

I give Alfredo the benefit of the doubt that his concept lies somewhere within a structure that may be a composite of what I describe above. There is, after all, an obvious language barrier. Any manipulation of octave width and/or temperament will have some kind of effect. It cannot be said that one idea is right and the other is wrong, only that any two ideas are different and produce differing effects as a result.

We should all try to understand and be able to try and implement these differences depending on the goal of the tuning at hand. Passing the PTG tuning exam is one goal but creating a more pleasing effect for a different kind of circumstance is another.
Posted by: Jerry Groot RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/21/09 03:33 PM

Nicely put Bill.

Gadzar,

I do not recall ever saying that I did not understand the ET. ET is what I've used my entire life until I purchased RCT. Even now, the best tunings that I have saved on RCT are all my own tunings done using ET. I can tune a piano very well with that method thank you very much. smile

I am just NOW learning Bill's method or, trying too. That to me, is a bit more confusing, learning a completely new and different method after 40 years of tuning one way... Perhaps that is to what you are referring.

What does not matter to me at all, is the mathematical theoretical stuff (that I do not bother to read by the way) that is spewed back and forth in here. All that is to me, is one person trying to impress another and it does not impress me one bit. It doesn't mean that one can hear it... All that shows is one can talk it... wink Proving it is an entirely different matter.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 07:27 AM

Gadzar:

I think Alfredo is sincere, but thought he knew more than he actually did. He may be able to learn more here. By reviewing his paper and looking at the subject from a different angle I have learned some things. So I don’t think it is a waste.

My personal, practical definition of ET is where all M3s and M6s beat progressively faster. On poorly scaled pianos, I don’t think this is possible. One or the other (or both) will have a jump in beat speed across the break. Even on a well scaled piano, it may be difficult to achieve if the pinblock and rendering are poor. Each note must be within 0.2 cents of an ideal frequency!
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 07:36 AM

Bill:

I am not sure if anyone has noticed, and I have delayed bringing up the subject because it may be moot, but your mindless octaves are a slightly different thing than Alfredo’s equal beating 12ths and 15ths. His have a common note on the bottom, your's are on the top. I am still mulling over what this might signify.

You’ve mentioned before that Steve Fairchild demonstrated that 5ths and 12ths become wide. How did he do this?
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 11:21 AM

Ok. Guys, all of you know what is ET, but none of you tune it the same way. Saying that all intervalls must be tempered equally is easy but how to tune all intervalls equally tempered is not that clear!

What happens if I tune the fundamentals of all strings to their theoretical frequencies? The piano will sound untuned because of iH, but you can not say the intervalls are not equally tempered, in fact the intervlalls at the fundamentals will be equally tempered, thus it will be ET.

The question for me is to find a way to tune ET taking into account iH, and that's here exactly where beats come into play. Some say to tune a smooth progression of M3rds and M6ths (M10 and M17 where 3rds and 6ths are no more audible), others say to tune by octaves, double octaves or even triple octaves, using different types of octaves along the scale, others use to tune 12ths and 15ths (as I do), or a combination of the above, and all of us can get a well tuned piano, but what is then ET? Is there a true ET?

You can tune M3 and M6 progressively faster, but then what about 5ths, do they will beat also progressively slower? No way! You must include them in your tuning, as you must also include 4ths, it is not enough to test M6ths thinking of them as M6 = P4 + M3, you must control explicitly P4s. Do the P5s become beatless at some point and then become wide? Probably, but some of you say yes, some say who knows?

Nobody can for sure tell us how to spread the temperament accross the scale, each of you use a different way to do it. That is why I say you are not able to give a true definition of ET.

The pianos we all tune can sound great, no doubt, some of you are great tuners, but is there a true and unique ET? Or iH makes it impossible to find a UNIQUE WAY to tune those progressively faster beats?

Now, what is CHAS? Is it a mathematical model that can not be landed in a real piano tuning?

How does Mr. Carpuso effectivelly tune CHAS? The sequence he posts here, is as I've said before, one more of those sequences of 5ths and 4ths arbitrarily tempered to what sounds "slightly" wide and narrow.

That is not a new system! How is he choosing the values of the variables delta, s and s1? How is he puting these values in his way of tuning? He says nothing about that.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 12:32 PM

Mr. Melo,

Great post.

It is my position that theoretically there are an infinite number of equal temperaments, because any interval can be divided any number of times to create a temperament that qualifies an an equal temperament.

In practice, committees of 3 RPT piano techs regularly reach consensus on an optimum piano tuning for PTG's exam pianos.

I believe the practical application of equal temperament allows certain irregularities in the progressions of beat rates in order to preserve the overall progression of all the tuning intervals. In other words, the tuner makes the best compromise possible among all the tuning intervals, and the result is still an equal temperament. Inharmonicity-induced irregularities in the beat rate progressions, in practice, does not preclude calling a tuning equal temperament.

It is extremely interesting to me that as we go forward, it appears that talk of optimum stretch preferences seem to be giving way to talk of the optimum intervals on which to bass the underlying mathematical model of equal temperament. This is why the present discussion is of interest to me.

But either way, the underlying impetus for stretched tunings or tunings based on different equally-tempered intervals is the same -- inharmonicity.

I also await the answer to the question you ask, that is, "How does Mr. Carpuso effectivelly tune CHAS?" Presently, I doubt that "CHAS" actually exists as a successful piano tuning system, but, though I am skeptical, I believe I am still open to CHAS.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 12:38 PM

Gadzar:

I have had the same questions, but have only come up with some answers.

I have thought of a number of ways to define the stretch of an aural tuning. We can choose a non-beating intervals such as 2:1, 4:2, 4:1, 8:2, 3:1, 6:2 etc and let it define the stretch. Or we can choose one of these intervals to beat at a certain rate such as 2:1 @ 1/2bps. Or we can have a chosen interval beat progressively faster such as doubling every octave. Another scheme is equal beating between 2 intervals, such as mindless octaves, but then we also have to define an initial stretch. One that I like in the midsection is a beat ratio between fourths and fifths. And there can be a progression of a beat ratio between two intervals that changes. There can be hybrids of all these definitions, and any others that I have not mentioned. And if we know the iH of every string on a piano, we can calculate the frequencies of these intervals and then perform some kind of non-linear interpolation to calculate the other frequencies. When I think of doing this, I remind myself to call anyone that has written a tuning program “Mister”!

But can a tuning only be defined by some aural standard? I don’t think so. Each partial for the notes of a piano can be defined mathematically by a curve; even fractions of partials. Perhaps, even negative partials. A mathematical curve can be generated for any of these partials, or the difference between partials, or a ratio between partials. Calculations can then be performed to determine the other partials and a piano can be tuned either electronically or by using beats.

So where to go from here? I think back to the purpose of tuning to begin with: Preparing the piano to make music. And I think of real pianos and a story about rabbit stew. (True story.) A family was eating rabbit stew and someone asked who put a raison in it. It did not matter that there was only one “raison”, dinner was over! With pianos I think the important thing is that no intervals sound bad. That is the outside of the envelope. Everything else is OK, including non-ET.

Then what is ET? The broadest definition is that all keys sound the same. This would include, for most people, mild well-temperaments. But what about us folks that are going to listen to progressive beat rates? Can all intervals be progressive on a piano that has iH? I am not so sure. There are many descriptions of tunings where fifths become wide in the high treble. Well, unless they also become fast in the bass, this would not be progressive. Unless we want to say that a curve that goes from slow to fast to slow to stopped to wide is progressive…

And then there is the practicality of tuning a real piano. I asked a while back and it seems that the strings on a piano can only be set stabile within 0.3 cents. Why that is not even enough to guarantee that M3s and M6s beat progressively let alone 4ths! That is why I choose my practical definition of M3s and M6s beating progressively. If they do not, then an error that can probably be corrected can be pointed out.

Well, I’ve rambled enough. One person’s perfect ET is another’s poison. Because of iH, and personal preference, ET is region not a location.

And I have already asked questions about how a piano is tuned to CHAS and cannot understand the answers.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 12:59 PM

You guys are thinking about this too much. You tune a piano so that it sounds like it is equal tempered. That is all there is too it. Someone else can come along with a frequency counter and debate what it actually is, but as Duke Ellington put it, "If it sounds good, it is good."
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/22/09 01:50 PM

Originally Posted By: Kent Swafford
.....
But either way, the underlying impetus for stretched tunings or tunings based on different equally-tempered intervals is the same -- inharmonicity......


I am not so sure. I tend to think that the piano is so popular because of the stretch that iH naturally gives to octaves. But also, additional stretch can be given in an attempt to satisfy the well-documented human ear's desire to hear stretched octaves. The use of 12ths, not necessarily pure, is a great tool in controlling this additional stretch.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 06:27 AM

Mr. Swafford, I’m with you, iH makes the difference.

Without iH all is easy, if we assume that

1 There is no iH
2 ET is defined as the division of the octave in 12 semitones equally tempered and
3 The octave's ratio is established to be 2:1

Then, the simple mathematical model of ET where

Semitone = 2^(1/12)


would perfectly do the work. Even if the “well-documented human ear's desire to hear stretched octaves” (by Tooner) is not satisfied.

With the presence of iH in pianos that model is no more applicable; we must tweak the frequencies calculated by this model in order to get acceptable tunings. That distorts what we understand by ET.

We lack a new mathematical model which includes iH and solves the problem of having several incompatible kinds of an interval. How can we tune an octave if there are 2:1, 4:2, 6:3, 8:4, 10:5, 12:6 octaves, and they are incompatible with each other? How can we tune a 5th if there are two distinct incompatible kinds of them?

Models like semitone = 3^(1/19) or even more complicated, like the formulas used by Mr. Capurso, don't solve the problem because iH is not directly addressed.

We need a new mathematical model where the octave's ratio is no more a constant but a variable value that will fit the piano's iH all along the scale. A new mathematical model where there will be only one type of each interval to tune, namely only one kind of octave, fifth, fourth, etc. A new mathematical model where the iH of each one of the strings in the piano will be taken into account.

Pianos are scaled in a discrete way, there is no continuous curve that can describe iH along the scale. We have sometimes six contiguous unisons using the same size of wire and then the next unison has another size, here we have a jump in iH. The same happens when we get to the wound strings, and to the doublewound strings. Our new mathematical model can not just ignore this, because our ears do not.

I can not think about such a new mathematical model without thinking at Virgil Smith's concepts explained in his book "New Techniques For Superior Aural Tuning".

In that book he says:

"Every note is composed of several different pitches called partials or overtones, but the ear does not normally hear these partials as separate pitches because of its unique ability to combine all the partials of a note into one sound and pitch.
However, many tuners have trained their ears to hear these partials as separate pitches, and to actually hear the beat between two specific partials. Many have found this helpful in learning to hear beats.
However, it is not necessary to hear the pitch of single matching partials to hear beats for aural tuning, because of the ability of the ear to combine all the partials of a note into one pitch.
When an interval is expanded or contracted to produce beats, the ear (when listening to the two notes normally) combines all the partials of both notes into two single pitches, just like it does with one note alone. In addition, it combines, all the beats between the partials into one beat. The beat then comes from all the partials instead of one set of partials.
This beat can be tuned to the desired speed or eliminated completely. This means that beats can be heard two different ways: between single matching partials, and between notes as the ear hears them naturally with all the partials of each note sounding. They both originate with partials, but they are heard in two different ways.
For clarity, one will be referred to as “partial beats”, and the other as “natural beats”. It is important that every tuner clearly understand this, for failure to understand this has lead to much confusion in the past.
Even though each note contains many different partials, the ear hears the note naturally as one pitch and sound. This sound is referred to as the “whole sound”.
Every tuner should be able to hear the beat by listening to the notes naturally – it is the way musicians and listeners hear them – and hearing them this way is the way the final quality of the tuning is judged. The finest quality aural tuning can be accomplished by dealing only with natural beats.
No mater what method of tuning is used, the final evaluation must be with the natural beat heard when listening to the whole sound of each note with all the partials of the notes contributing to the sound. In Some cases, the beat at the single matching partial level is different when all the partials are contributing to the one beat.”

------------- End of Mr. Virgil Smith's citation ---------------

So the “natural beat” concept can be applied to solve the problem of several kinds of intervals sounding simultaneously. It is then necessary to translate mathematically the ability of the human ear to combine all the partials into one sound and one pitch.

If we can translate to a mathematical formula this way of combining several “partial beats” into one “natural beat” there will be only one unique type of interval to tune. No more 2:1, 4:2, 6:3 octaves, nor 3:2, 6:4 fifths, nor 4:3, 8:6 fourths, but only one kind of octaves, fifths, etc.

In that respect I think Mr. Capurso's research although being a true effort to find a new model, missed the target by confusing iH with stretch. I think the solution to this problem is working on iH not stretching the width of the semitone in an arbitrary manner.

I think people like the late Dr. Albert Sanderson (Accutuner), Mr. Robert Scott (Tunelab), Mr. Dave Carpenter (Verituner), Mr. Dean Reyburn (Cybertuner), and Mr. Bernhard Stopper (Onlypure), who have developed the theory, algorithms, software and some of them even hardware of the most advanced ETDs available today, are aware of the lack of such a mathematical model.

The goal is to developpe a mathematical model which is able to combine all the inharmonic partials produced by two notes sounding together into one unique "natural beat". That will give us the ability to calculate real frequencies to tune the piano.

And then Tuning will be no more an Art but a Science.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 08:08 AM

Gadzar:

I understand how you are looking at the tuning puzzle, but do not know if it is workable. Here are some thoughts and observations on the subject.

First, thank you for quoting Mr. Smith’s book. I had read a number of discussions on this “natural beat” object, but since there was not agreement on what they were, I decided not to buy the book. By Mr. Smith’s description, it is something that can only be experienced, not calculated. But maybe it is not really an aural illusion. I have ideas what this might be.

Just as two frequencies produce a sum and difference beat (the difference beat is used for tuning) I believe that the two different beats also produce sum and difference beats. So if a fifth is beating at, say, 1 bps at the 3:2 partial match and 2.2 bps at the 6:4 partial match, there is also a beat at 1.2 bps and 3.2 bps. Would there be even more beats of beats of beats like a house of mirrors, I think so.

Something I have noticed is the smoothness of an octave when the wide 4:2 partials beats at exactly the same speed as the narrow 6:3 partials. (I can only get an octave tuned precisely this way by ghosting the partials directly.) The beats are there and can be determined with test notes, but I do not hear them. I believe this is because the sum of the beats is exactly twice the beat and the difference is zero. The beats of beats of beats are whole number multiples just like partials of non-iH tones.

I suspect that the purity of Mr. Stopper's tuning is due to getting as close to whole number beat of beat ratios as possible. And how he does this electronically without first sampling the iH of many notes is intriguing. Perhaps his tuning is the model you are looking for. After all, there is no choice for different stretch preferences.

But tuning to satisfy beats, or beats of beats of beats, will only produce a harmonically sounding tuning. There are other things that the human ear hears. For sure, it hears melodically and I believe also musically. The melodic octave seems to be 1220 cents wide, and is impossible to fully satisfy and still have good harmony. But compromises can, and I believe should, be made for it. And the musical way that the ear hears does not seem to be defined or discussed much. I believe it is simply whether a note is at the frequency that the ear expects it to be after hearing other notes. I have found that 8:2 double octaves work very well to satisfy my musical ear, but they are not always the best compromise harmonically. I believe the human ear does detect iH in single notes and expects other notes to be in the inharmonically correct place when listening musically.

I guess we can recognize each other as “math people” and I understand the need to quantify everything. When I think of my nautical background, the ship could only be in one place at one time, there could only be so many barrels of oil in a cargo tank, and if the ship displaced so much water it weighed so many tons. A place for everything and everything in its place. But is tuning the same thing? Like you, I want it to be, but it is not.

If nothing else, consider two identical pianos. One is tuned favoring harmonic tuning, the other favoring melodic. When sitting down and playing you may prefer the harmonic piano, but at a crowded restaurant (where the beats cannot be heard) you may prefer the melodic piano.

If everyone tuned the same and everyone liked the same tuning, this would indicate there is a universal optimum, and through empirical observation a model could be developed. But since this is not the case, this indicates that there is no mathematical model that will satisfy everyone.

So I have a question. Why do you think we are in need of a “new mathematical model”?
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 12:54 PM

You are maybe right and we do not need a new model to tune, but when we talk about something like ET or the stretch on a piano or something else about piano tuning, we do need a reference in order to speak about the same thing. And the reference we have now is 2^(1/12) which does not correspond to what we are really doing.

The same thing happens with the Braid-White Sequence: everybody uses it to tune but each one in a different way, so the true BW sequence has now disapeared and what survives is a tweaked form of it.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 01:05 PM

Tooner,

I think a little: Why Einstein developped his Relativity theorie? Because there were things unexplained by Newton's theories.

In the same way, why a new tuning model? Because the existing one doesn't describe the real world as it is.

We human beings are like this. We must know!
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 01:12 PM

Originally Posted By: Gadzar
You are maybe right and we do not need a new model to tune, but when we talk about something like ET or the stretch on a piano or something else about piano tuning, we do need a reference in order to speak about the same thing. And the reference we have now is 2^(1/12) which does not correspond to what we are really doing.

The same thing happens with the Braid-White Sequence: everybody uses it to tune but each one in a different way, so the true BW sequence has now disapeared and what survives is a tweaked form of it.


Equal temperament is not 2^(1/12) except in some unworkable theory. For all practical purposes, it remains: Beatless octaves, and geometrically increasing beat rates of all other intervals.

If you really want to understand tuning, start by getting to know beats.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 01:17 PM

BDB, I don't want to be unpolit but let me tell you that I understand tuning and beats more than you can ever dream to understand yourself.

Tooner, here is the proof! When I said the model of ET is semitone = 2^(1/12) there was immediately someone who came tell me I was wrong. You see?

We definitely need a new model!
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 01:49 PM

Gadzar:

So, we need a new model so that everyone will agree? It matters very little to me if people agree on anything…

I find that the BW beat rate model that is based on 2^(1/12) to be very workable because the beat rates, when applied to non-iH tones, produce a variable semitone ratio. BW did the right things for the wrong reasons. It is also interesting to consider that he used 6:3 octaves in the bass, 4:2 in the tenor and 4:1 in the treble and 2:1 in the high treble. This might be a bit conservative nowadays, but not anything like tuning to theoretical pitches.

Here is something you might find interesting. Maybe you already know this, but it still may spark something that we can discuss until Alfredo returns. The emphasis seems to be on the semitone ratio. A little while back I thought erroneously that any ratio larger than 2^(1/12) would produce a Railsback curve. I now realize that any fixed ratio will produce a straight line on a Railsback diagram, not a curve. The semitone ratio defines the slope of the curve at any point. So, for a tuning to produce a Railsback curve, the semitone ratio must be least at the midsection and greater at the extremes.

This makes me think more about Mr. Stopper’s use of a cent based on the 3^(1/19). It seemed silly at first to me to have an entirely new value for a cent, but if it is decided to base calculations on 3^(1/19) being a slope of zero, other calculations would be simpler and there may be less error when interpolating between 12ths.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 01:57 PM

Tooner,

When you say:

Originally Posted By: UnrightTooner
A little while back I thought erroneously that any ratio larger than 2^(1/12) would produce a Railsback curve. I now realize that any fixed ratio will produce a straight line on a Railsback diagram, not a curve. The semitone ratio defines the slope of the curve at any point. So, for a tuning to produce a Railsback curve, the semitone ratio must be least at the midsection and greater at the extremes.

This makes me think more about Mr. Stopper’s use of a cent based on the 3^(1/19). It seemed silly at first to me to have an entirely new value for a cent, but if it is decided to base calculations on 3^(1/19) being a slope of zero, other calculations would be simpler and there may be less error when interpolating between 12ths.


Aren't you in search of a new mathematical model that describes the correct way of tuning ET in a piano?
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 01:58 PM

Originally Posted By: Gadzar
BDB, I don't want to be unpolit but let me tell you that I understand tuning and beats more than you can ever dream to understand yourself.

Tooner, here is the proof! When I said the model of ET is semitone = 2^(1/12) there was immediately someone who came tell me I was wrong. You see?

We definitely need a new model!


Of course you meant to be impolite. It is easier for you just to claim that you understand something better than me than it is for you to actually prove it.

The twelfth root of two is not a good model for equal temperament because nobody can hear a twelfth root of two. It is no more than a mathematical approximation. Tuning is about what you hear, not about numbers. There are other models of equal temperament that describe what you should be hearing when a scale is tuned that way, and they are equivalent. But the twelfth root of two is a derivation which is not useful in tuning practice, although it may be for setting frequencies.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 02:02 PM

BDB,

That is exactly what I am saying. ET is not described correctly by the model of semitone = 2^(1/12).

Now, we are searching a new model that does it! Haven't you red all what I've said in this thread?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 02:22 PM

BDB:

I consider the twelfth root of two as vital in aural tuning. Raising it to the power of 4 gives the ratio of CM3s, to the power of 2 - the difference in the beat rate for the M3-M6 test for a tempered 4th. And just like it is - the ratio of the beat rate of chromatic intervals. I admit that I do not hear this in decimal format, though.

Gadzar:

No, I am looking at different ways to describe tunings in general, not to define a correct tuning. I don’t believe there is such a thing, not even a “good” tuning. When someone says something is “correct” or “good” it just means it is what they prefer. It is not like the Deity declaring something to be “Correct” or “Good”.

Oh, and BDB may appear as “dumb as a fox” sometimes….
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 02:31 PM

Jeff D.,

I am looking not for a "good" tuning. I'm sure BDB can do a good tuning without knowing what he is doing. But I am in search of a true standard for piano ET.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/23/09 03:03 PM

Jeff,

Originally Posted By: UnrightTooner
A little while back I thought erroneously that any ratio larger than 2^(1/12) would produce a Railsback curve. I now realize that any fixed ratio will produce a straight line on a Railsback diagram, not a curve. The semitone ratio defines the slope of the curve at any point. So, for a tuning to produce a Railsback curve, the semitone ratio must be least at the midsection and greater at the extremes.


That's what I am talking about when saying that the model must take into account iH all along the scale. It can not be a constant ratio it must be a variable ratio. And this variable ratio will depend on the iH of the string being tuned in order to come out with Railsback curve. Such a model can be easyly implemented in an ETD like Verituner which already measures 8 partials of each string while tuning.

But the problem with Verituner is the lack of a mathematical model wich dictates the apropiate strech needed for each note. Verituner solves the problem by using what is called "Styles". One can define a tuning style by stating differents amounts of stretch all along the scale by means of defineing cents or beats in a given type of interval in a given point of the scale. I think there is a limit of 8 stretch points for a customized Style.

So, it is the responsability of the human tuner to design a Style which matches or fits exactly the piano's scale being tuned, if he fails to do a good match, the tuning won't be good.

That is where a new mathematical model can come to help the tuner, the human tuner, so he has not to tweak intervals, stretch, cents and beats.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/24/09 08:35 AM

Gadzar:

Then it seems that what you are looking for is an ETD that does things differently. It also seems that you see the selection of different Styles as being necessary due to different scalings, not for a preference of how a piano will sound. I am not sure what to think. (Was just given a large project with a deadline, so I am a bit distracted.)

But a tuning model does not have to use semitone ratios at all. From a Railsback curve, the cents deviation can be applied directly to the theoretical frequencies. It seems a much simpler way to calculate the frequencies. But then I wonder which partial should be used. The first partial is used for fewer intervals than the fourth partial. Why not just use the piano’s iH to create a Railsback curve for the 4th partial and go from there? I am just musing. This has been figured out by ETD programmers already.

So what do you think this new model would do or be like?
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/24/09 11:57 AM

I am looking for a standard mathematical definition of ET which includes iH.

Of course, with that definition one can program an ETD.

I don't see the selection of different Styles as being necessary due to different scalings, it is necessary with the actual model because it is not perfect so we need to change our claculations for each different scale, it wouldn't be necessary with a better model.

The goal of course should be the way the piano will sound in ET.

I don't think that choosing a unique partial, the fourth partial for instance or the third partial, which is also used by 5ths, 8 ves, 12ths, etc., will work. iH can be estimated but there will be strings with irregularities in iH, even partials with negative iH where some partial may have a lower frecuency than its corresponding theoretical harmonic, so implementing a model based on estimations of the iH calculated from a single partial does not seem to be adecuate.

When tuning aurally we hear at the real partials, I think the model can do the same and it can determine tuning frecuencies from all the real partials being heard when playing a note. Just as the "natural beat" of Mr. Virgil Smith does, combining them mathematically in a one beat.

Let me put it that way, when tuning CM3s you are hearing at the 4th and 5th partials, but then you need to tune a 4th in order to get to the following set of CM3s, you need to hear to the 3rd partial. And when you hear at the F3-F4 and A3-A4 octaves, you need partials 1,2,3,4 and 6 at least to achieve good octaves. So when tuning aurally the temperament octave or tenth in this case you are hearing at the 1, 2, 3, 4, 5 and 6th partials. You can not ignore, nor estimate, none of them.

You can not privilegiate a one single partial of any note because the others will beat in an uncontrolled way. So you need a model which treats iH as a whole phenomenom combining all of these partials into one unique result to calculate the frecuency at which a given note will be tuned.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/24/09 01:03 PM

Gadzar:

Maybe there is a vocabulary problem. When we are using the word “model” I am thinking of this definition: “A system of postulates, data, and inferences presented as a mathematical description of an entity or state of affairs.” (Merriam-Webster online dictionary)

A model that I am familiar with is the Hydrostatic Tables for a ship. Before leaving port I would compare my calculations with the ship’s draft and then sometimes ask the Chief Engineer where he is hiding the extra 500 tons of fuel oil! (Actually, I would already have a pretty good idea.)

When using the word "model" are you thinking of an algorithm?

I am not sure that a standard definition of ET, or even a mathematical model of ET, needs to include iH. If we say that the beat rate of intervals should do such and such, and apply this to tuning a piano, the iH will be the determining factor for the frequency of the notes. But the iH is not in the definition or the model, but in the application. The end result, regardless of the piano, would be the same. The intervals would beat such and such.

But this is talking about a model of a tuning, not a model of a piano. A model of a piano must include iH, preferably tabular and not approximated.

And if we wanted to tune a piano by calculating the frequencies of individual notes, we would need both models. Then we could tune by whatever partial is convenient. We would know where the other partials will be and what the beat rates will be.

But what do you think this “natural beat” is and how would it be calculated?
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/26/09 02:23 AM

I want all of the above. A model, an algorithm, a new ETD, etc.

The goal is to achieve accurate ET in pianos without having to tweak theoretical values when tuning a real piano because of iH.

About "natural beat", I am still wondering what Mr. Virgil Smith hears!

For me, there is a range, a vast range, when tuning an interval, where it sounds correct to my ears. It could be a little wider or narrower and still sound good. I must use a test to set it accurately where I think it must be.

For example when tuning A4 to the fork. Initialy I sound the fork and the center string of A4, and I tune to a clean, beatless sound. Then, I play F2-fork and compare the beats with F2-A4, and I tune equal beating. So where is the natural beat gone?

Again, for me it is way a large margin left, not accurate enough to make a tuning.

Maybe I am too young as a piano tuner to be able to identify such a "natural beat" with accuracy. But I am sure Mr. Virgil Smith can do it with the required accuracy.

Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/26/09 07:52 AM

Gadzar:

The “natural beat” seems to be heard only by those that hear it. It might as well be the Emperor’s New Clothes to me.

I know what you mean about tuning an interval that sounds good and then the tests show it not to be where I think it is. I think this is caused by multi-partial matches. It happens to me most with midrange octaves. Then I wonder that if a difference cannot be heard in the interval that I am tuning, then what does it matter? But when other intervals are played, then it does matter. So why not tune to make those other intervals sound right? More and more I have been just listening to the fourth and fifth to set the octave where I want.

Since you have an ETD that listens to all the partials, and it does not do what you want, perhaps an ETD cannot do what you expect. That leaves aural tuning, but many of the tests beat too fast or too slow in some parts of the piano to be useful. But if the test note is adjusted so that the beat rate was useable, then more accuracy would be possible.

I had played around with using a pitch source as a controllable test note and it seemed promising, but also kind of “propeller head-ish.” It wasn’t that easy to hear the beats, either. If I have time I’d like to try clamping some kind of transponder directly to a soundboard rib and give it another try. It seemed a good way to even set a temperament if the base pitch is offset correctly to accommodate iH.

Something that has worked very well for me to avoid multi-partial matches and their ambiguity is listening to the 12ths. The 3:1 match is so much stronger than the 6:2 match that there is no ambiguity. Also, the beat rate is very slow or even zero, and I think this is important.

If an interval’s difference in the nearly coincident partials is 10 cents and the interval beats at 10 bps, then a 1 cent change will change the beat rate by 1 bps, or by 10 percent and hopefully be noticeable. But if the interval is 2 cents and beats at 2 bps, then a 1 cent change will still change the beat rate 1 bps, but will be a change of 50% and certainly will be audible. 12ths beat very slow, if at all, and can be used for very accurate aural tuning. This is one more of the reasons I prefer SBIs to RBIs in general….
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/26/09 01:09 PM

OK, I guess we can continue talking about this endlessly, whithout coming to an answer.

So let's wait for Mr. Capurso's return to hear what he has to say.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/28/09 02:13 PM

Thank you all for your elaborations.

Kent, back some posts you say:

“I had hoped to hear exactly what intervals or chords sound harsh in a pure 12th tuning…”

In practice, tuning pure 12ths (theorethical 19th root of 3) means beatless 12ths, so moving away from 3ths (theoretical 4th root of 5/4), octaves (theoretical 12th root of 2), 10ths (theoretical 16th root of 5/2) and so on intervals harmonic ratios.

”...and what can alleviate that harshness, according to those who hear such harshness.”

To alleviate that harshness we can distribute the tare, i.e. the difference, on 12ths (narrowing tare) and 15ths (widening tare).

“...I can't hear such harshness so I haven't a clue where harshness might be coming from.”

In order to really understand what you can hear we should talk in front of a real piano. As I could tell you, I do not know what is the final result after using Stopper’s device, nor I know if you can aurally ascertain whether you have tuned and stabilized pure (beatless) 12ths or what. Has Stopper told you, at the end of your tuning, you should find (aurally) beatless 12ths? What 12ths do you find in the piano you have just ETD-tuned? Are they supposed to beat? Do they beat?

“...Actually, my hypothesis is that my execution of Stopper's implementation of a pure 12th tuning would not sound "harsh" to any listener.”

Please, first let’s try to understand what tuning you are fixing with your tuning device, so we’ll know what we are talking about.

...” Stopper has made some rather bold claims; the difference is, however, that unlike Capurso, he has made it possible for us to duplicate his tuning and verify the claims.”

To duplicate one’s tuning (litterally) we should have that one’s aural power and that one’s wrist, to this extend no ETD can give us that chance. As a matter of facts, no duplication, no verification. What’s more I’ve already tried to make a distinction between Chas theory’s model and tuning devices, there lies the difference.

Bill, thank you for the master tunings figures.

Talking about beats, also in my tuning 5ths became faintly wide between C5 and C6 (when tuning middle string), while 12ths never invert with 15ths. Moreover I need to conferm that 4ths and 5ths have there regular, precise beat-proportions and beat-curves. By managing 4ths and 5ths progressive beat/rate, together with RBIs, no iH degree can impede the finding of the correct beat-form. You may conferm this, by comparing beats you too can find again and again your favorite tuning-form, on average scaled pianos.

About tuning the highest octave, I’m used to checking with lower 15ths and 12ths, for example: for tuning A7, besides A6, I play together A5 (lower 15th) and D6 (lower 12th), and plucking A7’s middle string I can fine-tune for no-beating.

You kindly say: ...”Therefore, it begs the question whenever any one of these new and improved ETs come out which claim to have the ultimate answer to universal beauty, "Just where is that sweet spot?" If we don't hit it this time but get it the next, will anyone really ever notice or care?”

Firstly, generally speaching, with wrong teachings one may never hit a sweet enough spot (many of us still think that 4ths, 5ths, octaves may invert due to iH); secondly, I’m sure I’m not the only one who, by comparison, can distinguish a sweet spot from a sweeter one.

You also say: ...”While I agree with both Jeff and Kent that one would need to compare the actual partials, they are unknown and can't be determined without actually measuring them. Once the partials reach into the 8th octave and higher, that is impossible. They also cannot be heard, so it does not matter that 5ths and 12ths are wide.”

Could you please explain me this one point? I’m not sure of what you meant.

Tooner, thanks for your sharing. You commented Bill:

...“Very excellent point! Who really notices or cares? Probably only the person that did the tuning.”

I’m sure you do not really think that.

...”I sometimes think that many people could hear what I hear, but it is the caring that makes the difference. I have resigned myself to the burden of never being quite satisfied.”

I’m sure your resignation will never reduce your commitment.

Previously you said: ...“You do not ignore iH in your paper. You mention it a number of times. That does not mean you are applying it correctly.”

I’m simply not applying iH because iH does not have effects on the Chas-form I’m describing. Also in my practice, I only refer to beats and “normal” degrees of iH never disarrange the interval's beat-form. I think Bill finds the same with its tunings.

...“You referenced Young’s paper, but seem to either not understand it, or believe that iH is different than described.”

I think that string’s iH has been calculated only approximately and therefore iH calculation (at least on pianos) can be improved.

...”Young’s paper describes how iH is calculated and how it affects the frequencies of partials.”

We’d better say ”...how it affects the expected theoretical frequencies of partials”.

...”You have not used those calculations in your paper.”

Nobody imposed me that.

...“I don’t know how I can be more - responsible and utterly precise when I say your paper is in error.”

I still think you have got no reasons for saying that. I do not mind your severe look, only I can not stand an error when talking about error. In my opinion the Chas article is defective (?), but only because it could not be a treatise. In the Chas article’s economy, it was enough proving that (from section 1.6):

“The chas octave deviation curve is in line with the Railsback curve, as shown below (section 4.2)”.

By doing so, I’m proving that a relevant part of the frequencies deviation should be referred – actually it does depend - on the reference model, what I’ve already mentioned as being the bottom question.

You say:...” [EDIT]: Ok, I think I know what you mean, now. Rather than considering the semitone as being an interval that is tuned, it should be considered as a note on the piano. The iH doubles every 8 notes in this example. iH is a function of piano scale design, not tuning.”

You say your self that iH is a function of piano scale design, so it depends on the piano scaling. Depending on the piano, we may then find different partial frequencies values. Chas model does not approach scale frequencies, it approaches ratios differencies and intervals beats. In practice this means that, despite whatever actual frequencies values you will determine, you can still go for the one most correct interval’s beat progression (what most aural tuners do). When piano scaling will be improved...

Robert, thanks for your answers.

When I wrote: “Have I talked about iH definition? No, I have and I am talking about the reference model on which basis iH’s effects have been calculated.”, you answered:

...”You may not have explicitly talked about your definition of IH. But one must learn to walk before one learns to run.”

On the iH’s ground I have not walked nor ran yet.

...”How can you make accurate statements about IH if you don't understand what it is at a more fundamental level?”

Is there a more fondamental level than what has been shown so far? Is then iH a deep notion or not? Also, if I may re-ask, are you an aural tuner? Could you please tell me, when you say about pipe organ “...It has harmonics…..that is the definition of zero inharmonicity...”, what are the harmonics values you are/they were referring to, when fixing zero iH?

Kent, thanks for improving. You say:...” I believe it is true that pianos tend to be scaled in such a way that beat speeds taken from the mathematical model of equal temperament can be used in the mid-range of pianos with only relatively small modifications due to inharmonicity.”

This is the crucial point: Chas theory’s model proves that those “relatively small modifications” are not only due to iH, they are the direct and natural consequence of interweaving partials 3 and 4. This leads to Chas theoretical stretched and natural-beating octaves.

Then you say:...” There probably are still contributions to be made concerning "the proper consideration of inharmonicity", but my opinion is that these contributions should be made with great respect for the coherent effort and considerable intellect that has already been devoted to this subject. After all, many pianos already sound very well tuned.”

I totally agree.

Tooner, you say: “I think the problem that is underlying this discussion is bruised egos.”

In my opinion there may be also threaten egos, due to transversal interests. Anyway, I do not look at it as a problem, it is a quite human interfering theme.

...” It is interesting that the octave tests that were used were appropriate even though they were not designed in "consideration of inharmonicity." Just musing...”

They could represent one more relevant clue, besides talking about iH you can always listen to and control beats, and go for a precise beat-form.

Bernhard, thanks for your references. You say:...” An ET based on the theoretical model of the 19th root of three has been proposed by me in euro-piano 3/1988.”

You see, I do not look at 19th root of 3, 31st root of 6 and so on, as if they were theoretical models; like 12th root of 2, for me they are only the expression of pure ratios logarithmic progressions, adopted on the base of an ancient fashination for “pure ratios”, a way to easly describe nature with small integer numbers. In other words, those ratios represent a quite banal use of a powerful algebraic instrument. When we fine-tune a string, we go for the nth decimal tension’s degrees, what we can not represent with 2*2 nor with 3*3.

Moreover, I’m quite sure you know what is the difference between theoretical models and theoretical pure ratios, like the ones mentioned above.

You say:...”Inharmonicity consideration is targeted by using inharmonicity affected partials through the use of beats when tuning aurally.”

I find your detail-giving way mysterious, and I wonder: is it me? Let me also ask you: would’nt it be nice, in 7/2009, if you thoroughly specified how your ETD works and how iH consideration can be targeted by using beats? Don’t you think it’s time you shared your actual foundings?

You kindly wrote:...“A general mathematical model of tunings has been proposed by Guerino Mazzola 1989 in his book "Geometrie der Töne". In section 2.3.3,

a "convention dependant linear function of the form
Y = uX + v, where u and v are not constants,
with X = ln(f) " is proposed, what finally includes all possible theoretical ET and non ET models.”

Well, I can not see anything that recalls theoretical frequencies ratios nor practical piano tuning. I can not even see Bill’s EBVT model. Chas ET EB dynamic algorithm improved:

(3 – (Δ*s1))^(1/19) = (4 + (Δ*s))^(1/24)

approaches the scale’s incremental ratio through beats, exactly what we do when we tune aurally. This makes Chas theory simply unique and may explain why this model could be extracted only from tuning experience. In Chas algorithm, the deltas represent the pure ratio’s differencies, i.e. beats, s1 and s represent the two variable discretional values that can change that curve and draw infinite curves. This means that deltas can work as a retractor, s1 and s work as regulators, say trimmers. Do you think Chas model may somehow upset Guerino Mazzola?

Gadzar, good you managed to join in. It was interesting how it seemed to me that you jumped out of a top hat...so I’m still wondering about Chas illusionistic power. Fortunately I could somehow fall into line with you, by gobbling down a couple of sangrias. In a short while I’ll hopefully find a way to discuss, in the correct order, your points too. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/29/09 07:18 AM

Alfredo:

I cannot “connect the dots” to see how what you are saying fits together. Your CHAS ratio will produce neither the graphs nor the aural tuning that you described. I am going to just read the responses from other posters for a while.

Also, consider making separate posts when replying to different posters. I think it would be easier to follow the discussions that way.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/29/09 09:07 AM

Alfredo:

Oh well, I guess I couldn’t sit on the sidelines after all. Here is a specific error that you can consider:

In section 1.6 “STRING INHARMONICITY” you state:

“The chas octave deviation curve is in line with the Railsback curve, as shown below (section 4.2).”

And in section 4.2 “COMPARISON BETWEEN RAILSBACK CURVE AND CHAS OCTAVE CURVE”, Table 3 contains:

EQUAL VALUES / CHAS VALUES / CHAS DEVIATION (Hz) / CHAS DEVIATION (CENTS)
55.0000000 / 54.956192929 /-0.0438070708 /-1.37
110.0000000 / 109.941582816 /-0.0584171842 /-0.91
220.0000000 / 219.941575058 /-0.0584249421 /-0.45
440.0000000 / 440.000000000 / 0.0000000000 / 0.00
880.0000000 / 880.233761848 / 0.2337618481 / 0.46
1760.0000000 /1760.935171585 /0.9351715846 /0.92

The Chas deviation (cents) describes a straight line, not a curve. So this shows that the Chas octave deviation IS NOT in line with the Railsback curve. At least not in how you describe it.

Like I said, I just cannot “connect the dots.”
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/30/09 12:55 PM


This post needs to be an unscheduled tutorial, so it may not be of general interest.

Gadzar, let’s look at your points.

In your nine-days-ago first post you start with: “I don't believe my eyes!”

Well, neither would I, you could read my surname Carpuso when it happens to be Capurso.

You say: ...”So you guys can not come out with a definition of ET? In this topic I've seen the most experienced people...confused by Mr. Carpuso to the point of admiting they don't know what ET is for sure?”

At some point you must have gone off a tangent. What had you had for breakfast? The colleagues you’ve mentioned do not seem to be confused by me at all.

...”Mr. Carpuso doesn't give an answer with all that math stuff. He only plays with some clever equations which relate to nothing in piano tuning's real world.”

I think this is uncommentable. Anyway, you’d better know that, while trying to take your wordy, unjustifiable philippic easy, I’m not playing at all. What you could here define in some lines with your freshest neural effort, is the result of hard work, many seriously committed people’s hard work.

You write:...”His sequence is the same sequence up a fifth, down a fourth taught by Randy Potter in his course...”

Not exactly. In Step 3 and 4, tuning D4 and E4, I temporarily draw up the stretch for two pure-directing 5ths (A3-E4 , D4-A4), two wide-directing 4ths (A3-D4 , E4-A4) and A3-A4, just on the wide beating-soil. Then, as I’ve said, I categorically make use of SBI and RBI and, depending on how flat the piano, I draw a variable stretch-curve.

...”and he does not explain how to exactly temper fourths and fifths.”

Once you believe your eyes you can go back reading. Anyway, my sequence as I could say is nothing special, although it may help to lay down a correct 4ths and 5ths overcrossing and beats-curve proportions (beats progression), the base structure that (amongst others) in my experience can lead to progressively stretched intervals and to Chas ET EB form. Going from step 1 to step 4, I can establish my hypothesis of temperament foundation, and avoid being misled by iH.

...”It is unbelievable that Mr. Carpuso dares to claim that he has constructed a new model in which there is a variable called 's'...”

Well, you could believe that. I’m trying to share a new ET EB dynamic model that uses a variable called “s”. Now you only need to calmly breathe through your nostrils and wait for your self to click.

...”(Mr. Carpuso confuses iH with stretch)...”

I do not think I confuse iH with stretch, nor I think Mr. Scott does, when he says:...”more IH will cause any implementation of ET to have more stretch than it would if there were less IH.”.

Instead, I’m still trying to explain why I can not agree with Mr. Deutschle when he says:….” The octave is tuned wider than theoretical due to iH.”

In fact what I’m saying (since I can prove it) is that, with or without iH, we need to stretch octaves. Why? Because also partial 2, with the other partials, through stretching can practically contribute to hold up a resonant beating-whole system. Negating the beat’s value (or relevance), we would never get to the Chas concept of a beating-whole.

Mr. Bremmer pictures: "...new and improved ETs come out which claim to have the ultimate answer to universal beauty...”

Maybe in that kind of effort, still today we see more ET with “pure” scale’s incremental ratios be supported, and this happens when not considering the beats potential value and while still thinking in static terms. What I think is that “universal beauty”, if anything, is dynamic. Chas model approaches a dynamic beating-whole, that can be qualified as “pure” in that all partials are theoretically – as in practice - involved in the beating-whole’s form (section 2.0).

...”And it turns out in this thread that he doesn't even know what inharmonicity is...”

Generally speacking, don’t stay to what may turn out, make use of your own elaborations and conclusions. So doing, you’ll defend yourself from cello-syndrome.

...”And here we have gentlemen like Mr. Jeff Deutschle and Mr. Robert Scott explaining him what iH is! Or brave Mr. Bremmer trying to find wide fifths and 12ths in a Master tuning in an effort to understand what Mr. Carpuso says! No way!

My impression is that something with you may not be always fiting. You seem to understand if people are gentlemen, or brave, without realizing how you may sound like a rash, impudent and spoilt child. You see, when I mention “wrong teachings in tuning” I also refer to behaviour, since sound is the severest mirrow of our being. Tuning, in the way I look at it, may also be the ultimate expression of the finest and most correct evaluation, the result of maximum control and self-control.

You write:...”Don’t get confused by all this bla, bla, bla.”

I’ll understand this as a silly fear that may find room in your cockiness. For this, commit yourself completely and trust time.

...“At first I was impressed by this thread when reading Mr. Carpuso's paper...It impresses people by using confusing math, which leads nowhere.”

I do not know about you, I could normally study that “impressing/confusing” math (i.e. proportions, fractions and roots) when I was about twelve. Will it lead nowhere? ((((You never know)+(you may change your mind))*(speack for yourself))^(do your best))/(and doubt).

...”All that math isn't required to tune an octave slightly wide, a fifth beating...”

I agree, remember though that our system in not Gadzar-centric, so that basic math may interest someone else. In your case, if you were polite, I’d probably ask: are you trying to tune piano aurally? Did you know that narrow progressive 5ths invert and very smoothly direct to pure?

…”Sorry Mr. Carpuso but I think you are an excellent illusionist.”

I’ve decided to thank you for mentioning excellence, only avoid saying “sorry” just when you congratulate me, Ca pu r so. For the future, also Alfredo will do. a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/30/09 01:38 PM

Tooner,

the way you keep on doing, writing: "...Here is a specific error that you can consider:..."..."...The Chas deviation (cents) describes a straight line, not a curve. So this shows that the Chas octave deviation IS NOT in line with the Railsback curve. At least not in how you describe it.",

ever alarming for errors that are not errors is only spreading an unconfortable cello-syndrome, as you could see with Gadzar, and this does not help at all.

What I suggest is that you walk in any math or phisics department and ask someone you trust for the explainations you need, exactly the way I did. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 06/30/09 02:17 PM

Alfredo:

Maybe the problem is that you walked into the wrong math department!

But you are side-stepping the issue. Your table describes a straight line. The Railsback curve does not. Is that why it is in tablular and not in graph form? The difference is less noticable that way.

And after all, you are the one that did not like a statement of general errors. And now you simply declare that any errors I point out are errors on my part, without explanation from you!

The cello remark is insulting.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/02/09 11:39 AM

OK Mr. Capurso. You are right. I am really sorry about what I have said and the way I have said it. I apologize; I was in a bad day. Sorry also about misspelling your name.

I see your system as a real effort to know more about piano tuning.

I disagree with your system. The tuning sequence you've posted here is like a dozen I know, tuning fifths and fourths. And in fact I don't believe you are really applying CHAS model in your tunings.

Furthermore, as I've said before, CHAS works by stretching intervals. I don't see how CHAS deals with iH, if it does at all. For example, if the lower note of an interval has more iH than the upper note then the interval should be stretched, but if the lower note has less iH than the upper then the interval should be shrunk. In a real piano both cases are present along the scale and I don't see how CHAS can deal with this fact if it does not take into account iH for each note. You can say that the values of delta, s and s1 could cope with it, but in that case your model is incomplete as you don´t relate them with iH.

I believe the only way to implement a new mathematical model for tuning pianos is by working on iH.

It is iH that creates different types of octaves and other intervals. Without iH you would successfully tune ET by tuning 2^(1/12) ratio semitones. Thus the new mathematical model should take iH of each note into account.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/02/09 11:49 AM

Gadzar:

I have bad days, too. Today is not one of them, yet...

I have some comments on what you said about the amount of iH of lower vs higher notes of intervals determining whether the interval is stretched or not. Are you interested?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/02/09 01:05 PM


Tooner, you say:

“The cello remark is insulting.”

Once again, if I can respectfully say, you happen to be in error.

Can the “cello” remark be insulting? Just for clearity, let’s see.

The “cello story” starts with your third post in this Topic (05/07/09), when you write: “I still have not finished reading your paper...But there is also a math error.”...

Just following your erroneous statement, Mr. Stopper could quickly write his first post, (05/07/09):

“Well observed Jeff. Maybe a kind of scientific hoax of the category "cello scrotum".”

For what I could understand, that was a banal and vulgar insinuation. In this respect, I use the verb “to cello” meaning “to insinuate”. So it can not be insulting.

As Mr. Stopper confermed with his second post (05/20/09): “...even if something has been published in a serious scientific medium, we have to be very careful about the content.”..., with “cello scrotum” category he meant to warn against “scientific hoaxes”.

So I use “cello-syndrome” referring to that level of “superficial, prejudicial suspiciousness” that will not help to distinguish a “scientific hoax” from a conceptual study and a reliable numerical evidence.

Maybe now you can better understand that neither this neologism is meant to be insulting.

Why do I talk about “prejudicial suspiciousness”?

Because neither you nor Mr. Stopper had finished reading the Chas article.

Now you are saying:

“Maybe the problem is that you walked into the wrong math department!”...

For what I can understand, you are again insinuating, I’d say “celloing”, about the reliability of those phisics and maths university lecturers involved in Chas theory. This, for me, is rude, unjustifiable and unprofitable. Since I’m not interested in triviality, I can not discuss on this ground.

...”But you are side-stepping the issue.”

By now, you may well know that it is not my style.

You write: ...”Your table describes a straight line. The Railsback curve does not. Is that why it is in tablular and not in graph form? The difference is less noticable that way.”...

Again, you are celloing (i.e. insinuating). In Chas article, believe me, you will not find any trick. This is what I would call cello-syndrome, i.e. over-distrustfulness.

...” And now you simply declare that any errors I point out are errors on my part, without explanation from you!”...

So far, you have had theoretical and numerical explainations from bobrunyan (05/07/09), from Robert Scott (05/12/09), from ROMagister (05/29/09) from Roy123 (05/31/09) and from me, during all this discussion.

For what I’ve just explained above, while thanking you for your precious contributing, I friendly and peacefully suggest that you ask someone you trust for the math explainations you may need. As for Chas theory’s conceptual and practical issues, I’ll be pleased to contribute.

Regards, a.c
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/02/09 01:19 PM

Gadzar, thanks for your outlook on Chas model V tuning. I'll contribute asap.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/02/09 02:34 PM

Alfredo:

I do not need anyone to explain to me the difference between a straight line and a curve.

The folks from institutions of higher learning do not impress me at all when it comes to practical matters. Remember the Cold Fusion debacle?

Now as far as your paper, I have read through it a number of times, but since understanding each step depends on understanding, and accepting all the previous steps, I do not get very far with it. Your statement that this straight line shows that Chas is “in line” with iH as shown on a Railsback curve allows me to go no further in understanding or accepting. Even if this line was curved, and not straight, (by using a variable ratio) that would still not indicate that Chas is “in line” with iH. Only by including a discussion of iH could it be shown that Chas is “in line” with iH.

Now let me take three steps back and look at this from a broader perspective. Here you are declaring that there is a new, improved way to tune and offering a paper to prove that it is a better way. Now I agree completely that when tuning aurally, the beat rates determine the tuning, and that any theory of iH means very little. The effect of iH adjusts the frequencies so that the desired beats are heard. And if someone prefers the beats to be a certain way, they can tune a piano that way and it will sound that way. But then what is the purpose of the paper?

So I take another three steps back and look at how things are said, rather than what is said. When confronted with apparent errors, you dismiss them, not explaining deeper meanings. And you send the questioner to ask someone that knows what they are talking about, inferring that the questioner does not (but do you?). When information that you admit that you do not know is presented to you, you dismiss it as unimportant, yet it was important enough for you to use it as point in your paper…. When asked about the process involved in having the paper published, the answers are rhetorical, like: what do you think, and how could anyone argue with it??? When someone disagrees with your paper you try to degrade their importance, but when they say agreeable things, you are condescending.

There is a saying that if it walks like a duck, talks like a duck and acts like a duck: it is a duck. This sure seems like a scam to me. What the purpose is, I don’t know.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/08/09 11:45 AM

What’s the purpose? To become the USA’s President.

I had to look up my dictionary to know what “scam” means and now I can reassure you, if it means cheat Chas model is not a scam.

Chas theory’s model is simply meant to tranlate my outlook on tuning sound’s intervals in a scale, together with my professional experience of a progressive and EB (equal beating) temperament, where “progressive” is referred to beating 3ths, 4ths, 5ths, 6ths, octaves, 10ths, 17ths and so on, and EB is referred to 12ths (narrow-beating) and 15ths (wide-beating). Consequently, the Chas article is meant to share Chas theory’s model.

The novelty regards the concept of purity and the idea of a dynamic beating-sound-whole.

In section 2.0 you read: ...“The chas model approach starts from the traditional chromatic scale, but brings innovation to the theory and practice of tuning by recognising that beats are as natural for octaves as they are for the other intervals.”

So in Chas theory, natural beats represent the key, foundamental phenomenon.

...“Octaves, too, can and must be tempered, exactly as fifths and thirds have been. Thus the need arises to combine partials 2, 3 and 5 in a new set.”

Chas model draws indeed a new set of sounds, a set of frequencies that, for the first time, derives from proportional beats and from the interweaving of partials.

...“Purity no longer derives from a single combination or from a pure ratio, but from a new set which is pure because it is perfectly congruent and coherent.
The sounds in the scale all give up a small part of their pure partial value for the benefit of this set which is now harmonic and dynamic since it is the result of a natural, intrinsic correlation between frequencies and beats frequencies.”

So, Chas model first establishes the beats proportions for partials 3 and 4 relative scale’s values, then it gains the scale’s incremental ratio. So doing, Chas model enlarges the traditional 12 semitones module (section 3.4) and theorizes an intermodular temperament.

Why do I like the idea of sharing Chas model?

Maybe to pass on my satisfaction in aural tuning, maybe to leave behind unconvinient historical heritages and to correct the usual approach to tuning, maybe for those and more reasons taken as a whole.

Tooner, about Chas you say: …“Your statement that this straight line shows that Chas is “in line” with iH as shown on a Railsback curve allows me to go no further in understanding or accepting. Even if this line was curved, and not straight, (by using a variable ratio) that would still not indicate that Chas is “in line” with iH.

To be precise, this is what I would say: the Railsback curve is not “iH”. The Railsback curve is a first representation of the iH’s effects on some attempts of traditional ET pseudo-model tuning.

In section 1.6 you can read:

“The term inharmonicity describes the deviation of partial frequencies from the natural values of the harmonic series. String rigidity is one of the causes of this phenomenon.
String length, diameter, density and tension all contribute to calculating inharmonicity. The phenomenon, discovered last century, obliges the 2:1 octave ratio to be stretched.
Railsback measured average deviation from the 2:1 ratio in the pianoforte; from the lower sounds, the curve gradually flattens toward the middle sounds, where the degree of inharmonicity is slight, and again grows as the notes become higher.”

So, the point is not if Chas theoretical values deviation draws a curve or a line, the point is that Chas values deviation, likewise in Railsback’s representation, grows as the frequencies depart from the mid-range.

Why is this relevant? You said it yourself (06/23/09):

...“I now realize that any fixed ratio will produce a straight line on a Railsback diagram, not a curve. The semitone ratio defines the slope of the curve at any point. So, for a tuning to produce a Railsback curve, the semitone ratio must be least at the midsection and greater at the extremes.”

You see, “to produce a Railsback curve” should be our goal only if traditional ET pseudo-model was theoretically reliable and only if the Railsback curve was identifiable with iH. But careful, the Railsback curve is not iH, it is a representation of iH’s effects, and traditional ET pseudo-model is not reliable since there is no logical reason for fixing 2:1 octaves.

You then wrote: ...“This makes me think more about Mr. Stopper’s use of a cent based on the 3^(1/19). It seemed silly at first to me to have an entirely new value for a cent, but if it is decided to base calculations on 3^(1/19) being a slope of zero, other calculations would be simpler and there may be less error when interpolating between 12ths.”

Now, if you are on the point of understanding an “entirely new value for a cent” and “3^(1/19) being a slope of zero”, it should’nt take you long to understand Chas model’s “(3 – (Δ*s1))^(1/19) = (4 + (Δ*s))^(1/24) being a slope of zero”, and Chas cent = 100.038318440222…

When did I move away from all theoretical “pure ratio”, like 12th root of 2, or 19th root of 3? Once I’ve accepted a foundamental dynamic notion: in zero terms (theoretical zero-beating) you can describe nothing.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/08/09 02:32 PM

Alfredo:

You must have a different meaning than I do for progressive when you describe what your 5ths do as being progressive.

Because of iH, there is no such thing as a beatless octave, although they may sound that way. If the second partial of the lower note is at the same frequency of the first partial of the upper note, then the fourth partial of the lower note will not be at the same frequency as the second partial of the upper note, and visa versa. Although it may sound like the 2:1 partial matches are beatless, the octave is probably tuned wider than that, and always has been. Tuners have been tuning “stretched octaves” all along without any iH or stretched octave theory being involved.

...“Purity no longer derives from a single combination or from a pure ratio, but from a new set which is pure because it is perfectly congruent and coherent. The sounds in the scale all give up a small part of their pure partial value for the benefit of this set which is now harmonic and dynamic since it is the result of a natural, intrinsic correlation between frequencies and beats frequencies.”

The above is just another way of saying to tune for the best compromise. Unless there are no cross checks involved, any aural tuning scheme will make compromises. It is this kind of “hype” that makes me suspicious…

I may or may not be starting to understand what you are trying to say about your Chas deviation being a straight line verses the curved line of the Railsback diagram. Do you think that if the frequencies of your tuning were actually measured that they would produce a straight line of deviation? If you believe this, you should have the frequencies measured to find out.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/11/09 01:35 PM

Tooner, thanks for your answer.

About Chas 5ths you are right, although in my tuning 5ths invert, as you said, I still call them "progressive". Maybe there exists a better word. While from C3 to mid-range 5ths get narrower, at one point they invert, so you will have two 5ths - at a tone distance - with the same beat/rate.

The inversion of 5ths opened me the way to Chas model. The old teaching was "narrow 5ths" and described a monotone curve. Early on I realized that 5ths curve mast have been duale, i.e. 5ths should invert, so to avoid narrower 12ths, 19ths and so on.

You say: ..."Tuners have been tuning “stretched octaves” all along without any iH or stretched octave theory being involved."

You are thinking of real, actual frequencies values, is it not? When I think of tuning I still think in terms of beats.

You write: ..."The above is just another way of saying to tune for the best compromise. Unless there are no cross checks involved, any aural tuning scheme will make compromises. It is this kind of “hype” that makes me suspicious..."

Althoug I keep on reading this, I do not understand what makes you suspicious, and I wish I did.

In that paragraph (section 2.0) I explain the conceptual approach that justifies theoretical "wide octaves" and the interweaving of partials, i.e. the use of two partials in Chas algorithm. In fact, with a single-partial formula we would not control the infinite possibilities to distance and combine partials. About "compromise" I hope I'll be able to write a dedicated post.

You say you also use SBI in your tuning. What do you do with your 5ths?

Regards
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/13/09 08:55 AM

Alfredo:

Hope you had a nice weekend.

”Althoug I keep on reading this, I do not understand what makes you suspicious, and I wish I did.”

Long or obscure words, and “fluffy” adjectives or adverbs make me suspicious. Plain talk does not make me suspicious. For instance, you used the term “monotone curve” when talking about what the beat rate of 5ths did in old teaching. Well, “monotone curve” is new to me so I looked it up. It describes what you say your new 5ths do, but not what old theory predicts. Your use of the term “interweaving partials” is another red flag for me.

The biggest red flag is a declaration of trustworthiness. Why would this ever be needed to be declared, and if it is, why should anyone believe it? Nixon’s statement of “I am not a crook.” is a perfect example. Your statement: “ I had to look up my dictionary to know what “scam” means and now I can reassure you, if it means cheat Chas model is not a scam.” makes me more, not less suspicious.

Another thing that makes me suspicious is when a subject is dealt with in detail that seems to have little, if anything to do with the subject at hand. As an example, let’s say that I am talking about the best type of glue to use for keytops. I could go on and on about something like a “Hindenburg Conundrum”, all of which may be true, but has nothing to with gluing keytops. Your math does not support how you tune because it is missing an equally detailed connection to the effect of iH on beat rates.

And finally, I have learned to be naturally suspicious when it comes to what I choose to believe. This has served me very well. Constant doubt has kept me out of trouble many, many times. But, I also recognize that some things can only be believed (or disbelieved...) by Faith.

”You say you also use SBI in your tuning. What do you do with your 5ths?”

Sometimes I wonder what the 5ths do to me!! smile I look for the best compromises, and which intervals can be used depends on the part of the piano that is being tuned. The ratio of the 4ths to the 5ths works well in the tenor and they both beat faster going up the scale, with the 4ths increasing speed more than the 5ths, depending on how the octaves, 12ths and double octaves sound. In the treble there is a point where I am not sure what the 4ths and 5ths do. The octaves, 12ths, double octaves and triple octaves are more important. Making evenly progressive 10ths and 17ths is critical, along with the proper ratio of their respective m3s and m6s in order to prove, again respectively, compromised 6:2 twelfths and 8:1 triple octaves. Going down in the bass the 4ths and 5ths beat slower until they become unusable and the 12ths and double octaves are more important, with progressive 10ths again being critical. When tuning the monochords the m10 – M6 test for 12:3 double octaves works very well, although usually only the bottom few notes are actually this wide. It depends on how the octaves and double octaves are sounding. But all these tests are just tools. What I strive for is for the entire piano to sound in tune with itself. To see how this is progressing, I will often listen “musically” to the major chord in the temperament section for the note that I am tuning in the treble to decide if the note is where it “should” be. Then I can adjust the compromises that I am making with my tests if needed.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/17/09 11:46 AM

Tooner,

I've been away but now I should have more time at home.

Thanks for telling me about how you got suspicious, I'll soon answer you. I'm sorry if, with what I said, I made you suspicious even more, and talking about "Long or obscure words, and “fluffy” adjectives or adverbs" or "declaration of trustworthiness", I understand you better.

Have a nice weekend, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/17/09 01:07 PM

Alfredo:

You have a nice weekend, too.

I understand replies better when they are plain.

I am looking forward to your answers. I have been looking deeper into how iH affects 12ths differently than 15ths. We may end up with a very interesting discussion.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/21/09 01:56 PM

Bill, you kindly wrote (06/02/09): “5ths become wide on PTG Tuning Exam Master Tunings in the 6th octave.”…”It must be close to 20 years ago that I saw Steve Fairchild demonstrate that 5ths do become wide. He also said that 4ths become narrowed.”

So we agree in saying that, somewere, 5ths do invert, now the questions may be: can or should 5ths be progressive or can 5ths have casual beats/rate? And what about 4ths?

Answering Mr. Melo, with your 06/21/09 post, you say: ...“Regarding whether 5ths and 12ths become wide or not, they do, I am convinced of that. I learned that very long ago from Steve Fairchild who demonstrated it at a PTG convention.”...

Could you precise whether you are talking about actual frequencies values or beats?

Then you say: ...“If you adjust F5 on the ETD so that the pattern rolls equally sharp and flat when F3 and A#3 are played alternately, you will find an ideal spot for F5 to be tuned. The double octave will be slightly wide and the 12th, slightly narrow, each by a very small amount, nearly imperceptible to the ear.”...

Actually, this is what I do aurally, using SBI and RBI, and what I’m sure any tuner could do by aurally controlling beats.

So you say: ...“That is why I dubbed the concept as "mindless" because if either the double octave or the 12th beats, it sounds "wrong" but when there is that exact compromise between the two, it sounds "right". This proved to be true for me even when tuning an unequal temperament.”...

What you say is true for me too, only I do not call that a compromise, I call it an equal, precise distribution of a tare, i.e. the differencies from partials 3 and 4, that Chas theory can describe mathematically.

Then you say: ...“When you continue upwards, you will find the exact opposite of what you found at F5. When the 12th stops the pattern, the double octave will be wide, when the double octave stops the pattern, the 12th will be narrow, still each by a very small amount.”...

But what you are saying here is not the opposite of what we found at F5, it is exactly the same: pure 12ths will produce wide double octaves, pure double octaves will produce narrow 12ths.

You say: ...“I have now long taken to the practice of tuning pure double octaves and 5ths from F6 to the top.”...

I do not, my goal is 12ths (narrow) and 15ths (double octaves - wide) equal beating all along.

You say: ...“I have met and had discussions with Bernhard Stopper and have also heard his tuning. It has a remarkably clear character to it. While I still do not fully understand it, I did gather from what he has said that the 12ths also become wide at some point in his tunings as well.”...

Don’t you think it is time to get out from doubting and to understand exactly what we (including Stopper) are talking about? Beat wise, i.e. listening to beats, in my tuning 12ths never get wide; listening to beats, 12ths and 15ths can be equal beating (EB), and this is what I’m mathematically describing with Chas theory’s model.

You kindly say: ...“I give Alfredo the benefit of the doubt that his concept lies somewhere within a structure that may be a composite of what I describe above. There is, after all, an obvious language barrier.”...

While I thank you, in my hart I really hope that, by now, Chas structure will not be so obscure anymore. It will definatelly take some time but, after all, Chas model is only the rigorous description of a precise ET EB dynamic temperament, were ET is justified by our natural way to evaluating sounds with a logarithmic approach, and 12ths-V-15ths EB is justified by the relevance of a symmetric-beating resonant whole. In other words, Chas model is showing to be a synthesis between those claimed models with a “pure ratio” greater than 12th root of 2 and your mindless-octaves. More than a language barrier I’m experiencing different kinds of mental reservations, all very human and understandable in that Chas theory violates two ancient theoretical dogmas, the octave module and the pure octave’s ratio. Time its self will do.

You finally say: ...“It cannot be said that one idea is right and the other is wrong, only that any two ideas are different and produce differing effects as a result.”...

In my opinion, if some ideas were to gain wrong conclusions we are obbliged to say that.

Tooner, thanks for your contributing.

Every time I think about it, it seems very strange to me how you can still refer to traditional ET pseudo-model and yet have strong resistence for Chas theory's model, which can theoretically include our old model (see the kite analogy, posted 06/04/09).

I needed to go back when you asked (06/04/09): “But once you have the frequencies, what do you do with them? (Your devil's advocate is asking this.)”

The answer is: exactly what we have been doing with traditional ET pseudo-model’s frequencies.

The same day you kindly wrote: ...“Alfredo: We cross posted...if we take the beat rates (or at least the ratio between beat rates, including equal beating) that are predicted from a frequency ratio (such as 2^1/12) that does not take into account iH, and then tune a piano with iH using the beat rates we end up with a decent tuning, but a different frequency ratio, one that is non-linear. So on the one hand, the frequency ratio is wrong, but on the other, the beat rates are correct. And since when tuning aurally, we listen to beat rates, the model works even though it is incorrect.”...

There you talk about “decent tuning”, I’m telling you about a unique tuning; you say “the model works even though it is incorrect.”, I’m trying to share a correct and comprehensive model.

With one post (06/06/09) I’ve shown you a “a non linear octave difference-ratio”, you have’nt answer to that, could that be more clear?

Mr. Robert Scott kindly wrote (06/07/09): ...“The definition of the inharmonicity (which Jeff correctly cited) is an intrinsic property of a string, like the "length" or the "thickness". It does not depend on which tuning system is being used (ET or Chas or anything else). It affects the outcome of the tuning, but the tuning does not affect the inharmonicity. So when you say that the "inharmonicity constant or inharmonicity coefficient may need to be corrected", I must disagree.”...

Is it true that “the tuning does not affect the inharmonicity”? Robert, are you saying that the string’s tension does not affect iH? Is it true that, depending on the expected/desired frequency, you chose the string’s length and thickness, and those variables, with string’s tension, do influence iH? Does Chas model change the frequencies to be expected?

Jerry Groot RPT, as you joined in (06/21/09) you wrote:

...“What does not matter to me at all, is the mathematical theoretical stuff (that I do not bother to read by the way) that is spewed back and forth in here.”...

I’m sure you could have conveied your thinking in a less hasty and repulsive way. Nobody in here is spewing maths, although your math fobia could make you sick. Speacking for my self, I’m trying to share a new approach to the semitonal temperament, through a model that can also be described mathematically. As I could say to Gadzar, there may be someone amongst us (tuners and/or composers) who wants to elaborate on Chas numerical evidencies.

You say: ...“All that is to me, is one person trying to impress another and it does not impress me one bit. It doesn't mean that one can hear it... All that shows is one can talk it... Proving it is an entirely different matter.”...

If your interest in maths were to grow you would understand Chas theoretical model, also that I can talk about it only because I can hear it and that nobody could ever dream to impress another by explaining such basic figures.

Tooner you wrote to Gadzar (06/22/09): ...“I think Alfredo is sincere, but thought he knew more than he actually did. He may be able to learn more here.”...

I realy thank you for your friendlyness and yes, I can confirm, I’m learning more. Forth tuning is quite more difficult than Piano tuning.

You say: ...“My personal, practical definition of ET is where all M3s and M6s beat progressively faster.”...

My personal definition of traditional ET pseudo-model includes 10ths, 17ths and so on, but Chas ET EB theoretical definition is where all intervals are progressive, including 4ths, 5ths and octaves, with the only two exceptions of constantly equal beating 12ths (narrow) and 15ths (wide).

You say: ...“On poorly scaled pianos, I don’t think this is possible. One or the other (or both) will have a jump in beat speed across the break. Even on a well scaled piano, it may be difficult to achieve if the pinblock and rendering are poor.”...

In my experience, listening to beats, I’ve found astonishing how, even on poorly scaled pianos, it is possible to establish the ET EB beat-form I’m trying to share. If I loose control of the beats, iH may durty my job even more but if I keep controlling beats, almost any degree of iH can be tamed.

In the same day (06/22/09) you wrote to Bill:

...“I am not sure if anyone has noticed, and I have delayed bringing up the subject because it may be moot, but your mindless octaves are a slightly different thing than Alfredo’s equal beating 12ths and 15ths. His have a common note on the bottom, your's are on the top. I am still mulling over what this might signify. You’ve mentioned before that Steve Fairchild demonstrated that 5ths and 12ths become wide. How did he do this?”...

Has Bill answered you? Was he talking about 5ths and 12ths beats/rate or actual-frequencies values?

Recently you wrote about what makes you souspicious, let’s see:

...“Long or obscure words, and “fluffy” adjectives or adverbs make me suspicious. For instance, you used the term “monotone curve” when talking about what the beat rate of 5ths did in old teaching. Well, “monotone curve” is new to me so I looked it up. It describes what you say your new 5ths do, but not what old theory predicts.”...

If you look in the Chas article, section 4.3, you will be able to ascertain that, considering the scale as a set, on 4ths and 5ths + octaves traditional ET pseudo-model produces differencies that can only double, what makes a monotone curve.

You then say: ...“Your use of the term “interweaving partials” is another red flag for me.”...

Even today I can not think of a better immage to describe the effects of the two partials (3 and 4) treated by Chas algorithm. Also when I tune, in my mind, I’m weaving together the threads of a story, the story of partial sounds. Would you have a better word?

You say: ...“The biggest red flag is a declaration of trustworthiness.”...” Nixon’s statement of “I am not a crook.” is a perfect example.”...

If you took that declaration of mine so seriously, you may get ready for the party, when I become the USA’s President… No, I was simply trying to take your questionable statement in a friendly and easy way.

You also say: ...“Your math does not support how you tune because it is missing an equally detailed connection to the effect of iH on beat rates.”...

Well, what about 12 root of 2? When I think how we have had to go by with our unjustified, lame traditional ET pseudo-model, I tend to think Chas theory as a real improvement, and anything that may be missing in the Chas article can always be added.

I am looking forward to your elaborations on how iH affects 12ths differently than 15ths. Thank you (my devil's advocate) this discussion is being - for me - interesting.


Regards, a.c.
Posted by: Robert Scott

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/21/09 11:02 PM

Originally Posted By: alfredo capurso

...Is it true that “the tuning does not affect the inharmonicity”? Robert, are you saying that the string’s tension does not affect iH? Is it true that, depending on the expected/desired frequency, you chose the string’s length and thickness, and those variables, with string’s tension, do influence iH? Does Chas model change the frequencies to be expected?

This was a bit of an over-simplification on my part. Tuning does affect inharmonicity, but only slightly. Only very large changes in tuning/tension will affect inharmonicity measureably - such as you would get during a pitch-raise. For any reasonable tuning, the inharmoncity can be considered to be practically a constant for each note.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/22/09 11:03 AM

Alfredo:

When you and I refer to the “ET model,” things get confusing. There are a number of different definitions for this term, and we both seem to change the definition we are using when making a point. Your answer to what you do with Chas frequencies being to do the same thing as what is done with ET frequencies adds to the problem and is a non-answer.

Perhaps you meant that the Chas frequencies can be used to predict Chas beat rates, even though they will not be the actual frequencies that are tuned. I described the same “dong the right thing for the wrong reason” happening with traditional ET.

But maybe the difference in how we look at traditional ET is that, for you, the problem is that the 2:1 octave ratio is used. While, for me, the problem is that a 2:1 partial match produces a different octave ratio than a 4:2 partial match. I have mentioned this sort of thing before, but have not gotten a response that shows understanding from you as to why this is so. This is probably why we do not communicate very well about the problem with traditional ET.

In the 6-6-09 post you included the following table:

ET octaves Cents - Chas octaves Cents - Chas-ETdifferences
1200 1200,45982128266 0,45982128266405
2400 2400,91964256533 0,91964256532810
3600 3601,37946384799 1,37946384799216
4800 4801,83928513066 1,83928513065621
6000 6002,29910641332 2,29910641332026
7200 7202,75892769598 2,75892769598431
8400 8403,21874897865 3,21874897864836

I see this as linear octaves. Each octave is the same number of cents wide. Therefore each octave ratio is the same, also. (By the way, I enjoy seeing commas used in place of decimal points. It brings back memories of dealing with European methods.)

Yes, you certainly can have equal beating 12ths and 15ths regardless of changes of iH, but that does not mean that all other intervals will be progressive. In fact, it does not even mean that the equal beating will be progressive, just that it will be equal. (More on this, below.) The same thing can be done with octaves and 5ths, by the way. My definition of a poorly scaled piano is that you cannot have both M3 and M6 intervals progressive.

I should thank you also for this continuing discussion. There are a number of things that were unclear in my mind, that are much clearer now. You have challenged me. The biggest is the idea of iH causing 5ths to become wide while not changing the octave type. By this I mean that they beat wide, not just that the number of cents is greater than 702. To understand this phenomenon it is easier to consider iH values and iH slope rather than dealing with frequencies. For that matter it is easier to deal with 12ths and 15ths than it is to deal with 5ths and octaves.

But first let’s consider how aurally tuning equal beating 12ths and 15ths can be accomplished. If a tuner wished, he could pick any 12th or 15th and tune it to beat at whatever rate they choose. Then from either the upper or lower note, tune the other interval to beat at the same speed. The result will be a fourth between the two notes. Or, a tuner can start by setting a 12 note temperament, tune octaves until a 15th is tuned and make it beat the same rate as the 12th having either the upper or lower note in common. The width of this 15th will be determined by the width of the 12th and the width of the fourth between the two notes. Or, a tuner could construct a 25 note temperament, and while doing so, set the fourths, 12ths and 15th to whatever widths, speeds and common note they choose.

None of these methods will define or require progressively beating M3s and M6s, nor guarantee 12ths and 15ths that beat progressively, although the 25 note temperament could be used to come very close and further refinement could be made as more 15ths are tuned.

But will the 12ths and 15ths beat progressively; beat about twice as fast for each octave going up the scale? Non-iH theory says yes, iH theory says no. And since your 5ths, Alfredo, are not progressive in this way, you could not expect your 12ths to be so either. If your 12ths and 15ths “inverting” is what you mean by “interweaving partials,” you have not made it clear.

I rarely get positive feedback when displaying math, so I will try to explain this with just concepts.

The reason it seems that 12ths must always be narrow and 15ths must be wide in order to be equal beating is because a 12th, like a fifth, is tempered narrow by 2 cents. But this is ignoring iH, which effects the beat speeds but not the temperament. So when we consider the partials of a 12th being 3:1 and spanning 19 notes, but the partials of a 15th being 4:1 and spanning 24 notes, there is enough difference for iH to effect the beat rates of the 15th more than the 12ths at some point in the scale that the difference is more than the 2 cents required for tempering the 12ths. This also explains how 3:2 fifths and 4:2 octaves can also invert.

Putting the EB 12ths and 15th aside for a moment, another way of looking at this is the effect on stretch when tuning P12s compared to tuning P15s. A P12 tuning will have more stretch in the bass than P15s, but P15s will have more stretch in the treble than P12s. This is another indication that these intervals invert.

But there is no reason that a tuner cannot decide on an even wider stretch where the 12ths (and 5ths) will invert lower in the scale. Considering the part of the scale that your 5ths invert, your 12ths must be inverting also, and you are tuning much wider than the Chas ratio.

From your paper:

“4.3 – Comparison between equal temperament and chas differences for ratios 4:3 and 3:2 In the equal temperament scale, based on a ratio of 2, octave intervals have zero differences. As a direct consequence, the differences for partials other than 2 have ratios which are multiples of 2. The differences, divided by themselves, have a quotient of 2 for combinations 0-12, a quotient of 4 for combinations 0-24, and so on. With the exclusion of partial 2 and its multiples, the difference curves relating to all the other partials move away from each other exponentially in a monotone curve.

(Graph that shows a zig-zag line)

In the chas frequency scale the differences curves describe the exact form ordered by the incremental ratio and by the difference ratio. This substantiates the optimisation of beats and the absolute coherence of the chas form.”


I cannot follow your paper. You mention differences for ratios, then differences of partials, then differences divided by themselves (which would equal 1 unless the difference is 0?) having a quotient of 2 (which means that the numerator must be 2?) Please do not quote your paper unless you include better explanations. In your post, you mention a monotone curve, but what difference does that make?

And also in your last post:

”You also say: ...“Your math does not support how you tune because it is missing an equally detailed connection to the effect of iH on beat rates.”...

Well, what about 12 root of 2? When I think how we have had to go by with our unjustified, lame traditional ET pseudo-model, I tend to think Chas theory as a real improvement, and anything that may be missing in the Chas article can always be added.


Well, come on now! This is like saying that since tuning theory based on the 12th root of 2 is flawed, a theory based on the 12th root of the Chas ratio is better. But if improvement is needed it should be made to the 12th root of Chas theory rather than the 12th root of 2 theory!

Alfredo, my suspicion has gone back and forth between this being a scam to this being a misguided effort. I am back on the side of misguided effort, which means that I do not think you are deliberately trying to mislead. But if you want to convince me that you know what you are talking about (and perhaps become US President) consider the challenge I gave you on 6-7-09 (or 7.6.09 DIN):

Very well then, assuming an iH constant of 0.1 for C3 that doubles every 8 semi-tones (and to make it easy, lets continue this down to A0) and I desire a tuning that results in all octaves beating ½ bps wide at the 2:1 partial match, how would the CHAS algorithm be used to determine the fundamental frequencies of the tuning?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/22/09 02:06 PM

Robert, thanks for answering.

Tooner,

even in my last post I wrote: ..."my goal is 12ths (narrow) and 15ths (double octaves - wide) equal beating all along."

Why do you talk about 12ths and 17ths (double-octave + 3d)?

I'll read your post and answer you.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/22/09 02:31 PM

Sorry for the typos. I changed 17ths to 15ths in the post. Thanks for pointing it out in time for me to edit it!
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/28/09 01:44 PM

Gadzar,

thanks, you wrote (06/22/09): “...how to tune all intervalls equally tempered is not that clear!”...”...what is then ET?

I think we always need to precise if we want to talk about theoretical frequencies, or actual frequencies, or iH, or beats, or tuning sequence, i.e. tuning procedure.

Any numerical sequence with a fixed incremental ratio could represent an ET sequence, i.e a set of numerical values equally distant, but this does not mean that beats will be progressive for all frequencies intervals (i.e. all combinations of sounds). The point is that, in a sounds scale those equal distancies (our semitones) must represent all of the string’s vibrating lenths, and the latter are submultiples of prime numbers (sections 1.1 and 1.2).

As a matter of fact, our traditional ET pseudo-model utilizes an algebraic instrument that can indeed equalize those semitonal distancies, but in fixing the 2:1 octave ratio as a datum point, it damages those partial frequencies (the invers of the string’s lenths) that are not multiples of 2.

You then ask: ...“Is there a true ET?”...

For me the question should be: is there an ET frequencies sequence that works for our semitonal scale? I.e.: is there an ET sequence that does not damage any partial frequence?

You write: ...”you must control explicitly P4s. Do the P5s become beatless at some point and then become wide? Probably, but some of you say yes, some say who knows?”...

I can perfectly understand your frustration, actually I appreciated your great honesty in denouncing the dark areas where many tuners, still today, may often get lost (please, take this as referred only to my teaching experience). Anyway, no problem if some positions are not univocal, actually it makes our professional experience intriguing and fashinating. You will have to elaborate your own position, be it someone else’s too or not. For istance, I find 4ths, 5ths and octave’s progression as foundamental as RBI, 5ths must invert in the mid-range and need to go wide (tuning high notes middle-string), and about 5ths it seems that Mr. Bill Bremmer quite agrees.

You ask:...” Or iH makes it impossible to find a UNIQUE WAY to tune those progressively faster beats?”...

In my experience, if we do not manage to get progressive beats it will not be because of iH. For to long this has been an alibi, a sort of excuse for poor tuning, in some cases passed from the teacher to the apprentice. In aural tuning, iH is not a problem at all, the problem being our lame traditional ET pseudo model that fixes 2:1 octaves. Do you know what I thins is (in my experience) the main question? Pich stability, related to string’s, bridge’s, sound-board’s and pin’s elasticity. Being able to stabilize a frequency is like being able to walk, and only then you can look for your favorite temperament and perfect your (aural or ETD) tuning, otherwise you’ll keep on crawling.

You ask: ...”Now, what is CHAS? Is it a mathematical model that can not be landed in a real piano tuning?”...

Chas theory describes a new temperament’s model that comes from a new conceptual approach and from my practical tuning experience. Concepts and practical experience summed together have gained a new algorithm that represents no more no less than a new geometrical entity.

You say. ...”How does Mr. Carpuso effectivelly tune CHAS? The sequence he posts here, is as I've said before, one more of those sequences of 5ths and 4ths arbitrarily tempered to what sounds "slightly" wide and narrow.”...

So far you could only read (and learn) that 4ths and 5ths can be tuned with “similar” beats/rate. I’m stating that octaves, 4ths and 5ths can - and need to - be progressive, like 3ds, 6ths, 10ths and so on. The first 4 steps of my sequence (like any other sequence could do) establishes “inverted 5ths” and wide octaves, the beat/rate is then described in “wider and narrower” terms because in aural tuning all intervals are related to each other, and it would be pratically of no use talking of very slow beat/rates. In fact, for the octaves, the only way to make them progressive, in my experience, is to calculate the time needed for the beat to rise, a question of very very small variations.

Let me ask you: did you know that 5ths invert and can be progressive? And that, despite iH, 4ths and octaves can be progressive too?

You say: ...“That is not a new system! How is he choosing the values of the variables delta, s and s1? How is he puting these values in his way of tuning? He says nothing about that.”...

The Chas delta is the unknown variable, so you can not choose its value. Chas basic algorithm (3 - ∆)^1/19 = (4 + ∆)^1/24 represents a “dynamic ideal”, i.e. a 49-sounds beating-whole, that is made perfectly stable by synchronic and symmetric beats (section 3.4); “s and s1” are discretional variables, i.e. you can choose their values to modify the temperament’s beat-curves. In this thread the kite analogy (posted 06/04/09) shows you that, by choosing s1=1 and s=0 you find our traditional ratio 12th root of 2, while if choosing s1=0, s=1 you find 19th root of 3 ratio. This shows how, from a new concept of “dynamic beating-set” we can gain an ET scale’s incremental ratio that could variate, from any theoretical “pure ratio” to any mixed ratio, depending on your harmonic taste. This proves that Chas, simply coming from a new approach, is an entirely new system. In its ideal form, Chas model uses a 1:1 delta proportion for the differences on partial 3 and 4. While traditional ET pseudo-model uses a zero-beating constant, Chas model uses a double-difference constant, i.e. constant equal beating (equal difference = ∆) for 12ths and 15ths.

It seems that theoretical frequencies can be usefull for beat/rates and beats are foundamental in aural tuning procedure. I ask you: when does it become relevant considering iH? When you tune aurally? No, because you have to listen to beats, no matter what real/actual frequencies values you will get.

Considering iH becames foundamental when you want to define the piano’s strings scaling, and to define the strings scaling, like in a circle, together with other parameters you must consider the final expected frequencies, this is why we should refer to a reliable theory.

Tooner,

you kindly write: “When you and I refer to the “ET model,” things get confusing.”...

I do not think so. Traditional ET pseudo-model fixes 2:1 ratio for the octaves (i.e. a double numerical value for the 12th semitone). As I say to Gadzar, any fixed incremental ratio may be called ET but not all equal ratios will manage progressive beats for all intervals. I think you well know that.

You write: ...“Your answer to what you do with Chas frequencies being to do the same thing as what is done with ET frequencies adds to the problem and is a non-answer.”...

That was meant to be a clear and linear answer. So far you have been able to calculate so much, having to be happy with a lame pseudo-model, now that Chas theory’s model is conceptually and mathematically correct you may refine your calculations and finally translate theory into practice.

You write: ...”Perhaps you meant that the Chas frequencies can be used to predict Chas beat rates, even though they will not be the actual frequencies that are tuned. I described the same “dong the right thing for the wrong reason” happening with traditional ET.”...

Yes, with Chas frequencies you may predict Chas model’s beat rates. The difference is that Chas theoretical frequencies are finally the result of a rigth and correct “difference-ratio”.

Finally, with Chas algorithm (sections 3.1, 3.2, 3.3) partials are theoretically melted - hope this word is ok - in a single scale’s incremental ratio. The novelty is that Chas incremental ratio works – proportion wise - for both frequencies and beats (i.e. differencies from integer partials values, like 2, 3 and so on).

While so far – for the last 2500 years (?) - beats have been the consequence of somehow or “equalized”, proportional frequencies, today Chas model’s frequencies (and Chas incremental ratio) are the direct consequence of proportional beats (i.e. proportional differences on partial 3 and 4). As you can read in the Chas article (section 3.0), the scale’s frequencies (the foreground) are determined by proportional differences on partials (the background).

You write: ...“But maybe the difference in how we look at traditional ET is that, for you, the problem is that the 2:1 octave ratio is used. While, for me, the problem is that a 2:1 partial match produces a different octave ratio than a 4:2 partial match. I have mentioned this sort of thing before, but have not gotten a response that shows understanding from you as to why this is so.”...

You are right, for me the problem comes from using any “pure ratio”, like 2:1 or (3/2):1, or 3:1 or 5:1. For me, we needed to conceptually include partial 2 in our theoretical tempering, and we needed to fuse all “pure” partials effects into a single scale’s incremental ratio, the way Chas model does. From here we may take up iH again for more precise calculations.

You write: ...“In the 6-6-09 post you included the following table:…. I see this as linear octaves. Each octave is the same number of cents wide. Therefore each octave ratio is the same, also.”...

If you consider octaves differences – in the second table – you’ll see that they go esponentially.

You say: ...“But will the 12ths and 15ths beat progressively; beat about twice as fast for each octave going up the scale? Non-iH theory says yes, iH theory says no. And since your 5ths, Alfredo, are not progressive in this way, you could not expect your 12ths to be so either.”...

I’d better make it clear that in my practical tuning (as in Chas model), equal beating for 12ths and 15ths is constant, i.e. 12ths and 15ths are not progressive, all along have the same beat/rate, 12ths being narrow, 15ths being wide. To gain this, depending on the piano strings V sound-board settling expectation, I tune a more accentuated beat-curve for the whole.

You wrote: ...” I cannot follow your paper.”...

You tell me where, I can help.

(section 4.3, graph 5)...”You mention differences for ratios,”...

Yes, differences calculated on 4:3 and 3:2 ratios, 4th and 5th’s ratios.

...”then differences of partials,”...

Carefull, I talk about differences for partials other than 2.

...”then differences divided by themselves (which would equal 1 unless the difference is 0?) having a quotient of 2 (which means that the numerator must be 2?)”...

It means divided by themselves in sequence, i.e. one difference divided by the previous one.

...”you mention a monotone curve, but what difference does that make?”...

In practice, that monotone curve (section 4.3, graph 5) means that when you play, for example, C3 together with its 12th and its 19th, the structure load (i.e. differences, i.e. beats) rests all on the latters. For our ear that is flat.

When you said: ...“Your math does not support how you tune because it is missing an equally detailed connection to the effect of iH on beat rates.”...I ansered: “Well, what about 12 root of 2?”

Now you say: ...“Well, come on now! This is like saying that since tuning theory based on the 12th root of 2 is flawed, a theory based on the 12th root of the Chas ratio is better. But if improvement is needed it should be made to the 12th root of Chas theory rather than the 12th root of 2 theory!”...

In my opinion we’d better try to improve anywhere it may be needed. If you can improve Chas theory, please (sincerely) go ahead. Only, please, acknoledge that Chas semitone’s incremental ratio is not conceived as a 12th root, it is conceived as (3 - ∆)^1/19 = (4 + ∆)^1/24, i.e. as a beat-ratio resulting from “proportional differences”, i.e resulting from a tare = ∆ that you can regulate with the “s1” and “s” discretional variables.

You friendly say: ...“Alfredo, my suspicion has gone back and forth between this being a scam to this being a misguided effort. I am back on the side of misguided effort,”...

I’d almost prefere if you remained souspicious...I'm left without words, what will I say in the day of swearing-in ceremony?

You end up writing: ...“But if you want to convince me that you know what you are talking about (and perhaps become US President) consider the challenge I gave you on 6-7-09 (or 7.6.09 DIN):

Very well then, assuming an iH constant of 0.1 for C3 that doubles every 8 semi-tones (and to make it easy, lets continue this down to A0) and I desire a tuning that results in all octaves beating ½ bps wide at the 2:1 partial match, how would the CHAS algorithm be used to determine the fundamental frequencies of the tuning?”...

I’m sorry, I'm not used to starting a calculation on the base of approximated standards (i.e. standards that have been fixed on wrong theoretical frequencies and relative strings scaling), like those in use for strings iH calculation, and this is why I was generally underlining “proper consideration of iH”. Moreover, Chas algorithm is not meant for this kind of use in that, as you may understand, Chas is dealing with and featuring a proportional beating-whole. Nevertheless, wanting to honour your curious challenge, I conferm you that, if you consider a theoretical ET sequence, to get this octave’s theoretical bps of yours you’d need a theoretical ratio for each octave, what you could still find with Chas “s” variable. But let me ask you: is this an aural tuning’s point? I guess this desired tuning of yours could be the goal for a composer, for someone who may want to explore unusual harmonic combinations, it definatelly would not be the goal for a tuner in need to satisfy himself and his customer.

Thanks a lot and regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/28/09 02:01 PM

Alfredo:

This reminds me of a story. An American was working in Japan as an engineer. He was have a difficult time coming to an agreement with a Japanese engineer. While discussing a project they finally did agree that they were thinking along parallel lines. But the next day they were working against each other again. So the American asked the Japanese if they hadn’t agreed the day before that they were thinking along parallel lines. The answer was: Yes, parallel lines never meet.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/28/09 02:55 PM

Originally Posted By: alfredo capurso

It seems that theoretical frequencies can be usefull for beat/rates and beats are foundamental in aural tuning procedure. I ask you: when does it become relevant considering iH? When you tune aurally? No, because you have to listen to beats, no matter what real/actual frequencies values you will get.


I think iH has more to do with aural tuning than you say. You don´t tune a concert grand the same way you tune a small spinet. The difference is iH. No matter if you tune them aurally or using an ETD, iH is there and it must be taken into account.

When tuning aurally you are listening to beats, not frequencies OK. But beats produced between inharmonical partials. So iH changes the actual frequencies in your aural tuning.

I see a black spot on CHAS because there is no formula that contains iH data in an explicit way.

By the way, I have read a number of times that fourths and fifths must progress. I tune different types of octaves all along the scale favoring high partial octaves in the bass and low partial octaves in the treble, not that octaves exactly progress, but I don´t tune the same octaves all along the scale. But I confess that this is the first time I hear about tuning inverted fifths intentionally.

When ascending in the scale there is a point where beats in 5ths and 4ths become inaudible. My way of aural tuning makes me test larger intervals, say 10ths, 12ths, 15ths, 17ths, and even 19ths in the treble. So I don't know if my fifths become inverted at some spot in the treble. I tune equal beating 12ths and 15ths, so 12ths are narrow and 15ths are wide. But that doesn't mean 5ths are narrow, it will depend on how wide is the octave.

Another question: You named your system CIRCULAR HARMONIC but I don´t see why circular and why harmonic. Can you explain more about this?
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/28/09 03:07 PM

Quote:
When tuning aurally you are listening to beats, not frequencies OK. But beats produced between inharmonical partials. So iH changes the actual frequencies in your aural tuning.

Beats are produced by the interference of two periodic wave sources. Each string produces one wave, regardless of its form. There are not a bunch of different waves from each string corresponding to each partial.

People get led astray by the picture of a string vibrating in primary mode on top of a string vibrating in secondary and in tertiary mode. That is not what happens. The string only vibrates in primary mode, but the shape of it is not the sinusoidal curve that is depicted. (For that matter, even if it were, it would only be that shape for one moment in its motion.)

That only makes sense. A string is not going to bend itself in the middle for no good reason.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/28/09 03:20 PM

Originally Posted By: BDB
Quote:
When tuning aurally you are listening to beats, not frequencies OK. But beats produced between inharmonical partials. So iH changes the actual frequencies in your aural tuning.

Beats are produced by the interference of two periodic wave sources. Each string produces one wave, regardless of its form. There are not a bunch of different waves from each string corresponding to each partial.

People get led astray by the picture of a string vibrating in primary mode on top of a string vibrating in secondary and in tertiary mode. That is not what happens. The string only vibrates in primary mode, but the shape of it is not the sinusoidal curve that is depicted. (For that matter, even if it were, it would only be that shape for one moment in its motion.)

That only makes sense. A string is not going to bend itself in the middle for no good reason.


If I don't see it here I didn't believe it!

So strings only vibrate in primary mode?

That´s why you can't tell the difference between a 2:1 and a 4:2 octave.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 07/28/09 03:27 PM

BDB,

Let me tell you that strings not only vibrate in primary, secondary, terciary,...., n_ary modes simultaneously. That is transversal vibrations. But they also vibrate longitudinally at several modes all at the same time.

Please study a little!
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/01/09 02:07 PM

Gadzar, you say: ...“I think iH has more to do with aural tuning than you say.”...

Ok, let’s see why you think so.

You write: ...“You don´t tune a concert grand the same way you tune a small spinet.”...

Actually, I do. Whether I tune a modern concert grand or a small Wurlitzer, an old clavicord or a cabinet or a harpsichord, I can go for the same beat-form, ET EB Chas form. And if I were to tune a pipe organ, I would try to do the same.

You say: ...“The difference is iH.”...

Yes, iH is a difference, but where? iH may differentiate the final frequencies values, but since I’m listening to beats, it does not make any difference at all.

You add: ...“No matter if you tune them aurally or using an ETD, iH is there and it must be taken into account.”...

When I’m tuning, what I take into account are beats.

You explain: ...“When tuning aurally you are listening to beats, not frequencies OK. But beats produced between inharmonical partials.”...

Ok, what’s the problem with “inharmonical partials”? They do not make me deaf. One by one, I go for the beat I want, and every time I recognize that it is not the beat I wanted I correct it.

...”So iH changes the actual frequencies in your aural tuning.”...

This is true, iH can change our tuning’s actual frequencies, but so what? When tuning, we have to order intervals beats.

You write: ...“I see a black spot on CHAS because there is no formula that contains iH data in an explicit way.”...

You see a black spot there, I see a blanc, mainly because iH data are not needed for aurally tuning Chas (nor for understanding Chas approach and maths), therefore it was not a matter of primary importance. Anyway, once we’ll be able to calculate the real incidence (the impact) of iH on Chas theoretical frequencies (in circle with strings scaling), it will not be difficult to formulate it.

For the time being, I’m happy to share a new outlook on tuning that finally makes theory applicable into practice, and get reed off theoretical wrong, damaging axiomes: the octave module, the 2:1 octave’s ratio and any “pure” partial’s incremental ratio. Would you please give me a good reason for theoretical zero-beating octaves (12th root of 2)? Or for “pure” (i.e. zero-beating) 12ths? Or “pure” 5ths or what’s so ever “pure” zero-beating ratios? Chas model’s theoretical key is “beating” in any dynamic-form, a form that can be said “ideal” when it can beat an yet remain perfectly stable.

You write: ...“By the way, I have read a number of times that fourths and fifths must progress.”...

Ok, but you red they must progress how?

You say: ...”I tune different types of octaves all along the scale favoring high partial octaves in the bass and low partial octaves in the treble, not that octaves exactly progress, but I don´t tune the same octaves all along the scale.”...

Chas octaves have a very very slow beating in the middle register, but are more and more beating when going towards bass and treble, always wide and always progressive.

...”But I confess that this is the first time I hear about tuning inverted fifths intentionally.”...

Good, you are one of the few that explicitly admit it.

You say: ...“So I don't know if my fifths become inverted at some spot in the treble.”...

Try to intentionally invert 5ths between A3 and A4.

You say:...”I tune equal beating 12ths and 15ths, so 12ths are narrow and 15ths are wide. But that doesn't mean 5ths are narrow, it will depend on how wide is the octave.”...

When in the treble you can check octaves, 10ths, 12ths, 15ths, 17ths and 19ths you do not need to check 5ths anymore. Check for progressive octaves (check with middle string only), tune middle string a bit higher, i.e. make your check-intervals a bit wider, so that when you join left and right strings you can get stable and constant 12ths and 15ths equal beating.

You write: ...”Another question: You named your system CIRCULAR HARMONIC but I don´t see why circular and why harmonic. Can you explain more about this?”...

Thanks, I had to give this answer to ROMagister too. I named Chas system “harmonic” because it deals with partials.

The word “circular” can paint Chas model’s soul, and it has conceptual, semantic, geometrical and numerical relevance. In a way the circularity concept, common to many cultures, substitues the usual “pure” founding concepts, those related to integer partial value’s incremental ratios.

Thinking “circular” has helped me to come over the theoretical dichotomy between “consonance” and “dissonance” (section 1.3), and suggested me to look at harmony in terms of consonance-within-dissonance. So conceptually, Chas model gains purity through a circular function, i.e. the strict relation between frequencies and beats. Beats themselves are determined by the interrelations amongst all intervals, and therefore amongst all theoretical partials, and little non-pure partials values can determine a pure-whole (section 2.0).

While “circular”, in its definition, can generally describe interrelation, this word may well evoke the continuous flowing of beats, what we should never try to stop (in theory as in practice) since on “beats-flowing” depend the natural dynamism of any frequency-whole. Chas model substitues the traditional theoretical concept of static “zero-beating” intervals – the product of any “pure”, integer ratio - with a dynamic and yet perfectly stable “beating-whole” (section 3.4). So again, the dichotomy between “static” and “dynamic” is overtaken by the concept of “stable-within-dynamic”.

In adopting the word “circular” I also meant to refer to what can be calculated in tables, as for ephemeris, and to what - based on beats - can geometrically be represented through circonferences (section 3.5), where a vector returns to the same point in a precise lap time. In fact “time”, perceptible as beat-rate, is the one Chas model’s root (section 1.2).

Also, “circular” as referred to the geometrical ideal where all points are equidistant from a centre, in the way the scale’s values relative to partial 3 and 4, in Chas model, are equidistant from there pure ratio, due to Chas delta in 1:1 proportion. From “delta” equidistance, with Chas algorithm, we could reach any difference/beating order to the nth decimal point, i.e. we could cut - precisely to infinite – any difference value (as demonstrated with the kite analogy).

To appreciate the uniqueness of Chas model and have one more numerical proof of what I’m stating about circularity, let’s find together Chas 4th centre, i.e. the numerical middle distance between Chas scale’s values relative to intervals 12th and 15th:

- From Chas incremental ratio, we gain Chas scale’s values,

Chas incremental ratio = (3 - ∆)^1/19 = (4 + ∆)^1/24 = 1.0594865443501...

- subtract the scale’s value n. 19 (Chas 12th = 2.997874610034...) from the scale’s value n. 24 (Chas 15th = 4.002125389964...), and divide by 2.

The result is: 0.502125389964...

Chas 4th centre, i.e. the middle distance from Chas 12th and 15th shares Chas 15th’s decimal value. This explains also why Chas ratio, while resulting as well from a square root (octave ratio’s square root), has been described as deriving from a “circular” system.

Tooner, thanks for that story, good fun. Did you know that, once they left Euclide’s geometry behind, they went out together?

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/02/09 05:02 AM

"Tooner, thanks for that story, good fun. Did you know that, once they left Euclide’s geometry behind, they went out together?"

Yes, they went out together to the parking lot and settled it with their fists! smile
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/02/09 07:56 AM

Originally Posted By: alfredo capurso
Actually, I do. Whether I tune a modern concert grand or a small Wurlitzer, an old clavicord or a cabinet or a harpsichord, I can go for the same beat-form, ET EB Chas form. And if I were to tune a pipe organ, I would try to do the same.


I doubt you do. With no iH, i.e. in a pipe organ, fifths must remain 2 cents narrow and will beat progressively faster as you go up the scale, you will never tune a piano this same way.

In a concert grand you can tune 6:3 octaves in the temperament section and they will sound good, if you do the same on a small spinet you get horrible soundig octaves. Why? Because partial 6 is much higher on a spinet than in a concert grand so you can not tune them in the same way, with the same stretching.

You say you only hear to beats and thus iH is mindless. I disagree. You tune adjusting beat rates but the way these beat rates progress is drastically affected by iH for some intervals more than for others. In order to get clean octaves you must tweak the beat rate progression of the other intervals:fifths, fourths, etc...

I am still confused by the inverted fifths. If we tune wide fifths then there is no equal beating possible between 12ths and 15ths. Because if the octave is wide and the fifth is inverted, i.e. wider than pure, then the 12th will be also wide and will beat at a slower rate than the 15th. Does CHAS tune equal beating 12ths-15ths?

Originally Posted By: alfredo capurso
I can go for the same beat-form, ET EB Chas form.


I suppose ET EB Chas form means: Equal Temperament Equal Beating Chas form.

Equal Beating in what intervals?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/03/09 10:35 AM

Alfredo:

I have been trying to think of a single point of discussion that might be productive. I think your 5ths becoming wide between A3 and A4 may be one.

Can you show step by step how your CHAS ratio predicts wide 5ths between A3 and A4?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/03/09 12:40 PM

Gadzar, I'll soon answer to all you wrote in your post. Meanwhile, you say:

..."I suppose ET EB Chas form means: Equal Temperament Equal Beating Chas form.

Equal Beating in what intervals?...

Is this an attempt to drive me insane (with laughing)? ET = equal temperament, EB = equal beating, ever equal (and constant, i.e. non-progressive) 12ths (narrow) and 15ths (wide).

Tooner, you write: ..."I think your 5ths becoming wide between A3 and A4 may be one."...

Please, acknowledge that when I say 5ths invert I do not mean "5ths become wide", I mean that 5ths stop going progressively narrower and start going progressively less and less narrow, i.e. once 5ths invert (between A3 and A4), 5ths go progressively towards pure-crossing. Moreover, I talked of wide 5ths (untill you can ear 5ths) when tuning middle string only (check in my past posts), this to compensate string's, bridge's and sound-board's elasticity. In fact, when you unison left and right string, high notes (especially) will flatten a little bit, so wide 5ths (and generally wider check intervals) to counterbalance that flattening phenomenon.

Regards, a.c.
Posted by: Gadzar

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/03/09 01:43 PM

Mr. Capurso,

For me:

Pure fifth = beatless fifth with frequency of 3rd partial of lower note of fifth equal to frequency of 2nd partial of upper note of fifth.

Narrow fifth = beating fifth with frequency of 3rd partial of lower note greater than frequency of 2nd partial of upper note.

Wide fifth = beating fifth with frequency of 3rd partial of lower note of fifth being inferior to frequency of 2nd partial of upper note of fifth.

I am talking about 3:2 fifths type of course, not 6:4 fifths type.

I have understood that: inverted fifth = wide fifth.

But you say:

Originally Posted By: Mr. Capurso
Please, acknowledge that when I say 5ths invert I do not mean "5ths become wide", I mean that 5ths stop going progressively narrower and start going progressively less and less narrow, i.e. once 5ths invert (between A3 and A4), 5ths go progressively towards pure-crossing.


So, bellow inversion point, were fifths going progressively narrower? And at the inversion point they begin to stretch (go progressyvely towards pure)? Do they become pure at some point? Do they cross pure point and become wide at some other point? In the low bass, are they pure or even wide?


So, to put it in my own words:

You say that in E.T. the width of the fifths is no constant all along the scale but it progresses from pure, to narrow, to pure? And maybe even: it progresses from wide, to pure, to narrow, to pure, to wide? Is that it?

Note that I am not talking about beat rates, but about the width of the fifths. We can have constant width of the interval with progressing beat rates, because beat rates increase with frequencies of partials involved. But if the width is constant the beat rates can only progress in one direction, i.e. becoming faster all along the scale (or slower all along the scale), If width is constant there is no inversion point.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/04/09 01:56 PM

Gadzar thanks, I had prepared my answer to your second last post, hope it is still relevant.

To me saying: ...” Whether I tune a modern concert grand or a small Wurlitzer, an old clavicord or a cabinet or a harpsichord, I can go for the same beat-form, ET EB Chas form. And if I were to tune a pipe organ, I would try to do the same.”...

You answer: ...”I doubt you do. With no iH, i.e. in a pipe organ, fifths must remain 2 cents narrow”...

You say this because you are considering traditional ET pseudo-model.

...”and (fifths) will beat progressively faster as you go up the scale,”...

This is why, as a consequence, 12ths and 19ths are generally unbearable.

...”you will never tune a piano this same way.”...

Exactly, I’d never tune a piano nor a pipe organ that way. On both, I’d stretch octaves by inverting 5ths, and find ET EB Chas form.

You say: ...”In a concert grand you can tune 6:3 octaves in the temperament section and they will sound good, if you do the same on a small spinet you get horrible soundig octaves. Why? Because partial 6 is much higher on a spinet than in a concert grand so you can not tune them in the same way, with the same stretching.”...

In your way, iH can determine your octave. I do not let iH dictate me the amount of octave stretch in any section, as I’ve said I go for the beat I want, be it a concert grand or a spinet. If anything, in the temperament section, inverted 5ths determine octave stretch and vice-versa (sequence’s steps 1 – 4), i.e. by stretching the octave I can set up inverted 5ths.

You say:...”In order to get clean octaves you must tweak the beat rate progression of the other intervals:fifths, fourths, etc..."...

What do you mean with “clean octaves”? About having to “tweak the beat rate progression of the other intervals”, this happens becouse you do not invert 5ths. Try inverting 5ths as I suggest, then you can tell us.

You say: ...“I am still confused by the inverted fifths. If we tune wide fifths…”...

I hope you are not confused anymore. Inverting 5ths does not mean making 5ths wide, it means that 5ths go progressively towards pure-crossing, as I wrote in my last post.

Now you ask: ..."So, bellow inversion point, were fifths going progressively narrower?"...

Yes.

..."And at the inversion point they begin to stretch (go progressyvely towards pure)?"...

Yes.

..."Do they become pure at some point?"...

In Chas final tuning form, high 5ths will sound pure, although I think 5ths sound pure there because the beat-rate is to slow to get it in the sound duration. As I've said, to gain this final form, when tuning middle string you need to cross 5ths pure-point and make 5ths progressively wide.

...Do they cross pure point and become wide at some other point?...

Yes, when tuning middle strings. The point is between C5 and C6, but consider a very very slow progression.

..."In the low bass, are they pure or even wide?...

No, in low bass 5ths are narrow. I'll complete my answer with my next post, this i.point is shuting down.

Regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/08/09 06:09 AM

Gadzar, to complite answering your questions, you ask:

..."the width of the fifths is no constant all along the scale but it progresses from pure, to narrow, to pure?"...

In my final tuning, from bass to mid-range 5ths go progressively more and more narrow, between A3 and A4 5ths invert and progressively direct to pure. Notice that, when tuning high notes middle-string, it is necessary to cross 5ths pure-point (i.e. you need to stretch more than 3/2^(1/7) and therefore more than 19th root of 3) so that, after unisons and settlings, you'll be able to get Chas ET EB form.

You ask: ..."And maybe even: it progresses from wide, to pure, to narrow, to pure, to wide? Is that it?"...

No, 5ths invert only between A3 and A4. I'm going to replay to some other interesting post of our colleagues and of your's, thanks.

Regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/15/09 06:02 PM


Kent, on the 06/22/09 you kindly wrote:

...” it appears that talk of optimum stretch preferences seem to be giving way to talk of the optimum intervals on which to bass the underlying mathematical model of equal temperament.”...

So far, there were not many options for ET models, actually we could only choose amongst single “optimum intervals” pure ratios, more precisely 2:1, 3/2:1 or 3:1. Today, Chas model uses two intervals, the 12th relative to partial 3, and the 15th relative to partial 4. In this way, it is possible to proportion also partial 2 and partial 5, the latter being semitone n. 28, the 7th third. With any other ET mathematical pure ratio, where single partial’s ratio (2:1, 3/2:1, 3:1) determine the temperament, the other non-determining partials are bound to be damaged.

To Gadzar you wrote: ...”I also await the answer to the question you ask, that is, "How does Mr. Carpuso effectivelly tune CHAS?"...

This was my answer (in case you missed it): So far you could only read (and learn) that 4ths and 5ths can be tuned with “similar” beats/rate. I’m stating that octaves, 4ths and 5ths can - and need to - be progressive, like 3ds, 6ths, 10ths and so on. The first 4 steps of my sequence (like any other sequence could do) establishes “inverted 5ths” and wide octaves, the beat/rate is then described in “wider and narrower” terms because in aural tuning all intervals are related to each other, and it would be pratically of no use talking of very slow beat/rates. In fact, for the octaves, the only way to make them progressive, in my experience, is to calculate the time needed for the beat to rise, a question of very very small variations.

Kent, may I ask you: did you know that 5ths can invert in their being progressive? And that 4ths and octaves can be progressive too? Have you been able to see what 12ths you get at the end of your ETD tuning? Are 12ths pure, a little narrow or what? How are 12ths supposed to be?

You say: ...”Presently, I doubt that "CHAS" actually exists as a successful piano tuning system,"...

I do not really understand what you mean. You could believe that 2:1 ET ratio exists, today you say you believe that 3:1 ET exists, although it uses one more zero-beating, pure-ratio. Now, Chas model is featuring an ET ratio, ET like those above, where the difference regards the scale ratio’s attribute. In fact Chas ratio is not a “zero-beating” ratio, it is an ET EB ratio. Way do you doubt it exists? If anything, I’d ask: can “zero-beating” actually exist? On the theoretical and practical grounds, could you give me/us one good reason for adopting a zero-beating theoretical ratio?

...”but, though I am skeptical, I believe I am still open to CHAS.”...

Thanks for being still open to Chas. After 300 years of 2:1 ET ratio and the latest 3/2:1 and 3:1 pure ratio, I’m sure it will not take long to appreciate Chas ET EB ratio, coming at last from partials combination and from a comprehensive algorithm that can calculate those pure ratios as well. By the way, how clear (or obscure) did you find the “kite analogy” (06/04/09)?

Tooner, you kindly wrote (06/22/09): ...”Can all intervals be progressive on a piano that has iH? I am not so sure.”...

I’ve already stated precisely that Chas model describes 12ths and 15ths constant-equal-beating intervals, so 12ths (narrow) and 15ths (wide) are not progressive, they have the same, constant beat-rate all along. All the other intervals are progressive.

You say:...”There are many descriptions of tunings where fifths become wide in the high treble. Well, unless they also become fast in the bass, this would not be progressive.”...

In my opinion, the practical question is not if 5ths in the high treble are to be tuned wide, the practical point is how to manage 4ths, 5ths and octaves, starting from the temperament section. Right from the beginning we must determine the 4ths-5ths-octave’s relation, i.e. we must set up widening 4ths, directing-to-pure 5ths and a wide octave. This is way, to begin with, I tune A3 (from A4) and D4 and E4 so that I can temporarily evaluate together two 4ths, two 5ths and one octave (this sequence is not a must).

You write: ...” I asked a while back and it seems that the strings on a piano can only be set stabile within 0.3 cents.”...

As I’ve have said, string’s stability is the tuning must and it is the only real challenge, if a tuner can not set a string stable, he’ll get nowhere.

You say:...”One person’s perfect ET is another’s poison. Because of iH, and personal preference, ET is region not a location.”...

In my opininion, those conclusions are misleading. From what you say, it seems that nothing should ever go beyond iH (you were talking of “iH on everything”, remember?) or personal preference. I do not know what standards you are used to or referring to, but I can tell you that a truly good tuning will never be a poison for anybody. As for our practice, before surrendering to iH or talking about personal preferences, before saying that there is no reason for improving tuning theory and no way to improve tuning practice, I would take the chance to invert 5ths and go for their smoothest progression too. If I’m trying to share Chas ET EB model is because, in my professional experience, Chas is an absolutely precise ET location, inside the ET region. Way? Because Chas draws the precise ET form deriving from constant-equal-beating 12ths and 15ths, i.e. deriving from two opposite constants with the same one beat/rate all along.

May I ask you as well: could you give me/us one good reason for using an ET zero-beating (2:1, 3:1,…) theoretical ratio?

You write: ...“And I have already asked questions about how a piano is tuned to CHAS and cannot understand the answers.”...

Please, tell me if I can explain more about Chas tuning.

BDB, you wrote (06/22/09: ...” You guys are thinking about this too much. You tune a piano so that it sounds like it is equal tempered. That is all there is too it. Someone else can come along with a frequency counter and debate what it actually is, but as Duke Ellington put it, "If it sounds good, it is good."...

With this same approach to human progress, we would still light a fire by hands. Sure, also simplicity can be a life-key.

Tooner, the same day you wrote: ...”additional stretch can be given in an attempt to satisfy the well-documented human ear's desire to hear stretched octaves.”...

Not only, on my part there is also the desire to correct a wrong theoretical assumption, that you can base a tuning model on a zero-beating ratio, be it 2:1, 3/2:1 or 3:1.

You say: ...”The use of 12ths, not necessarily pure, is a great tool in controlling this additional stretch.”...

Even more, opposite-constant-equal-beating 12ths and 15ths, in practice, gives us a double errorless tool and a perfectly stable beating-whole (section 3.4).

Gadzar, you wrote (06/23/09): ...”Without iH all is easy, if we assume that

1 There is no iH
2 ET is defined as the division of the octave in 12 semitones equally tempered and
3 The octave's ratio is established to be 2:1

Then, the simple mathematical model of ET where

Semitone = 2^(1/12)

would perfectly do the work. Even if the “well-documented human ear's desire to hear stretched octaves” (by Tooner) is not satisfied.”...

Live human ear’s desire alone, all the above premises of yours are discretional, when not wrong:

1 If iH is relative to the mean, with different degrees iH is ever present.
2 Octave-based ET was an arbitrary and unlucky choice, the octave can not be cloned.
3 There was no logical reason for establishing 2:1 octaves.

You say: ...“With the presence of iH in pianos that model is no more applicable; we must tweak the frequencies calculated by this model in order to get acceptable tunings. That distorts what we understand by ET.”...

I look at it the other way around: 2:1 octaves lame ET model distorts the relation between ET and iH, in other words, we must tweak 2:1 ET’s frequencies firstly because those wrong values can not fit a combined-partials whole.

You add: ...”We lack a new mathematical model which includes iH and solves the problem of having several incompatible kinds of an interval. How can we tune an octave if there are 2:1, 4:2, 6:3, 8:4, 10:5, 12:6 octaves, and they are incompatible with each other? How can we tune a 5th if there are two distinct incompatible kinds of them?”...

The intervals incompatibility is firstly due to prime numbers 2, 3, 5 and their multiples. Theoretically, we needed to unite (mathematically) those “prime ratios” in one single ET ratio, what Chas model has finally done.

You say: ...“Models like semitone = 3^(1/19) or even more complicated, like the formulas used by Mr. Capurso, don't solve the problem because iH is not directly addressed.”...

The scale’s problems derive from non-combined partials. Firstly, we have to renounce “pure” ratios, then we can address iH. In fact, with Chas algorithm we could tweak frequencies the way we prefere, since “s” and “s1” variables work as a fine trimmer.

You say: ...“We need a new mathematical model where the octave's ratio is no more a constant but a variable value that will fit the piano's iH all along the scale.”...

If this was the problem, it would be solved. In fact, as I’ve said, Chas model’s algorithm allows you to variate the scale’s ratio as you like.

You ask for: ...“A new mathematical model where there will be only one type of each interval to tune, namely only one kind of octave, fifth, fourth, etc.”...

In Chas model each beat works for the others, each interval’s beat is solidly behind the beat of any other interval, due to a proportional “difference” ratio that involves all partials. Therefore, Chas intervals are of one precise mathematical type (the precise location). Also in practice, in addiction to usual check-intervals, as my constant-reference (the datum) I use just one type of beat (one beat/rate), 12ths (narrow) and 15ths (wide) constant-equal-beating all along. Traditional ET pseudo-model gave us a zero-beating constant that no tuner has ever been able to use, Chas ET EB model gives us two constants and the same one beat, so describing a tuning-form that we can always achieve into practice.

You say: ...“We have sometimes six contiguous unisons using the same size of wire and then the next unison has another size, here we have a jump in iH. The same happens when we get to the wound strings, and to the doublewound strings. Our new mathematical model can not just ignore this, because our ears do not.”...

Our ears can suffer jumps in iH and strings scaling, what a correct mathematical model can improve. Way? Because our expected frequencies (and our expected beats) affect strings scaling, and the latter affects iH.

Commenting Virgil Smith's concepts, you write: ...“So the “natural beat” concept can be applied to solve the problem of several kinds of intervals sounding simultaneously. It is then necessary to translate mathematically the ability of the human ear to combine all the partials into one sound and one pitch.”...

I can say that I use those “natural beats” for tuning Chas-form, “partial beats” for unisons. I think that listening to “partial beats” may expose you to iH. Chas model mathematically combines all partials into one natural ET semitone’s ratio, in the way our ear could do on larger intervals by using “natural beats”, with or without iH.

In my opinion though, one Smith’s line contains a partial information when, describing natural beats, he says:

...“ it combines, all the beats between the partials into one beat. The beat then comes from all the partials instead of one set of partials.
This beat can be tuned to the desired speed or eliminated completely.”...

Can we eliminate a beat completely? How long does it take the slowest beat to rise? If we do not hear any beat it is because we are in the partials leeways, where beats are going to define. Then, is it better counting beats or zero-beating? Shoud’nt we manage partials leeways? In my experience, that leeway allows me to define the beat’s rising and launch my tuning-form.

You say: ...“If we can translate to a mathematical formula this way of combining several “partial beats” into one “natural beat” there will be only one unique type of interval to tune. No more 2:1, 4:2, 6:3 octaves, nor 3:2, 6:4 fifths, nor 4:3, 8:6 fourths, but only one kind of octaves, fifths, etc.”...

Well, in Chas model each interval’s beat-curve is a function of all the others, i.e. partial beats (related to partial strings lengths) are expressed as a combined function. You will not find a zero-beating curve, nor an arbitrary “pure” ratio, you’ll find a purely dynamic form, so called “chorale” because deriving from all partials flows of beats combined. In Chas algorithm, delta variable determines an “ideal” beats-flowing, s1 and s variables open to the “real” infinite variety of flows.

You say: ...“In that respect I think Mr. Capurso's research although being a true effort to find a new model, missed the target by confusing iH with stretch.”...

I’m afraid I’ve to say that confusion took place when referring iH to wrong scale’s frequencies, those deriving from 2:1 octave’s ratio, a scale with a zero-beating-constant ratio that does not combine partials.

You say: ...“I think the solution to this problem is working on iH not stretching the width of the semitone in an arbitrary manner.”...

Somehow I agree, let’s combine partials in absolute terms and get the most natural (and logical) semitone’s ratio, then we can work on iH.

You say: ...“I think people like the late Dr. Albert Sanderson (Accutuner), Mr. Robert Scott (Tunelab), Mr. Dave Carpenter (Verituner), Mr. Dean Reyburn (Cybertuner), and Mr. Bernhard Stopper (Onlypure), who have developed the theory, algorithms, software and some of them even hardware of the most advanced ETDs available today, are aware of the lack of such a mathematical model.”...

I think so too.


Thanks amd regards, a.c.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/15/09 07:17 PM

I believe there is a wide world of tuning expertise in existence that is more sophisticated than you give it credit for.

I am sorry. I would really like to communicate meaningfully with you, but your language is impossible for me to follow.

You write:
"Today, Chas model uses two intervals, the 12th relative to partial 3, and the 15th relative to partial 4. In this way, it is possible to proportion also partial 2 and partial 5, the latter being semitone n. 28, the 7th third. With any other ET mathematical pure ratio, where single partial’s ratio (2:1, 3/2:1, 3:1) determine the temperament, the other non-determining partials are bound to be damaged."

I tune intervals. I don't understand your term, to "proportion partials."

You write:
"I’m stating that octaves, 4ths and 5ths can - and need to - be progressive, like 3ds, 6ths, 10ths and so on."

The definition of equal temperament is that temperament in which all intervals progress smoothly. You cannot possibly make the claim that this characteristic is unique to Chas.

I respectfully suggest that you need to fully study the existing literature on the subject of tuning equal temperament.

I repeat my suggestion that you make available recordings of your tuning.

I repeat my suggestion that you make available a coherent set of directions for executing your tuning.

Kent Swafford
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/20/09 02:04 PM

Kent, you kindly say: ...” I believe there is a wide world of tuning expertise in existence that is more sophisticated than you give it credit for.”...

It would be interesting to know exactly what you are referring to. This would give depth to your convictions and may enlarge our horizons. If you wanted to tell me/us more about the world of tuning expertise and how it is more sophisticated, the time may be now.

You say: ...“I am sorry. I would really like to communicate meaningfully with you, but your language is impossible for me to follow.”...

Do not give up, on my part I’m trying to improve my language.

You say: ...“I tune intervals. I don't understand your term, to "proportion partials."...”...

Well, no problem. The history of temperaments is centred on the scale’s partials proportions. Briefly, the word “partial” generally refers to the vibrating string’s frequencies (section 1.1): the string vibrates at a foundamental frequency, say = 1, together with partial frequencies = 2, 3, 4, 5, and so on. Those integer numbers can determine an ET scale’s proportion, expressed by the semitone’s incremental ratio. In fact, traditional ET pseudo-model adopted the 2:1 partial proportion for dubbling the octave’s value.

“Chas model uses two intervals, the 12th relative to partial 3, and the 15th relative to partial 4. In this way, it is possible to proportion also partial 2 and partial 5…”... this is meant to explain how Chas theory proportionally combines – in its ratio - also scale’s partials 2 and 5. When in practice you tune intervals, you do proportion beats/rates and those are the direct product of partials proportional matching.

You write: ...“ The definition of equal temperament is that temperament in which all intervals progress smoothly. You cannot possibly make the claim that this characteristic is unique to Chas.”...

I’d never make that claim, since I would have no reason. In fact, Chas 12ths and 15ths do not progress at all, remember? In Chas model, 12ths (narrow) and 15ths (wide) are the two system’s constants, i.e. 12ths and 15ths have opposite, constant-equal-beating all along. So, if anything, Chas is unique in that it theoretically justifies octaves progression as the result of combined partials. Traditional ET pseudo-model fixes zero-beating 2:1 octaves, so our ET reference model leaves octaves out of any theoretical beat-progression, therefore it does not fit that definition either. Anyway, my goal is not ET’s definition, and what makes Chas model unique has been listed in this Topic’s first post.

You kindly say: ...“I respectfully suggest that you need to fully study the existing literature on the subject of tuning equal temperament.”...

Thank you for your suggestions. I can peacefully state that in equal temperament’s literature you will not be able to read about Chas tuning yet, and that is way I’m sharing Chas ET EB theory’s model and tuning practice here and now.

You say: ...“ I repeat my suggestion that you make available recordings of your tuning.”...

In a few days traditional ET and Chas ET EB digital comparison will be available. Meanwhile, in case you kindly wanted to contribute, I’ll list my latest questions:

1 - did you know that 5ths can invert in their being progressive?
2 - and that 4ths and octaves can be progressive too?
3 - have you been able to see what 12ths you get at the end of your ETD tuning? Are 12ths pure, a little narrow or what? How are 12ths supposed to be?
4 - way do you doubt Chas model exists?
5 - on the theoretical and practical grounds, could you give me/us one good reason for adopting a zero-beating theoretical ratio?
6 - how clear (or obscure) did you find the “kite analogy” (06/04/09)?

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/20/09 02:16 PM

Alfredo:

Consider this zero beating interval, 8:2 double octaves. In some parts of the scale (but not all) it will produce EB 12ths and 15ths. The test for the 8:2 double octaves is when the minor 6th above the lower note beats the same as the major 10th below the upper note. This will also produce octaves between 4:2 and 6:3. I have been striving for this on unwound strings as high as I can discern the beats. Wound strings are another matter…
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/20/09 07:43 PM

Let me tell you what I think may be happening here.

In order for there to be a discussion of scientific/technical nature, the participants must share a common knowledge of the given subject along with a common vocabulary relating to that subject.

You and I simply do not have the required knowledge in common that is prerequisite to carrying on discussion on the subject of tuning theory.

For 30 years I have been reading the Piano Technicians Journal. It does not purport to be a scientific journal, but it is a technical Journal and it does contain the wealth of expertise that is the basis of my considerable understanding of piano tuning.

If you wish to participate in a discussion of piano tuning theory beyond your knowledge base, then I would suggest that you familiarize yourself with the Piano Technicians Journal, which is available on CD-ROM from the Piano Technicians Guild, www.ptg.org.

Take some advice, pending your completion of reading the Piano Technicians Journal. You simply, absolutely must drop your term "ET pseudo-model" in relation to the 12th root of two model of ET. Frankly, this term makes you look like a fraud and/or a clown to those who are familiar with the piano tuning theory to which I am accustomed.

When you are able to describe your ideas in terms of accepted piano tuning theory, or when you can communicate your ideas well enough for another tech to describe your ideas in terms of accepted piano tuning theory, then I look forward to continuing this discussion.

I hesitate to mention this; perhaps it will just open me to more of your insults; but I have published many articles on tuning theory in the Piano Technicians Journal, I have been an administrator of the PTG's tuning exam for some 25 years, and I am a past president of the Piano Technicians Guild. I state respectfully: I know what a partial is. <grin> You need to withdraw and learn the language of piano tuning theory. Once we have a common vocabulary, I suspect you would find me able to be of assistance to you in disseminating any good ideas you may have among the piano techs of the world.

This will be the last of my comments here. Farewell.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 08/24/09 05:28 PM

Kent, you kindly say: ...“In order for there to be a discussion of scientific/technical nature, the participants must share a common knowledge of the given subject along with a common vocabulary relating to that subject.”...

In this Topic, I do not see any “common knowledge” problem, nor a vocabulary problem. And if there were any, it could be easily solved by adding one more line.

You state: ...“You and I simply do not have the required knowledge in common that is prerequisite to carrying on discussion on the subject of tuning theory.”...

About knowledge in common, I do not understand what makes you so negative. In my professional experience, I have been able to debate tuning theory and practice with colleagues and pianists coming from all countries. What did we have in common? Our will to explain each other and to know from each other.

You say: ...“For 30 years I have been reading the Piano Technicians Journal. It does not purport to be a scientific journal, but it is a technical Journal and it does contain the wealth of expertise that is the basis of my considerable understanding of piano tuning.”...

I could then believe that your understanding of piano tuning, in terms of general knowledge, is not much different from mine.

You say:...“If you wish to participate in a discussion of piano tuning theory beyond your knowledge base, then I would suggest that you familiarize yourself with the Piano Technicians Journal, which is available on CD-ROM from the Piano Technicians Guild,www.ptg.org.”...

Kent, I’m sure you know that there is plenty of alternative literature available on temperament theory and tuning practice. Sure, I could familiarize with the Piano Technicians Journal, but this would not make our knowledge more common. In my opinion, the available literature on this subject, in any western language, is quite univocal.

You write:...“You simply, absolutely must drop your term "ET pseudo-model" in relation to the 12th root of two model of ET.”...

Here, I may think we do not have the same kind of knowledge, but it may not be a problem. I’m saying “pseudo-model” meaning that 12th root of two lacks in consistency, i.e. 12th root of two does not correspond tuning reality.

You say:...“Frankly, this term makes you look like a fraud and/or a clown to those who are familiar with the piano tuning theory to which I am accustomed.”...

I do not understand why you talk about frauds and clowns. In my opinion, many of us have long ago realized that 12th root of two, as a model, can not be put into practice, and I’m trying to explain why it is so. What 12th root of two should then be called, if model, proto-model, super-model or what, this may be an academical matter; my actual goal is sharing Chas approach, Chas theory’s model and its numerical evidencies. You see, in Chas article’s conclusions you can read: …“the numbers dispel all doubt concerning the simplicity and power of this long-awaited entity”.... so, since you can always check Chas numbers, you will never think in terms of clowns. Instead, you could now acknowledge that Chas scale’s ratio combines all partials, included partial 2, and derives from an algorithm that can proportion partial differencies, and therefore beats.

You say:...“When you are able to describe your ideas in terms of accepted piano tuning theory, or when you can communicate your ideas well enough for another tech to describe your ideas in terms of accepted piano tuning theory, then I look forward to continuing this discussion.”...

Well, I think it'll be up to you. I’m writing here about a new ET EB dynamic theory and, as you know, it may take some time for its terms to be accepted.

You write:..“I hesitate to mention this; perhaps it will just open me to more of your insults;”...

Maybe this is a problem of yours, you feel insulted but in this Topic nobody has insulted you.

...“but I have published many articles on tuning theory in the Piano Technicians Journal, I have been an administrator of the PTG's tuning exam for some 25 years, and I am a past president of the Piano Technicians Guild.”...

For what you have experienced, I’m sure you’ll soon appreciate the difference between traditional ET and Chas ET EB model.

You say: ...“You need to withdraw and learn the language of piano tuning theory. Once we have a common vocabulary, I suspect you would find me able to be of assistance to you in disseminating any good ideas you may have among the piano techs of the world.”...

If I had some vocabulary problem, I would like to find you able to assist me then, especially since you yourself think that vocabulary may help. Anyway, Chas model is not really a good idea only, it is a new ET theory that finally improves our traditional 12th root of two. The sooner you realize it, the sooner you’ll be ready to help. Thanks anyway.

Tooner, thanks for your post, I'll soon answer you.

Regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/02/09 07:23 AM

Dear colleagues, here

http://www.chas.it/ContentPartViewer.aspx?ID=confronto

you can listen to 24 digital tracks – it takes about 5 minutes - and compare four chords, formed with 12th root of two ET frequencies (odd tracks) and Chas frequencies (even tracks). Those are the notes that are played: A 3 - A 5 - B 5 - C# 6 - E 6.

This is the chords order:

From track 01 to 06 → A 3 - E 6
From track 07 to 12 → A 3 - A 5 - E 6
From track 13 to 18 → A 3 - A 5 - C# 6 - E 6
From track 19 to 24 → A 3 - A 5 - B 5 - C# 6 - E 6

Each chord is played using three kinds of wave (square, sawtooth, triangular). These are the notes and their frequencies:

Notes - 2:1 ET frequencies
A 3 - 220.00
A 5 - 880.00
B 5 - 987.76
C# 6 - 1108.73
E 6 - 1318.51

Notes - CHAS frequencies - offset in cents
A 3 - 220.00 - 0.0
A 5 - 880.46 - 0.9
B 5 - 988.33 - 1.0
C# 6 – 1109.41 - 1.1
E 6 - 1319.41 - 1.2

Those notes have been chosen to keep Chas frequencies approximation within +/- 0.2 cents, while 12th root of two ET frequencies could have no approximation (100 cents/semitones). For converting Chas frequencies in cents we have used Peterson’s software, available here:

http://www.petersontuners.com/index.cfm?category=15

For recording we have used: DIGIDESIGN, Mod.: PRO TOOLS HD 3 ACCEL
Interface: 192 I/O - Signal Generator: DIGIDESIGN
Software: PITCH DIGIDESIGN

This work has been done to aurally evaluate 12th root of two ET model and 2:1 theoretical octave ratio’s practical efficiency, when applied on digital sounds.

I need to precise that Chas model’s algorithm, while describing an ideal geometrical entity, can also provide infinite ET scale’s incremental ratio (section 3.3), in departure from any cultural, “euphonic” approach and any “pure” theoretical ratio (section 2.0).

In fact, Chas theory grows from the concept of dynamic and beating affinity in the sound-whole.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/04/09 02:00 PM

Alfredo:

Yes, I hear a difference, but without a musical comparison, I would not be able to say which I prefer. For piano tuning there would need to be the same music played on the same piano tuned differently in order to determine a preference.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/15/09 01:54 PM

Dear Colleagues,

today, in my mail box, I found a message from Mr. Kent Swafford. It was sent on the 08/25/09 and, since it is about Chas, it may as well be discussed.

Here you'd have read the message but, as Tooner suggests, I'll ask for Mr. Swafford's permission.

Kent, can we publicly discuss your outlook on Chas theory?

Tooner, thanks for your feed-back, I'll answer you asap. Would you please tell me what you found "private" in Mr. Swafford's message?

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/15/09 02:09 PM

Alfredo:

The message that Mr. Swafford sent you seems to be a private message to you, and you alone. I think you should remove it from this Public Forum. You can do this in the near future by clicking the "edit" button on your post.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/17/09 06:10 PM

Kent,

you tell me that all of the matters Chas theory raises have been dealt with and settled over the years in alternative and superior ways.

On the one hand I could be quite glad because that “common vocabulary” is not a problem anymore and you could evaluate Chas theory’s matters. On the other hand I do not know how to take your latest understanding. Should I think that you were right, that there is a “common vocabulary” problem? Should I think that confusion is taking place? For both cases, I need to further precise a few points.

I’m not competing with theoretical or practical solutions that may represent alternative or superior ways for tuning pianos. In fact, I’m only trying to share what I think is a new theoretical approach to the sound scale, strictly related to my practical (aural) tuning experience.

Chas theory approaches the sound scale as a “beating-whole”. This is why in Chas algorithm (3 - ∆)^1/19 = (4 + ∆)^1/24 you find partials 3 and 4 together with ∆ (delta). In other words, Chas algorithm gains a basic scale incremental ratio = 1.0594865443501… with a delta beat-factor (section 3.1). So doing, Chas model can express an ET proportionate frequencies scale as the result of proportionate beats. Chas algorithm, in its basic form, describes then a precise geometrical entity (section 3.5) where 12ths (narrow) and 15ths (wide) are the equal-beating scale’s constants. This is why Chas is a precise ET EB theoretical model, what in my practical experience I consider the most euphonic and resonant semitonal sound-set. How Chas model combines partials is another matter and it is a mathematical evidence, as it is an evidence how, from Chas improved algorithm (3 – (∆*s1))^1/19 = (4 + (∆*s))^1/24 all ET conventional pure ratios can be gained.

If you can now evaluate the matters Chas theory raises and if you had heard of Chas model called in an other way, please let me/us know, we shall call it with the right name. Otherwise, try to take Chas model as it is, in its theory as in my practice, with no need to elbow.

Since my questions were serious, I need to go back there.

Question n. 1: did you know that 5ths can invert in their being progressive?

For the temperament, common teaching says: narrow and similar 5ths. In my experience, to get opposite and constant-equal-beating 12ths (narrow) and 15ths (wide), I have to stop 5ths from getting narrower in the temperament section, and I have to progressively drive 5ths towards no-beating. So, 5ths inversion = when tuning centre strings, invert 5ths in the temperament and let them go from progressive narrow to progressive wide.

Question n. 2: and that 4ths and octaves can be progressive too?

In my experience, to get progressive octaves I need to invert 5ths.

Question n. 3: have you been able to see what 12ths you get at the end of your ETD tuning? Are 12ths pure, a little narrow or what? How are 12ths supposed to be?

Answering this question is of general interest.

Question n. 4: on the theoretical and practical grounds, could you give me/us one good reason for adopting a zero-beating theoretical ratio?

I’ve also asked this question to Tooner and Gadzar, I’ve got no answer. What is a “zero-beating ET theoretical ratio”? It is a theoretical scale proportion that referes to any of those ET integer ratios: 2:1, 3/2:1, 3:1, 5:1, and so on. In my opinion, thinking in terms of “zero-beating ratio" is an abstraction that negates a dynamic reality.

Question n. 5: how clear (or obscure) did you find the “kite analogy” (06/04/09)?

Answering this last question may suggest to find a better way for sharing Chas model. Thanks.

Tooner,

hopefully in a short while I'll be able to add some recordings of Chas piano tuning. Remember though to separate what our musical taste can be - in my case Chas basic ET EB form - from a comprehensive model, what Chas can express with its "s" variable. In other words, I would not make it a question of personal preferencies. Thanks.

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/21/09 08:56 AM

Alfredo:

I don’t think you understood that Kent has said that he will not be replying to you anymore.

You have not shown a mutually exclusive relationship between zero beating tuning and equal beating tuning. So, the failure to prove the validity of one does not prove a validity of the other.

But here is something that you have not answered:

How does the Chas theory predict that fifths “invert”? The answer is that it does not predict this. It is an effect of inharmonicity when choosing a particular stretch style. Although you think that your theory is coherent, it is not because it is incomplete. It does not predict what actually happens.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/22/09 02:22 PM

Tooner,

You kindly say: “I don’t think you understood that Kent has said that he will not be replying to you anymore.”...

Thanks but yes, I understood what Kent wrote at the end of his last post, nevertheless Kent may change his mind and, rather than privately, we may discuss publicly about Chas, that is what I would prefere.

...“You have not shown a mutually exclusive relationship between zero beating tuning and equal beating tuning.”...

On the theoretical ground, I have not shown a “ mutually exclusive relationship between zero beating tuning and equal beating tuning” merely because I can prove the opposite, i.e. Chas ET EB theory can also comprehend any zero beating ratio (section 3.3 - THE S VARIABLE). Let’s see together one example:

From Chas algorithm (3 – (∆*s1))^1/19 = (4 + (∆*s))^1/24

If s1 = 1 and s = 0

we can find a delta value that makes our equation true:
Δ = 0.0033858462466

In fact:(3–(0.0033858462466*1))^(1/19) = (4+(0.0033858462466*0))^(1/24) =
= 1.059463094359 = 2^(1/12) = 12th root of 2 zero beating ratio.

As for 12th root of 2 octave’s zero beating ratio, Chas model does not exclude any theoretical ratio, zero beating or what ever. This is why, at the end of section 2.0 you can read: “In conceptual terms, the model is trans-cultural; it also responds to a new requirement on the contemporary music scene, by providing an algorithm which can give form to all kinds of microtonal sound structures.”

On the practical ground, in my experience, 12th root of 2 zero beating ratio is useless, 7th root of 3/2 and 19th root of 3 are unnecessarily extreme as a final tuning form, although not “sharp” enough when tuning mid-strings in the treble. In fact in my practice, when tuning centre strings in the treble I go beyond the 7th root of 3/2 pure fifths ratio and I stretch progressive wide fifths. As I’ve said, this is done to compensate sound-board, bridge and strings elasticity, so considering my tuning settling-down.

You say:...“So, the failure to prove the validity of one does not prove a validity of the other.”...

True, then the question may be: what makes Chas beating-whole’s ratio better than any zero beating ratio? Leave all previous reasonings aside, you find an answer reading Chas article’s section 4.5 - Sequence of quotients..., and Table 6: Comparison between quotients deriving from ratios 3:2, 5:4, 3:1 and 5:1. There you’ll be able to evaluate Chas validity only with the help of numbers.

...“But here is something that you have not answered: How does the Chas theory predict that fifths “invert”? The answer is that it does not predict this. It is an effect of inharmonicity when choosing a particular stretch style.”...

I'd put it in a slightly different way: when choosing a particular stretch style, we need to invert fifths. In other words, inverting fifths in the temperament is the technique I use for achieving Chas form.

12th root of 2 predicts zero beating octaves, 7th root of 3/2 and 19th root of 3 predict respectively zero beating fifths and zero beating 12ths, Chas model’s basic form predicts opposite and constant 12ths and 15ths equal beating. How to get to Chas basic form in practice is a different matter, and Chas algorithm, with its s variable, allows you to make use of all ratios you need.

...“Although you think that your theory is coherent, it is not because it is incomplete. It does not predict what actually happens.”

I would not fuse coherence with completeness and we can still discuss both concepts. Chas theory describes a beating-whole and it gains the scale’s frequency values with a delta (∆) beat-factor. I can not immagine anything more coherent, can you?

About prediction, fifths inversion and Chas theory’s completeness, maybe this can help. In section 3.4 - CHAS SET…and section 3.5 - EFFECT OF ±DELTA…, you can read that the Chas model opens up a module of 49 sound elements, in a semitonal order, from 0 to 48, whose scale ratio is (4 + Δ)^2. If 12th root of two zero beating symmetry encompasses 8 scale’s degrees (one octave), Chas model’s beating symmetry encompasses 29 degrees of our semitonal scale. Now, in section 4.3 (graph 5, 6) you can compare 12th root of two and Chas difference curves for ratios 4:3 and 3:2, and in section 4.8 (graph 10) you can see what happens: within Chas compass, the beat curve for ratio 3:2 inverts its progression. Using your words I ask you: is this an effect of inharmonicity? I do not think so but, as you'd say, I think it is the effect of a particular stretch style, in this case the (predicted ?) effect of Chas ET EB.

Thanks a lot and regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/23/09 07:45 AM

Alfredo:

Can you explain this in more detail:

“Now, in section 4.3 (graph 5, 6) you can compare 12th root of two and Chas difference curves for ratios 4:3 and 3:2, and in section 4.8 (graph 10) you can see what happens: within Chas compass, the beat curve for ratio 3:2 inverts its progression.”

You mention “ratios 4:3 and 3:2” but do not explicitly say what the ratios are of. Perhaps you mean partial matches? That is where beats come from, not ratios. Are you saying that using a semi-tone ratio of 1.0594865443501 will produce fifths that “invert”, in other words, beat faster and then slower when progressing up the scale?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/24/09 07:10 AM

Tooner,

...“You mention “ratios 4:3 and 3:2” but do not explicitly say what the ratios are of. Perhaps you mean partial matches? That is where beats come from, not ratios.”...

True. In section 4.3, at the bottom of graphs 5 and 6 you find the scale degrees that can be compared; as well as in section 4.8, were you also find Table 9 listing Graph 10 values.

...“Are you saying that using a semi-tone ratio of 1.0594865443501 will produce fifths that “invert”, in other words, beat faster and then slower when progressing up the scale?”

No, as I’ve said inverting fifths in the temperament is the technique I use for achieving the ET EB tuning form that Chas model describes. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/25/09 07:11 AM

Alfredo:

Then, your paper is incomplete as I said before. It does not predict the change in the beat rates of fifths that you find necessary for equal beating 12ths and 15ths on a real piano.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/25/09 05:21 PM

Tooner thanks, more often you help my mood.

Before you were talking about Chas theory being incomplete, now you are saying that my paper is incomplete. What happened in the meantime?

Could you tell me what can be average deducible from Graphs 5, 6 and 10? Could you get the partials-matching involving beats?

Maybe you know this:

http://www.pykett.org.uk/temperament_-_a_study_of_anachronism.htm

I'd like to know your opinion sometime. a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 09/29/09 07:34 AM

Alfredo:

Because your paper is the only place that Chas theory is explained, and the only topic in your paper is Chas theory, I see them as one thing: A problem with one is a problem with the other. But I understand that you would view them as separate things. So, where do you think the problem is between how your beat rates are when you tune and how your paper explains what the theory predicts beat rates should be?

I have pondered far too many times what graphs 5, 6 and 10 are indicating and still have no idea. I do not know what you mean by degrees. The only mention of degrees in music that I can think of is in the Psalms, which may have indicated ascending stair steps while singing. But your graphs supposedly indicate “differences”, which should mean that one value is subtracted from another, but I have no idea what the values are.

Sorry, I have very little interest in unequal temperaments. A disturbing reason I have read for tuning unequal temperaments is that hardly anyone can tune ET anyway, so it is far better to tune an UT, especially one that is designed to be easily tuned. It reminds me of excuses for “new morality” which is really just old sin. If you started a new Topic on the paper that you provided a link to, there is sure to be a great deal of response.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/07/09 02:43 PM

Tooner, thanks for your reply, I came back yesterday and finally I can answer you.

You kindly say:...“Because your paper is the only place that Chas theory is explained, and the only topic in your paper is Chas theory, I see them as one thing: A problem with one is a problem with the other. But I understand that you would view them as separate things.”...

It is good that you understand me when I view Chas theory and the Chas article as separate things, yet I do not understand what benefit you get from viewing at them as one thing. While Chas theory and maths are solid, the Chas article can be improved for sure, so I’d rather keep them separate.

You ask:...“So, where do you think the problem is between how your beat rates are when you tune and how your paper explains what the theory predicts beat rates should be?”...

I can only think that the problem you are pointing out may derive from the way you look at Chas model, at its scale and at the scale’s values (to know more about scientific modelling and systems:
http://en.wikipedia.org/wiki/Model_(abstract)

Once you refere to general modelling, you will not expect Chas model to predict what beat rates should be on pianos.

Chas model’s aim is to represent an ET scale of frequencies related to partials 3 and 4 differences, i.e. an ET scale related to the Chas system’s constants, the 12th and the 15th intervals.

Chas model describes a beating-whole where partials effects are finally combined. In other words, in Chas scale no interval is pure; all intervals have proportional differences from their pure partial value, in an intrinsic correlation between frequencies and differences arising from the infinite combinations of the scale’s elements (section 3.0). As a result, in theory as in practice Chas octaves are progressively stretched (section 4.2, graphs 3 and 4), and 12ths and 15ths have opposite equal beating (section 4.1, table 2, graph 2).

You write:...“I have pondered far too many times what graphs 5, 6 and 10 are indicating and still have no idea. I do not know what you mean by degrees. The only mention of degrees in music that I can think of is in the Psalms, which may have indicated ascending stair steps while singing.”...

Here you can get an idea:
http://en.wikipedia.org/wiki/Degree_(music)

You then say:...“But your graphs supposedly indicate “differences”, which should mean that one value is subtracted from another, but I have no idea what the values are.”...

You can find graph 6 differences values in table 4 (section 4.4, graph 7), in table 8 (section 4.7, graph 9) and in table 9 (section 4.8, graph 10). Sorry if it comes out a bit confusing.

About Professor Colin Pykett’s paper you write:...“Sorry, I have very little interest in unequal temperaments.”...

I should have been more precise, in part 4 – Impure octaves, these are the lines I found intriguing:

“...For example, the beat rate of any interval played depends on the octave in which the interval resides. In other words, a fifth played in the third octave will beat faster than if it is played in the second octave, but slower than if it were to be played in the fourth octave. With any temperament which uses pure octaves, the ratio of these beat frequencies has a simple numerical relationship to the octaves considered...”,

“...Why not ease this problem a little by making the octaves themselves adjustable as well?...”,

“...Currently I have yet to decide on a definite road map for the study though. Doing it with the degree of emphasis on arithmetic and theory which constitutes current work on temperament is almost certainly debarred. It is debarred because pure octaves underpin the entire concept of temperament as it is understood today, and removing them will also remove the relative arithmetical simplicity of the subject. If the octaves are no longer pure, the subject could easily become theoretically anarchic and entirely experiential. Any note on the keyboard could in principle take any frequency value, and the frequencies actually chosen would then arise solely through empiricism – trial and error.”...

Tooner, would you say that Chas model is anarchic? That Chas values arise from error? By the way, last saturday I could record Chas tuning on a small grand, now I only need to make an mp3 of it.

Regards, a.c.
Posted by: RonTuner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/07/09 04:00 PM

About Colin Pykett’s paper - it seems to be heavily tilted to temperament and octaves in organs... There is only a passing explanation of inharmonicity's effect on piano tuning.

While "pure" octaves are the "norm", there have been plenty of piano tuners over the years adjusting the octave width to achieve a pleasing balance of sound across the range of the instrument. Even the pure fifth temperament as well as the pure 12th temperament stretches have been discussed. Bill Bremmer was one of the first I remember tempering his octaves to match the octave+5th in unequal temperaments.

Ron Koval
Chicagoland
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/08/09 07:17 AM

Alfredo:

I am going to take a break from your Topic. The idea of using the degrees (or steps) of a scale (a major scale presumably, although since it is not specified, it could be any number of scales…) as a basis for analysis is just too foreign to me. And when I try to make sense of it by examining paragraph 4.3 I again read “The differences, divided by themselves,…”. ANYTHING EXCEPT ZERO DIVIDED BY ITSELF EQUALS ONE!!!!!

I am going to take a break….
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/11/09 07:39 AM

RonTuner,

you kindly say:..."About Colin Pykett’s paper - it seems to be heavily tilted to temperament and octaves in organs..."

In one post (06/30/09) I wrote: "I’m still trying to explain why I can not agree with Mr. Deutschle when he says:….” The octave is tuned wider than theoretical due to iH.”

In fact what I’m saying (since I can prove it) is that, with or without iH, we need to stretch octaves. Why? Because also partial 2, with the other partials, through stretching can practically contribute to hold up a resonant beating-whole system. Negating the beat’s value (or relevance), we would never get to the Chas concept of a beating-whole."

One thing I find interesting is that Professor Colin Pykett is a pipe organ tuner and, as you can read, he admits octaves stretch.

Tooner, in one post I had already explained what "The differences, divided by themselves,…” means. Anyway, in section 4.5 you can always see what that means.

Thanks and regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/12/09 08:04 AM

Alfredo:

I have taken a fresh look at graphs 5 and 6 now that I understand what you mean by degrees. What they say to me is intuitively understood. The twelfth root of two predicts beat rates that double every octave. The twelfth root of a number greater than two (such as Chas) predicts beat rates of wide intervals to double more often than every octave and narrow intervals less often than every octave.

But there are a few problems. First is the way you present the information. You seem to have no misgivings from stating “The differences, divided by themselves…” This is a mathematical show-stopper (and there are others.) You expect too much from others to bend their vocabulary to match what you mean instead of what you say.

Likewise there is the form of graphs 5 and 6. The x axis is labeled 1,2,3,4 etc, but actually are values 1,4,5,8 etc for the degrees of a scale! And the y axis are logarithmic values, but indicate 0 and negative values! Logarithmic values do not reach zero let alone become negative! Again it is up to the reader to try to figure out what you mean instead of what you say.

The biggest problem is that all this is only pertinent to harmonic instruments and not pianos. This is obvious to you, and yet you still put forth your paper as something important to piano tuning. You should not be surprised that there is little interest in your theory and paper. It is not useful for pianos, and puts a great burden on the reader to understand what it is that you are saying.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/14/09 10:27 AM

Tooner,

you say:...“I have taken a fresh look at graphs 5 and 6 now that I understand what you mean by degrees. What they say to me is intuitively understood. The twelfth root of two predicts beat rates that double every octave.”...

Good, I’m glad you could intuitively understand. Consequently you may also understand why 12th root of two, based on the 2:1 ratio, is an unreal and unusable model (section 4.5).


...“The twelfth root of a number greater than two (such as Chas) predicts beat rates of wide intervals to double more often than every octave and narrow intervals less often than every octave.”...

Correct. The next question may be: why Chas? Then you may have to look at Chas system from two different perspectives:

1 – Chas as a dynamic and comprehensive theory, capable of gaining infinite logarithmic ratios, including 12th root of two and any other pure-interval based ratio;

2 – Chas as the model that describes a perfectly stable beating-whole, an ideal logarithmic scale where all intervals are non-pure (impure) and where beats give rise to a synchoronic event, i.e. the 12ths (narrow) and 15ths (wide) equal beating.

You say:...“But there are a few problems. First is the way you present the information. You seem to have no misgivings from stating “The differences, divided by themselves…” This is a mathematical show-stopper (and there are others.) You expect too much from others to bend their vocabulary to match what you mean instead of what you say.”...

I’m sorry, luckily though you could understand. Would you kindly suggest a better way?

...“Likewise there is the form of graphs 5 and 6. The x axis is labeled 1,2,3,4 etc, but actually are values 1,4,5,8 etc for the degrees of a scale!”...

This is written inside and below each graph.

...“And the y axis are logarithmic values, but indicate 0 and negative values! Logarithmic values do not reach zero let alone become negative! Again it is up to the reader to try to figure out what you mean instead of what you say.”...

Tooner, me too I’m trying to figure out what you mean with what you are saying. I can immagine how you feel, people move bits while you shovel mountains. Please, look at table 9 (section 4.8 – graph 10). Like those ones, all the values in the graphs are differences. For example, from table 9:

1.4985392354 – 1.5 = - 0.0014607646

You write:...”The biggest problem is that all this is only pertinent to harmonic instruments and not pianos.”...

What you are stating has a lot of implications. Are you aware of it?

You then say:...“This is obvious to you, and yet...”...

Please Tooner be carefull, never force your mind in someone else’s head. Overconfidence and conjectures may take you ill, this is why I ask you to always argue your positions.

...“you still put forth your paper as something important to piano tuning.”...

So doing you may be misleading and off-putting for your readers. Talking about temperament and piano tuning, Chas model simply derives from a new approach to the logarithmic scale and from the combination of partials 2, 3 and 5. I ask you: to which model are we refering nowadays? Is’nt 12th root of two a logarithmic model? And Cordier’s 7th root of 3/2?

...“You should not be surprised that there is little interest in your theory and paper.”...

Are you talking about yourself? Was it the rattling of your pc’s keeboard that interested you during this last five months? No, I guess this may have been you in a bad day, nothing to do with my sharp devil's advocate. Anyway, about general interest in Chas theory and practice, for the time being I should be quite happy and there is no point in me saying why.

...“It is not useful for pianos, and puts a great burden on the reader to understand what it is that you are saying.”

Why and how Chas model can be useful, in its theory and in our practice, has extensively been written in this thread. Actually, with Chas Topic I’m trying out an alternative way for sharing the theoretical and practical results of my professional experience. Like any reader can do, I’ve freely chosen my burden and, if needed, I’m still willing to add more explainations.

About tuning practice, have you tried inverted 5ths? Have you tested constant equal beating 12ths (narrow) and 15ths (wide)?

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/15/09 08:09 AM

Alfredo:

Just got handed a time-critical project, so please excuse my brevity. I will not address all that you posted.

I do not think that I have assumed too much. I am the only one that has shown continued interest in your Topic. And you have admitted that Chas does not explain how you tune. Perhaps this will:

The effects of iH cause higher partials to be at higher frequencies than theoretical. This causes octaves to be wider than they are, and frequency ratios to be greater than the 12th root of 2, or even the Chas ratio. (I know this has been said many, many times, but needs to be restated as a review for the following which is not said.) It is easily assumed that this would cause wide tuning intervals to beat faster and beat more than twice as fast for each octave, and for narrow tuning intervals to do the opposite. Oddly, this is not true, because iH increases logarithmically. With one notable exception, the opposite is true. The exception is the 3:2 partial match of the 5th. [Edit] The beat rates of fifths progress differently than other narrow intervals.

So, if we start with an equal temperment octave tuned on an actual piano within a 4:2 octave width so that the lower fourth beats at the same rate as the upper fifth, and tune upward, always keeping the fourth and fifth beating at the same rate for each octave, the following will happen: For a while both the fourth and fifth will beat faster, but not twice as fast each octave. Then they will beat at the same speed from one octave to the next and then start beating slower. (This is the “inverting” that you mention.) If they could be heard, they would eventually both become beatless and then the fourths would become narrow while the fifths become wide!

I don’t think this is well known for a number of reasons. First, 4:2 octaves will cause audible beating if continued too far up. Second, the higher partials become harder to hear. Third, tuners listen to other things in the high treble. I can hear the fourths and fifths speed up and then slow down as I tune, but not become beatless. By then there are other, more important things, to listen to.

So what happens if octaves are tuned wider than 4:2? Well, the fourths will beat faster, the fifths beat slower and what you call an “inversion” will happen lower in the scale. But also, if the octaves are tuned much wider than 4:2, this will be too wide for equal beating 12ths and 15ths with the 12ths being narrow. However, it can cause 12ths and 15ths to beat at the same speed (or at least seems to) but with the 12ths being wide! It would seem impossible for this to happen, but the effects of iH, being logarithmic, do unexpected things. I think this is how you are actually tuning because of where in the scale your fifths are “inverting.”

Must go now. Btw, most people have good days and bad days; I have good moments and bad weeks.
Posted by: JDelmore

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/15/09 11:09 AM

I've been following with interest. I just haven't had time or energy to try to decipher the meaning.
Posted by: Jim Moy

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/15/09 04:55 PM

Originally Posted By: JDelmore
I've been following with interest. I just haven't had time or energy to try to decipher the meaning.

Well, I wouldn't go as far as saying "full of sound and fury signifying nothing," but it sure feels that way.

But maybe I should clarify in advance, when I say "feels that way," I mean the personal, subjective, and emotional response coming from a human when they don't understand the rationale behind what they've experienced, LOL.

[That's an attempt at a joke, up there, is what that is...]
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/17/09 06:47 AM

Tooner,

Thanks for contributing, you are helping me a lot, and thanks for your continued interest.

To be precise, I complain when you assume to little, in terms of responsibility. In other words, I do not like when you offer hasty judgements or conjectural evaluations because to me that sounds illogical and superficial, what may result in being misleading, that’s all.

Next, you say:…”you have admitted that Chas does not explain how you tune.”…

I’ve explained that Chas is a model. Do you find hard to simply take note of it? What is that you do not understand about past and present models?

Thanks for describing iH’s effects in piano tuning. I also hope you take note of another undeniable fact: the actual approach to iH (on pianos) is still referred to pure-octave tuning.

You kindly say (from -/ to +// = skip):

...“The effects of iH cause -/ higher partials to be at higher frequencies than theoretical. This causes octaves to be wider than they are, and frequency ratios to be greater than the 12th root of 2, or even the Chas ratio. (I know this has been said many, many times, but needs to be restated as a review for the following which is not said.) It is easily assumed that this would cause +// wide tuning intervals to beat faster and beat more than twice as fast for each octave, and for narrow tuning intervals to do the opposite.”...

In your previous post, about the Chas graphs, you wrote:...“The twelfth root of a number greater than two (such as Chas) predicts beat rates of wide intervals to double more often than every octave and narrow intervals less often than every octave.”...

To me this sounds the same, and yet you talk about iH’s effects, Chas graphs show you theoretical values.

You then say:...”Oddly, this is not true, because iH increases logarithmically. With one notable exception, the opposite is true. The exception is the 3:2 partial match of the 5th.”...

So, “the opposite is true” means that 5ths progress like narrow intervals. But then you say:

...[Edit]“The beat rates of fifths progress differently than other narrow intervals.”...

Here, I need you to conferm that: at first 5ths progress like narrow intervals, but then 5ths will progress differently than other narrow intervals.

I’d stop here and wait for your answer (take your time). You may also check 2ds (section 4.6) and maybe notice how they progress.

JDelmore,

Thanks for your interest, let me know if I can help you to decipher Chas model’s meaning or if it is only a question of time and energy.

Jim Moy,

Thanks for joining, I’m getting emotional responses too. About Chas rational understanding, is there anything I can do? Do you think another way would be clearer? For istance:

Chas model describes infinite scales of proportional frequencies deriving from all partials proportional differences.

Does Chas model deny previous ET models? No, actually Chas model includes any conventional ET pure-ratio based model, like 12th root of two.

What makes Chas a new model? A new theoretical approach to the sound scale.

What’s new about Chas approach? The way Chas gets to the frequencies. Chas gains the scale’s frequencies with a “difference” factor.

Why a “difference” factor? Because a difference factor can determine differences on any partial’s ratio.

Would this way be any better?

Have a nice w.e.,regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/18/09 04:32 PM

Dear colleagues,

Some of you asked for a recording of Chas tuning. Here,

http://www.megaupload.com/?d=QAVZ7RLE

you find the first of a series of recordings in order to demonstrate only one fact: no matter the size of the piano and despite the usual iH’s degrees, we can find our favorite form again and again, in my case Chas basic ET form, together with its intervals progression (on demand) and its constants, opposite equal beating 12ths (narrow) and 15ths (wide). This is why I’m choosing this conditions: small pianos, non-professional recording, non-professional playing. Finally, this is one of the ET forms that Chas theoretical model can mathematically describe.

This was recorded at Alessandro Petrolati’s lab (many thanks). On this Steinway S (5’ 1”, 155 cm), last month he put new strings, new hammers (still to be voiced) and new pins. For recording he used a 250 Euro device that he positioned about two meters away. I then asked him to kindly play whatever he liked to.

Ah, I hope you do not mind mega-muscles. Actually, I asked one of my sons to help me put this recording in the web. If you can suggest another place, so much the better.

Regards, a.c.
Posted by: Grandpianoman

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/18/09 04:41 PM

I use www.box.net It's very good and there are never any problems with it.

I have not been following this thread, as it is very technical and beyond my expertise.

Mr. Carpurso, I just listened to your recording, and to my ear, it's very pleasing. I would not hesitate to use your tuning stretch in my piano. smile
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/18/09 04:50 PM

grandpianoman,

nice to get an immediate help! Are you the pianist that can tune his own piano?
Posted by: Grandpianoman

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/18/09 05:11 PM

Yes.....I do try! smile But, I am not a pianist, my 2 player systems are the ones that 'play' my piano. wink
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/18/09 05:28 PM

grandpianoman,

I liked your ETD tuning, thanks a lot for your sharing. I'll tell you more tomorrow.

Regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/19/09 05:52 PM

Granpianoman,

In my opinion your recordings have well demontrated that with an ETD you can tune your piano. In the case of professional tuning, pin’s control and tuning stability are also very important. In fact, a pro tuner should never spoil the pins block while ensuring a stable, long lasting tuning. In my experience, my aural skill and my wrist’s sensitivity developed together, although implying opposite conditions: deepest relaxation and best body tension.

About Chas, the all question is much easyer than pro tuning and more handy than what it may seem to be. Today we are still referring to 12th root of two, so theorizing a pure, beat-less 2:1 octave. When I’ve asked why today should the octave be theoretically pure, I did not get any answer. Yet today, all the maths calculations for piano scaling, for piano iH and beats are based on 12th root of two, a model that nobody has ever been able to put into practice.

Chas model describes an ET scale where neither the octave is theoretically pure, a scale that is not based on a pure-octave module, nor a single pure ratio. In fact Chas finally combines all ratios like 2:1, 3:1, 5:1 and can theoretically describe a euphonic beating-whole. In stead of adopting the 2:1 zero-beating octave constant, Chas ET adopts two opposite equal beating constants, 12ths (narrow) and 15ths (wide), the two constants I can relate to my practical tuning.

If you would like to know more about Chas, you’ll be welcome with any question.

Regards, a.c.

Chas tuning on a Steinway S:

http://www.megaupload.com/?d=QAVZ7RLE
Posted by: Grandpianoman

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/19/09 08:28 PM

Mr.Capurso, thank you for your thoughts on a consumer being able to tune their own piano with an ETD. I have proven that it is possible. smile

Am I the only one here that likes your CHAS tuning...that can't be. My ears do not lie, and I find your tuning on the Steinway to be very pleasing to the ear. The whole piano sounds good!

I am really not technically minded to understand your CHAS tuniing theory, but I do know what I like, and I like your tuning. Is there a way I can implement your tuning into my Reyburn Cyber Tuner? I would like to try it on my grand piano. smile
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/20/09 08:21 AM

Grandpianoman,

thank you for your feedback. I do not use a Reyburn Cyber Tuner so unfortunately I can not help you. Maybe some expert will read this, or you could start a Topic, or we could ask the developer, Mr. Reyburn. About others who may like or dislike Chas tuning, let’s wait. After all, I’m not urging my colleagues to say how they like it and why I’m doing all this is because I would like to honestly contribute with a new approach to the sound whole, what has helped me in my tuning practice.

Dear colleagues,

now you can find the first recording of Chas tuning on a Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Yesterday I wrote: “When I’ve asked why today should the octave be theoretically pure, I did not get any answer.”

In my opinion, one of the reasons why 12th root of two ET theorized a pure 2:1 octave is because, at that time, the pure octave was a dogma, or maybe because nobody knew how to mathematically combine also the 2:1 ratio in an ET scale.

So again, I ask: how can you think 12th root of two as an "ideal" when no one can give a logical theoretical reason for a zero-beating octave?

And yes, I would also like a technical comment about iH limits on the recorded baby grand. Last but not list, does this kind of effort make sense? Should I try out my Japanese?

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/20/09 08:31 AM

Alfredo & GPM:

I will listen to your recordings and reply, within a week.

GPM:

Your Avatar makes me think of a slice of mincemeat pie. smile
Posted by: Grandpianoman

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/20/09 03:56 PM

Alfredo, You're welcome. I understand, and I also hope at some point, your Chas system can be repeated in an ETD.

Jeff, lol...mincemeat pie?!!...you know, I tried to make that picture of my grand piano larger...I do have a large appetite! wink
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/21/09 01:07 PM

Alfredo:

I have decided that I will no longer discuss nor study your paper for personal mental health reasons. It does not seem that I have been helpful, anyway.

We can discuss any and all other aspects of tuning and tuning theory, though.

Our discussions have caused me to look deeper into tuning and tuning theory, and also how to talk about them. I did not make myself very clear on the beating of fourths and fifths. Let me try again.

If theoretical, harmonic tones are tuned to 12th root of 2 pitches, the beat rate of fourths and fifths will double every octave. Also, the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave.

If a typical piano with inharmonic tones is tuned so that the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave (this being a definition of a 4:2 octave) the beat speed of the fourths and fifths will less than double every octave. Specifically, in the fifth octave they will stop increasing in beat speed, and start decreasing in beat speed. In the sixth octave they will become beatless and then start beating again, but with the fourths being narrow of just and the fifths being wide of just.

Of course, if the piano is not typical or the octaves are not 4:2, the beat rates may do something else.

Sometimes I ask too many questions in one post, or I forget that I had already asked a question. Here are just a few that I don’t think I have asked before:

You said that your 12ths and double octaves all beat at the same speed. What speed is this?

You said that your 12ths are narrow and your double octaves are wide. How do you know this for sure?

PLEASE, lets talk about other things and not your paper.
Posted by: RonTuner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/21/09 03:40 PM

GP, the Verituner should be able to replicate this type of tuning. I haven't kept up with this discussion, but are we talking about balancing between two or more intervals as opposed to just working with the octave? Someone give me a short version.... tuning up, the note to be tuned (call it C6 two octaves above middle C)is compared/balanced between which notes?

You mentioned 3:1, which would be the octave+fifth below (F4) 4:1 double octave (C4), single octave types (C5), octave+fifth (G4)? Fourths and fifths? (F5, G5)?

RCT can be "pushed" to kindof replicate this type of tuning approach - but it is limited by only having a few inharmonicity measurements to project across the range of the keyboard. It's a great project for you because you are only concerned with one instrument. You'd create multiple tuning files aiming for specific interval matches - using the custom equalizer. Then you can try averaging the files, or flipping back and forth in the graph mode to find a happy medium... I think I still have a working copy of RCT on an old laptop - if I get the how-to specifics, I might be able to come up with a work-around.

Ron Koval
Posted by: Grandpianoman

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/21/09 10:32 PM

Thanks Ron, that sounds like a great idea. Let me know if you get the how-to specifics and work them with your copy of RCT.

I don't know anything about the 3:1 ratios etc that you mention. Perhaps Mr. Capurso can chime in.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/22/09 07:18 AM

Tooner,

You say:...“I have decided that I will no longer discuss nor study your paper for personal mental health reasons.”...

What’s this, the beginning of a bad week? If you were joking I’d answer: do not worry, it’s to late.

...“It does not seem that I have been helpful, anyway.”...

You well know that most of the time you are helpful, in this Topic like in many others, so what is this about?

...“We can discuss any and all other aspects of tuning and tuning theory, though.”...

As you like. For me, it has been this way since my second post (05/08/09) when I wrote: “...then we'll talk about anything you like.” Has it been so? Should it be different?

...“Our discussions have caused me to look deeper into tuning and tuning theory, and also how to talk about them.”...

I’m glad you have not wasted your time. But you seem to be regretting something...

You kindly say:

...“If theoretical, harmonic tones are tuned to 12th root of 2 pitches, the beat rate of fourths and fifths will double every octave. Also, the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave.
If a typical piano with inharmonic tones is tuned so that the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave (this being a definition of a 4:2 octave) the beat speed of the fourths and fifths will less than double every octave. Specifically, in the fifth octave they will stop increasing in beat speed, and start decreasing in beat speed. In the sixth octave they will become beatless and then start beating again, but with the fourths being narrow of just and the fifths being wide of just.”...

I’ve gone back in this Topic to see if I had said the same and yes, have a look at what I posted on 05/20/09.

You say...“Of course, if the piano is not typical or the octaves are not 4:2, the beat rates may do something else.”...

In my tuning, A4-A3 (A3 flat) goes together with A3-D4 (4th) beating at about 1 bps, D4-A4 (5th) beating slower at about 1/3.5 bps, A3-E4 (5th) about 2/3 bps, E4-A4 (4th) about 2 bps. As you notice, 4ths beat progressively faster but they sort of collapse at G4-C5, i.e. 4ths beating is not discernable anymore.

...“You said that your 12ths and double octaves all beat at the same speed. What speed is this?”...

On 05/23/09 I posted my sequence (please mind: wide = sharp, narrow = flat, but for any tuner this is obvious). At the bottom you also read: Chas delta-wide 15ths and delta-narrow 12ths beat’s rate is about 1b/3s.

...“You said that your 12ths are narrow and your double octaves are wide. How do you know this for sure?”...

Because to eliminate the 12ths beat I need to turn my tuning hammer clockwise. The opposite for double octaves.

About tuning and tuning theory, I ask: why should the octave be theoretically pure? Why should not we theoretically combine also 2:1 ratio in our ET scale, like we have done with 3:2 and 5:4 ratios?

RonTuner, GPM,

I need to read your post again and try to understand what the question is, maybe RonTuner you can kindly word it in a different way. Thanks a lot.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/22/09 09:02 AM

Alfredo:

No regrets, just pruning.

Your 5/20/09 post does mention what you call the “inverting” of fifths, but not fourths. (I have to be careful with your use of that term, it means something different to me.) And then mentions that the answer is in the Chas model. Sorry, I choose not to go there.

You mention a fault in tuning using a 2:1 ratio. I agree that there is a fault, but not with using the ratio of 2:1 instead of another number. The fault is in using a ratio at all. Tuning is done by matching partials so that they are either beatless, or beat at a specific (usually progressive) rate. Before inharmonicity was understood, it was believed that if an octave was beatless the frequency ratio was 2:1. After frequency measuring devices were available it was discovered that a 2:1 partial match does not mean a 2:1 frequency ratio.

Now if what you mean is that an octave should not be tuned to a 2:1 partial match, apparently they are not tuned that way, anyway. They are tuned closer to a 4:2 partial match, regardless of what people thought. This is shown to be true when considering the P4-P5 test for a 4:2 octave, which was a standard octave check before inharmonicity was well understood.

So what happens when 4:2 octaves are tuned? I wasn’t quite sure when listening to my tunings. But when calculating the beat rates, taking into account inharmonicity, it turns out that the double octave beats wide and about the same speed as the 12ths beating narrow in the mid section! I did not expect this. Equal beating 12ths and 15ths, at least in the mid section, seem to have been the norm all along.

But you are saying that your octaves are audibly beating, and your P4-P5 test confirms that you are tuning wider than 4:2 octaves. But the speed of your 12ths and 15ths are in range for 4:2 octaves in the mid section. This is just an objective analysis. What the discrepancy may be, I do not know.

Another discrepancy is that your 12ths and 15ths all beat at the same speed. Well, they may seem to. Mathematical analysis does not agree; they are progressive. I have proved this to myself when tuning this way and using a “drone tone” to make sure that the 12ths were narrow and the 15ths wide.

There are those that say all fourths should beat at the same speed and the beat rate of fifths is barely discernable according to modern tuning theory. I hear something different and mathematical analysis supports what I hear. I have no explanation for this discrepancy either.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/23/09 10:06 AM

Tooner very good, you have written a very well pruned post. May I seriously invite you to come and help me with my olive trees? Anytime, from now 'till next March.

You say:...“Your 5/20/09 post does mention what you call the “inverting” of fifths, but not fourths. (I have to be careful with your use of that term, it means something different to me.)”...

How would you call the 5ths fenomenon, so that we can relax? About fourths, I thought it may have been to much although, asking Bill Bremmer about what he had heard, I wanted to introduce the question. This is what Bill wrote (06/02/09):

“5ths become wide on PTG Tuning Exam Master Tunings in the 6th octave.”…”It must be close to 20 years ago that I saw Steve Fairchild demonstrate that 5ths do become wide. He also said that 4ths become narrowed.”

I answered: “So we agree in saying that, somewere, 5ths do invert, now the questions are: can or should 5ths be progressive or can 5ths have casual beats/rate? And what about 4ths?”

Bill had also written: “I have met and had discussions with Bernhard Stopper and have also heard his tuning. It has a remarkably clear character to it. While I still do not fully understand it, I did gather from what he has said that the 12ths also become wide at some point in his tunings as well.”
And: “I have now long taken to the practice of tuning pure double octaves and 5ths from F6 to the top.”

I could not understand how he can have pure double octaves and 5ths, together with wide 12ths and progressive RBI. I would have talked about it but, as a result, since then I have not heard from Bill. This may be how tuning can become an indulgent mystery.

You now say:...“You mention a fault in tuning using a 2:1 ratio. I agree that there is a fault, but not with using the ratio of 2:1 instead of another number. The fault is in using a ratio at all.”...

This was your conclusion also five months ago, but now there may be some good reasons to update it.

...“Tuning is done by matching partials so that they are either beatless, or beat at a specific (usually progressive) rate.”...

Yes.

...“Before inharmonicity was understood, it was believed that if an octave was beatless the frequency ratio was 2:1.”...

Yes. Using your words, it was also believed that the octave could and should be beatless. It was also believed that there was no need to combine 2:1 ratio with 3:1 and 5:1 ratios, and that we could get by with 12th root of two model. It was believed that the traditional octave module was correct, that the temperament could be referred to 13 notes and that the octave module could then be copyed. All these being unjustified theoretical premises, all wrong beliefs.

You say:...“After frequency measuring devices were available it was discovered that a 2:1 partial match does not mean a 2:1 frequency ratio.”...

Yes, and a heavy curtain was drawn over those unjustified premises. In other words, not only a 2:1 partial match does not mean a 2:1 frequency ratio, a 2:1 partial match is a wrong target in that it comes (with Kent’s permission) from a lame model.

...“Now if what you mean is that an octave should not be tuned to a 2:1 partial match, apparently they are not tuned that way, anyway. They are tuned closer to a 4:2 partial match, regardless of what people thought.”...

Exactly. You say that nobody goes for a 2:1 partial match. You conferm that we are not referring to 12th root of two model, so we are lacking for a reliable model. You say that we are going for a greater ratio and you refer this to iH. Actually I do not. In fact, any ratio greater than 2:1 combines – perhaps you’d say compromises - partial 2 with partials 3 and 5. If this combining is ideal or not, this is another question and Chas is the answer.

..."So what happens when 4:2 octaves are tuned? I wasn’t quite sure when listening to my tunings. But when calculating the beat rates, taking into account inharmonicity, it turns out that the double octave beats wide and about the same speed as the 12ths beating narrow in the mid section!"...

Very good indeed. We may be on the right track.

..."I did not expect this. Equal beating 12ths and 15ths, at least in the mid section, seem to have been the norm all along.”...

You say the norm, I’d say the “about” tendency. In my opinion, all tuners have been and are seeking the model and the practical way that combines all partials and all relative ratios.

..."But you are saying that your octaves are audibly beating,”...

Yes, and progressive.

...“and your P4-P5 test confirms that you are tuning wider than 4:2 octaves.”...

Wider than 2:1 for sure, how much wider I’ve never measured. Can you calculate what the theoretical ET scale’s incremental ratio is with 4:2 octaves?

...“But the speed of your 12ths and 15ths are in range for 4:2 octaves in the mid section. This is just an objective analysis. What the discrepancy may be, I do not know.”...

Would you be more precise on this? I can not get the point.

...”Another discrepancy is that your 12ths and 15ths all beat at the same speed. Well, they may seem to. Mathematical analysis does not agree; they are progressive. I have proved this to myself when tuning this way and using a “drone tone” to make sure that the 12ths were narrow and the 15ths wide.”...

From your analysis, would 12ths and 15ths be convergent or divergent?

...”There are those that say all fourths should beat at the same speed and the beat rate of fifths is barely discernable according to modern tuning theory. I hear something different and mathematical analysis supports what I hear. I have no explanation for this discrepancy either.”...

In my tuning fourths are progressive. Around C5 they seem to collapse, so I can not say if they invert in the high section. I will have a go plucking the strings. The beat rate of fifths is very well discernable, like RBI, and fifths too in Chas tuning are progressive, first going narrower, then toward pure.

Tooner, thanks for your efforts and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/24/09 05:29 PM

RonTuner,

The constants of Chas tuning’s form are 12ths (octave+fifth) and 15ths (double octave). In other words, 12ths (narrow) and 15ths (wide) have the same opposite beat-rate all along the keyboard.

How to get this final tuning form is a different matter and it may depend on the piano’s conditions and settling.

I first tune only middle strings from C3 to C6. Generally speacking, on normally-flat pianos, when I go up the temperament section, i.e. from A#4, I stabilize (on mid-string) a preparatory wider stretch for all notes (and all check intervals) so to obtain - at list - chromatic pure 12ths. This is to say at list 3:1 matching. So, from A3-E5 12th up, all chromatic 12ths will be - at list - beatless (on mid-strings). All double octaves will then beat (on mid-strings) about 3/2 bps, so that after unisoning left and right strings I get (hopefully) equal beating 12ths (narrow) and 15ths (wide). If the piano was very flat I may tune it twice, anyhow I would stretch all check intervals so to obtain chromatic little-wide 12ths, with maybe a wide ¼ bps.

Is this of any help?

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/26/09 09:02 AM

Alfredo:

When I read something like “The fifths are …..” what I imply is “The [beat rate] of the fifths are …” So when I read something like “The fifths invert ….” I imply “The [beat rate] of the fifths invert [from narrow to wide] …” But I understand that you mean “The [progression of the beat rates] of the fifths invert [from beating faster to beating slower] ….” It is not a big problem. I understand what you mean. But I want to make it clear in my mind and in the mind of other readers. If you choose, you could include the phrase “beat rate progression" when appropriate, such as "The beat rate progression of the fifths invert from becoming faster to becoming slower."

I will not speak for other posters including Bill and Kent. I do love your term “indulgent mystery”, though. It is how I look at what you say about your tuning.

We seem to be covering old ground (going in circles) in the discussion of frequency ratios, partial matches and inharmonicity. Perhaps we can break free of this if I play the Devil’s Advocate again. You ask: “Can you calculate what the theoretical ET scale’s incremental ratio is with 4:2 octaves?” Yes I easily can and will, but you must first tell me what you will do with the answer.

Now also consider this: You say your fourths beat progressively faster. When a 12ths is tuned up from the upper note of a fourth so that the resulting 15th from the lower note of the fourth is wide and beats at the same speed as the narrow 12th, the beat speed of the 12th and 15th will be ½ the beat rate of the fourth. Yet you say that all your 12ths and 15ths beat at the same speed. I consider this to be an “indulgent mystery.”
Posted by: RonTuner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/26/09 10:27 AM

Thanks - that's what I was looking for. That should be easy to program into the Verituner. (Only down to D2- then I lose the 3:1 as an option...)

Ron Koval
chicagoland


Originally Posted By: alfredo capurso
RonTuner,

The constants of Chas tuning’s form are 12ths (octave+fifth) and 15ths (double octave). In other words, 12ths (narrow) and 15ths (wide) have the same opposite beat-rate all along the keyboard.


Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/26/09 11:30 AM

Ron,

excellent! I really hope you can manage. Only remember that, if you go straight for the Chas final form, the piano's settling may cause a lowering of the frequencies, especially from mid-high section (C5) up.

This is why I mention a preparatory wider stretch.

Tooner,

thanks, I shall reply asap.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: RonTuner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/26/09 11:46 AM

I'll try it out - in my experience, the Baldwin school upright pianos prove to be the death of many of these types of tuning schemes. The added width to the octave by referencing the octave+5th as a basis leaves the single octave with a problematic, obvious beat. I'll see if tempering the 12th with the 15th works with these types of pianos. I'm confident it will be fine with larger instruments.

Ron Koval
chicagoland
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/26/09 07:27 PM

Tooner,

You kindly say:...“If you choose, you could include the phrase “beat rate progression" when appropriate, such as "The beat rate progression of the fifths invert from becoming faster to becoming slower."...

Thank you, I’ll include the phrase you are suggesting.

...“I will not speak for other posters including Bill and Kent. I do love your term “indulgent mystery”, though. It is how I look at what you say about your tuning.”...

Ok, I feel the blow.

...“We seem to be covering old ground (going in circles) in the discussion of frequency ratios, partial matches and inharmonicity.”...

I do not think so, and you yourself talk about some fresh understanding.

...“Perhaps we can break free of this if I play the Devil’s Advocate again. You ask: “Can you calculate what the theoretical ET scale’s incremental ratio is with 4:2 octaves?” Yes I easily can and will, but you must first tell me what you will do with the answer.”...

This sounds more like the Devil’s Ambassador. Anyway ok, with that answer I’ll turn hell into heaven. Only then you yourself may be able to approach the only one ratio greater than 12th root of two that can straighten the beat rate progression of the 12ths and 15ths, in the way Chas model describes its equal beating constants. It can not be now though.

You write:...“Now also consider this: You say your fourths beat progressively faster.”...

Not exactly. I say that Chas fourths beat progressively faster up to G4-C5. I’ve also said that, tuning up the scale, I use many other check intervals and that I do not use 4ths because they seem to collapse. To be more precise (it may be useful), I do not even exclude that the beat rate progression of the 4ths invert also going down the scale, after C3-F3, from becoming slower to becoming faster.

You then say:...“When a 12ths is tuned up from the upper note of a fourth so that the resulting 15th from the lower note of the fourth is wide and beats at the same speed as the narrow 12th,”...

Is this to say: when you tune a 12th and a 15th opposite equal beating?

...“the beat speed of the 12th and 15th will be ½ the beat rate of the fourth. Yet you say that all your 12ths and 15ths beat at the same speed. I consider this to be an “indulgent mystery.”

To me this looks like a tuning experience gap. Will a recording of 4ths with relative 12ths and 15ths be enough?

I also asked you: From your analysis, would 12ths and 15ths be convergent or divergent? Also this may be useful.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/27/09 08:23 AM

Alfredo:

The reason I like your term “indulgent mystery” is that I consider it to be a term of acceptance, not of rejection. An acceptance that people do not hear things the same, and come to different conclusions. I did not mean it as a jab, but more as a prod. Indulge is a warm, not a cold, word. Again, I am not your enemy.

Since we do hear things differently, if we want to reach a consensus, we need an objective way of looking at tuning. The most objective way that I know of is to look at beat rates. Especially beat rates that are tests: tests where a third note is used. The third note is used to understand what is happening with another interval or intervals. This is my basis of bringing up the conflicting statements of progressive fourths and unprogressive equal beating narrow 12ths and wide 15ths. But maybe we should put this aside for now, and hopefully bring it up again later. Other tests have been mentioned and you have not responded in a way that shows understanding.

I enjoy a challenge, and it continues to be a challenge to communicate with you. There is some vocabulary that we should define. (Remember, I have travelled, and am used to those that English is not their primary language. And why should it be!)

When the terms “narrow” interval or “wide” interval are used, it refers to the relationship of the nearly coincident partials that cause beats. If the nearly coincident partial of the lower note is higher than the nearly coincident partial of the upper note the interval is ”narrow.” So, if the upper note of a “narrow” interval is raised in pitch (or the lower note is lowered in pitch) the beat rate decreases. And if the nearly coincident partials are at the same pitch, the interval is “just” and beatless.

You asked “From your analysis, would 12ths and 15ths be convergent or divergent?” I am not sure what you mean by 12ths and 15ths being convergent or divergent. I will certainly answer your question, but need to understand it first. Could you give examples?

The reason I asked what you would do with the incremental ratio value for theoretical ET 4:2 octaves is to know what your understanding of the subject is. I think much of the communication problem is that we each assume that the other already knows certain things, looks at things a certain way, or is asking a question for a certain reason. And when we assume incorrectly we end up talking about two different things.

This reminds me of a story. A young boy asked his Mother where he came from. The mother wanted to be truthful, but did not know where to start. So stalling for time, she asked her son why he wanted to know. The son said that Suzie next door said she came from New York but he didn’t know where he came from.

So, I want to try to talk with you about just one or two things at a time, and make sure we understand what each other are saying before moving on.

So what significance do you think the incremental ratio of theoretical ET 4:2 octaves could have?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/28/09 08:41 AM

Tooner,

I well know you are not my enemy, I was joking. The same word can result in having different meanings though, I do not need to tell you and vice-versa.

You talk about “conflicting statements of progressive fourths and unprogressive equal beating narrow 12ths and wide 15ths.”

The beat rare progression of fourths also invert, for sure in the mid-high section, maybe also in the bass section. Will this be any better? You did not like the word "interweaving" but this is what 4ths and 5ths do with their beat rate progressions.

About “narrow” and “wide” intervals, and how they refer to coincident partials, I think I’m ok, maybe you wrote about this for the latest yung reader.

...“I am not sure what you mean by 12ths and 15ths being convergent or divergent…Could you give examples?"...

Yes. When you wrote:...”Another discrepancy is that your 12ths and 15ths all beat at the same speed. Well, they may seem to. Mathematical analysis does not agree; they are progressive. I have proved this to myself when tuning this way and using a “drone tone” to make sure that the 12ths were narrow and the 15ths wide.”

I answered: From your analysis, would 12ths and 15ths be convergent or divergent?

So, you say they are progressive. If it is so, how do they progress? Overcrossing, like 4ths and 5ths? Getting further apart?

...“So, I want to try to talk with you about just one or two things at a time, and make sure we understand what each other are saying before moving on.”...

Good idea, me to, I would like to know what you have understood so far about Chas model and Chas algorithm. You have written somewhere that delta is superfluous and that Chas algorithm could be an equation, when in fact it is an equation.

...“So what significance do you think the incremental ratio of theoretical ET 4:2 octaves could have?”...

Sorry, I should have written: the incremental ratio of 4:2 octaves using your theoretical iH tables. What significance? I told you, you may soon or later discover that there is a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB. You need to discover that yourself though.

Thanks and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/28/09 10:15 AM

Alfredo:

I know you know what narrow and wide intervals are, but there was a problem with the vocabulary when you described your tuning sequence, so I wanted to makes things very clear. Thanks.

You wrote: “Sorry, I should have written: the incremental ratio of 4:2 octaves using your theoretical iH tables. What significance? I told you, you may soon or later discover that there is a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB. You need to discover that yourself though.”

Thank you, I think I know where to go from here. With inharmonic tones, the octave (and incremental) frequency ratio changes from note to note when the octave type is constant. This is why the Railsback curve is not a straight line. So there cannot be “…a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB.” For a given iH curve any octave ratio for a 4:2 octave can be determined, but is of very limited value. The twelfth root of this ratio cannot even be used to determine the temperment because the next octave chromatically will have a different ratio.

But a picture is worth a thousand words. Here is a link with some graphs that show beat rates, cents deviation (Railsback curve) and the iH curve for a simulated studio sized upright piano:

http://www.box.net/shared/rxb631v2yz

The tuning was done note by note to try to have all 12ths beating narrowly at the same speed as the 15ths beating widely. It could be refined some but cannot be made perfect. For the upper notes, the common note must be on top. For the lower notes, the common note must be on bottom. For the middle notes, there had to be compromises, because the intervals do not all beat at the same speed; for the given beat speed of a 15th, it does not work to have both the upper and lower 12ths beat the same.

The graph that shows the beat rate for the 12ths and 15ths predict that a piano tuned this way will have these intervals beat progressively faster, then slower, then beatless and finally faster again with the 12ths being wide and the 15ths being narrow.

The graph that shows the 4ths and fifths show the fifths beating progressively faster and always being narrow, while the fourth beat progressively faster, then progressively slower, and finally faster again but being narrow.

But that is not how someone may actually tune when trying to tune with equal beating 12ths and 15ths. Nor is it how a person may think a piano sounds when tuned so that the 12ths and 15ths actually do beat equally. It can be an indulgent mystery.

As far as discussing the CHAS algorithm or model, sorry, NO!
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/28/09 01:06 PM

Tooner, you have done a very good job.

You say:...“With inharmonic tones, the octave (and incremental) frequency ratio changes from note to note when the octave type is constant. This is why the Railsback curve is not a straight line.”...

I would never say that the octave type should be constant. I was giving you an exercise.

...“So there cannot be “…a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB.”...

Really? Let’s see why.

...“For a given iH curve any octave ratio for a 4:2 octave can be determined, but is of very limited value. The twelfth root of this ratio cannot even be used to determine the temperment because the next octave chromatically will have a different ratio.”...

I agree. In fact I’m talking of a ratio that is not only the octave’s ratio. Chas ratio is a combined ratio, i.e. it is a “difference” ratio that stands for all intervals.

You say:...“But a picture is worth a thousand words. Here is a link with some graphs that show beat rates, cents deviation (Railsback curve) and the iH curve for a simulated studio sized upright piano:

http://www.box.net/shared/rxb631v2yz

To be sincere, I think that you (if it was you) could have called it “ET-EB 12ths and 15ths test on a simulated studio sized upright piano”. We will be able to call it “Chas tuning” when we will get the right chance with a real Chas tuning. Anyway I’m not disappointed, mainly because ET-EB is what Chas is describing. I also think the results are surprisingly favorable.

...“The tuning was done note by note to try to have all 12ths beating narrowly at the same speed as the 15ths beating widely. It could be refined some but cannot be made perfect.”...

Nevermind.

...“For the upper notes, the common note must be on top. For the lower notes, the common note must be on bottom. For the middle notes, there had to be compromises, because the intervals do not all beat at the same speed; for the given beat speed of a 15th, it does not work to have both the upper and lower 12ths beat the same.”...

Do not worry, let’s look at the results together.

...“The graph that shows the beat rate for the 12ths and 15ths predict that a piano tuned this way will have these intervals beat progressively faster, then slower, then beatless and finally faster again with the 12ths being wide and the 15ths being narrow.”...

What I can see is that 12ths (3-1) and 15ths (4-1) are straighten, although not perfectly straght.

...“The graph that shows the 4ths and fifths show the fifths beating progressively faster and always being narrow,”...

Also in my final tuning form, as I’ve said, I do not think 5ths get wide. And in the highest section I tune octaves and check 10ths, 12ths, 15ths and 17ths. Listening to plucked strings (so with different iH) 5ths are narrow.

...“while the fourth beat progressively faster, then progressively slower, and finally faster again but being narrow.”...

Yes, this is what I find in my tuning form, excluding the bass section. Also the octaves go in the way Chas describes them. BTW, what was the range in this test?

...“But that is not how someone may actually tune when trying to tune with equal beating 12ths and 15ths.”...

In fact, to gain Chas form I invert the 5ths beat rate progression (on mid-strings preparatory tuning).

...“Nor is it how a person may think a piano sounds when tuned so that the 12ths and 15ths actually do beat equally. It can be an indulgent mystery.”...

In fact, Chas 5ths and 4ths do not sound flat at all, yet if you pluck the strings in the highest section you can hear them flat, 4ths more than 5ths. And this mystery is disclosed every time I tune.

...“As far as discussing the CHAS algorithm or model, sorry, NO!”

Ok, but you will avoid saying nonsense then, like “it could be an equation”. So, live the test’s imperfections and compromises alone, what are your latest temporary conclusions? Oh, have you heard Chas recording? Would you complain for iH’s effects?

Thanks a lot and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/28/09 01:24 PM

Alfredo:

One step at a time.

You posted: “I agree. In fact I’m talking of a ratio that is not only the octave’s ratio. Chas ratio is a combined ratio, i.e. it is a “difference” ratio that stands for all intervals.”

Please explain this more. I do not know what you mean by a combined ratio or a difference ratio. Examples are probably needed for me to understand you.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/29/09 08:17 AM

Tooner,

I’m going to answer you questions, then it will be your turn. Careful not to leave to many question marks behind.

You wrote:...“For a given iH curve any octave ratio for a 4:2 octave can be determined, but is of very limited value. The twelfth root of this ratio cannot even be used to determine the temperment because the next octave chromatically will have a different ratio.”...

I replyed: “I agree. In fact I’m talking of a ratio that is not only the octave’s ratio. Chas ratio is a combined ratio, i.e. it is a “difference” ratio that stands for all intervals.”

Now you kindly ask:...“Please explain this more. I do not know what you mean by a combined ratio or a difference ratio. Examples are probably needed for me to understand you.”...

Ok. What does “combined ratio” mean? In the way nature combines oxygen and hydrogen to make water, Chas combines two intervals ratios to make one scale ratio. You well know, so far ET scale’s incremental ratio could only be referred to a single interval’s ratio, in the way 12th root of two is referred to the octaves ratio, 7th root of 3/2 refers to fifths ratio and 19th root of 3 refers to 12ths ratio. Chas ratio is now composed of two combined ratios, 3:1 and 4:1 ratios.

This may explain more. When I realized the antagonism amongst 3ds, octaves and 5ths I guessed I had to look for a trivalent scale ratio, i.e. for a ratio that could represent the game I was playing with those three intervals beats, trying to find the best beat rate progressions. After a while, I understood that 3ds and octaves were on the same beat line, so that octaves could well represent 3ds (and vice versa). Since then I knew that 5ths and octaves were the two original stretchers, and that the correct scale incremental ratio would have had to combine those two intervals ratios in a balanced “beat regulator”. Then it could only be a beat-ratio.

If you wanted to cut a straight furrow, two oxen would do much better than one.

If anythyng then, I would have had to look for a double constant, i.e. a constant for two beating-intervals, so to gain the difference ratio that could balance the two original stretchers. Once the beats progression of all intervals could restore the same euphonic form again and again, the scale double constant could be extracted, and how 12ths and 15ths delta-differencies could include all intervals ratios, was quite evident.

Thanks and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/29/09 06:34 PM

Bill,

You say: "Alfredo, take Kent up on what he offers. You will benefit from it greatly."

May I ask you to tell me precisely what you mean? Do you mean Kent's open invitation to visit Kansas City?

Regards, a.c.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/29/09 11:12 PM

Originally Posted By: alfredo capurso
Bill,

You say: "Alfredo, take Kent up on what he offers. You will benefit from it greatly."

May I ask you to tell me precisely what you mean? Do you mean Kent's open invitation to visit Kansas City?

Regards, a.c.



Yes, I do.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/30/09 08:12 AM

Alfredo:

Ok, a combined ratio is one that is calculated from two different ratios. I do not know of any intrinsic value that the ancestry of a ratio could have. What is important is what the ratio will accomplish on its own.

Maybe we can jump ahead a bit and see what such a ratio will accomplish. Let us use the Beat Analyzer as a simulator to see the results of using such a combined ratio. You would need to give me the ratio and directions in how to use it. Also, you should predict what the results will be ahead of time, so as to have an objective evaluation. I can also provide a table showing all the numerical values, for verification of the calculations and construction of the graphs.

You asked: “BTW, what was the range in this test?” All the graphs are for the full range of the piano, 88 notes. This is shown by the scale at the bottom of the graphs, A0-A7. The Beat Analyzer also provides mid-range (C3-C5) graphs for RBIs and SBIs but I did not include those.

Sorry, I did not listen to your recording. When I clicked on the link a number of questionable pop-ups appeared. I did not want to take a risk by downloading the file. And I am not sure how my opinion of the tuning would effect our present discussion, anyway.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/30/09 09:33 AM

Bill,

thank you very much, I will treasure your suggestion. At the moment I’m quite confused.

Do you think it would sound less subversive and maybe closer to actual tuning if I presented Chas model as (thanks to Robert) a variant of ET where the octave ratio is something a little bigger than two?

Kent,

It's kind of you to invite me to Kansas City and, if this gave us all a better chance, I would consider your offer. But tell me please, is it sincere?

Tooner, thanks, I'll reply asap.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/31/09 08:56 AM

Quote:
Kent,

It's kind of you to invite me to Kansas City and, if this gave us all a better chance, I would consider your offer. But tell me please, is it sincere?


Of course it is. We have the headquarters of the Piano Technicians Guild here with its piano museum which you might find interesting. The headquarters has more than just museum pieces; there is at least one modern 7' grand available there.

At my university, there are many very fine pianos that would be available, and we have professional audio recording equipment available there as well.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/31/09 11:26 AM

Originally Posted By: alfredo capurso
Bill,

thank you very much, I will treasure your suggestion. At the moment I’m quite confused.

Do you think it would sound less subversive and maybe closer to actual tuning if I presented Chas model as (thanks to Robert) a variant of ET where the octave ratio is something a little bigger than two?

Kent,

It's kind of you to invite me to Kansas City and, if this gave us all a better chance, I would consider your offer. But tell me please, is it sincere?

Tooner, thanks, I'll reply asap.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871



Yes Alfredo, the larger than theoretical, larger than typically chosen temperament octave size and consequent octaves is always whet I have understood your basic idea to be. It is my idea as well, as it is Herr Stopper's but each person has their own unique concepts.

To me, when any of these ideas was presented with ET alone, it represented only a subtle difference yet that difference has always seemed very important to those who advocate it as it does with you. Recently, I was impressed by the difference I heard between the Reyburn default stretch and the Stopper tuning from posts by Grandpianoman. Being able to compare the two tunings easily with each other, the difference proved to be more than subtle.

I have always placed fundamental importance on just how I stretch my octaves when I tune the EBVT. If I use my ETD's calculated program and the stretch it provides, even if I adjust it here and there a bit and use the device's "DOB" function to boost the high treble, the tuning just does not quite have the magic it has if I tune by direct interval. I don't get the signature "pipe organ effect" I am looking for. It doesn't sound bad and it has much of what I want but it is not the same.

I think everyone has given you the benefit of the doubt that you have discovered something that really appeals to you and you are very eager to let everyone know about it. The problem has always been in communication. Many of us, myself included, do not understand or use advanced mathematics in our thinking and reasoning. That does not mean that such a way of defining your concept is not valid. Perhaps it will ultimately be essential.

The developers of all of the ETD programs certainly did use advanced mathematics and have kept those secrets as protected information. Nevertheless, I know just from comments I have heard that they have had to struggle with each other over who had the right to use whatever findings they developed and how they came to know that information.

In your case, you have eagerly presented what you know for all to consider but very few if any could really grasp what it is that you are trying to say. That, I well understand is frustrating. I have encountered the same kind of frustration, for sure.

I have known Kent Swafford for many years, close to 25 in fact. We have always had respect for each other as piano tuners who seek excellence above and beyond the typical and ordinary. Kent is known to virtually every PTG member because he served the organization as its president and vice president and also a very long tenure as one of its examiners. That service had nothing directly to do with advancements in tuning concepts but because it was he who chose to serve PTG, the direction PTG took under his leadership has his own mark of distinction on it.

Kent is not the type of person to easily dismiss an idea. He listens. He will discuss and he will give his opinion but he also does not present himself as the final authority. He is also a very skilled pianist.

I could not imagine a better opportunity for you than what he has offered and I will make that offer even better. Kent has also expressed the desire to witness the way I tune in its best light. Presently, you have the advantage in currency exchange. The Euro is worth about $1.50. During winter, airplane flights from Europe to the USA are typically at their lowest.

I will also offer to meet with you and Kent to show you what I can do and listen to your idea as well. At the university where Kent is employed, you could also have the benefit of having skilled pianists play pianos tuned the way you advocate, the way I advocate and compare those two with standard practice. You can listen, Kent can listen and the pianists can comment on the effects of each.

It can be both a "blind" study, where the pianists do not know which piano is which and then an informed trial where they do. The feedback (our often used term for results or reaction) from the pianists would be interesting and informative in each case.

I can go even further in extending a welcome hand to you: My aunt lives in Kansas City and is always receptive to hosting guests such as for a week. If you do not have the means to pay for a hotel, you can be assured to have a place to stay at no further cost to you. I can meet you at the airport, have a car and take you to the PTG Home Office and the university on each of the planned days of your visit.

I would suggest planning a trip in February for a week. Arrive on Sunday and leave again the following Saturday or Sunday. I would imagine you could find a flight schedule for only a few hundred Euros and as far as I know, you would not need a visa, only a passport.

I would be willing to take the time off from my business for this and I am sure that Kent would make every accommodation he could as well. The only thing that would make it even more interesting would be to have Herr Stopper come too but let's not go too far.

I would mention with respect to Herr Stopper that he also encountered much negative resistance to his ideas and also had a similar problem in communication due to the difference in our languages. Kent was singly responsible for inviting Herr Stopper to the PTG convention one year and I am glad he did. Previous to that, I had not been very impressed with Herr Stopper's concepts through writing and a recorded sample alone. It took actually hearing his tuning and meeting Herr Stopper to gain a respect for him that I will always have.

So, you see, Kent is a man of his word, he has made such an offer before and he has made it again to you. So, I suggest that you take the offer as your very best opportunity ever to present your concepts. The language in your papers can be worked out so that a more general audience can understand and appreciate what you offer.

PTG has people with unique areas of expertise. Not all of the finest contributions come from RPTs either. People with writing skills like me, mathematics and editing skills like Jerry Viviano who is an Associate member and mathematics skills like Robert Scott who is also and Associate member can all possibly work together to help you come up with an English language paper which is presentable. It could be published in the PTG Journal or even be made into a book. Nothing is beyond possibility.

It would indeed be an example of where a PTG non-member, RPTs and Associates all can and do work together cooperatively with mutual respect. It does happen and is not uncommon. Advanced concepts of all kinds are what I believe should be presented at PTG conventions and those are sometimes the focus of the event. Your concepts could be presented in the future and you could have substantial numbers of people interested in them if you take the right course. Accepting Kent's and my offer would be an important step towards that.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/31/09 12:55 PM

Bill provides one good, possible scenario. We have had "tuning summits" before of course, but they usually happen at the annual PTG convention, which is not the best environment for such things due to the crowded schedule.

I would love to put together something. Of course, it is hard to say how much subsidy would be available for participants. It would depend upon the specific negotiated arrangements, who we could interest in being a sponsoring body, and I suppose, whose aunts happen to live nearby. <grin>

Let's see what we can make happen.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/31/09 05:43 PM

Thanks, Kent, I, for one, do not need any subsidy and I don't need to stay at my aunt's house but I was going under the assumption that such a trip and a hotel stay for Alfredo might be more than he could afford. I can say this: my aunt would be as thrilled to host a visitor from Italy for a week as she would me. She had been used to hosting visitors in earlier years and used to rent her upstairs boarders who were students at Rockhurst University which is just steps from her home on the 51st block of Virginia avenue. A few years ago, she hosted one of our Chapter members who could not afford the hotel stay at the convention. She was happy to do so.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/31/09 05:50 PM

There is, by the way, a good example of an American taking a work written in English by a German whose English was nearly incomprehensible. Del Fandrich saw the value in it and took it upon himself to edit and annotate Piano Tone Building.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 10/31/09 07:29 PM

Bill, Kent,

Thank you very much for your generous offer and the spirit of cooperation. I will be very happy to personally know you as also our colleagues, and one week in February would be the right time for me too. You'll receive my e-mail address, so to define all the details.

Also thanks for your warm welcome hand, Bill, it will be very nice to meet your haunt in any case...She is not a piano tuner, is she?

Have a nice Sunday, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/02/09 07:35 AM

Alfredo:

I suppose I should wait until you answer my last post, but I came up with something you ought to be interested in and is pertinent to our present discussion.

You seem to have derived your CHAS ratio (1.0594865443501…) from the ratios of 3^1/19 and 4^1/24 (which is the same ratio as 2^1/12, btw) by using the CHAS algorithm. But there is a simpler, more “elegant” way that does not pretend to use other ratios. This means that the CHAS ratio need not be considered a “combined ratio” at all.

The desire is to have an incremental (semi-tone) ratio that produces equal beating 15ths and 12ths.

If “x” is this ratio,

and the multiplier of the fundamental frequency to determine the beat rate of a 15th is:

(x^24)-4.

and the multiplier of the fundamental frequency to determine the beat rate of a 12th is:

3-(x^19)

then for equal beating 15ths and 12ths with the common note on the bottom:

(x^24)-4 = 3-(x^19)

or:

(x^24) + (x^19) = 7

and, by trial and error:

x = 1.0594865443501…

If the common note is on the top, it is a different equation and a different ratio.

This could be looked at as a “combined ratio” when your algorithm is used (which is not necessary for this purpose), but I cannot look at it that way.

I wonder if I will regret posting this. I want to avoid getting drawn into a “tar-baby” discussion about your paper.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/02/09 04:28 PM

Tooner sorry, I have been away.

You kindly wrote:...“Ok, a combined ratio is one that is calculated from two different ratios. I do not know of any intrinsic value that the ancestry of a ratio could have.”...

I’d say that the intrinsic value of a ratio is its ancestry or, a ratio can have an intrinsic value depending on its ancestry. Chas ratio returns the qualities of two algebraic magnitudes, 19 root of three and 24 root of four.

...“Let us use the Beat Analyzer as a simulator to see the results of using such a combined ratio. You would need to give me the ratio and directions in how to use it.”...

Yes, any direction you need, could you be more precise on this?

...“Also, you should predict what the results will be ahead of time, so as to have an objective evaluation.”...

Do you mean results in terms of beats?

...“I can also provide a table showing all the numerical values, for verification of the calculations and construction of the graphs.”...

This will be relevant too.

About me you say:...“You asked: “BTW, what was the range in this test?” All the graphs are for the full range of the piano, 88 notes. This is shown by the scale at the bottom of the graphs, A0-A7. The Beat Analyzer also provides mid-range (C3-C5) graphs for RBIs and SBIs but I did not include those.”...

Very good indeed. Are you using that iH constant, the one that doubles every 8 semitones? Do you think there is a way to reduce the approximations?

...“Sorry, I did not listen to your recording. When I clicked on the link a number of questionable pop-ups appeared. I did not want to take a risk by downloading the file. And I am not sure how my opinion of the tuning would effect our present discussion, anyway.”...

If you like I can always e-mail you that recording. I did it also because you suggested me to do so (05/20/09). You also wrote (06/23/09): ...“If everyone tuned the same and everyone liked the same tuning, this would indicate there is a universal optimum, and through empirical observation a model could be developed. But since this is not the case, this indicates that there is no mathematical model that will satisfy everyone.”

I prefere to think that a universal optimum does exist, although it may not be needed. You'll find a curious analogy in "Pareto (improvement)" linked below:

http://en.wikipedia.org/wiki/Pareto_efficiency

Then you said that ET is a region, and I think I understood what you meant, but in another reasonable way Chas ET EB is a precise location, and that recording is meant to be the first demonstration. I would also like to demonstrate that small-pianos iH does not impede the finding of Chas ET EB form.

Now you say:...“You seem to have derived your CHAS ratio (1.0594865443501…) from the ratios of 3^1/19 and 4^1/24 (which is the same ratio as 2^1/12, btw) by using the CHAS algorithm. But there is a simpler, more “elegant” way that does not pretend to use other ratios. This means that the CHAS ratio need not be considered a “combined ratio” at all.”...

No, it does not need to, but this is what it can be said wanting to use descriptive terms. Numbers themselves do not call for any kind of consideration, yet we can give attributes to them. You say “...that does not pretend to use other ratios”.... Then you would have to explain what 3, 19, 4, 24 and 7 are. Btw, about 7, this is were the gem sparkles.

You say:...“The desire is to have an incremental (semi-tone) ratio that produces equal beating 15ths and 12ths.

If “x” is this ratio,

and the multiplier of the fundamental frequency to determine the beat rate of a 15th is:

(x^24)-4.

and the multiplier of the fundamental frequency to determine the beat rate of a 12th is:

3-(x^19)

then for equal beating 15ths and 12ths with the common note on the bottom:

(x^24)-4 = 3-(x^19)

or:

(x^24) + (x^19) = 7

and, by trial and error:

x = 1.0594865443501…

If the common note is on the top, it is a different equation and a different ratio.
This could be looked at as a “combined ratio” when your algorithm is used (which is not necessary for this purpose), but I cannot look at it that way.”...

You can look at it the way you prefere, as long as we can share it as much as possible. As for simplicity, I was quite impressed by ROMagister when, commenting Chas model, he could explain it in (about) ten lines, five months ago.

ROMagister(05/29/09): “...it IS Equal Temperament, but with another ratio: not the classic one where 12 semitones = 1 octave of exactly 2:1 (Pythagorean octave still accepted as axiom in classical ET).
The basic version (s=1) makes an equal compromise between the 'justness' of 3rd and 4th harmonics (octave+fifth vs 2 octaves). "s" is just the compromise parameter which says how important is the error in the 3rd harmonic compared to the error in the 4th harmonic. It can be set "politically" as we want, and the Delta results as a solution of the (implied) equation, also the practical frequency ratio that results.”

You say:...“I wonder if I will regret posting this. I want to avoid getting drawn into a “tar-baby” discussion about your paper.”...

Do not worry, as I told you it’s up to you. You had written (10/28/09): ...“As far as discussing the CHAS algorithm or model, sorry, NO!”

You see, If you get steady, it’ll be easyer for me too. Now you talk about Chas algorithm and elegance, and yet you do not want to discuss it. Anyway, I prefered a delta-difference in Chas equation so that I am always able to recall beats and refer to my/our tuning practice.

Nevertheless I find (x^24) + (x^19) = 7 very much appealing. A "7th day" is when I was born.

Oh, in all this you are the oxygen, thanks.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/03/09 07:31 AM

Alfredo:

Interesting about Pareto efficiency. I believe in any human exchange, if it is not to the benefit of both, it is not to the benefit of either.

Let me continue to try to steer clear of your paper for the benefit of both of us.

Originally Posted By: alfredo capurso
...

...“Let us use the Beat Analyzer as a simulator to see the results of using such a combined ratio. You would need to give me the ratio and directions in how to use it.”...

Yes, any direction you need, could you be more precise on this?

...“Also, you should predict what the results will be ahead of time, so as to have an objective evaluation.”...

Do you mean results in terms of beats?

...“I can also provide a table showing all the numerical values, for verification of the calculations and construction of the graphs.”...

This will be relevant too.

About me you say:...“You asked: “BTW, what was the range in this test?” All the graphs are for the full range of the piano, 88 notes. This is shown by the scale at the bottom of the graphs, A0-A7. The Beat Analyzer also provides mid-range (C3-C5) graphs for RBIs and SBIs but I did not include those.”...

Very good indeed. Are you using that iH constant, the one that doubles every 8 semitones? Do you think there is a way to reduce the approximations?

...


The iH curve was shown on the last graph. It closely approximates the iH curve for a well scaled studio upright, and I suggest we use this for the simulation. Instead, I could use actual observed iH values, but since there are little deviations in the curve (perhaps due to instrumentation errors) for a simulation I think the computer generated curve would be better. But a curve for a smaller or larger piano could be chosen instead. The table that I will include will list the iH for each note. Yes, if you predict what the beat rates will do before seeing the graphs, this would be an objective comparison.

Yes, I need some direction from you, I guess to be your lab assistant. After all, it would do no good for you to predict the results for a different simulation than the one being conducted. Do you want me to use the CHAS ratio of 1.0594865443501…, or some other ratio? Do you want the same ratio used for all notes? (I ask this because if the semi-tone ratio is constant, the Railsback curve will be a straight line.) Is the same iH curve that was used for what I called “ChasTuning” satisfactory, or would you prefer a different iH curve, or no iH at all? What would you like to call the tuning results? May I make a suggestion? Since you are striving for “straightened” 12sth and 15ths how about a Straight we are both familiar with? “The Straight of Messina”

Oh, and the most important thing for any lab assistant to know, how do you take your coffee?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/03/09 03:47 PM

Bill, I wrote you and Kent an Email Post (I could not remember how to send a PM).

Did you get it?

Regards a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/04/09 01:34 PM

Tooner,

You kindly say:...“The iH curve was shown on the last graph. It closely approximates the iH curve for a well scaled studio upright, and I suggest we use this for the simulation."...

I agree, for a simulation the computer generated curve works fine.

...“But a curve for a smaller or larger piano could be chosen instead.”...

I suggest to try to refine one simulation at the time, but also the curve for a smaller or larger piano may be interesting eventually. Any idea about how to reduce approximations?

...“The table that I will include will list the iH for each note. Yes, if you predict what the beat rates will do before seeing the graphs, this would be an objective comparison.”...

Ok, I agree. The only thing is that I normally fix 4ths 5ths and octaves from C3 to C6 in my preparatory tuning (using RBIs too), and after unisons I check RBIs and 12ths and 15ths. Nevertheless I’ll give you the beat rates and I’ll try to be as precise as possible.

...“Do you want me to use the CHAS ratio of 1.0594865443501…, or some other ratio?”...

Let’s use Chas ratio first, then we’ll see.

...“Do you want the same ratio used for all notes?”...

Yes, please.

...“Is the same iH curve that was used for what I called “ChasTuning” satisfactory, or would you prefer a different iH curve, or no iH at all?”...

Yes, that curve was ok.

...“What would you like to call the tuning results?”...

Chas ET EB simulation of...on...with...using.... Thinking of Messina we should call it Bridge... all the administrations here talk about it when they want to poll more votes.

...“Oh, and the most important thing for any lab assistant to know, how do you take your coffee?”

Possibly, in good company.

Thanks Tooner and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
_________________________
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/04/09 02:30 PM

Alfredo:

I have the graphs and values ready to publish when you predict the beatrates. The values are on an embedded worksheet. I hope you have Excel on your computer. If not, I can add some pages with the values printed out. Let me know.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/05/09 07:28 AM

Alfredo:

You should also predict the beatrate of the 12ths and 15ths.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/08/09 12:15 PM

Tooner,

I'm waiting to tune a reliable piano, so that the beat rates I'll talk about may be reliable too. You'll be given the beat rates relative to the preparatory tuning, i.e. from C3 to C6 on mid-strings only, separate from the after-unisons beat rates.

I had asked you: Are you using that iH coefficient that doubles every 8 semitones? Do you think there is a way to reduce the approximations?

About your own maths work, I think you are now very close to the Chas partials-combination key. May I supply one more hint?

You kindly wrote:

The desire is to have an incremental (semi-tone) ratio that produces equal beating 15ths and 12ths.

If “x” is this ratio,

and the multiplier of the fundamental frequency to determine the beat rate of a 15th is:

(x^24)-4.

and the multiplier of the fundamental frequency to determine the beat rate of a 12th is:

3-(x^19)

then for equal beating 15ths and 12ths with the common note on the bottom:

(x^24)-4 = 3-(x^19)

Let's stop here. Now that you can find a value for "x", you can substitute that value and see what you get. I'm asking you to do it yourself only because I think that this may help other colleagues too. Would you be willing to put your figures processing in words?

Regards, a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/09/09 07:53 AM

Alfredo:

You wrote: “I'm waiting to tune a reliable piano, so that the beat rates I'll talk about may be reliable too. You'll be given the beat rates relative to the preparatory tuning, i.e. from C3 to C6 on mid-strings only, separate from the after-unisons beat rates.”

You may wait if you wish, but I do not see a real reason to. You have tuned enough pianos that I thought you would know about what the beatrates will be. What I am trying to do is what I mentioned on Oct 26, 2009. “We seem to be covering old ground (going in circles) in the discussion of frequency ratios, partial matches and inharmonicity.” I am hoping that comparing the simulation with your experience will shed light on this.

And you wrote: “I had asked you: Are you using that iH coefficient that doubles every 8 semitones? Do you think there is a way to reduce the approximations?”

No, the iH curve that I chose does not double every 8 semitones. The curve was smoothed by me from actual measurements of a well scaled studio upright and that was provided to me by another. The measurements given me were the actual frequencies of partials that I then calculated an iH curve from, which then was smoothed. For what I am hoping to accomplish, this seemed a good way. Since we are not dealing with the same piano in any case, reducing approximations could do more harm than good.

Here is a link to a website with many iH curves that you might find interesting: http://www.goptools.com/gallery.htm

These curves are from measured string data and not calculated from frequency measurements.

And you also wrote: “About your own maths work, I think you are now very close to the Chas partials-combination key. May I supply one more hint?

<SNIP>

(x^24)-4 = 3-(x^19)

Let's stop here. Now that you can find a value for "x", you can substitute that value and see what you get. I'm asking you to do it yourself only because I think that this may help other colleagues too. Would you be willing to put your figures processing in words?”

I would rather continue to the next step that I mentioned before and put all the terms containing x on one side and all the terms that do not on the other side. The equation is then simplified by combining like terms. Finally x is solved. That is how algebra is performed. I am not looking for things that do not really exist!

The minor point that I was trying to make is that the idea of this combined ratio is a mental fabrication.

The major point that I am trying to make is the relationship between frequency ratios, partial matches and inharmonicity. I have been trying to make this point for quite some time, and am hoping that a simulation will help.

But in an attempt to give you something philosophical to nibble on: Why is it preferred to have equal beating 12ths and 15ths with a common note on the bottom rather than the top? Or would it be better for these two sets of intervals to be non-equally beating by the same amount? But wouldn’t the non-equality need to be defined not linearly, but logarithmically? But the problem with all this is when inharmonicity comes into play, we find that the “earth is round.” A straight line on a nautical chart is rarely the shortest distance between two points on the earth. And a frequency ratio will not give the expected results when inharmonicity is included.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/10/09 05:07 PM

Tooner,

You write:...“You may wait if you wish, but I do not see a real reason to.”...

Strange, I think I’ve told you the reason. I normally check and fix the intervals beat rate progressions during my preparatory tuning, i.e. while I tune mid-strings only. These beat rates need to be different from the final tuning form’s beat rates, i.e. from the after-unisons intervals beat rates. More precisely, in my preparatory tuning, wide intervals need to be wider and 5ths need to invert their beat-rates progression.

So, I have already been able to tell you and all colleagues what my preparatory-tuning intervals beat rates are, and when it came to my practical final tuning form I could confirm exactly what Chas ET-EB model predicts, i.e. Chas theoretical ET EB constants, i.e. equal beating 12ths (narrow) and 15ths (wide) with all the other RBI check-intervals being progressive.

Now, if you are interested in my final tuning form’s beat rates in precise terms, you only need to wait for something I’ve never measured before. For me, tempering is done on mid-strings all along, while working on unisons results more into voicing. But is that what you are asking for? I may have misunderstood.

Still today, my impression is that you are not happy with Chas algorithm, a fairly simple equation that makes an equal compromise between the “justness” of the 3rd and the 4th partials (octave+fifth Vs double octaves – ROMagister), and still today I wonder how you could get by with 12th root of two.

You said:...“I am hoping that comparing the simulation with your experience will shed light on this.”...

In my opinion our goals may diverge, but in saying this I’m not even sure about your goals. What I understand is that the two of us move with quite different approaches, different experiences and different beliefs.

For istance, you say that any piano tuning theory’s usefulness is limited in aural tuning, I do not. In my opinion, a correct and reliable temperament theory will address aural tuners towards a practicable and euphonic model, and I must say that 12 root of two ET needed to be improved. Today Chas ET-EB model can well represent an improved variant of ET (thanks Robert), a stretched-octave ET-EB variant that is in line with what (experienced) aural tuners may already be doing in their practice or be looking for. Not to mention that a temperament theory is not only directed towards pianos, but towards all keyboard instruments and more generally towards all musical instruments.

You say that we need to stretch octaves because of iH, I do not. I think that in aural tuning - i.e. when it comes to beats - iH is neutral, and I think we need to stretch octaves simply because the fairest compromise is to be found between two roots, 19 root of 3 and 24 root of 4; as a consequence, ET theoretical octaves should not and will not be in 2:1 ratio.

You want iH to come into play, I do not. Chas theoretical model, being a temperament model, does not consider whether you are tempering iH tones or what. Actually, I’m trying to share Chas theory because this model can describe our actual tuning, and because its constants agree with my practical results.

You kindly answer:...“No, the iH curve that I chose does not double every 8 semitones. The curve was smoothed by me from actual measurements of a well scaled studio upright and that was provided to me by another. The measurements given me were the actual frequencies of partials that I then calculated an iH curve from, which then was smoothed. For what I am hoping to accomplish, this seemed a good way. Since we are not dealing with the same piano in any case, reducing approximations could do more harm than good.”...

Maybe I did not get what you are hoping to accomplish. Could you also tell me more about that well scaled studio upright? Was it a real piano? Which temperament was used and how? What standard did you smooth the curve by? Do you know the approximation degrees, just to have an idea? In my tuning experience, reducing approximations has been and still is the real challenge.

About your maths work, you say:...“The minor point that I was trying to make is that the idea of this combined ratio is a mental fabrication.”...

For me this is disappointing, firstly because I’ve said that “combined ratio” is used as a descriptive mean, secondly because I was expecting you to understand the difference between using a one-ratio formula, like 12 root of two, and using a two-ratios equation. Differently also from the latest ratios, Chas ET incremental ratio is a new logarithmic average between two ET ratios.

You say:...“The major point that I am trying to make is the relationship between frequency ratios, partial matches and inharmonicity. I have been trying to make this point for quite some time, and am hoping that a simulation will help.”...

Yes, I understand that your interest is on the relationship between frequency ratios, partial matches and inharmonicity, then not only our goals may diverge, but also our methods. In fact, if I seriously wanted to make a comparison I would not use a somehow-approx-simulation, I would go for the real thing.

You say:...“But the problem with all this is when inharmonicity comes into play, we find that the “earth is round.”...

I can not understand how you understand logarithms. With 12 root of two ET the earth is flattened on every zero-beating octave. With Chas ET-EB the earth is rounded octave after octave.

Should I understand that you would like to find a way to improve ETDs? You see, I’m too keen on aural tuning and I do not see the point unless we try to do that with the least possible approximations.

About “something philosophical to nibble on”, I thank you and I’ll reply soon.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/11/09 08:01 AM

Alfredo:

You wrote: “So, I have already been able to tell you and all colleagues what my preparatory-tuning intervals beat rates are, and when it came to my practical final tuning form I could confirm exactly what Chas ET-EB model predicts, i.e. Chas theoretical ET EB constants, i.e. equal beating 12ths (narrow) and 15ths (wide) with all the other RBI check-intervals being progressive.”

Okay, I guess I really just wanted to confirm this. Here is a link to the data. I included an embedded spreadsheet in addition to hard copy tables and graphs. I will let you have first say about the results of using the CHAS incremental ratio on a piano with iH. I am not trying to evade your other questions. These can be dealt with. But I am hoping that you will see what a moot point much of what we try to discuss is when including iH into tuning theory.

BridgeTuning
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/12/09 12:24 PM

Thanks very much Tooner for your elaborations.

You kindly say:...“I will let you have first say about the results of using the CHAS incremental ratio on a piano with iH.”...“But I am hoping that you will see what a moot point much of what we try to discuss is when including iH into tuning theory.”...

What can be said about your generous elaborations? I can say that Bridge tuning shows not only the results of using the CHAS incremental ratio on a piano with iH, but more generally the effects of iH on beat rates, if we were to use any set of theoretical frequencies values on a piano.

We may as well consider one evidence: 12th root of 2 predicts zero-beating octaves, 7th root of 3/2 predicts zero-beating 5ths, 19th root of 3 predicts zero-beating 12ths. Then I may ask you: when including iH, is it of value to theorize a zero-beating ET incremental ratio? In other words, taking your latest analisys to extremes, can an infinitesimal degree of iH agree with ET theoretical zero-beating choromatic intervals?

I do not think it is including iH into tuning theory that is making much of what we try to discuss a moot point, but a kind of deafness (what a nightmare), in my opinion that kind of deafness deriving from different ways of looking at the same issue, in our case being the way we look at piano tuning and temperament theory.

Four points, in my opinion, are mainly causing this phenomenon:

The relevance of iH in piano tuning
The relevance of tuning theory in piano tuning
The relevance of Chas theory as an improved ET-EB temperament variant
The relavence of zero beating octaves and more generally of zero beating intervals

It was June when we long discussed about iH, when you kindly posted some calculations (06/11/09) that again could prove to all readers how iH effects the actual frequencies values of partials, and how the actual frequencies will be different from theoretical values. That calculation could well prove that theoretical frequencies values and the relative theoretical beat rates will not correspond in practice, due to iH. That is to say that, theoretical beat rates can be fixed in practice, but the actual frequencies will differ from theoretical frequencies values.

Then iH was not a moot point, though our goals were already diverging, you wanting to prove the limited usefulness of any tuning theory in aural tuning, me wanting to say that iH effects could be re-calculated, so to reduce approximations. None of us could doubt about iH’s effects on actual frequencies values. Both of us had good reasons for going back to iH calculations: (posted in 06/16/09) ” Certainly you will have read where our cello-expert writes: “…the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.” What do you think he meant, saying “…proper consideration of inharmonicity”?”
You answered: “I am not certain what Mr. Stopper means. It may be similar to what I call “the largely self-correcting effects of iH on beat rates”.”

Where to go then to oxygenize this discussion? This is my proposal:

a)Let’s distinguish the general meaning and relevance of a temperament theory, what Chas ET-EB is, from what piano tuning’s issues are

b)Let’s analyse what logic is behind each one considered theory

c)Let’s separate iH issues from beats-control and tuning-form issues

d)Let’s see if we can address aural piano tuners towards a more reliable and practicable model

e)Let’s try to clear up how SBI and RBI intervals should go, and get rid of all misteries

f)Let’s evaluate if today there is a way to reduce approximations relative to iH and piano scaling

In your previous elaboration,

http://www.box.net/shared/rxb631v2yz

when you added iH to Chas theoretical frequencies values, you/we all could see how 12ths (narrow) and 15ths (wide) beat rates can be straighten in terms of opposite equal beating. In my opinion, there you find Chas theory’s usefulness in aural tuning, without forgeting that yes - due to iH - actual frequencies are different from theoretical values.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/12/09 12:35 PM

Alfredo:

Let's first finish what is going on now.

Please answer YES or NO.

Is an incremental (semi-tone) ratio useful for predicting the beat rates of intervals of inharmonic tones?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/12/09 12:54 PM

I had just written:

In your previous elaboration,

http://www.box.net/shared/rxb631v2yz

when you added iH to Chas theoretical frequencies values, you/we all could see how 12ths (narrow) and 15ths (wide) beat rates can be straighten in terms of opposite equal beating.

Tooner, you ask: "Is an incremental (semi-tone) ratio useful for predicting the beat rates of intervals of inharmonic tones?

The answer is: Yes, if you add iH to the theoretical values. No, unless you add iH to the theoretical values. a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/12/09 01:45 PM

Alfredo:

You posted: "when you added iH to Chas theoretical frequencies values, you/we all could see how 12ths (narrow) and 15ths (wide) beat rates can be straighten in terms of opposite equal beating."

No, I did not add (a better term may be apply) iH to "Chas theoretical frequencies" in the tuning that showed equal beating 12ths and 15ths. I determined through trial and error what frequencies are required on a simulated piano with iH so that 12ths and 15ths would beat equally.

However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally.

This is why I see no usefulness in frequency ratios for predicting beatrates. The desired beatrate is determined, and then the frequencies are calculated. Finally, if wanted, the frequency ratios can be ascertained. But they are a byproduct, not used in the calculations.

Perhaps you misspoke.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/12/09 04:38 PM

Tooner,

you say: "No, I did not add (a better term may be apply) iH to "Chas theoretical frequencies" in the tuning that showed equal beating 12ths and 15ths. I determined through trial and error what frequencies are required on a simulated piano with iH so that 12ths and 15ths would beat equally."...

Sorry, I misunderstood.

..."However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally."...

And what happened? Could you tell me more?

...This is why I see no usefulness in frequency ratios for predicting beatrates....

When I wrote (06/04/09): ”There is a fenomenon that I do not really understand, how is it possible to take a lame theory inside and out, one minute referring to it and the minute after negating it. Now theoretical wrong value from traditional ET have a sense, the minute after they do not.”

Your answered:..."Because if we take the beat rates (or at least the ratio between beat rates, including equal beating) that are predicted from a frequency ratio (such as 2^1/12) that does not take into account iH, and then tune a piano with iH using the beat rates we end up with a decent tuning, but a different frequency ratio, one that is non-linear. So on the one hand, the frequency ratio is wrong, but on the other, the beat rates are correct. And since when tuning aurally, we listen to beat rates, the model works even though it is incorrect."...

So, you try to agree with yourself.

Now you say:..."The desired beatrate is determined, and then the frequencies are calculated. Finally, if wanted, the frequency ratios can be ascertained. But they are a byproduct, not used in the calculations."

This is what I would do in practice too, I'd tune Chas form and then I would ascertain the frequencies values and ratios.

Granpianoman,

many thanks for converting the .rar file in an MP3 file, for using your site and for thinking that other people may prefere this or want a quicker way to hear the file.

Regards, a.c.

First recording (.rar) of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 http://www.box.net/shared/od0d7506cv
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/13/09 07:32 AM

Alfredo:

We may come to a mutual understanding yet!

You posted: “"However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally."...

And what happened? Could you tell me more?


The graphs show what happened. Even though the incremental ratio was greater than 2^1/12, it was not enough to make even the 2:1 partial matches beat wide, let alone the 4:1 partial matches to even be just. Equal beating 12ths and 15th did not happen.

And you posted: “Now you say:..."The desired beatrate is determined, and then the frequencies are calculated. Finally, if wanted, the frequency ratios can be ascertained. But they are a byproduct, not used in the calculations."

This is what I would do in practice too, I'd tune Chas form and then I would ascertain the frequencies values and ratios.”


Now you really have my interest. How did you ascertain the frequency values and ratios? If it was by using the CHAS algorithm, then this shows that the algorithm is inadequate. We have already looked at the results.

As was shown above, the CHAS theoretical frequencies, when iH is applied, do not have the expected results. In fact, even without including iH there is a problem. The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate.

I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless. I believe the simulation is accurate and this is also similar to what I hear when tuning this way. On the average the beatrate is less than 1 bps.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/13/09 09:52 AM

Alfredo:

My apologies. I miss read ”This is what I would do in practice” as ”This is what I did in practice”.

But you still have my interest! So what would you then do with the frequencies and the ratios?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/13/09 05:32 PM

Tooner,

I hope you’ll still play the Devil’s Advocate, despite our mutual understanding.

We wrote:...“However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally....

And what happened? Could you tell me more?

The graphs show what happened. Even though the incremental ratio was greater than 2^1/12, it was not enough to make even the 2:1 partial matches beat wide, let alone the 4:1 partial matches to even be just. Equal beating 12ths and 15th did not happen.”...

Going back to
http://www.box.net/shared/rxb631v2yz

I could again confirm that your ET-EB simulation do make the 2:1 partial matches beat wide. True?

Then you say “Equal beating 12ths and 15th did not happen”, but you have managed to straighten 12ths and 15ths. In my opinion, to get EB 12ths and 15ths you/we may have to adjust iH.

You say:...“The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate. I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless.”...

What you are saying may well be possible. I can not be 1000 % sure about choromatic equal beating, although I trust my sense of rhythm.

...“I believe the simulation is accurate”...

I think it is accurate only to some extent. Not because of you though.

...“and this is also similar to what I hear when tuning this way.”...

You prove to be a very good Advocate, and surely you are a very good tuner.

...“On the average the beatrate is less than 1 bps.”...

Yes, I'd say between 1/2.5 and 1/3 bps.

...“But you still have my interest! So what would you then do with the frequencies and the ratios?”

I would pass you those values and invite you to point f).

Regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/16/09 10:19 AM

Alfredo:

Have no fear. Even with a mutual understanding on the relationship of frequency ratios, beat rates and inharmonicity there is plenty that we can disagree on.

The next step is to discuss the nature of iH itself. But first, since you and I have different native tongues, allow me to summarize the mutual understanding:

Frequency ratios are useful in predicting beat rates of intervals made from harmonic tones, but are not useful in predicting beat rates of intervals made from inharmonic tones.

Before we continue on the nature of inharmonicity, perhaps you could acknowledge this mutual understanding, just to be sure.

Oh, and if you choose to give me the frequencies of a tuning that produce a certain set of beat rates, please include the inharmonicity so that all other beat rates can be calculated. If the iH is not included, then it would be best to send me the frequencies on very soft paper, so that I could find some use for them
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/17/09 06:00 PM

Tooner,

You let me know about your preferencies in terms of paper and that’s ok, I understand that as being pretty original for you.

Then you say:...“The next step is to discuss the nature of iH itself.”...

For me, the next step is going back to what you/we have already stated. In fact, lots has already been written about the nature of iH and, above all, I would like to pursue my actual aim, I would like to share Chas model as a variant of ET where the octave ratio is something a little bigger than two, in the way we are actually tuning.

You wrote (11/11/09):...“I am not trying to evade your other questions.”...

So, the questions I posted on the 10th and the 12th of November are, in my opinion, quite crucial and may well represent the central issues of this discussion, better than us having different native tongues.

You say:...“Frequency ratios are useful in predicting beat rates of intervals made from harmonic tones, but are not useful in predicting beat rates of intervals made from inharmonic tones. Before we continue on the nature of inharmonicity, perhaps you could acknowledge this mutual understanding, just to be sure.”...

Instead, I think you should explain what you meant when you wrote (06/04/09):..."Because if we take the beat rates (or at least the ratio between beat rates, including equal beating) that are predicted from a frequency ratio (such as 2^1/12) that does not take into account iH, and then tune a piano with iH using the beat rates we end up with a decent tuning, but a different frequency ratio, one that is non-linear. So on the one hand, the frequency ratio is wrong, but on the other, the beat rates are correct. And since when tuning aurally, we listen to beat rates, the model works even though it is incorrect."...

And what you mean when you write (11/13/09)...“The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate.”...

Now, if frequency ratios are not useful in predicting beat rates of intervals made from inharmonic tones, why do you raise the question for Chas model? And again I ask you: how could you use 12th root of two?

You say:...“I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless.”...

Although I do not know whether you are talking about 12th root of two or what you are referring your "truth" to, I can confirm that, in my tuning form, 12ths and 15ths are opposite equal beating all along the scale. But maybe you have one more model in mind.

I asked you: “Could you also tell me more about that well scaled studio upright? Was it a real piano? Which temperament was used and how? What standard did you smooth the curve by? Do you know the approximation degrees, just to have an idea?

And I would now add a very simple question too: do you realize that, while 12th root of two ratio is a compromise between 3ds and fifths, Chas ratio is a compromise between 3ds, 5ths and octaves?

You asked for a Chas tuning example. Have you heard Granpianoman's MP3 conversion? Then we can talk about iH's effects on small pianos.

Thanks and regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/18/09 10:17 AM

Alfredo:

Even though it was believed (and apparently still is by you) that octaves on a piano are tuned (aurally) by frequency ratios, this is not true. They are tuned by partial matches. So I just cannot contribute to a discussion about tuning with frequency ratios. I cannot pretend “the earth is flat”.

Ok, on 10 Nov you asked: ”Could you also tell me more about that well scaled studio upright? Was it a real piano? Which temperament was used and how? What standard did you smooth the curve by? Do you know the approximation degrees, just to have an idea?

The model piano is a Charles Walter upright. The length of A0 is 48 inches. A file was generously provided to me that included the frequencies of the partials of an actual tuning. Given the frequencies and the partial numbers, I was able to calculate the inharmonicity for each note, which is affected very little by the actual tuning. Unfortunately, the file included only one partial frequency for the top octave; it takes the frequencies of two partials to calculate iH. Nonlinear extrapolation was used to estimate these iH values. It really does not matter much, since only the first partial is usable in the top octave.

The resulting curve was “V” shaped on a logarithmic graph with the left arm shorter than the right. Also the left arm, being wound strings, was “squiggly”. So by taking the value of iH for note 1, note 88 and the value of the lowest iH with its note number; an idealized “V” could be constructed using logarithmic interpolation. But this would produce an uncharacteristic sharp point to the “V”. By using a computer subroutine, this sharp point was rounded off by using increasing fractions of the slope for the eight notes centered on the point of the “V”.

I am not sure how to explain the approximation degrees in terms that would be valuable. But the purpose of the simulation was to show the general effect of iH on beat rates, and the approximations made this clearer than raw values would. Any piano’s iH values would have shown the same general effect.

The temperament was as equal as I thought practical. I had to start with a slightly wide 15th, calculate what beat rate this would produce in a 12th, and adjust back and forth. When the 3rds and 6ths were progressive, I decided that this was good enough for the simulation.

And on 12 Nov you asked: ” We may as well consider one evidence: 12th root of 2 predicts zero-beating octaves, 7th root of 3/2 predicts zero-beating 5ths, 19th root of 3 predicts zero-beating 12ths. Then I may ask you: when including iH, is it of value to theorize a zero-beating ET incremental ratio? In other words, taking your latest analisys to extremes, can an infinitesimal degree of iH agree with ET theoretical zero-beating choromatic intervals?”

“Then I may ask you: when including iH, is it of value to theorize a zero-beating ET incremental ratio?” My answer is: Yes there is no value. (“Yes, we have no bananas!”) When tones are inharmonic, octaves cannot be zero-beating. If one set of coincident partials are at the same frequency, none of the others will be. At some level the octave always beats, although it may not be noticeable. This is the case regardless of the accuracy of the iH.

Now, I have defined a limit to what I am able to discuss and have answered your outstanding questions. I am going to stop here. I am hoping that you will make shorter posts and try to deal with single subjects.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/18/09 05:34 PM

Tooner, thanks for your answer.

You say:...“Even though it was believed (and apparently still is by you) that octaves on a piano are tuned (aurally) by frequency ratios, this is not true. They are tuned by partial matches.”...

I aurally tune octaves, as well as all other intervals, by beats, like all aural tuners. Beats result from partial matches, so I can not disagree with you.

You write:...“So I just cannot contribute to a discussion about tuning with frequency ratios. I cannot pretend “the earth is flat”.”...

I’m trying to share Chas ET-EB theory because it is an improved ET model, in fact it is the model that can finally compromise all intervals, octaves included, into a beating-whole. And I could tell you about the relevance of a general temperament/tuning theory (11/10/09): “...a correct and reliable temperament theory will address aural tuners towards a practicable and euphonic model...”

Thanks for telling me more about your simulation. From what you say, I understand that there might be a chance to reduce approximations. Anyway, you've done a great job.

You say:...“I am not sure how to explain the approximation degrees in terms that would be valuable. But the purpose of the simulation was to show the general effect of iH on beat rates, and the approximations made this clearer than raw values would. Any piano’s iH values would have shown the same general effect.”...

I agree.

...“The temperament was as equal as I thought practical. I had to start with a slightly wide 15th, calculate what beat rate this would produce in a 12th, and adjust back and forth. When the 3rds and 6ths were progressive, I decided that this was good enough for the simulation.”...

I hope one day we’ll be able to work directly on Chas tuning, that day we’ll also make sure that 4ths, 5ths and octaves are progressive.

I asked you: “...when including iH, is it of value to theorize a zero-beating ET incremental ratio? In other words, taking your latest analisys to extremes, can an infinitesimal degree of iH agree with ET theoretical zero-beating choromatic intervals?”

You answer: “Yes there is no value. (“Yes, we have no bananas!”) When tones are inharmonic, octaves cannot be zero-beating. If one set of coincident partials are at the same frequency, none of the others will be. At some level the octave always beats, although it may not be noticeable. This is the case regardless of the accuracy of the iH.”...

Then you may agree on one issue (I’m asking you): an infinitesimal degree of iH makes any zero-beating theory no value.

...“I am hoping that you will make shorter posts and try to deal with single subjects.”

I’ll shorten. T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/19/09 07:25 AM

Alfredo:

You posted: “I’m trying to share Chas ET-EB theory because it is an improved ET model, in fact it is the model that can finally compromise all intervals, octaves included, into a beating-whole. And I could tell you about the relevance of a general temperament/tuning theory (11/10/09): “...a correct and reliable temperament theory will address aural tuners towards a practicable and euphonic model...”

That’s odd. You used the word “compromise.” I remember you taking exception when I described tuning as being about compromises…

You also posted: “Then you may agree on one issue (I’m asking you): an infinitesimal degree of iH makes any zero-beating theory no value.”

Yes, and iH makes other theories invalid (of no value), also. But the word infinitesimal can mean immeasurably or incalculably small, so there is a point when iH can be so small that its effect is negligible. I don’t believe that it is this small on any string of any piano and maybe not on other string instruments either. (There is no doubt in my mind that iH effects the tuning of guitars.)
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/19/09 05:32 PM

Tooner and all Colleagues,

Often we read about having to compromise in tuning. I will treasure a concept, recently expressed by an American colleague I hold in high esteem. He makes a distinction between a “compromise” - what for me means “to make the best of a bad job” - and a “superior compromise”, what for all of us may represent an “optimum”. Today I get the chance to briefly write on this, the aim being to show how 12th root of two ET model results as a compromise, while Chas EB variant of ET can represent a superior compromise.

Most of you will know about the commas conflict. Also our practical experience confirms that if we tuned a pure interval all along the scale, this would be to all other intervals detriment and here is where the bottom problem lays.

In tuning - iH tones or non-iH tones - three contiguous pure 3rds will produce a narrow octave; in the opposite way, pure 5ths produce a wide octave. We all can experience the conflict amongst octaves, 3rds and 5ths.

Then, to get a zero-beating octave we have stretched – in theory and in practice - wide 3rds and narrow 5ths. In fact, this is what 12th root of two was meant for: this theoretical model stretches 3rds (wide) and 5ths (narrow) so to get a theoretical 2:1 pure octave. The compromise is then made between two intervals, 3rds and 5ths (considering 4ths as mirror-like 5ths).

Now, say that the octave module is a shelter, and that three 3rds are the three vehicles we room under our shelter. If we wanted to room three longer vehicles, shouldn’t we lengthen our shelter first?

So, considering a single 3rd as the octave’s sub-module, the question may be: Since three contiguous 3rds make an octave and we need to stretch 3rds, shouldn’t we stretch the octave?

As I say, this is where the bottom problem lays, the conflict that theoretical 12th root of two does not resolve in its entirety. This pure-octave ET model uses one single root, i.e. the “root of two”, so making a compromise between 3rds and 5ths, but crushes our choromatic stretched 3rds in an arbitrary 2:1 zero-beating octave. This theoretical, unpracticable and arbitrary constant (zero-beating octaves) has left tuners without a fair scale’s ratio and, above all, without reference. How could we ever go back home without a reference?

Chas ET-EB model, by using the root of 3 and the root of 4, manages to stretch 3rds (wide) and 5ths (narrow) by stretching octaves. Actually, what really happens is that 3rds, 5ths, octaves and all intervals stretch each other in a multiple function. No interval and no ratio are hold dearest, all intervals “compromise” in their own favor and in favor of a sound beating-whole. Our tuning form’s reference can now be double: 12ths and 15ths opposite equal beating in what Chas can describe as the practicable, optimized ET.

T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/20/09 08:16 AM

Alfredo:

Very nicely written. Your thoughts came through in an orderly, understandable way. Thank you.

The basis of the 12ths root of two is the idea that this will produce a temperament with all intervals being the same width (have the same frequency ratio), and at the same time octaves that are beatless. This is only possible with harmonic tones.

Keeping the discussion to harmonic tones (for now), the 12th root of any number will produce a temperament with all intervals being the same width. However, unless the number is two, the octaves will beat.

Now to have 12ths beat narrowly and at the same time 15ths beat widely the number has to be larger than 2 but smaller than (3^(1/19))^12 or 2.0014269… The compromise that is given results in equal beating 12ths and 15ths when these intervals have a common note on the bottom.

But it is not clear why this compromise is necessary at all, let alone why a superior compromise results with this sort of equal beating. Not to mention how the roots of any other numbers are needed to calculate this compromise. In fact, they are not needed nor actually used although it could seem that way.

The argument could easily be made that the common note should be on the top, or the 15th should beat faster than the 12ths that has a common note on the bottom, but slower than the 12ths than has a common note on the top. Another argument could be made that if anything should be equal beating, it should be the single octaves beating the same as the 5ths. But then the question again arises as to why, which note should be common, or should they actually beat equally?

Things get difficult when trying to use the 12th root of any number to describe the tuning of inharmonic tones. But rather than go into the difficulties, let’s look at what actually happens when a piano, with inharmonic tones, is tuned.

Oddly enough the tendency when tuning beatless sounding octaves is that the effects of inharmonicity produce narrowly beating 12ths and widely beating 15ths throughout much of the scale. The tuning can be adjusted so that these intervals beat equally in any or all parts of the piano, or unequally in any or all parts of the piano.

This is the true value of these intervals. They are a tool that the tuner can use to make compromises that are more important than arbitrarily equal beating intervals. They can be used to make and define compromises between melodic, harmonic and musical priorities in the tuning.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/21/09 02:24 PM

Tooner, this is not very short, I must apologize.

...“The basis of the 12ths root of two is the idea that this will produce a temperament with all intervals being the same width (have the same frequency ratio),”...

Maybe you meant “semitones being the same width”. Yes. Like for any geometric progression’s term, the semitones do have the same incremental ratio but, if we were to make a staircase, each frequency value could give us the step’s depth, or lenth, and each step would proportionally differ from the next one.

...“and at the same time octaves that are beatless.”...

Yes, and this is one of The Problems. A theoretical beatless octave is a wrong assumption, although 300 years ago it was in line with the common approach to temperament theories.

You say:...“This is only possible with harmonic tones.”...

Not true. You say that also with iH tones, on single partial matchings, we may go for beatless octaves. In any case, this is only temporarly and apparently true, any beatless interval will end up beating in a beat-flow. This is not to be understood, this is to be acknowledged or, once you acknowledge it you may understand.

...“Keeping the discussion to harmonic tones (for now), the 12th root of any number will produce a temperament with all intervals being the same width. However, unless the number is two, the octaves will beat.”...

Not correct. Like any interval, octaves will beat anyway, since 12 root of two is only an abstract case. Also “purely harmonic tones” is abstract thinking, the "pure" attraction again, it is forcing an abstract zero-iH concept into a model.

...“Now to have 12ths beat narrowly and at the same time 15ths beat widely the number has to be larger than 2 but smaller than ((3^(1/19))^12 or 2.0014269…”...

Correct.

...“The compromise that is given results in equal beating 12ths and 15ths when these intervals have a common note on the bottom.”...

If you approach the scale in terms of mirror-like order, you will not need to discriminate between top and bottom anymore.

...“But it is not clear why this compromise is necessary at all,”...

I wrote about this in my previous post. This compromise is necessary in that all intervals, with their stretch, can now contribute to the tonicity of the tuning form.

...“let alone why a superior compromise results with this sort of equal beating.”...

Opposite equal beating 12ths and 15ths results in a superior compromise for three reasons: firstly because it involves all intervals, wich are now beating intervals; secondly because the set gains stability by opposing a constant counter-beat, so all intervals compromise now for determining a perfectly stable, counter-balanced beating-whole; thirdly because the 15th encloses two octaves, what is needed to gain and ensure the intermodular quality. So, from one zero-beating octave block we progress to a two octaves beating matrix.

...“Not to mention how the roots of any other numbers are needed to calculate this compromise. In fact, they are not needed nor actually used although it could seem that way.”...

Please argue this last statement and be aware that you are getting into maths details, so before I answer please confirm you will not regret.

...“The argument could easily be made that the common note should be on the top,”...

No need. Anyway, show me please how you’d build a house starting from the roof, then I’ll follow you.

...“or the 15th should beat faster than the 12ths that has a common note on the bottom, but slower than the 12ths than has a common note on the top.”...

Ok, we both may be keen on break-dance, but this is not the place.

...“Another argument could be made that if anything should be equal beating, it should be the single octaves beating the same as the 5ths. But then the question again arises as to why, which note should be common, or should they actually beat equally?...

You try then: tune EB 5ths and octaves and then tell me how you like it. If you really like it, you can still refer to Chas algorithm:

((3/2) – Δ)^(1/7) = (2 + (Δ*s))^(1/12)

s = 1

Δ = 0.001178134272…

Scale ratio = 1.05951508823057…

...“Things get difficult when trying to use the 12th root of any number to describe the tuning of inharmonic tones.”...

Thinks get difficult only if or when you expect to find the theoretical frequencies values on iH tones. As for describing, Chas model is derived from a precise beats order and therefore can faithfully describe our actual tuning.

...“The tuning can be adjusted so that these intervals (12ths and 15ths) beat equally in any or all parts of the piano, or unequally in any or all parts of the piano. This is the true value of these intervals. They are a tool that the tuner can use to make compromises that are more important than arbitrarily equal beating intervals.”...

I hope you can better understand now the value of EB-ET and why it results in a superior compromise.

And do not worry, there will always be room for melodic, harmonic and musical priorities. Instead of calling it compromise, we'll call it knowledge.

T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv


Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/23/09 09:01 AM

Alfredo:

So, you see the problem of equal beating octaves and 5ths as one of ascetics, that it would not sound good. But the advantage of equal beating 12ths and 15ths is that it makes a wonderful mathematical model! They both make a wonderful mathematical model, and they both can be ascetically pleasing.

But we have a bigger problem. We continue to be unable to have a productive discussion about frequency ratios, beat rates and inharmonicity. We do not agree on the same concepts. I guess I will try again.

If the semitone interval is equal (frequency ratio is equal) then all interval ratios will also be equal.

If inharmonic tones are used, and an octave partial match is tuned to be beatless for all octaves (or tuned for equal beating intervals), the semitone intervals will not be equal.

I hoped that the simulations would make these concepts clear to you. Apparently they did not.

There is another concept that we disagree on. You do not understand that beat rates are generally progressive. There can be no “mirror image.” If a 12th and 15th beat equally with the bottom being common, the 12th that is a 4th higher will have a different beat rate.

Yes, I could show you the mathematics to calculate the frequency ratio to produce equally beating 12ths and 15ths with the common note on top. But instead, let me show you a way to tune them aurally starting with any 4th. Use the example of G3-G5 15th and C4-G5 12th. We will use D#3 as a test note. This test note is used for its 5th partial which is G5 and need not be tuned precisely, only so that it produces a beat at a useable speed. G3 and C4 have already been tuned. The 12th and 15th will beat equally when the difference in beat rate between the D#3-G3 M3 and the D#3-G5 M17 has the same difference as between the D#3-C4 M6 and the D#3-G5 M17. You may want to try this test on one of your tunings to see whether your 12ths and 15ths beat as you think they do.

I am tempted to discuss your CHAS algorithm on purely mathematical terms if we can agree to limit the discussion to just the mathematics involved and not the (mis)use in tuning. But I am not sure you understand what happens with your algorithm to begin with. For instance, in the equal beating 5ths and octave calculation you obtained a delta of 0.001178134272… Do you realize that by simply adding 2 to this number, you have the octave ratio? And this octave ratio is larger than 2 but smaller than (3^(1/19))^12 or 2.0014269… and therefore will produce wide 15ths and narrow 12ths? It is not far at all from equal beating 12ths and 15ths. This small difference would probably not be discernable.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/23/09 02:23 PM

Originally Posted By: alfredo capurso
.....

You say:...“This is only possible with harmonic tones.”...

Not true. You say that also with iH tones, on single partial matchings, we may go for beatless octaves. In any case, this is only temporarly and apparently true, any beatless interval will end up beating in a beat-flow. This is not to be understood, this is to be acknowledged or, once you acknowledge it you may understand.

.....


…this is only temporarily and apparently true…

Truth has a life span?

…any beatless interval will end up beating in a beat-flow…

If an interval doesn’t beat when you want it to, it just beats in a beat-flow? Like an alternative universe?

… This is not to be understood, this is to be acknowledged or, once you acknowledge it you may understand...

You are asking me to believe by faith so that I may then have an experience in order to believe this greater truth through the experience. Sorry, I do not mix religion and piano tuning.

I am wondering why I should take you seriously at all. You may be convinced of what you say or you may be knowingly misleading. (I continue to debate this with myself…) Either way it does not form a train of logic. I am going to rethink continuing this discussion with you. I may decide that it cannot be constructive.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/24/09 04:56 AM

Tooner,

On my part there is no interest in talking about ascetics, nor about philosophy or religion, but about Chas EB-ET as an improved temperament model. So far I’ve tried to explain what a model is generally meant for, how Chas comes from my practice and why ET equal beating 12ths and 15ths can be a solid and reliable reference.

I understand that you would like a model for piano tuning, a different one for pipe organs, one for guitars and, why not, one for bagpipes. I’m sorry, I can not help you.

I do not doubt about the well known iH’s effects on actual frequencies and ratios but aural tuning is about beat-control, and it is about a beat form that we – aural tuners – can possibly find again and again, no matter the usual iH's degrees. This is what I’m saying, maybe this is what Bill says talking about EBVT and his tunings, this may be what other tuners have been and are talking about when it comes to aural tuning and temperament models.

In my opinion, how you keep on mixing aural beat-control and accuracy with iH, actual frequencies values, actual ratios, ETDs issues, piano imperfections, compromises, pinblocks, rendering points and passing thunderstorms does not help you either. Above all, it does not help young people who are approaching aural tuning, people that may like to refer to the most correct and practicable model (Chas or whatever), people that do not need to be pushed towards second-rate standards.

What I think is that our different pro experiences have matured us into different tuners with two quite opposite approaches.

To a novice you choose to say “A piano being an imperfect instrument cannot be tuned perfectly. So I try to tune pianos perfectly out-of-tune”…, I’d say: I know I’m not perfect, so I tune pianos at my best. You say “at a certain point it actually sounds worse and worse to me”…, I’d say: point after point, it has to sound right. You say “The fine little imperfections come out and all I end up doing is trying to make things sound less bad”…, I’d say: I tend to refine any little imperfection and all I keep on doing is trying to make things sound at their best. You say “And then when it is played by someone else, it sounds wonderful”…, I’d say: And then when it is played by someone else, I hope he/she’ll like it too.

You ask for short posts and single subjects, so I shall stop here. But this time, I’d like to tell you more about aural tuning, maths, undiscernible small differencies, knowingly misleading ghosts, trains of logic and constructive discussions, and I shortly will. Many scaring pop-ups on Chas tuning MP3?

R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/24/09 08:14 AM

Quote:
So, you see the problem of equal beating octaves and 5ths as one of ascetics, that it would not sound good.


You meant "aesthetics"?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/24/09 08:44 AM

Yes, aesthetics is what I meant. A spell checker thing. It did seem odd that there was not a "h". But now I know a new word!
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/24/09 11:37 AM

Originally Posted By: alfredo capurso
.....

Chas model is derived from a precise beats order and therefore can faithfully describe our actual tuning.

.....


Really!?!?!?

The Chas model predicts 12ths and 15ths with a beat rate that doubles about every octave. You say your actual tuning does not. The simulations that I provided do not. So, the Chas model does not faithfully describe actual tuning.

What I continue to wonder is if you conveniently forget this fact or just hope that everyone else will.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/24/09 05:51 PM

Kent,

Thank you for checking. Not only there was the "h", but also the random t and e.

Tooner,

I wrote “describe”.

I didn’t write: you can use theoretical values on pianos, or for amateur simulations. What’s wrong with you? Do you still want to talk about having to apply iH on theoretical values?

Recently you wrote:...“The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate. I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless.”...

My reply: Although I do not know whether you are talking about 12th root of two or what you are referring your "truth" to, I can confirm that, in my tuning form, 12ths and 15ths are opposite equal beating all along the scale. But maybe you have one more model in mind.

More recently you wrote:...“For instance, in the equal beating 5ths and octave calculation you obtained a delta of 0.001178134272...Do you realize that by simply adding 2 to this number, you have the octave ratio? And this octave ratio is larger than 2 but smaller than (3^(1/19))^12 or 2.0014269… and therefore will produce wide 15ths and narrow 12ths? It is not far at all from equal beating 12ths and 15ths.”...

So you say “Do you realize...It is not far at all from equal beating 12ths and 15ths.”

Tooner, what can I tell you? Only what I’ve already told you: one minute you state in a sense, the minute after you negate it. To me, you may be raving on everything and its opposite.

Still today 12th root of two describes our world-wide tunings, and 19th root of three could then describe pure 12ths tuning. Maybe you prefere the way 12th root of two describes equal temperament or, refusing Chas, you may prefere ET 19th root of three, this is up to you and I will not blame you.

...“What I continue to wonder is if you conveniently forget this fact or just hope that everyone else will.”

I conveniently stick to practical Chas tuning and I have academic theoretical reasons and evidencies for hoping to share Chas EB-ET model.

R. a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/24/09 09:07 PM

Quote:
in my tuning form, 12ths and 15ths are opposite equal beating all along the scale.


As measured in cents, in your "tuning form", how expanded is the double-octave (in cents) and how contracted is the twelfth (in cents)?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 07:15 AM

Alfredo:

It probably seems that I am being contradictory because you do not understand the concepts that I am talking about. And I am unable to explain them to you.

After today I doubt that I will be posting for close to two weeks. Besides Thanksgiving, I will be at deer camp. When I get back I probably will not post on this Topic anymore. I don’t see how it can be productive.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 10:13 AM

Kent,

You ask: “As measured in cents, in your "tuning form", how expanded is the double-octave (in cents) and how contracted is the twelfth (in cents)?”

You may be talking about actual pitches, i.e. real Chas tuning frequencies. If it is so, I have not been able to elaborate on them, since I do not have any adequate device.

By using the conversion program linked here:

http://www.sengpielaudio.com/calculator-centsratio.htm

I have been able to elaborate only on theoretical Chas values, so I could calculate theoretical Chas semitone’s cents value = 100.0383184402... - with quite little approx., around the 10th decimal point - and the offset in cents from A4 (440 Hz) to A6. Let me know if those figures can be of some use.

I would be very happy to be able to make all sorts of measuring on the real Chas tuning form.

Tooner,

What I have understood is that, basing on iH issues, you may like a different model for every musical instrument that produces iH tones. Or maybe a specific model for pianos.

In my opinion, we can do well with tuning, both in aural and ETDs cases, referring to 12th root of two ET model, although we could never tune its pure octaves. Now we have a chance to do even better with a reliable and practicable ET model and its two tuneable constants for reference, equal beating 12ths and 15ths.

I’ll be looking forward to hearing from you and don’t think about productivity, it is more than that, it’s breathing.

Have a good time.

Regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
_________________________
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 11:47 AM

Alfredo:

I can hardly even breathe. The air is so foul.

But let me try to give you and Kent a helping hand while I hold my breath.

Take the cent value of 100.0383184402. This means that each semitone is 0.0383184402 wider than theoretical. So multiplying this by 24 gives the cent deviation for 15ths and by 19 gives the cents deviation for 12ths. Rounding off, makes the 15ths 1 cent wider than theoretical and the 12ths ¾ cent wider than theoretical.

For iH tones rather than measure the deviation from theoretical, it is better to measure it from beatless intervals. This will be about 1 cent wide for 15ths and about 2 minus ¾ or 1-1/4 cents narrow from just for 12ths.

For practical, aural tuning this will change the M3-M17 4:1 15th test so that the M17 should beat as fast as the M3 one semitone higher than the test. And will change the M6-M17 3:1 12th test so that the M17 should beat as fast as the M6 one semitone lower. Or to be even more practical, with the common note on top, the M17 should beat faster than the M3, but slower than the M6. I do not think there is any practical difference between Chas and mindless octaves.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 11:56 AM

Quote:
Besides Thanksgiving, I will be at deer camp.

Does that mean you are going to a stag party?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 11:57 AM

Originally Posted By: BDB
Quote:
Besides Thanksgiving, I will be at deer camp.

Does that mean you are going to a stag party?


In a way, but it certainly is not the stag's idea of a good time.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 12:20 PM

Be sure to bring along the correct supplies, so you can demonstrate to the to the new guy how you can tell that a deer was there by using your sense of taste!
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 12:37 PM

My favorite gag is shoving a dead mouse in the toe of someone's boot. Well, I think its funny.

The other one is the fake phone on the wall. Pick it up, you get a dial tone. Dial any muber and 10 seconds later you get a busy signal. Of course the closest phone line is miles away. Designed and made the circuits myself.
Posted by: Jim Moy

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 02:05 PM

Hey, careful, or we'll have another thread devolving into an "OT Paging Jerry Groot"-fest, lol.
Posted by: BDB

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 02:10 PM

I thought it already had!
Posted by: Jim Moy

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 02:13 PM

Not even close, that one went by the 1600-post mark a while ago!
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/25/09 04:41 PM

Oh, for a few minutes it was like being at a party.

Tooner, it'll be nice if you come back with some antlers pictures. a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/26/09 07:10 AM

Kent, Bill, all Colleagues,

Tooner kindly writes: “I do not think there is any practical difference between Chas and mindless octaves.”

If only we could have said this six months ago! Actually Bill, reading about your tuning’s effects, the pipe organ, the crystal clear, made me think that our “sweet spot” must have been very close indeed. Will it be so?

If really Tooner were right, I could state:

So far you/we have been looking at equal beating 12ths and 15ths only as a technique, now you/we may acknowledge that it can also be the expression of the 12th root of two ET model’s improvement.

We may share the reason for discarding two unjustified theoretical assumptions - that the range of the scale module must be 12 semitones, and that the octave must be theoretically pure - and appreciate the practical outcome of a theoretical superior compromise.

All together, we may now justify a revisory effort and evaluate iH’s effects in their true light.

In fact, Chas model stretches the octave and all intervals not only because of iH, but for gaining a precise sound whole, and corrects the theoretical semitone’s value so that it can be a function of our two practicable ET constants.

This makes me convinced that Chas model’s “compromise” may one day represent the temperament reference for other (western traditional) orchestral instruments, what may finally take to a superior harmonicity degree.

I also think that ET teachings may improve in terms of clarity and exactitude, and maybe this would make our aural tuning experience shareable even more.

Regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 11/29/09 01:00 PM

All Colleagues and Tooner,

You wrote (10/30/09):...“I do not know of any intrinsic value that the ancestry of a ratio could have. What is important is what the ratio will accomplish on its own.”...

My reply (11/02/09): I’d say that the intrinsic value of a ratio is its ancestry or, a ratio can have an intrinsic value depending on its ancestry. Chas ratio returns the qualities of two algebraic magnitudes, 19 root of three and 24 root of four.

You also wrote (11/20/09):...“But it is not clear why this compromise is necessary at all, let alone why a superior compromise results with this sort of equal beating.”...

More recently you wrote (11/23/09):...“So, you see the problem of equal beating octaves and 5ths as one of ascetics, that it would not sound good. But the advantage of equal beating 12ths and 15ths is that it makes a wonderful mathematical model! They both make a wonderful mathematical model, and they both can be ascetically pleasing.”...

Then you said you were talking about aesthetics.

I reply so that some unfortunate statements will not result in being misleading.

You have already acknowledged that using a Chas-type equation with two partials values makes a “wonderful mathematical model”. Now I may better explain what the “intrinsic value of a ratio” can be, and why Chas ET 12ths and 15ths equal beating is a “superior compromise”, not only compared to any pure-ratio model, but also compared to equal beating octaves and 5ths.

Let’s see. The ratio for 12ths is 3:1, the ratio for 15ths is 4:1. So, Chas EB-ET equation combines the values 3 and 4.

5ths and octaves involve ratios 3/2:1 and 2:1.

The two means/average-values tables below is worth for comparing and simply noticing what happens.





The values 3 and 4, together, can average couples (dyads) of ratios and include from 1 to 9 and up to 16.





The values 3/2 and 2 can average couples (dyads) of ratios and include only from 1 to 5.

Can we now compare 2:1, 3/2:1, 3:1 only “pure ratio” models?

Can we share a good reason for a combined "two ratios" ET model?

Can we say that ET 12ths and 15ths equal beating represents a superior compromise?

Could we ever say that it is only a question of aesthetics?

Regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/01/09 12:28 PM

I must apologize, there must have been a problem with the server and the first means/average-values table disappeard. Here it is again, I hope it'll last.






Regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/06/09 01:09 PM

All Colleagues,

Before starting with Chas ET-EB system, I needed to be sure it was not a subjective preference, so I kept all my tunings (and my customers) under observation for more than 20 years.

So doing, I have also been able to confirm that, in our practice, iH is not a problem at all. Grand pianos or small spinets, ‘800 cabinets, clavichords or harpsicords, on all of them it was and it is possible to define the ET-EB form I am talking about, no matter the usual iH’s degrees. Thus I decided that there may have been a problem with the overall approach, and so I started to check our 12th root of two’s theoretical premises.

I noticed that all temperaments have one assumption in common: all theoretical models try to gain the scale by working on the scale’s mere numerical values. In other words, the beats order was/is thought as having to depend on the tones size, but in fact it (the beats order) depends on partials matchings and therefore on proportional differencies.

So, if anything, it should be the other way around: beats, i.e. differencies, should determine the scale frequencies values, and this is how I matured Chas approach and started looking for a difference-constant.

Once I experienced that octaves and fifths are the two stretchers, i.e. that the compromise had to be found between these two intervals, I got ready to notate a double difference-constant. In fact, all partials had to contribute to the optimum compromise, all intervals had to give up a small part of their “pure” value, and the correct way would have involved both stretchers, fifths and octaves.

Therefore we could say that, with 12th root of two, fifths, thirds with the other intervals are stretched in favor of only theoretical 2:1 octaves, but in fact 12ths and 15ths can stretch all intervals, in theory and in practice, in favor of a beating whole. All intervals can actually be thought as equal rank stretchers, all together sharing in the most intrinsic correlation amongst scale frequencies and beats.

Chas ET-EB 12ths and 15ths can express a synchronic beat order. This means that tones do not produce out of time beats, nor unconvenient contrasts and so beats and partials together can gain the greatest overtones outcome, in synergistic terms. In other words, proportional beats can turn into energy and enhance the sound-whole with more sounds. This is what I hear, this is what I experience every time I tune.

Chas system translates my aural tuning experience and it is meant to share an assumption-free, correct and reliable temperament model, together with its two practicable EB constants. It numerically proves the ultimate proportion between all partials differencies and the scale foundamental frequencies, it proves the symmetries within four octaves and the whole’s stability deriving from ET-EB 12ths (narrow) and 15ths (wide).

In terms of means, Chas model uses our own (aural tuners) “key”: beats, i.e. differencies. This explains the delta and s difference-factors inside the Chas formula.

Chas algorithm is so powerful as to enable any frequencies logarithmic progression, even pure-ratio-based scales like 2:1 or 3:1 ETs, so in a way it may be considered “dogma-free”, but it features finally a reachable target for all tuners, the precise equal-difference that will open to a perfectly resonant beating whole.

CHAS article - G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

Regards, a.c.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/11/09 04:36 PM

Originally Posted By: alfredo capurso
All Colleagues,

Before starting with Chas ET-EB system, I needed to be sure it was not a subjective preference, so I kept all my tunings (and my customers) under observation for more than 20 years.

So doing, I have also been able to confirm that, in our practice, iH is not a problem at all. Grand pianos or small spinets, ‘800 cabinets, clavichords or harpsicords, on all of them it was and it is possible to define the ET-EB form I am talking about, no matter the usual iH’s degrees. Thus I decided that there may have been a problem with the overall approach, and so I started to check our 12th root of two’s theoretical premises.

I noticed that all temperaments have one assumption in common: all theoretical models try to gain the scale by working on the scale’s mere numerical values. In other words, the beats order was/is thought as having to depend on the tones size, but in fact it (the beats order) depends on partials matchings and therefore on proportional differencies.

So, if anything, it should be the other way around: beats, i.e. differencies, should determine the scale frequencies values, and this is how I matured Chas approach and started looking for a difference-constant.

Once I experienced that octaves and fifths are the two stretchers, i.e. that the compromise had to be found between these two intervals, I got ready to notate a double difference-constant. In fact, all partials had to contribute to the optimum compromise, all intervals had to give up a small part of their “pure” value, and the correct way would have involved both stretchers, fifths and octaves.

Therefore we could say that, with 12th root of two, fifths, thirds with the other intervals are stretched in favor of only theoretical 2:1 octaves, but in fact 12ths and 15ths can stretch all intervals, in theory and in practice, in favor of a beating whole. All intervals can actually be thought as equal rank stretchers, all together sharing in the most intrinsic correlation amongst scale frequencies and beats.

Chas ET-EB 12ths and 15ths can express a synchronic beat order. This means that tones do not produce out of time beats, nor unconvenient contrasts and so beats and partials together can gain the greatest overtones outcome, in synergistic terms. In other words, proportional beats can turn into energy and enhance the sound-whole with more sounds. This is what I hear, this is what I experience every time I tune.

Chas system translates my aural tuning experience and it is meant to share an assumption-free, correct and reliable temperament model, together with its two practicable EB constants. It numerically proves the ultimate proportion between all partials differencies and the scale foundamental frequencies, it proves the symmetries within four octaves and the whole’s stability deriving from ET-EB 12ths (narrow) and 15ths (wide).

In terms of means, Chas model uses our own (aural tuners) “key”: beats, i.e. differencies. This explains the delta and s difference-factors inside the Chas formula.

Chas algorithm is so powerful as to enable any frequencies logarithmic progression, even pure-ratio-based scales like 2:1 or 3:1 ETs, so in a way it may be considered “dogma-free”, but it features finally a reachable target for all tuners, the precise equal-difference that will open to a perfectly resonant beating whole.



Regards, a.c.


Alfredo,

your theory does not answer why not taking instead for example a model like: (This is not the model i personally prefer, just an alternative to the CHAS model)

(6 − ∆*s1)^ (1/31) = (4 + ∆*s)^ (1/ 24)

with a solution of ∆ = 0.00230684644393...
and an slightly smaller incremental ratio than CHAS of: 1.059488545...
(which would result with better thirds for example...)

Analog to section 3.4 of the CHAS paper, we would have 63 steps (compared to only 49 in CHAS, i guess the CHAS model wants to appear superior because it provides 49 steps compared to only 13 in the standard ET model),
with a scale ratio of:

(6 + ∆ )^2

We also have a perfectly balanced system around step 31:

0*(4+∆)→24 → 31 → 38*(4+∆)→62
0→7 *(4+∆) → 31 *(4+∆)→55→62
0*(6-∆) → 31 *(6-∆)→ 62

ETC.

As i already mentioned, Guerino Mazzola has provided a general formula which can take every form of ET and non ET for the harmonic tones, CHAS does not provide something new here.

An "s" term alike element, which enables the CHAS model to take every size, is generally implemented in tuning programs by additional terms to hold for stretch caused by inharmonicity for example, also nothing new here.

Finally describing the standard ET in CHAS terms, with s=0 we get:
(3 − ∆*s1)^ (1/19) = (4)^ (1/ 24)
and a solution for ∆*s1=0,0033858462466..., the standard ET incremental factor, 1.059463094359295..

We can get this equation into another form by replacing 4^(1/24) with 2^(1/12):
(3 − ∆*s1)^ (1/19) = (2)^ (1/12)

If we multiply by ^19 and ^12 we get:

(3 − ∆*s1)^12 = 2^19
now we multiply out the left term and bring the difference to the right side:

3^12=2^19+∆*s1

if we express the difference as a factor (which has several advantages, when calculating with frequencies), we can write:

3^12=2^19*∆' (note*)

and we divide by 2^7:

we get:

3^12/2^7=2^12*∆'

which is simply our well old fifths circle with ∆' for the pythagorean comma. CHAS model is simply another form of the fifths cirlce, (as is my model too) not a spectacular new academic model as it may appear.

This is my last post to this thread.

All the best,

BS

(note*) you can find this "natural" form of the fifth circle in my paper:
http://www.piano-stopper.de/dl/PTG2008_StopperTemperament.pdf
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/11/09 04:59 PM

Originally Posted By: alfredo capurso
Tooner, this is not very short, I must apologize.

...“The basis of the 12ths root of two is the idea that this will produce a temperament with all intervals being the same width (have the same frequency ratio),”...

Maybe you meant “semitones being the same width”. Yes. Like for any geometric progression’s term, the semitones do have the same incremental ratio but, if we were to make a staircase, each frequency value could give us the step’s depth, or lenth, and each step would proportionally differ from the next one.

...“and at the same time octaves that are beatless.”...

Yes, and this is one of The Problems. A theoretical beatless octave is a wrong assumption, although 300 years ago it was in line with the common approach to temperament theories.

You say:...“This is only possible with harmonic tones.”...

Not true. You say that also with iH tones, on single partial matchings, we may go for beatless octaves. In any case, this is only temporarly and apparently true, any beatless interval will end up beating in a beat-flow. This is not to be understood, this is to be acknowledged or, once you acknowledge it you may understand.

...“Keeping the discussion to harmonic tones (for now), the 12th root of any number will produce a temperament with all intervals being the same width. However, unless the number is two, the octaves will beat.”...

Not correct. Like any interval, octaves will beat anyway, since 12 root of two is only an abstract case. Also “purely harmonic tones” is abstract thinking, the "pure" attraction again, it is forcing an abstract zero-iH concept into a model.

...“Now to have 12ths beat narrowly and at the same time 15ths beat widely the number has to be larger than 2 but smaller than ((3^(1/19))^12 or 2.0014269…”...

Correct.

...“The compromise that is given results in equal beating 12ths and 15ths when these intervals have a common note on the bottom.”...

If you approach the scale in terms of mirror-like order, you will not need to discriminate between top and bottom anymore.

...“But it is not clear why this compromise is necessary at all,”...

I wrote about this in my previous post. This compromise is necessary in that all intervals, with their stretch, can now contribute to the tonicity of the tuning form.

...“let alone why a superior compromise results with this sort of equal beating.”...

Opposite equal beating 12ths and 15ths results in a superior compromise for three reasons: firstly because it involves all intervals, wich are now beating intervals; secondly because the set gains stability by opposing a constant counter-beat, so all intervals compromise now for determining a perfectly stable, counter-balanced beating-whole; thirdly because the 15th encloses two octaves, what is needed to gain and ensure the intermodular quality. So, from one zero-beating octave block we progress to a two octaves beating matrix.

...“Not to mention how the roots of any other numbers are needed to calculate this compromise. In fact, they are not needed nor actually used although it could seem that way.”...

Please argue this last statement and be aware that you are getting into maths details, so before I answer please confirm you will not regret.

...“The argument could easily be made that the common note should be on the top,”...

No need. Anyway, show me please how you’d build a house starting from the roof, then I’ll follow you.

...“or the 15th should beat faster than the 12ths that has a common note on the bottom, but slower than the 12ths than has a common note on the top.”...

Ok, we both may be keen on break-dance, but this is not the place.

...“Another argument could be made that if anything should be equal beating, it should be the single octaves beating the same as the 5ths. But then the question again arises as to why, which note should be common, or should they actually beat equally?...

You try then: tune EB 5ths and octaves and then tell me how you like it. If you really like it, you can still refer to Chas algorithm:

((3/2) – Δ)^(1/7) = (2 + (Δ*s))^(1/12)

s = 1

Δ = 0.001178134272…

Scale ratio = 1.05951508823057…

...“Things get difficult when trying to use the 12th root of any number to describe the tuning of inharmonic tones.”...

Thinks get difficult only if or when you expect to find the theoretical frequencies values on iH tones. As for describing, Chas model is derived from a precise beats order and therefore can faithfully describe our actual tuning.

...“The tuning can be adjusted so that these intervals (12ths and 15ths) beat equally in any or all parts of the piano, or unequally in any or all parts of the piano. This is the true value of these intervals. They are a tool that the tuner can use to make compromises that are more important than arbitrarily equal beating intervals.”...

I hope you can better understand now the value of EB-ET and why it results in a superior compromise.

And do not worry, there will always be room for melodic, harmonic and musical priorities. Instead of calling it compromise, we'll call it knowledge.

T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv




Hello, the logic escapes me and is certainly good and true, but musically it does not suit my ear. Indeed larger tunings in the medium range are appreciated for jazz harmony with lot of sevenths but the treble lack liveness, and the basses are too tight to me, with too slow beats in the low medium.

Certainly practical, but too close to me and too "straight" where is the contrast and tonality ? it need more stretch in the beginning of the treble and bass.

I don t want to be short there, I simply say what my ears tell me. could be a question of habit, but in the end we need to play and to play together with other instruments as well.

the thirds are all but lively, you may not notice because you listen to the tuning while playing.

I feel more musical to "follow the piano" more than to use a layer of theory? Thanks for the recording anyway.

I send a friend which is very good to tune the "Cordier" temperament method, to tune a small C2 Yamaha at a pianist customer of mine (leaving far from my place). They finally told me that they called another tuner to tune the piano "normally" after him.
Others appreciate his tunings, and the C2 is a small piano hence that large tuning decompress the harmonic content so it make sense. It simply did not fit the pianists ears and he noticed that soon.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/12/09 02:31 PM



Kamin, thank you for commenting. For me it will be interesting to know about your tastes and tunings, I will soon reply to you.


Originally posted by me:

The article describes the theoretical and mathematical foundations that can enumerate ETs, including 19th root of three, and the basic 12ths and 15ths opposite equal beating ET. This is why I say: now, 12ths and 15ths opposite equal beating ET model has got a precise name.
As simple as that.

Bernhard Stopper, you wrote:

..."Exactly for this, we have Mazzolas formula/model already. And his model can not only enumerate any ET model but also any possible non-ET model, thus beeing superior over CHAS, which is limited to ET. As you are very talented in twisting the meaning of critical reviews about your CHAS model into the opposite and constantly ignoring existing, predating, valuable and even superior models to CHAS, i will stop the conversation here again.”...

Professor Mazzola’s formula does not isolate with figures, nor evidenciate 12ths and 15ths opposite equal beating ET. Probably, not being a piano tuner, he could not have that kind of clue, nor that standard urge.

This is probably why he himself, wrote back to me (months ago) and warmly suggested to go for a pubblication in the JMM.

Chas limited to ET? You are wrong, the algorithm can define the precise ratio also for single tones.

Bernhard Stopper, you started your "conversation" with your “cello scrotum” rotten insinuation (05/07/2009).

Now, about ignoring other predated models, go and read. None of them is Chas EB-ET. And I could never have immagined your way through false and defamatory statements.



a.c.



.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/14/09 06:59 AM

Originally Posted By: alfredo capurso

you started your "conversation" with your “cello scrotum” rotten insinuation (05/07/2009).
Now, about ignoring other predated models, go and read. None of them is Chas EB-ET. And I could never have immagined your way through false and defamatory statements.


As i mentioned already in my post from May, 20 2009, the cello scrotum publication is a brilliant article proving deficits of peer review processes of scientific journal articles. I was just guessing if your Chas paper could have been of this category after the missing references to existing state of the art developed in the recent years to overcome the limits of stand ET in piano tuning and the offensive language you was using. This isn´t a rotten insinuation at all.

Instead, you should welcome to get critical reviews, which could help you to eventually refine your paper if you plan for a publication in a scientific journal, which i would recommend you warmthly too, like Prof. Mazzola did.

Again the link to the cello scrotum article for those who are interested:
http://www.timesonline.co.uk/tol/life_and_style/health/article5601050.ece


Originally Posted By: alfredo capurso

Chas limited to ET? You are wrong, the algorithm can define the precise ratio also for single tones.

Of course, if the s variable is not only a scalar variable, but a vector variable with different values/coefficients for every note. In that case, any figurative aspects of the Chas model are lost and one can use the simplest form of describing tones, namely by a general vector variable containing the frequencies of every single tone´s partial(s).

Originally Posted By: alfredo capurso

Professor Mazzola’s formula does not isolate with figures,

Mazzola´s model allows for visualization of tunings in the three dimensional Euler space for example.

Originally Posted By: alfredo capurso

nor evidenciate 12ths and 15ths opposite equal beating ET.

The evidence of 12ths and 15ths opposite equal beating is not yet proved to be necessary, and rather an arbitrary and subjective selection as i figured out in my recent posts.

Originally Posted By: alfredo capurso

Now, about ignoring other predated models, go and read. None of them is Chas EB-ET.

Bremmers mindless octaves ET model is exactly Chas EB-ET.

Originally Posted By: alfredo capurso

This is why I say: now, 12ths and 15ths opposite equal beating ET model has got a precise name.
As simple as that.

12ths and 15ths opposite equal beating tuning method and ET model (Bill Bremmer described his method in words, it has become a model then, it must not necessary be described by mathematical formalism) has got a precise name already, called "mindless octaves".
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/14/09 02:37 PM


Stopper,

In my opinion you are offending also PW colleagues and readers.

..."The evidence of 12ths and 15ths opposite equal beating is not yet proved to be necessary, and rather an arbitrary and subjective selection as i figured out in my recent posts."...

This is written in Chas research report, section 2.0.

..."Bremmers mindless octaves ET model is exactly Chas EB-ET."...

What is Chas EB-ET model is explained in the research report. What is Bill's, is very well written in PW.

CHAS article - G.R.I.M. (Department of Mathematics, University of Palermo, Italy):
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

In my opinion, you and Tooner are now mystfying the truth, and I find this sickening. Fortunately, Bill and I have long written to each other about his mindless octaves technique, EBVT and Chas model. Bill would never dare to pretend what it can not be.

Originally Posted By: alfredo capurso

This is why I say: now, 12ths and 15ths opposite equal beating ET model has got a precise name.
As simple as that.

Stopper, you write:..."12ths and 15ths opposite equal beating tuning method and ET model (Bill Bremmer described his method in words, it has become a model then, it must not necessary be described by mathematical formalism) has got a precise name already, called "mindless octaves".

Ok Stopper, look at methods, words and models the way you prefere.

In practice, this is one example of how Bill Bremmer tunes 12ths and 15ths (Chas Topic - 06/02/09): “I routinely see them invert themselves: the 12th becomes wider than the double octave”.

This is the tuning Bill may like the best, call it words, method or technique, but it is not Chas EB-ET theoretical model nor Chas practical equal beating tuning form (12ths and 15ths equal beating along the whole keyboard). And Bill in his honesty would never dare to state lies.

Chas is a new temperament theory derived from my practice.

http://en.wikipedia.org/wiki/Theory

Since all this is quite common knowledge, I can only think of yours as a deceitful intention. Your dwelling then is sick, therefore I have nothing more to tell you.

Bill, all this may be only market strategy, I'm sorry too.
I apologize with all readers, I never meant to share this odious burden.

Posted here on 5/11/2009: About theory and tuning - you may ignore theory and tune aurally in a casual or personal way, or use an ETD without careing what’s behind it. Maybe a simple question of knowledge and consciousness that is up to you.

a.c.

.

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/16/09 04:46 AM

Dear Colleagues,

I’m trying to turn the recent falsifications in a sort of challenge, in a chance to grow in terms of consciousness.

Only now I understand how my detractors have tried, for seven months, to mystify and reduce the value of the Chas research report, with out-of-theme subjects, insinuations, with insults and absurd arguments. The idea of having taken part in a devilish project makes me shiver.

Eventually, these posters could not hide anymore the value of Chas theory but they have attempted to mix things up again. All their lies and all their ambiguities are written though, they have been well traced too, and paradoxically they themselves are confirming now the relevance of what I am trying to share.

With their devious deeds, by rereading their posts and/or noticing their silence, we may keep on training our insight power and our comparing skill, what is useful (in my opinion) for tuning too.

On your part, you may now open more to a new approach to the temperament and, maybe in a while, try the gaining of Chas beating whole.

Now you may appreciate the difference between having to refer to 12th root of two ET ratio, and having a correct ET theory’s ratio as a solid reference, as well as a tool for gaining any ET ratio.

We may be able to share also Chas Preparatory Tuning, the SBI's (slow beating intervals) control and progression, how they can make “that little difference” and how they could help me gain Chas pure tuning form.

Quite recently, an American colleague wrote to me kindly asking to know more about me. Today, since my professional route may not be understood as an incentive nor as ostentation, I can post my reply.

Regards, a.c.


CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv


.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/16/09 08:07 AM

Alfredo:

You could counter the posts of the detractors of your theories (including me) by referencing posts of the supporters (I do not remember any).
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/16/09 07:17 PM


Dear Colleagues,

I’m trying to turn the recent falsifications in a sort of challenge, in a chance to grow in terms of consciousness.

Only now I understand how my detractors have tried, for seven months, to mystify and reduce the value of the Chas research report, with out-of-theme subjects, insinuations, with insults and absurd arguments. The idea of having taken part in a devilish project makes me shiver.

Eventually, these posters could not hide anymore the value of Chas theory but they have attempted to mix things up again. All their lies and all their ambiguities are written though, they have been well traced too, and paradoxically they themselves are confirming now the relevance of what I am trying to share.

With their devious deeds, by rereading their posts and/or noticing their silence, we may keep on training our insight power and our comparing skill, what is useful (in my opinion) for tuning too.

On your part, you may now open more to a new approach to the temperament and, maybe in a while, try the gaining of Chas beating whole.

Now you may appreciate the difference between having to refer to 12th root of two ET ratio, and having a correct ET theory’s ratio as a solid reference, as well as a tool for gaining any ET ratio.

We may be able to share also Chas Preparatory Tuning, the SBI's (slow beating intervals) control and progression, how they can make “that little difference” and how they could help me gain Chas pure tuning form.

Quite recently, an American colleague wrote to me kindly asking to know more about me. Today, since my professional route may not be understood as an incentive nor as ostentation, I can post my reply.

Regards, a.c.


CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/16/09 07:24 PM


Bill, thank you.

I do think that your tuning are different from Chas EB-ET, and I'm looking forward to appreciating them on a real piano.

Actually, I think that the recent vicious attempts to squeeze Chas temperament into your technique have damaged your image too, regarding both your favorite ET routine tuning and your EBVT model.

These have been some of my consequent readings:

http://en.wikipedia.org/wiki/Moral_turpitude

http://en.wikipedia.org/wiki/United_States_defamation_law#Defamation_per_se

http://en.wikipedia.org/wiki/Intent



Regards, a.c.


Kent,

You ask:..."Why do you feel the need to use flowery analogy in speaking with experienced professional tuners?"...

Because the use of an analogy may help, be him/her a pro tuner or an apprentice.

Quote: So, I never go directly for the Chas form.

..."Specifically, why not?
Are you claiming that your temperament and/or stretch level of your tuning requires some special technique that must be followed in order for a professional tuner to execute your tuning with stable results?"...

Why not, it is explained in the same post, because the piano, like a bow, will adjust on an overall "at rest" condition. About my technique, I do not think it is special, and I'm actually trying to make it more common.

If that execution is required? Yes, and it gets stable results. Could you please answer me about the possible date of our meeting?

Thanks and regards, a.c.

.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 07:35 AM

And here is something I was reading:

http://en.wikipedia.org/wiki/Martyr_complex
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 08:11 AM


Kamin,

yes, tuning mid-strings I go for an accentuated raise of pitches and for faster beat rate progressions, so that the piano can then settle on Chas form.

By readind this thread you may find out that many issues have already been discussed. I sincerely apologize, but I do not have the time to write about the same issues again and again. I'd rather suggest a more attentive reading, if there is a real interest.

For istance, about my need for a preparatory tuning I wrote in May 23, 2009 and August 01, 2009:

"When in the treble you can check octaves, 10ths, 12ths, 15ths, 17ths and 19ths you do not need to check 5ths anymore. Check for progressive octaves (check with middle string only), tune middle string a bit higher, i.e. make your check-intervals a bit wider, so that when you join left and right strings you can get stable and constant 12ths and 15ths equal beating."

Chas ET Temperament Theory has two mathematical constants: 12ths and 15ths opposite equal beating all along the keyboard. This strictly requires progressive octaves and progressive SBI.

About what's happening and about the latest lies, you may get your own opinion. Please mind, I'm not pushing you towards any of this reading.

Bill Bremmer (June 21, 2009):..."I also learned from Kent that it doesn't matter how wide or narrow the octave is. Since he had explained it to me, I heard it from other sources as well.

Regarding whether 5ths and 12ths become wide or not, they do, I am convinced of that. I learned that very long ago from Steve Fairchild who demonstrated it at a PTG convention. Now, I take what Kent said about my post to heart but the figures as I posted them still suggest as much.

Now, if you continue this technique upwards, you will inevitable find a point where both the double octave and the 12th will both stop the pattern and to the ear, both will sound perfectly in tune. When you continue upwards, you will find the exact opposite of what you found at F5. When the 12th stops the pattern, the double octave will be wide, when the double octave stops the pattern, the 12th will be narrow, still each by a very small amount.

This means (at least by my reasoning), that the 12th has become wide and therefore the 5th as well. However, at this point, the coincident partials for the 5th may well be out of hearing range and therefore, however wide they may be won't matter because they cannot be heard. In any case, a slightly wide 5th is not unpleasant to the ear, especially that high up where the sustain is so short."...

My detractors could well understand and remember thas Chas EB-ET Theory is then original.

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 08:31 AM

I find Capurso´s statement (last line) not to be evident:
Originally Posted By: alfredo capurso
Bill Bremmer (June 21, 2009):..."
Regarding whether 5ths and 12ths become wide or not, they do, I am convinced of that. I learned that very long ago from Steve Fairchild who demonstrated it at a PTG convention. Now, I take what Kent said about my post to heart but the figures as I posted them still suggest as much.

Now, if you continue this technique upwards, you will inevitable find a point where both the double octave and the 12th will both stop the pattern and to the ear, both will sound perfectly in tune. When you continue upwards, you will find the exact opposite of what you found at F5. When the 12th stops the pattern, the double octave will be wide, when the double octave stops the pattern, the 12th will be narrow, still each by a very small amount.

This means (at least by my reasoning), that the 12th has become wide and therefore the 5th as well. However, at this point, the coincident partials for the 5th may well be out of hearing range and therefore, however wide they may be won't matter because they cannot be heard. In any case, a slightly wide 5th is not unpleasant to the ear, especially that high up where the sustain is so short."...

My detractors could well understand and remember thas Chas EB-ET Theory is then original.


Bill is describing in the posts you are quoting here a "tweaked" METHOD of his initial idea (The MODEL) of opposite equal beating duodecimes and double octaves. And the MODEL he described, is equivalent to the CHAS ET-EB MODEL form.

Quote from Bill Bremmers own words: (http://www.ptg.org/pipermail/pianotech/2002-January/101241.html)

"Essentially, it is an *Equal Beating* compromise between the Double Octave and the Octave and Fifth."


Also, i don´t find this statement to be evident:
Originally Posted By: alfredo capurso

Actually, I think that the recent vicious attempts to squeeze Chas temperament into your technique have damaged your image too, regarding both your favorite ET routine tuning and your EBVT model.

My statements about all harmonic ET models is, that all (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid.

What i am saying is, that there is no evidence of "Chas EB-ET 12ths and 15ths opposite equal beating" form over other forms of equal temperament, by the arguments from section 4 given in the Chas research report. (Falsification to follow)


Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 08:47 AM

not enough brain time to dive into the understanding now (bu I will try) but large fifths are eventually ugly in the high treble as in the medium range (I mean no precise limit but probably above the minor third major sixth equivalence)

Is the reasoning true to remember that partials only modify the global tone pitch perception), I truly try to avoid listening to partials particularly when tuning fifths.

And I can promise you that I have seen thirds progression that wher giving the impression of a nice progressiveness, while the partials where not at all following the same speed progression if I listen to them used to test more precisely. So what ... ?
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 08:52 AM

Originally Posted By: Bernhard Stopper
.....

My statements about all ET models is, that all (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid.

.....


For me, only piano tuning models that address iH are mathematically valid. This is not the case in models that produce a single semitone or incremental ratio, such as the models you mentioned. As an aural tuning model, they are valid. But not as a mathematical model. There is no application for the resulting ratio in inharmonic theory.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 09:00 AM

Originally Posted By: UnrightTooner
Originally Posted By: Bernhard Stopper
.....

My statements about all ET models is, that all (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid.

.....


For me, only piano tuning models that address iH are mathematically valid. This is not the case in models that produce a single semitone or incremental ratio, such as the models you mentioned. As an aural tuning model, they are valid. But not as a mathematical model. There is no application for the resulting ratio in inharmonic theory.


I agree that this statement needs correction for the piano models. I will correct my statement into "...all harmonic ET models..."
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 09:30 AM

Bernard:

Good enough, although I was hoping to see the math that applies iH. Perhaps the way your program does... (Just Fishing)
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 03:03 PM


This is for sharing a part of my experiences, matured through my professional route. Nobody is forced into this reading. Also, I apologize for my style and please, acknowledge most part of the content as representing only very personal opinions.

I was living in Padua when I stopped my law studies to go to Milan at the age of 21. There I started my technical apprenticeship and, two years later, I started collaborating with the Italian dealer of Yamaha pianos. I could then improve my skills in Paris and Hamburg, with the teachings of the most motivated German and Japanese technicians, so that I could hold for 15 years the position of Yamaha chief technician for all of Italy. Ten years ago, my companion, our two children and I moved back to where I was born, close to the sea and to a very generous nature.

What I happen to have found is a solid theoretical reason for stretching the octaves together with 3ds, 5ths and all intervals, this reason being having to interrelate all partials effects in a practicable ET scale. Today it is Chas Theory's mathematical rule.

All started when I asked myself: why should octaves be theoretically pure? Actually, why should any interval be theorized as beatless? Can zero-beating be reasonable? Yes, I could understand why the ancients could strive for “purity”, how that concept could represent the ultimate inspiration for human being, how it could well express perfection, so much so that we could not resist the application of this concept to numbers and models, as well as to waters, stones, metals, down to purebred animals and human race.

Also the most recent ET models promote a “pure” concept, pure 5ths or pure 12ths, this is fair enough. I’m actually describing what I simply call a pure beating-whole.

Tuning may be looked at as a dilemma, i.e. a circumstance in which a choice must be made between two or more alternatives that seem equally undesirable. In my opinion also, having to choose between pure 3ds, pure 5ths, pure octaves or pure intervals can only end up in a dilemma, but if we think in terms of “pure whole" we may appreciate Chas Theory's approach.

Octaves and 12ths are the real scale’s stretchers, i.e. partial 2 and 3 and they, together with all the other intervals, are able to hold the semitone scale. Now we may well consider a set that is pure, in that all partials and all intervals, in theory as in practice, can contribute to the tonicity and the stability of a beating-whole.

Maybe this concept itself is the most delicate issue, what makes this transition difficult, going from a pure-interval set to a purely beating-whole.

12 root of two ET model did not resolve the conflict known as the comma simply because, keeping up the tradition, it fixed a pure octave. Chas ET Theory resolves this conflict at its roots, and this can be said metaphorically as well as mathematically: I needed to renounce any “pure interval” root, and I needed to use a Delta Difference for partials 3 and 4 roots. This is why in Chas equation you now find 19 root of 3 and 24 root of 4, two roots instead of one single root, so to intermodulate one octave with the next one.

I think that in the comma conflict iH may as well result neutral. What we know for sure is that iH is responsible for the increase of the actual frequencies values. This does not mean that, due to iH, we must stretch octaves, nor that we should have beatless octaves, but only that a beatless octave will not be an actual 2:1 frequency ratio. In other words, we could tune beatless octaves, in which case we would not get the so called ET progression, and iH would still be there, neutral.

Chas Theory describes and supports all this with logic and faultless mathematical evidences. The traditional concept of a pure interval, be it the octave or what ever interval, has litterally kept all this secret, it has obscured the possibility to conceive the 88 sounds set purely as a beating-whole, it has forced us into thinking that there was no way to resolve the partials conflict, that there was no way to simply relate all partials effects in a sound scale, that tuning can only be a compromise.

Some say they manage to use iH for stretching octaves correctly and this may confirm that iH is not a problem, that if anything iH can be of aid. So iH may not be thought as a foe. The “against each other” factors were the partials, since they all can produce the commas, so all together they call for the fairest interrelation.

This is where SBI and RBI took me with their smoothest beat-rate progressions. This is when I finally trusted my musical ear. For many years, while tuning mid-strings, I had used my sense of rhythm for checking beats and my taste, but after unisons and some playing my musical ear was not satisfied.

Then I decided to go for a preparatory tuning and a wider stretch-curve untill, after unisons, my ear could discover a wonderful effect, like if a chorus of hundreds of people was there, all singing an astonishing amount of overtones, and the clearest sound, what in my experience only the clearest water can recall.

A great opportunity was given to me for years, when I had to follow some piano competions, and for up to 15 days I could see the day by day effect of time and heavy playing on my tunings. I had to work at night but that silence was gold. Then I could well evaluate the piano settling down tendency and adjust my preparatory tuning. Then I started thinking in dynamic terms and I accepted the necessity for a wider stretched preparatory tuning, despite what my musical ear would suggest.

Since then, when tuning mid-strings, I may have to go even beyond an aural pure 5th (on mid-string), so that after unisons I can gain Chas tuning form. Some questioning would be fully justified: Which Theory results in an "optimum"? Where is the "sweet spot"?.

In my opinion, the answer has to be given at two different levels, one being relative to us, the other being theoretical. At a relative level, subjective preferences may play an important role. This is why, musically, all temperaments can be valid. At a theoretical level, Chas is the ET Temperament Theory that mathematically relates all partials effects, so putting an end to the commas conflict, the temperament’s age-old problem.

a.c.

.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 05:41 PM

Hello, Alfredo,
I think I begin to get it.

To say it short, what I understand is that you let the justness of the piano be driven by the partials series, which gives a lot of air to the tuning (decompression of the spectra as I call it !).

The defect is that this will depart from justness by a variable point depending of the piano (while indeed giving a guarantee that the partial mix is well evened along the scale, with the same relation reproducing itself.

some scales have moderate ih others no, and one some instruments the progression is not that perfect

I have to hear again that piano, to be honest I have left after the second time because I was not convinced by the harmonic behavior (as said possible twisted hearing because of so called "perfect pitch")

I wish I could follow the maths but aint the case.

Did you try other approaches ?

As you said iH is an advantage, it allow to open the tuning.

I dont get your point with the central string, I guess you tune one doublet, then the other, indeed a guarantee against the usual drift, as it is done when tuning aurally, but the final pitch I am after is the one of the center string.

There are funny things in tuning , one of my colleagues produced much appreciated tunings when he was tuning with the right pedal engaged (as I trained a tad to do). I recall that one day that I had do that way, the piano was really very lively and singing as have been noticed by the engineer and pianist, who asked me what did I do to the piano.

But not a beats method there, out of thirds 4th and 5ths evened in the octave (same kind of thing as you, without the maths behind)

indeed a more sexy theory than any straight approach, but then, what music is it for ? when one need a close harmony dark piano for chamber music how is the harmony ? etc etc.

Your welcome if you come by, to show the way you lay that temperament/tuning, must be fun !

If you mostly worked on Yamahas I understand your need for more lively tunings !! (joking)





Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 05:50 PM

I don't know really if piano tuners worry about that comma (due to the piano iH)

But the basic Yamaha tuning also rely on theoretical basic frequencies,(or beats speed) if you do that too much you can have a bad sounding instrument without noticing as the ear accommodates with time.
I always wondered if the Yamaha approach was an intention to add some deepness to the instrument, by emphasis on close harmony and straight forward lining of 4:2 2:1.
Air quality in Yamaha tone it what misses indeed (comparatively with S....) I'll make friends at Yamaha !


Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/17/09 06:29 PM

I like the way the low basses align to mediums, but i think I understand what disturbs me (a little, in fact).

Tonality impression is not very present.


This is totally driven by the piano, the opposite of usual tuning to beat rates standards, but also the opposite of for instance "pure fiths" method where the fifth is made "pure" in intention to help the inharmonicity to be adbsorbed (if I get the point), hence a layer of justness added to the piano's own justness.

I believe the tuner have to provide a mix between what the instrument seem to want naturally and what he, the tuner, will accept as justness, contrast, harmony.
How to get there while passing by a mathematical model is above my understanding.

I for instance don't like so much the first harmonies at the beginning of the recording, (slow tenths, sing like chinese bells I believe that this low stretch medium lives room for the highly stretched treble - ) the FD sixth that also make that gamelan tone, then it gets better, (some unisons moaning) but no definitively the high treble is not even, when was the piano tuned ? the recording was in winter ?

D6 is low D7 as well, the A's don t line in the treble (out of sequence)

I like the high and medium bass C major harmony , moderate stretch, warm, nice and very well phasing for that small piano.

The harmonic progression generally speaking lack liveliness, (may be coherence) and that I cant get why. may be old hammers/strings and unisson attack quality impede the clarity so the tone is more nasal than you wish.

A professional tuning in anycase, thats not the question.

Indeed some "air" despite a fair use of the sustain pedal.

I am just trying to state what I hear/feel , sorry to forget convolutions, I for that try to avoid any personal attack that is far from my mind please believe that.


As a tuner I believe you get as much servicing the piano that you let the instrument dictate its own justness 100%. That is a very respectable approach, but is it practicable ?


Best regards.






Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/18/09 06:10 AM

Makes me think that the result may not be far from what I had with the tuning made with right pedal engaged (hence immediate stability, and tuning in the natural resonance of the instrument more)

When notes are played together, the partials literally seem to "jump of the box" and the piano sing as if the pedal was engaged. That is the effect I hear in your recording unless you use the pedal more than I think.

Explain also why FBI can be progressive while not responding to usual tests correctly.

The fact that my colleague (now retired, the one that tuned often with pedal engaged ) used a descending octave with a fourth within to place its bass justness (while comparing at each note with major/minor triad in the medium) seem to confirm the use of the double octave an the 12th partials to drive the justness.

btw he tuned his ascending octaves playing an octave containing the upper major third, this guarantee you that you have lively thirds all along the scale.

That said, I recall (on Steinway D) his high basses region was more active with faster tenths than I and others where using (then the low basses where tighter except the last 4-5 notes).

Best regards





Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/18/09 07:42 AM

Kamin, thank you very much for your comments and your deep analysis.

...“you let the justness of the piano be driven by the partials series, which gives a lot of air to the tuning (decompression of the spectra as I call it !).”...

Yes.

...”The defect is that this will depart from justness by a variable point depending of the piano (while indeed giving a guarantee that the partial mix is well evened along the scale, with the same relation reproducing itself."..."some scales have moderate ih others no, and one some instruments the progression is not that perfect”...

I agree, piano scaling can be improved. To improve it, we/they need the most correct and reliable references.

...“I wish I could follow the maths but aint the case.”...

Do not let maths embarrass you. Chas maths is quite straight, maybe a symbol free analogy wiil do (posted June 4, 2009).

...“Did you try other approaches ?”...

I guess you are referring to “justness” which I understand as “euphonicity”. I could compare mine with many other approaches, listening to other colleagues tunings with them there, I had no reason for departing from mine and those Colleagues happened to agree.

..."tuning with the right pedal engaged"...

Interesting, never seen, hope to know more. I do not engage that pedal for tuning, yes for checking the whole resonance effect.

...“I dont get your point with the central string… but the final pitch I am after is the one of the center string.”...

I mute from mid-bass strings-crossing to C6 and tune central strings, then unisons, then C#6’s central, C6’s right, C#6’s left, next central, previous right, next left and so on, up to B7.

...“But not a beats method there, out of thirds 4th and 5ths evened in the octave (same kind of thing as you, without the maths behind)”...

About SBI, in my tuning SBI are progressive and I invert the 5ths beat rate progression in my first sequence steps, so that A3-E4 beats narrower than D4-A4.

...“indeed a more sexy theory than any straight approach, but then, what music is it for ? when one need a close harmony dark piano for chamber music how is the harmony ? etc etc.”...

What a compliment! Let’s hope it will not compete with viagra...they’d kill me! About "a close harmony dark piano for chamber music" could you explain more? You know, as I write, I believe Chas Temperament Theory can represent the correct reference, in terms of partials interrelations, for eventually improve the tempering of other "fixed tempered" instruments.

...“Your welcome if you come by, to show the way you lay that temperament/tuning, must be fun !”...

I hope so, I look forward.

..."If you mostly worked on Yamahas I understand your need for more lively tunings !! (joking)"...

Well, actually you are quite right, some of them are a bit dull but you well know, standards variate from a single piano to another one, also for other makes. Generally speacking, I like St..depth, Bec…silvery, Bos…majesty, but then a Petrof exib. model happened to have much of these qualities...you never know.

...“I don't know really if piano tuners worry about that comma (due to the piano iH)”...

I wish they did not. In any case, in my opinion the approach to theory can be more correct.

..."But the basic Yamaha tuning also rely on theoretical basic frequencies,(or beats speed) if you do that too much you can have a bad sounding instrument without noticing as the ear accommodates with time.”...

If “ear accommodates with time” this may regard also our musical ear, mhhh...when, how, why does ear accomodation take place? Say we do not leave any "place", any reason, can that take place? I do not have the answer, but working with beats for me is like playing with rhythm, that does not tire me.

...“I always wondered if the Yamaha approach was an intention to add some deepness to the instrument, by emphasis on close harmony and straight forward lining of 4:2 2:1.”...

I do not really know but, say you are right, deepness is to any piano's advantage, do you agree?

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/20/09 07:44 AM


Kamin,

about the Chas recording you kindly write:...“Makes me think that the result may not be far from what I had with the tuning made with right pedal engaged (hence immediate stability, and tuning in the natural resonance of the instrument more)”...

I’m really curious about this technique, could you say more about it.

...“When notes are played together, the partials literally seem to "jump of the box" and the piano sing as if the pedal was engaged. That is the effect I hear in your recording unless you use the pedal more than I think.”...

It was not me playing that piano, I asked my colleague Alessandro Petrolati to play it, so that he could have a direct feel of Chas tuning. About the partials, that “jumping of” effect is what I hear too, and I relate it to the resonant power of the tuned frequencies. In other words, by ordering all the partials interrelations, it is like setting “entry frequencies”, so that any string is ready to resonate and give out the relative partials in a sort of coherent share.

...“Explain also why FBI can be progressive while not responding to usual tests correctly.”...

I’m not sure I’m getting your point, anyway...we needed to get the ET ratio for the scale’s 4ths, simply because 4ths with their wideness can proportionate the progressions of all intervals. You may have read the symbol-free analogy (06/04/09) I addressed you to. You notice that we can get all ratios of any kind simply by adjusting the arbitrary “s” variable, what determines the wideness of 4ths.

...“The fact that my colleague (now retired, the one that tuned often with pedal engaged ) used a descending octave with a fourth within to place its bass justness (while comparing at each note with major/minor triad in the medium) seem to confirm the use of the double octave an the 12th partials to drive the justness.”...

Yes, the use of the double octave an the 12th works for “justness” and, if you were to adopt Chas Theory’s rule, i.e. equal beating all along the keyboard, up to the 88th tone, it would work as a precise, solid and reliable reference for tuning an ever coherent and univocal SBI and FBI progression. This can now be called ET, in that it is what we mean when we say ET, what we needed for approving ET, both in terms of justness and practicableness.

You know Kammin, in my pro experience I have met so many tuners and very often felt their embarrassement and frustation, wondering about an abstract idea of ET and yet having to somehow argue and justify their quasi-ET tunings. Out of interest, about writing styles, octaves, ET definition and more, have a look in this thread, from 06/21/09, you only find confirmations of what I’m saying. Lots of confusion and approximation that turns in frustration and vivid wits. Little tuning, but lots of acrobatics.

About the recorded Steinway s, strings, pins and hammers (still to be voiced) had been changed three weeks earlier…when I knew that...I thought I was going to waste my time...and yet it did not go bad at all. My colleague was astonished for what he called “light”, brightness. Yet, on those small instruments, iH is still the argument for “relative”, quasi-ET tuning.


Best regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/22/09 05:57 AM

Tooner, you may understand why I do not feel that relaxed, why I do not manage to behave as I could and reply to you.

Leave your representation of Chas Theory aside, your personal understanding about theoretical models and tuning unsolved problems, how you and our cello expert have tried to banalize the Chas discovery and the research report, leave all this aside, I feel embarrassed mostly because I can not trust your intentions.

Nothing drammatic, but you may know the way it goes, when something you could never expect actually happens and it takes you beyond your immagination.

I would like to tell you how I’m grateful to you for giving me many (very many) opportunities to describe Chas, for helping me with my English, for offering me a chance to deepen many relevant tuning issues but, only considering my own limits, it happens that you (and Stopper) have gone over the top (my top).

Actually, if I were to reply to you I would have good reasons for asking myself: do I suffer from martyr complex? Am I looking for more insults? Is this me being a masochist?

Fortunally I feel quite balanced, with my family, my job, my friends and my hobbies. Also I deeply know that I’m trying to share Chas only because I sincerely believe in what this temperament may represent, how it resolves the partials order and how and why it accurately gains what other researchers and I have long been looking for, not an easy method but the scale partials interrelations. I have got no words to tell you how I feel about it.

I tend to believe that in the U.S. you may have different standards, when discussing with people you do not personally know. Maybe you can tell each other orrible things - look at the other thread - and go, the next day, for an aperitif, but it is not so where I live, nor in my environment. In my family we respect people’s intelligence, no matter the origins, and behave in the effort to be sincere, to result univocal, reliable and trusty.

I respect you (and Stopper) at a human level, only I do not manage to rely on my sporting spirit, to be casual when it comes to coherence and truth. Said that, I apologize for my limits and I sincerely wish you a Merry Xmas.

a.c.

.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/22/09 07:12 AM

Alfredo:

Sincere wishes for a Merry Christmas to you also.

My Post with the link about the Martyr Complex was in response to your post with links on Moral Turpitude, Defamation and Intent. What is sauce for the goose is sauce for the gander.

You can blame my actions on my former 24 year career of going to sea if you like. There were many times when we would want to kill each other one moment and then desperately need each other the next. (I have some interesting scars...)

You can also take criticism as constructional whether it is meant to be or not. If it is false criticism it can be ignored. If there is merit to a criticism, regardless of the intent or origin, it can be used in a positive way. But Pride must first be dealt with, which is much easier said than done for all of us.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/22/09 07:43 AM


..."You can also take criticism as constructional whether it is meant to be or not."...

I'm not referring to accidental constructural criticism, but to deliberate destructive actions.

..."If it is false criticism it can be ignored."...

I can not ignor a defamer, maybe you can and that is good for you.

..."If there is merit to a criticism, regardless of the intent or origin, it can be used in a positive way."...

Yes, to a certain extent though (in my case), then it turns into moral and intellectual damage and it can not go "regardless" anymore, the intent becomes foundamental.

About having to deal with Pride, by reading Chas Topic you may understand how I do not mind some training. As I tell you, I do not know anymore if you do not understand or if you pretend not to understand. But now that is only my dilemma.

..."What is sauce for the goose is sauce for the gander."...

Perhaps is that sauce I'm simply not interested in.

a.c.

.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/22/09 07:50 AM

(--------------- * MERRY XMAS *
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/22/09 07:59 AM

Alfredo:

"About having to deal with Pride, by reading Chas Topic you may understand how I do not mind some training. As I tell you, I do not know anymore if you do not understand or if you pretend not to understand. But now that is only my dilemma."

I patiently tried to show you some things but decided to stop when I got too aggravated. It is only normal for such frustrations to be felt personally - for both of us.

Yes, Merry Christmas and remember that this is a Season of Promise that one day All will be made Right.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/22/09 08:50 AM

Right.

a.c.

.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/23/09 10:56 AM

Alfredo:

Found a Proverb you can consider: "Faithful are the wounds of a friend; but the kisses of an enemy are deceitful." Proverbs 27:6.

I am not your enemy.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/23/09 06:01 PM


"Faithful are the wounds of a friend; but the kisses of an enemy are deceitful."

Thanks Jeff, I share the meaning of those words. What is there, in between a friend and a not-enemy?

a.c.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/24/09 08:29 AM

An acquaintance. I would prefer colleague, but it just did not happen.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/24/09 04:18 PM

I'll post the link in the Chas thread...

http://dl.free.fr/bbsIDOhsS (it is a wav file may be long to download, I will try to make a MP3 for the unisons tuning)

here is what I get , the basses I only used fourths, the treble only large fifths, the temperament I used a 2bps fourth and a low stretch octave, then I tried to stick to that interval size.

In the end some double octaves beats too much, the triples sounds Ok, it may be long from the Alfredo tuning but I forget the instructions at home !

Back to the champaign !
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/25/09 07:39 AM


Kamin, I can not listen to your recording due to a problem in the audio device on my PC.

Thanks so much for your trial, finally you are putting so many words into practice. I would not be happy with a short reply, so please wait...just a few bottles...

a.c.

.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/25/09 03:17 PM

Alfredo, it was not a real trial, I did not have the precise instructions so it stay where my understanding was, thats it, not much !

I have something I dont get with your method : the ih curve is raising in the low medium with the plain wire shortening, then the high basses have low ih that progressively goes up when we go down the scale.

is your beat "inversion" on the low slow beating intervals taking that in account, or is it a model based on theory ?


In the low medium area, I see no way to reconciliate intervals while sticking to a model unless you dont care with some of them. if you stick with an even progression of FBIO you get an eveness that is near theory.

If you stay within some partial reference, the progressivness change speed, more or less brutally depending of the way the tuner compromise to hide those scaling defects.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/26/09 02:32 PM

Kamin,

you kindly ask: ..."is your beat "inversion" on the low slow beating intervals taking that in account, or is it a model based on theory ?"...

About practice: the 5ths beat rate inversion (on centre strings), in between A3 and A4, is technically needed to gain Chas. From A0-E1 (were I avoid 5th’s beat) upward, 5ths get narrower and narrower, 4ths and octaves less and less wide. C3-F3 and C3-G3 (on centre strings) have the same beat rate, then 4ths get wider and wider, 5ths keep on getting narrower and narrower ‘till they invert (on centre strings). To invert 5ths, do so that G#3-D#4 and A#3-F4 have the same beat rate.

About theory: first I had to consolidate my tuning evidencies, then I kept those evidencies under observation for two decades, them being the result of a precise approach, on which basis I could elaborate Chas Theory. Finally I extracted the model's constants and gained the mathematical evidencies. Hardly ever iH has been a problem, never on concert pianos.

..."In the low medium area, I see no way to reconciliate intervals while sticking to a model unless you dont care with some of them. if you stick with an even progression of FBIO you get an eveness that is near theory. If you stay within some partial reference, the progressivness change speed, more or less brutally depending of the way the tuner compromise to hide those scaling defects...

In the low medium area, on badly scaled pianos you may need to favor some intervals, in that case I go for RBI’s smootheness and coherence. More often I go down the bass with 4ths, 5ths, octaves, 10ths, 17ths, 24ths (triple octave + 3rd) all together. If I’m forced to sacrifice one interval, it will not be a fifth. Mind you, I’m talking of very fine iH troubles and in those cases I prefere a wider octave to a narrow fifth.

I’m still waiting for an answer from Bill, in general about the existence of ET (and what all people is then talking about). Chas ET does exist, I can tune it on any size piano (what makes it shareable) and the resulting sound whole is so right, so resonant and musical that usually dissonant chords open to overtones in a way you could never predict (my judgement, but not only mine). Kamin, for me it is a true sound festival, it’s like a real overtones bath.

Bill writes:...“If I were only tuning ET...I might want to get some kind of variation from what would otherwise be a completely sterile sound.”...

What does Bill mean? What is a "completely sterile sound"?

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

.

Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/27/09 02:02 AM

Hello Alfredo

Thanks for your answer, indeed even on concert pianos there is an iH jump, even if smaller.

please confirm your idea the term narrow fifth , and wide octave. more than 1:2 et 2:3 or is it tuner's language (faster beating than usual) ?

I dont know what Bills mean with "sterile sound" , but it remains me some pianos that lack projection, then the right pedal may be on to blend a little, and hamony is poor.

I find the too precise progressivness tiring when it is done, I believe that the tuner follow the tone the instrument is giving at the moment of tuning,he modify the stretch demending of the room and the piano acoustics. That is giving some movment in the progression, tempered with the use of RBI's, I have tuned with a VT100for years, thet EDT is very healthy and his advantage is that he will compute the tuning based on many partials of the 88 notes. In the end it makes so much compromizing that the tuning is near the perfect moderate Yamaha concert tuning, but without the liveliness so the tuner have to keep his ears open and not rely 100% on the machine if he want to keep some (and for Steinwya he have to modify the program guidelines, in the end I only used it as an examinator, and when it have been robbed to me I just did not car eto buy another one it is a tool to make money, but I disovered I could make Pitch raises in an efficient manner because my ears had catched the kind of necessary raise ..

That said one partial driven tuning is also nice from the low partials harmony point of view, but the method is heard as a forced eveness, so I am unsure it is the best choice for a piano.

You can install on a piano any kind of method, that will be more or less transparent to the listener.
The piano have its voice, I believe it dictate what he can and what he cannot do, if forced in a differnet justness as pure fifth for instance, this is not transparent to the listeneer, it bring its musical ear in another concept of harmony.

Indeed sticking to the partial series will give a jump in resonance, and it is very pleasing. In the end the classical harmony is what may be the most difficult to attain, when the piano play with another instrument as a flute, or even a vilin (then the top treble may be high enough to please the vilolonist)

WHat makes Bill with the mindless octaves makes me think of the day where the conductor told me (half an hour before the rehearsal) that the pitch had to be 440 (while most of our concert pianos where tuned to 442 Hz).

I had to lower the A4 exactly to 440 , the A3 less, all the remaining temperament fifths less and less, then make a few octaves high and low until it reconciliates with the 442 tuning (at last it could be audible)

Nobody complained, and the tuning was strange but playeable !

I wonder if the "mindless octave" equity between 15th and double octave departs from the tempermanent, and even it a little when going to the treble and bass (may be no ?)

Back to eating now !!!


I'll give a try with the instructions, but I need to understand the logic too.!


What I agree with is that all octaves are false in our usual way of tuning !!


Best







Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/27/09 01:41 PM


Hello.

Kamin, thank you:....“please confirm your idea the term narrow fifth , and wide octave. more than 1:2 et 2:3 or is it tuner's language (faster beating than usual) ?”...

Octaves, 4ths and 5ths can be progressive and in Chas tuning they are progressive. I will explain in detail how to deal with these intervals in Chas Preparatory Tuning. Midrange octaves do not beat, they have a propensity (or propension?) to beat, in fact they roll, i.e. they open towards a beat like for a long long wave, what you hear as a mmmmuuuuoooooo coming in a shorter and shorter time, as you move toward the extremities. Octaves progression is so smooth and gentle that it takes (about) 40 notes to get 1/bps. You ask me to compare with usual, but I do not know if a “usual” exists, for what I’ve heard and from what I read there seems to be an infinite variety of ways.

...“I dont know what Bills mean with "sterile sound" , but it remains me some pianos that lack projection, then the right pedal may be on to blend a little, and hamony is poor.”...

I do not think Bill was referring to pianos, but tunings.

...“I find the too precise progressivness tiring when it is done, I believe that the tuner follow the tone the instrument is giving at the moment of tuning,he modify the stretch demending of the room and the piano acoustics.”...

Physically tiring or musically tiring? In the first case, precise progressions are a real challenge every time, as in any sport where the highest precision is required, here would be with beats and rate/rhythm. Musically, precise progressions are as tiring as perfect “in tune” can be, for me. I enjoy a perfectly “tuned” singer, violin and so on down (or up) to pianos. I enjoy being able to give myself up to a perfectly melodious voice, it goes straight to my deepest inwardness, it makes me feel in the most right and fair “place”, it makes me forget my physicalness, it opens to a dimension in which I wonder upheld and safe.

You say “...I believe that the tuner follow the tone the instrument is giving at the moment of tuning...”, I wish it was true. I tend to be more realistic and, for what I’ve seen and heard, I’d say: the tuner follows the tone given by his/her own chemistry and skill, he/she modifys the stretch depending on his/her inner room and the piano acoustics, for how he/she manages to care about.

...“That said one partial driven tuning is also nice from the low partials harmony point of view, but the method is heard as a forced eveness, so I am unsure it is the best choice for a piano.”...

Kamin, it will depend. One may prefere to look through a glass, I love looking through crystal. One may prefere potatos for a game of bowls, I prefere spheres.

...“You can install on a piano any kind of method, that will be more or less transparent to the listener.”...

You see, I love variety in timbre and colour, but it has to be in tune.

...“The piano have its voice, I believe it dictate what he can and what he cannot do, if forced in a differnet justness as pure fifth for instance, this is not transparent to the listeneer, it bring its musical ear in another concept of harmony.”...

Chas does not describe pure fifths nor pure 12ths. Chas Preparatory Tuning does (on centre strings) and that is the way you get there.

...“Indeed sticking to the partial series will give a jump in resonance, and it is very pleasing. In the end the classical harmony is what may be the most difficult to attain, when the piano play with another instrument as a flute, or even a vilin (then the top treble may be high enough to please the vilolonist)”...

Today I can state that the “in tune” line (or level) is perfectly shareable and, by comparison, nobody would go for a lame tuning, as original or unique as it could ever be. If the musician can not compare, then he/she will more or less accept what is given.

...“WHat makes Bill with the mindless octaves makes me think of the day where the conductor told me (half an hour before the rehearsal) that the pitch had to be 440 (while most of our concert pianos where tuned to 442 Hz).”...

Bill uses 12ths and 15ths equal beating for expanding his temperament, up to some point where he has to compromise, due to his own temperament in the midrange.

Chas equal beating 12ths (narrow) and 15ths (wide) is the sound whole’s symmetry Rule you can aim at, by adopting a preparatory tuning. So, Chas Theory’s final form and practical tuning is 12ths and 15ths all along the keyboard. You will not go for that though, because Chas tuning can only and will only result from your preparatory tuning.

...“I'll give a try with the instructions, but I need to understand the logic too.!”...

Thanks a lot. Tell me where I can help.

...“What I agree with is that all octaves are false in our usual way of tuning !!”...

That “false” is only the result of lack of reliable references and approximations, octaves simply need to be truly progressive.

Best regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

.


Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/28/09 12:54 AM

Hi ALfredo,

I'll send you links to boring tunings, some lack of stretch, some are too straight against the 12 th in the medium then change method so they have incoherence somewhere.

Yes working a tuning may be tiring, but rewarding as well, nothing new !.

In the end I have find the too much evened tunings boring, if the eveness is only due to a bunch of compromising it finally loose sense , and the ear better like a clear opinion as long as it is used on the whole instrument.

for instance the tuner is obliged to refrain to use extra large tuning in the medium range, because he will then have problems with the high treble, or he tunes very tight the medium and cant find an opening to tone higher.

Anyway, whatever way is used, if it is done in a musical way, the result will be nice (of course the piano may give the impression to be perfectly singing just)

Have a nice day !

I am impatient to read your instructions in the "preparatory" tuning, seem very esoteric to me up to now !
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/28/09 06:59 AM


Kamin,

I thank you for giving me an opportunity to look at the meaning of “esoteric” in depth. I understand your issue and this may explain my pleasure to make this concept clear, for me, you and our readers.

Is Chas Tuning esoteric? These are the common meanings:

(http://www.thefreedictionary.com/esoteric)
(http://www.answers.com/topic/esoteric)

1 Intended for or understood by only a particular group
2 Of or relating to that which is known by a restricted number of people.
3 Confined to a small group
4 Not publicly disclosed; confidential.
5 Restricted to or intended for an enlightened or initiated minority, esp because of abstruseness or obscurity
6 Difficult to understand; abstruse
7 Not openly admitted; private
8 Beyond the understanding of an average mind

These are my points and comments:

1 Yes, it is intended for tuners and musicians.
2 No, it is not restricted to aural tuners and musicians.
3 Large or small, it depends on what you compare the group with.
4 No comment needed.
5 Chas Theory and Tuning is not restricted nor exclusive at all.
6 How difficult to understand is subjective, it may also depend on knowledge, personal interest and prejudices.
7 No comment needed.
8 I never think in terms of average mind, in my opinion each single mind can open to new scenarios, then it may depend on the “key” and personal experiences, targets and chances.

Now, what does a tuner or a musician need to open to Chas “in tune” ET Temperament Theory and practice? He/She needs to acknowledge about:

-3 (optional) Partial lengths of a vibrating string
-2 (optional) Prime numbers and partials conflict (commas)
-1 (optional) Geometric progression (in relation to our aural system and the scale incremental ratio)

±0 Scale intervals and constants, i.e. Chas system's rule
+1 Piano adjustment following strings tensions and loading
+2 Strings adjustments following piano playing
+3 A method (any method) that uses both SBI and RBI to gain progressive SBI, RBI, progressive octaves and the two Chas system’s constants, i.e. 12ths and 15ths opposite equal beating all along the keyboard.

Now, consider what Bill Bremmer publicly reports:...“In the end, ET cannot exist either. The theoretical specification for each pitch is an irrational number which can never really be tuned. We have to round it off somewhere. The limitations of the piano and inharmonicity together with human variation and perception render every temperament inherently unequal.”

Consider also that the common approach to tuning enumerates these all variable factors:

The way the piano sounds (piano quality)
The acoustic of the room (air pressure)
The kind of music you play
Melodic and harmonic personal preferencies
Piano scaling and inharmonicity issues
Changes in climat

And that, due to these factors, we (aural tuners) could not share a reliable and practicable model*, I may ask you and all Colleagues to simply compare: which approach would you say is “esoteric”? Which approach ends up being more mysterious, or arcane or inscrutable?

*How do you tune 4ths and 5ths? How do you tune octaves? How do you tune 12ths and 15ths on 88 keys?

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 12/31/09 09:48 AM


What I wrote in November 12, 2009:

..."Where to go then to oxygenize this discussion? This are my proposals:

a) Let’s distinguish the general meaning and relevance of a temperament theory, what Chas ET-EB is, from what piano tuning’s issues are

b) Let’s analyse what logic is behind each one considered theory

c) Let’s separate iH issues from beats-control and tuning-form issues

d) Let’s see if we can address aural piano tuners towards a more reliable and practicable model

e) Let’s try to clear up how octaves, SBI and RBI intervals should go, and get rid of all misteries

f) Let’s evaluate if today there is a way to reduce approximations relative to iH and piano scaling"...

I still think this is what we could well do, we all together, by using our best intensions and by giving up all dogmas, all absolutist positions and all kind of prejudicial aversions.

This is amongst my hopes for the coming time. To All

-(-------------------------2009 * HAPPY NEW YEAR * 2010

.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/02/10 02:51 PM


Hello, I hope you all have had a good time!

Bill,

today I read your Topic "Aural Octave Tuning" where you write:

..."Sharpen F5 slightly until both the double octave and the octave and fifth have virtually the same quality. Neither interval will have much of any audible beat. They will both sound apparently or very nearly beatless yet the double octave will be slightly wide and the octave and fifth will still be slightly narrow. You can continue this very same procedure to C8….You may continue comparing the double octave and octave and fifth all the way to A0."...

I’m glad you are opening to the Chas ET "mathematical rule", opposite 12ths and 15ths equal beating all along the keyboard, from A0 to C8.

This somehow repays my efforts and my attempt to share what I think is the most natural and practicable ET scientif model. I dear to say “natural”, in that Chas resolves mathematically the age-old conflict amongst all partials, and I say “practicable” in that Chas is the result of a new theoretical approach and research, what took me to my practical tunings.

You talk about EB 12ths and 15ths as a "mindless procedure", so you write: ...“you CAN continue this very same procedure to C8”...

Chas research report describes how EB 12ths and 15ths can produce a perfectly resonant, stable and symmetric beating-whole. So I can write: My aim is Chas “optimum”, and I gain it by adopting a preparatory tuning. For me this is "mindful ET tuning", and maybe this is today a part of the distance between us.

I'm sure you will soon acknowledge that 12ths and 15ths never need to invert, that 5ths never need to become wide, that octaves need to be progressive and are strictly tied up (also) with the first octaves we tune.

I hope it will not take long for the PTG to acknowledge Chas ET scientific relevance, as I would really like to share this discovery with aural tuners first, i.e. our Colleagues first, all over the world.

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

.






Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/03/10 04:50 PM

Hello Alfredo,

I have tried to make a temperament and tune a few octaves with your method , and could not get thru.

Something I don't get with the size of the octave probably.

Which interval is determining the size of the first octave, is it the 4th/5th of the first step ? this will vary depending of the piano (as always)

what you call octave "barely beating is tested with which intervals ?)

the 4th and its relation to the 12th above is driving the tuning I suppose, then may be you should propose some checks that could be used as soon as possible to ascertain the primarly interval size (octave and 4th I suppose again).

Usually the iH tend to lower the speed of the thirds in the low temperament zone. it also may make the 5ths slower, I am unsure to understand if the inversion of the fifths beats relate to that or not.

I wonder also something, are you tuning pianos actually ?

If so ,could you post some recordings of the tunings (I mean intervals, not just music played as on your sample).
Even if you could record yourself tuning your temperament I am pretty convinced that it would be a big help in understanding. Do you have a little recorder ? (I have a Zoom H2 which is not expensive and is sufficient for that kind of thing )

BTW I dont understand which is that "grandpianoman" tuning of your sample. did you tune that piano yourself ?

Sorry but I am a little lost with the instructions.

I'll give another try when I'll be more sure to do what you propose.

Happy New year to you also, BTW !

Best








Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/03/10 05:29 PM

Hello Bernhard,

I have been looking for your temperament instructions , while I seem to have it at some moment, I cant find it actually.

Do you know a link where it is explained ?

I recall trying to tune based on pure twelves, (3:1) and I liked the 3-4 medium octaves, then I had problems to keep the progression and have the usual amount of stretch, or the relation with the medium disturbed me.

Thanks for the link, if any.

Best wishes for 2010

Isaac OLEG
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/03/10 05:36 PM

Originally Posted By: Kamin
Hello Bernhard,

I have been looking for your temperament instructions , while I seem to have it at some moment, I cant find it actually.

Do you know a link where it is explained ?

I recall trying to tune based on pure twelves, (3:1) and I liked the 3-4 medium octaves, then I had problems to keep the progression and have the usual amount of stretch, or the relation with the medium disturbed me.

Thanks for the link, if any.

Best wishes for 2010

Isaac OLEG




Isaac, i don´t want to hijack this thread, which is related to Chas.
Please mail me a pm.
regards,

Bernhard
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/03/10 06:10 PM

OK sorry, I did not realize !

Your explanations make sense, I did not understood the formulas in the Chas paper, but I understand your criticism if it rely to what is expressed in that paper.

I guess that wanting to mathematically put in formula what the tuner is doing is probably very difficult, and that beauty may be hidden in the contortions we do to get rid of that pythagorean thing (with the help of the instrument hopefully).

On more pure instruments as harpichords, or guitar, expressing differntly ET may well be interesting, as may be for low iH pianos.

I am pretty sure that the "Railsback curve only or mostly used Steinway's as references. As I tuned with an EDT many years I have seen the cts value of the notes, being the center octaves or the bass or high treble, always computed with the same model. They vary wildly as long as you want to keep the same beats relations.

Difficult to have the high treble of a Fazioli at more than 25cts or the one of a Bechstein at less than 35 cts.

Then, when tuning aurally, the 3 different kind of instruments will be tuned completly differently in fact, one with a large octave in the temperament, the other with a very tight octave because it get harsh very soon if not.

If that comma is solved on a larger distance, it decompress the ih of the piano and the eveness is easier to attain, but we are use to that high treble artificial stretch not for the piano acoustical reasons but because of our ears. SO to me, a good method may prepare the instrument to accept the stretch of the high treble so it is there while not too much out of justness with the center.

I cant understand how mathematically we can include that.

is not the spectra of the piano progressing more or less unevenly and asking for many compromising while tuning ?
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/03/10 06:29 PM

Originally Posted By: alfredo capurso

Chas research report describes how EB 12ths and 15ths can produce a perfectly resonant, stable and symmetric beating-whole. So I can write: My aim is Chas “optimum”, and I gain it by adopting a preparatory tuning. For me this is "mindful ET tuning", and maybe this is today a part of the distance between us.

I'm sure you will soon acknowledge that 12ths and 15ths never need to invert, that 5ths never need to become wide, that octaves need to be progressive and are strictly tied up (also) with the first octaves we tune.

I hope it will not take long for the PTG to acknowledge Chas ET scientific relevance, as I would really like to share this discovery with aural tuners first, i.e. our Colleagues first, all over the world.




As promised, more critical reviewing of Chas statements:

1. - Falsification of the evidence of Chas equal beating double octave - duodecimes (twelfths) equal temperament form over other equal temperament forms:

In section four of the Chas research report, difference quotients are calculated from differences of Chas octaves series which are compared with the harmonic octave series.
Other partial series like 3,9,27... are left out from the difference calculations, only partials related to the octaves are taken into account, so one can say "filtered" data has been used, to evidentiate the model.

Chas model compromises the pythagorean comma equally between the double octaves and duodecimes. That means octaves are half weighted by pythagorean comma distribution than duodecimes. Saying a natural interval ratio like 9/8 (which contains 2 duodecimes, (3^2) and three octaves (2^3)) is closer to the Chas ratio than to other ratios, does not mean more than saying the 12th root of two temperament is better than 19th root of three, because the octaves are better in 12th root of two.

The evidence of Chas equal beating form over other forms of equal temperament fails by taking differences of multiples of other harmonic series like 3,4,5… instead of only the second partial. In fact, some intervals are closer to the natural ratio in one equal temperament, while other intervals are closer to the natural ratio in other temperaments. This is all not new and also not solved with the Chas equal beating temperament.

2. Falsification of the statement (section 4.6) "This inversion* is determined by the s variable unique to this model"
(* inversion of interval progression, as observed by the Chas author when tuning real pianos, and "predicted" by the Chas model)

The s-variable has no evidence at all for this statement here, as the complete presented data tables presented in the Chas paper where this statement relates to, are all based on the harmonic case, where the s variable has a value of 1. Chas ratios for all intervals remain constant over the whole keyboard and do not "invert" somewhere when the s variable is 1 as with the presented data.

In fact, the predicted inversions are caused due to a wrong method to determine interval ratios by multiplying their numerator alone with the multiples of the second harmonic. To determine interval ratios an octave higher on the scale, one has to multiply the numerator AND denominator of an interval by two, not the numerator alone for example.

3. Falsification of the statement (section 4.5) "If in a different logarithmic scale, we wanted to favour partial 3 we would have to take value 3 and position 12+7 = ordinal 19, so the formula will be 3^(1/19) = 1.059526065... This ratio, too, in distances of octaves, (3*2)^(1/31), (6*2)^(1/43) etc, modifies towards 2^(1/12)."

Calculating higher harmonic degrees of an interval by multiplying with multiples of the second harmonic does not correspond to the interval ratios of a tuning at higher scale position. The correct formula for the 31st step is 3^(31/19) and not (3*2)^(1/31), and therefore does not modify towards 2^(1/12) but remains constant for the harmonic case. Same is for the wrong statement relating the partial 5 temperament.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/04/10 11:16 AM


Hello Kamin, you kindly ask:

..."Which interval is determining the size of the first octave, is it the 4th/5th of the first step ? this will vary depending of the piano (as always)...

...what you call octave "barely beating is tested with which intervals ?)"...

In the first four steps I determine the relations between two 4ths (A3-D4 and E4-A4), two 5ths (A3-E4 and D4-A4) and A3-A4 octave. So, all together they determine each other. In midrange iH is very low so it is no problem.

...the 4th and its relation to the 12th above is driving the tuning I suppose,"...

Going up, from A4, I use 4ths (going wider), 5ths (less and less narrow) and octaves (slowly progressive). On C#5, I can check five progressive octaves, fiths progression (less and less narrow) and the first 10th (A3-C#5). Then I use all intervals, slow and fast beating intervals.

..."then may be you should propose some checks that could be used as soon as possible to ascertain the primarly interval size (octave and 4th I suppose again)."...

You'll find that written in the sequence.

..."Usually the iH tend to lower the speed of the thirds in the low temperament zone. it also may make the 5ths slower, I am unsure to understand if the inversion of the fifths beats relate to that or not."...

I tune with 2:1, 3:2, 4:3, 3:1, 4:1, 5:2, 5:1 ratios, and iH has never been a problem. Remember that 5ths inversion is established on centre strings, so that A3-E4 is narrower than D4-A4, little less than 2/3 bps Vs 1/3.5 bps. I mute from mid-bass strings crossing up to C6.

..."I wonder also something, are you tuning pianos actually? If so, could you post some recordings of the tunings (I mean intervals, not just music played as on your sample).
Even if you could record yourself tuning your temperament I am pretty convinced that it would be a big help in understanding. Do you have a little recorder ? (I have a Zoom H2 which is not expensive and is sufficient for that kind of thing )"...

I may be in Paris by the end of this month, I'll let you know...would you be there? It would be much easyer to directly show you the procedure.

..."BTW I dont understand which is that "grandpianoman" tuning of your sample. did you tune that piano yourself ?"...

Yes, I tuned that Steinway s in front of our colleague Alessandro Petrolati, he could then play and record the piano. Grandpianoman was so kind to make an MP3 available, out of Chas Tuning .rar original version. This is why I mention him.

Best regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/04/10 04:16 PM


Bernhard Stopper,

I’m very happy with what I have described in Chas research report. Also I’m quite happy with your fast acknowledgement of Chas model’s relevance. In fact, only seven months ago you dared to write:

Bernhard Stopper on Chas Topic, 06/09/09:

...“My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not.

...“You WILL find symmetries of highest degree in the resulting beat tables, which are not present in other ET solutions (including yours). And THAT is a discovery (discovered in 2004, as is wrote in an earlier post of this thread).

...“Such relations are apparent in many ET solutions, but only in the case of 19th root of three this is true for all combinations of octaves fifths and fourths (with distances of octaves, fourths and fifths between them) etc.

...“The point is, that the acoustic effects caused by the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.

Two months ago you could write:

Bernhard Stopper on Chas Topic 12/11/09:

...“An "s" term alike element, which enables the CHAS model to take every size, is generally implemented in tuning programs by additional terms to hold for stretch caused by inharmonicity for example, also nothing new here.”...

What is new is that from Chas equation you can gain any ET scale's ratio and correction, and now you can “understand” that. Last month you wrote:

Bernhard Stopper on Chas Topic 12/17/09:

...”My statements about all harmonic ET models is, that all (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid..."...

You see Stopper, now you “understand” Chas mathematical validity. Soon you will also “understand” Chas model’s maximum coherence and congruence. I simply believe Time is enough to defeat your malignity and you yourself confirm that.

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/04/10 05:28 PM

I miss some substantial defense on my falsifcations of your paper.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/04/10 05:55 PM

I think you are missing your brain, Stopper, something you will not find in this thread.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 07:42 AM

Originally Posted By: alfredo capurso
I think you are missing your brain, Stopper, something you will not find in this thread.


Well, this clinches it for me, Alfredo. You are interested in appearance not substance. But then you are not alone.
Posted by: Kent Swafford

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 08:47 AM

Quote:
I hope it will not take long for the PTG to acknowledge Chas ET scientific relevance, as I would really like to share this discovery with aural tuners first, i.e. our Colleagues first, all over the world.


Quote:
Soon you will also “understand” Chas model’s maximum coherence and congruence. I simply believe Time is enough to defeat your malignity and you yourself confirm that.


Quote:
I think you are missing your brain,


You continue to expect acceptance, and yet you also continue to become abusive when questioned.

It is tempting to respond to you in the same manner as you yourself just responded. Time for me instead to simply log off.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 02:13 PM

Originally Posted By: alfredo capurso

Also I’m quite happy with your fast acknowledgement of Chas model’s relevance.

We probably have a different perception of the reality regarding this.
Relevance and validity are two slightly different things.

Originally Posted By: alfredo capurso

What is new is that from Chas equation you can gain any ET scale's ratio and correction, and now you can “understand” that.

You can gain this with Mazzola's general tuning formula as i mentioned already.
Again, Chas model is valid here, but not relevant as new.

Originally Posted By: alfredo capurso

Last month you wrote:

Bernhard Stopper on Chas Topic 12/17/09:

...”My statements about all harmonic ET models is, that all (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid..."...

You see Stopper, now you “understand” Chas mathematical validity.

That´s what i was saying already. Valid yes, new or relevant no.

I figured out too, that the model of the natural form of the fifth circle equation i am using to illustrate my tuning model can exactly take the Chas model form for the harmonic case. By replacing the constants with variables (which is equivalent to Mazzolas general model) every inifinite scale can be done as with Chas.

Originally Posted By: alfredo capurso

Soon you will also “understand” Chas model’s maximum coherence and congruence.

I have falsified your statements regarding this and your are obviously not able to defend against these falsifications.

Originally Posted By: alfredo capurso

I simply believe Time is enough to defeat your malignity and you yourself confirm that.

No comment.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 06:03 PM

Jeff, try to be happy with what follows and try to refrain from such comments: "You are interested in appearance not substance."...

You have had almost 8 months of "substance".

Kent, you write:

...“You continue to expect acceptance, and yet you also continue to become abusive when questioned.”...

Kent, I’m not expecting acceptance, I’m hoping to share Chas ET Theory and its tuning model, i.e. an ET temperament where octaves are a little wider than 2:1.

You say I become abusive when questioned but sorry, I can not agree, if anything the opposite is true. Chas Topic is full of examples where I’ve been questioned in abusive terms, nevertheless – to a certain extent – I’ve replied as usual.

This is so evident that I find difficult to justify your comment. Actually, since you could check what I’m stating - by simply reading this Topic - I would kindly and respectfully ask you to refrain from passing easy or superficial judgements. Otherwise, you could start a dedicated Topic.

In my opinion, and personally speacking, Stopper’s case is quite unusual. Leave all style factors aside, leave his off-putting manners aside (educational, cultural, moral barrier?), Stopper has been trying to banalize Chas ET Theory right from his first “cello” post, insinuating that Chas research report may be a scientific hoax. Read Stopper (05/07/09, page 1):...“Well observed Jeff. Maybe a kind of scientific hoax of the category "cello scrotum"...”.

Why in my opinion Stopper may be missing his brain (missing his elaborations) is quite simple: Stopper contradicts himself. Just read, this is not maths, but basic English:

STOPPER on Chas Topic 06/09/09:...“You WILL find symmetries of highest degree in the resulting beat tables, which are not present in other ET solutions (including yours). And THAT is a discovery (discovered in 2004, as is wrote in an earlier post of this thread).

The same day:..."Such relations are apparent in many ET solutions, but only in the case of 19th root of three this is true for all combinations of octaves fifths and fourths (with distances of octaves, fourths and fifths between them) etc...

..."My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not.

So, that was Stopper and his "...highest degree...",and "...only in the case of 19th root of three this is true..." and "...perfect symmetry...".

Now read this, STOPPER on Chas Topic 12/17/09:

...”My statements about all harmonic ET models is, that all (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid...”...

Even if I leave Stopper’s contradictions and style aside, there are three more points that reveal such a conversation partner’s unreliability.

1. Refuting Chas ET model’s maths and numerical evidences.
2. Confusing Chas approach and ET Theory with a technique to expand a temperament.
3. Claiming about who is first and what is new.

About point one, you may understand that Chas model’s maths is not different from any geometrical progression’s maths. Maybe you, Kent, do not know enough but Stopper may well know.

About point two, a technique can help you to get by, a solid and reliable ET Theory can represent a rational and precise reference. Again, you Kent may not be familiar with this, but Stopper may well be.

About point three, I have well explained what is new about Chas, and it is listed in my first post (page 1). But what is more important about Chas is the approach to what is Not New:

a) stretched octaves
b) stretch curves
c) pure intervals

Other models have already stretched octaves, nothing new.

What is new is the approach to the temperament and the resulting Chas "scale ratio": the reason for a mild (read mild) octaves stretch curve resulting from the interrelation of all partials.

Chas model theorizes a synchronic beating whole, where no interval is pure (sorry?). All intervals are stretched.

Then, all intervals can have their proportional beat curve, a proportional beat curve that you can aim at in practice, as it is drawn in theory. Difficult? Maybe you, Kent, could not realize but Stopper may well do.

For what is written above, I had already declared my lack of interest in discussing with Stopper, what I consider a perfectly admissible personal position. And please, do not take all this as a justification because I should not need any, take this as a mere description, my personal description. Then, stop insinuating, stop threatening, stop insulting, and if you have some spare time...enjoy yourself.

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.

Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 06:49 PM

I don't see a point where I contradicted myself. I did not took back any of my statements regarding the higher degree of symmetry which is causing purity in tempered chords in the temperament I prefer. (a purity you are avoiding in your form). Beside that I always said there is an infinite number with mathematical valid equal temperaments including yours possible. Where do you see a contradiction when I am falsifying the arguments you give to make your temperament more evident than others?

What caused me to assume your paper to be a scientific hoax where the sheer numbers of wrong statements inside your report (see my falsifications).
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 07:27 PM


Stopper, do you want me to start a "Stopper Vs Chas" Topic for you?

..."We probably have a different perception of the reality regarding this.
Relevance and validity are two slightly different things."...

Tell us Stopper, if you want to describe "only-pure" 19 root of three's relevance, why do not you do it?

..."You can gain this (ET scales ratio) with Mazzola's general tuning formula as i mentioned already.
Again, Chas model is valid here, but not relevant as new."...

Stopper, do not stop elaborating.

Basic Chas is: (3 - Δ)^(1/19) = (4 + (Δ*s))^(1/24)

For s = 1
Δ = 0.0021253899646...
Scale incremental ratio = 1.0594865443501...

Have you seen Chas ratio before?

You write:..."I figured out too, that the model of the natural form of the fifth circle equation i am using to illustrate my tuning model can exactly take the Chas model form for the harmonic case. By replacing the constants with variables (which is equivalent to Mazzolas general model) every inifinite scale can be done as with Chas."...

Good for you, you too have now figured out.

..."Your paper with it´s grossly wrong statements about interval progression inversion for the harmonic case and the valid but inevident form of the equal beating duodecime-double-octave equal temperament over other forms of equal temperament..."...

For me, you have lost your bearings. I talk about "inversion" for Chas preparatory tuning, and I talk about equal beating 12ths and 15ths as a result of partials interrelations.

...is a blame for yourself and for the the anonymous co-authors of the GRIM group, as they did not recognize them before making the paper publicly available."...

What is needed for interrelating all partials, say for playing all partials and coherently stretch all intervals, is a double-octave module. This gains Chas intermodular set, together with the proportional stretch curves for all intervals.

Where is the blame? Why do you blame "anonymous co-authors of the GRIM group"? This is the GRIM group:

http://math.unipa.it/~grim/Chi__siamo.htm

Prof.ssa Maria Elena Ajello Liceo Scientifico Cannizzaro Palermo tel. 091-6250651
marilina@katamail.com
Prof. Carmelo Arena Liceo Scientifico "Cannizzaro" Palermo tel. 091 347495
c.arena@libero.it
Prof.ssa Paola Brigalia Dottoranda tel. 3471353082
pbrigaglia@math.unipa.it
Prof. Benedetto Di Paola Assegnista MAT/04 tel. 091 23891053
dipaola@math.unipa.it
Prof.ssa. Maria Lucia Lo Cicero Dottoranda
locicero@math.unipa.it
Dott. Giuliano D'Eredità Dottorando
deredita@math.unipa.it
Prof. Santi George Dottorando
grpsanti@gmail.com
Prof. Luigi Menna Dottorando
luigimenna@yahoo.it
Dott. Mario Ferreri Membro Aggregato tel.091-6681188
mario.ferreri1@tin.it
Prof.ssa Daniela Galante PhD, Conservatorio di Musica di Stato V. Bellini di Palermo tel. 091 421405
danifranco@alice.it
Prof.ssa Brigida Grillo ITC "Libero Grassi" Palermo tel 091-587723
gribic@katamail.com
Prof.ssa Rosa La Rosa (Scuola Media "V.Emanuele" Palermo) tel.091-6681188
mario.ferreri1@tin.it
Prof.ssa Daniela LoVerde tel.091-6819342
Prof.ssa Elsa Malisani PhD, Scuola Media Ribera (AG) tel.0925-544006
schillacimalisani@tiscalinet.it
Prof.ssa. Gianna Manno PhD, Membro Aggregato tel. 328 7414678
giamanno@libero.it
Prof. Gaetano Militello Istituto Tecnico V.E. III Palermo tel. 091-307568
gaetanomilitello@libero.it
Prof.ssa Cristina Mostacci Membro Aggregato tel.0923-921064
cmostac@libero.it
Prof.ssa Francesca Niceta Membro Aggregato tel.091-6852255
fniceta@libero.it
Prof. Perez Emanuele Liceo Scientifico "Einstein" Palermo tel. 091-6823877
messier.104@tin.it
Prof.Francesco Pintaldi
Membro Aggregato tel.091-6523500
pintaldi@libero.it
Prof.ssa Marcella Profumo Liceo Scientifico "Cannizzaro" Palermo tel.091-543174
mprofumo@aliceposta.it
Prof. Aldo Scimone PhD, Istituto Magistrale "F. Aprile" Palermo tel.091-305324
aldo.scimone@libero.it
Prof.ssa Claudia Sortino PhD, Membro Aggregato tel 329 0903595
cla.noc@libero.it
Prof.ssa Natalia Visalli Liceo Classico "Garibaldi" Palermo tel.091-345669
natalia.visalli@gmail.com
Prof.ssa Teresa Marino Dipartimento Matematica Università di Palermo tel. 091-23891073
marino@math.unipa.it
Prof.ssa Grazia Indovina Dipartimento Matematica Università di Palermo tel.091-308016
indovina@math.unipa.it
Prof.Pietro Nastasi Dipartimento Matematica Università di Palermo tel.091-6477272
nastasi@math.unipa.it

Chas author, as you can read in Chas research report, is me. Now please go and enjoy yourself.

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/05/10 08:57 PM


Stopper, we cross-posted.

You were writing:

..."I don't see a point where I contradicted myself. I did not took back any of my statements regarding the higher degree of symmetry which is causing purity in tempered chords in the temperament I prefer."...

It was you and only you talking about "symmetries of highest degree...", Stopper, highest and not higher, like you are now saying.

..."(a purity you are avoiding in your form)"...

Exactly Stopper, what opens to a pure "intermodular set" where all partials are interrelated and all intervals have their precise beat curve.

..."Beside that I always said there is an infinite number with mathematical valid equal temperaments including yours possible."...

You have always said...? Nop, it was only last December.

..."Where do you see a contradiction..."...

I see a contradiction when you go from "My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not." last June, to "...all harmonic ET models (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid...”, last December.

..."when I am falsifying the arguments you give to make your temperament more evident than others?"...

Do you believe? I'm not interested in competitions with other researchers. I replied you about this long ago.

..."What caused me to assume your paper to be a scientific hoax where the sheer numbers of wrong statements inside your report (see my falsifications)."...

You see Stopper, now you would be admitting what you had assumed then, when you addressed your readers towards a scientific hoax...and only now you state about wrong statements...what is this?...oh, it is 2.57, goodnight.

a.c.
.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 06:11 AM

Originally Posted By: alfredo capurso


You were writing:

..."I don't see a point where I contradicted myself. I did not took back any of my statements regarding the higher degree of symmetry which is causing purity in tempered chords in the temperament I prefer."...

It was you and only you talking about "symmetries of highest degree...", Stopper, highest and not higher, like you are now saying.

If you find this a relevant contradicition, then interpret this difference as a typo and i mean of course highest degree.

Originally Posted By: alfredo capurso

..."(a purity you are avoiding in your form)"...

Exactly Stopper, what opens to a pure "intermodular set" where all partials are interrelated and all intervals have their precise beat curve.

All equal temperaments have precise beat curves, this is not an exclusive feature of Chas.
Purity is well defined as abscence of beats. You contradict yourself here.

Originally Posted By: alfredo capurso

..."Beside that I always said there is an infinite number with mathematical valid equal temperaments including yours possible."...

You have always said...? Nop, it was only last December.

Another wrong statement of your side.
Read my post from June, 9, 09:
"Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET).Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET)."

Originally Posted By: alfredo capurso


..."Where do you see a contradiction..."...

I see a contradiction when you go from "My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not." last June, to "...all harmonic ET models (including Chas model, Chas EB-ET form and mindless octaves, Cordiers pure fifth temperament, my own etc.) are mathematically valid...”, last December.


We are speaking in circles here. I told you about the the difference between mathematical validity and relevance of one form over others. (what i am referring here)

Originally Posted By: alfredo capurso

..."when I am falsifying the arguments you give to make your temperament more evident than others?"...

Do you believe? I'm not interested in competitions with other researchers. I replied you about this long ago.


I was not talking about competition between researchers here. I was questioning your argumentation about evidence of Chas form over other forms of temperament.

Originally Posted By: alfredo capurso

..."What caused me to assume your paper to be a scientific hoax where the sheer numbers of wrong statements inside your report (see my falsifications)."...

You see Stopper, now you would be admitting what you had assumed then, when you addressed your readers towards a scientific hoax...and only now you state about wrong statements...what is this?...

Defintively not a contradicition. I have taken notice that you denied that option and that your paper is not a hoax. So i had to precise out the wrong statements.


Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 08:06 AM

Originally Posted By: alfredo capurso
Jeff, try to be happy with what follows and try to refrain from such comments: "You are interested in appearance not substance."...

You have had almost 8 months of "substance".

.....


No, I have had almost 8 months of smoke and mirrors, but I was not decieved. I just have finally decided that your deception is deliberate; that you were not decieved, yourself.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 08:45 AM


Stopper, as I could explain, in my opinion you are not a reliable conversation partner. I take care about my face, so you can do about your face, but please stop arguing by insinuations.

...“All equal temperaments have precise beat curves, this is not an exclusive feature of Chas.
Purity is well defined as abscence of beats. You contradict yourself here.”...

What is the theoretical curve of 19 root of three on 12ths? And theoretical 12 root of two’s on octaves? You may well know about theoretical “abscence of beats”.

...“Another wrong statement of your side.
Read my post from June, 9, 09:
"Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET).Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET)."...

Yes, I confirm, I can not read about “validity”. There, you were only banalizing ETs “stretch”, while you may well understand that Chas ± Δ gains a very precise ET. This, in my opinion, may be what you do not like.

...“We are speaking in circles here. I told you about the the difference between mathematical validity and relevance of one form over others. (what i am referring here)”...

No circles but ways, Stopper. My way is describing Chas Theory’s relevance, a way that could also be your way for what concerns 19 root of three. But you have more interest in acting as a destroyer, as a detractor and as a defamer.

...“I was not talking about competition between researchers here. I was questioning your argumentation about evidence of Chas form over other forms of temperament.”...

I doubt you do not understand Chas relevance, and this doubting of mine suggests me to let TIME do his/her job. Nevertheless, yesterday I wrote you what you need for interrelating all partials and for intermodulating a 12 semitones set.

...“I have taken notice that you denied that option and that your paper is not a hoax. So i had to precise out the wrong statements. Although opting to the hoax variant could serve you well to keep your face.”...

How to tell you that I do not want any more of this staff? How to tell you that I find you too arrogant, deceitful and contorted. Please, keep on going for 19th root of three ET, as I shall keep on going for Chas ET Temperament and Chas Preparatory Tuning. Please write about your discovery, its relevance or supremacy or what ever you like in a different Topic.

a.c.

Jeff, you write:..."No, I have had almost 8 months of smoke and mirrors, but I was not decieved. I just have finally decided that your deception is deliberate; that you were not decieved, yourself."...

Good, at least you could make up your mind. Let me know when you make it up with Chas ET Theory's equation and with Chas ET ratio.

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.




Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 09:30 AM

Originally Posted By: alfredo capurso
.....

Jeff, you write:..."No, I have had almost 8 months of smoke and mirrors, but I was not decieved. I just have finally decided that your deception is deliberate; that you were not decieved, yourself."...

Good, at least you could make up your mind. Let me know when you make it up with Chas ET Theory's equation and with Chas ET ratio.

.....


Oh, I had made up my mind about that almost 8 months ago!
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 10:05 AM


Jeff, you write:..."Oh, I had made up my mind about that almost 8 months ago!"...

This is what makes me shiver, how you could be so shabby and yet double-headed.

a.c.

.
Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 10:44 AM

Get real. I have been critical of your paper all along. I posted in an effort to educate you.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 01:22 PM

Originally Posted By: alfredo capurso

Stopper, do you want me to start a "Stopper Vs Chas" Topic for you?


This is not necessary, thank you.


Originally Posted By: alfredo capurso

Tell us Stopper, if you want to describe "only-pure" 19 root of three's relevance, why do not you do it?

If have kindly invited you to participate at the italian convention last year, but you denied. There you have had the chance to listen what i was saying and presenting about, (sound, numbers and figures).
What i present in what form and when is my own decision. And the form of a public internet forum is definitively not what i will choose for.

Originally Posted By: alfredo capurso

..."You can gain this (ET scales ratio) with Mazzola's general tuning formula as i mentioned already.
Again, Chas model is valid here, but not relevant as new."...

Stopper, do not stop elaborating.

Basic Chas is: (3 - Δ)^(1/19) = (4 + (Δ*s))^(1/24)

For s = 1
Δ = 0.0021253899646...
Scale incremental ratio = 1.0594865443501...

Have you seen Chas ratio before?

Mindless octaves equal temperament model of Bill Bremmer has been described before Chas:
http://ptg.org/pipermail/pianotech/2001-December/098956.html

quote from link, Bremmer: "Essentially, it is an *Equal Beating* compromise between the Double Octave
and the Octave and Fifth."

It has been described whithout mathematical formalization, but that doesn´t matter. Did you know that many historical tunings where described without formalization and still were usually constributed to their inventors and not to their formal describers?
I recommend you for reading:
Andreas Werckmeister "Musicalische Temperatur" (Quedlinburg 1691)

Beside that i don´t see much need to advocate a tuning for someone else.
If you arrive to get this temperament contributed to you, so be it.


Originally Posted By: alfredo capurso

You write:..."I figured out too, that the model of the natural form of the fifth circle equation i am using to illustrate my tuning model can exactly take the Chas model form for the harmonic case. By replacing the constants with variables (which is equivalent to Mazzolas general model) every inifinite scale can be done as with Chas."...

Good for you, you too have now figured out.

Fine, so you can accept that your model is just a different form of the fifth circle.


Originally Posted By: alfredo capurso

..."Your paper with it´s grossly wrong statements about interval progression inversion for the harmonic case and the valid but inevident form of the equal beating duodecime-double-octave equal temperament over other forms of equal temperament..."...

For me, you have lost your bearings. I talk about "inversion" for Chas preparatory tuning, and I talk about equal beating 12ths and 15ths as a result of partials interrelations.

It seems your are not longer familiar with your own statements in your paper.

Quote from section 4.6 of Chas paper:
"The difference curve for these intervals inverts its progression at degree 51. This inversion is determined by the s variable, unique to this model. The same effect, we will see below, is found in degrees relating to ratios 3:2, 3:1, etc."

Originally Posted By: alfredo capurso

Your paper with it´s grossly wrong statements about interval progression is a blame for yourself and for the the anonymous co-authors of the GRIM group, as they did not recognize them before making the paper publicly available."...

What is needed for interrelating all partials, say for playing all partials and coherently stretch all intervals, is a double-octave module. This gains Chas intermodular set, together with the proportional stretch curves for all intervals.

Where is the blame? Why do you blame "anonymous co-authors of the GRIM group"? This is the GRIM group:

http://math.unipa.it/~grim/Chi__siamo.htm

Prof.ssa Maria Elena Ajello Liceo Scientifico Cannizzaro Palermo tel. 091-6250651
marilina@katamail.com
Prof. Carmelo Arena Liceo Scientifico "Cannizzaro" Palermo tel. 091 347495
c.arena@libero.it
Prof.ssa Paola Brigalia Dottoranda tel. 3471353082
pbrigaglia@math.unipa.it
Prof. Benedetto Di Paola Assegnista MAT/04 tel. 091 23891053
dipaola@math.unipa.it
Prof.ssa. Maria Lucia Lo Cicero Dottoranda
locicero@math.unipa.it
Dott. Giuliano D'Eredità Dottorando
deredita@math.unipa.it
Prof. Santi George Dottorando
grpsanti@gmail.com
Prof. Luigi Menna Dottorando
luigimenna@yahoo.it
Dott. Mario Ferreri Membro Aggregato tel.091-6681188
mario.ferreri1@tin.it
Prof.ssa Daniela Galante PhD, Conservatorio di Musica di Stato V. Bellini di Palermo tel. 091 421405
danifranco@alice.it
Prof.ssa Brigida Grillo ITC "Libero Grassi" Palermo tel 091-587723
gribic@katamail.com
Prof.ssa Rosa La Rosa (Scuola Media "V.Emanuele" Palermo) tel.091-6681188
mario.ferreri1@tin.it
Prof.ssa Daniela LoVerde tel.091-6819342
Prof.ssa Elsa Malisani PhD, Scuola Media Ribera (AG) tel.0925-544006
schillacimalisani@tiscalinet.it
Prof.ssa. Gianna Manno PhD, Membro Aggregato tel. 328 7414678
giamanno@libero.it
Prof. Gaetano Militello Istituto Tecnico V.E. III Palermo tel. 091-307568
gaetanomilitello@libero.it
Prof.ssa Cristina Mostacci Membro Aggregato tel.0923-921064
cmostac@libero.it
Prof.ssa Francesca Niceta Membro Aggregato tel.091-6852255
fniceta@libero.it
Prof. Perez Emanuele Liceo Scientifico "Einstein" Palermo tel. 091-6823877
messier.104@tin.it
Prof.Francesco Pintaldi
Membro Aggregato tel.091-6523500
pintaldi@libero.it
Prof.ssa Marcella Profumo Liceo Scientifico "Cannizzaro" Palermo tel.091-543174
mprofumo@aliceposta.it
Prof. Aldo Scimone PhD, Istituto Magistrale "F. Aprile" Palermo tel.091-305324
aldo.scimone@libero.it
Prof.ssa Claudia Sortino PhD, Membro Aggregato tel 329 0903595
cla.noc@libero.it
Prof.ssa Natalia Visalli Liceo Classico "Garibaldi" Palermo tel.091-345669
natalia.visalli@gmail.com
Prof.ssa Teresa Marino Dipartimento Matematica Università di Palermo tel. 091-23891073
marino@math.unipa.it
Prof.ssa Grazia Indovina Dipartimento Matematica Università di Palermo tel.091-308016
indovina@math.unipa.it
Prof.Pietro Nastasi Dipartimento Matematica Università di Palermo tel.091-6477272
nastasi@math.unipa.it

Chas author, as you can read in Chas research report, is me. Now please go and enjoy yourself.


So Chas author is you alone. Why do sign your paper with "CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy)" then?

Why do count up every member of the GRIM group here if not one person of the group was involved in your paper?

My statement about anonymous co-authors of the group was because not one person of the group co-signed your paper. You don´t want to say that every single member of the group has reviewed your paper?
And if so and they did not recognized the wrong statements they had blamed themselves.
And if they were not involved with your paper, you have blamed the group, but surely not me.



Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 01:44 PM

Originally Posted By: alfredo capurso

Stopper, as I could explain, in my opinion you are not a reliable conversation partner. I take care about my face, so you can do about your face, but please stop arguing by insinuations.

Let me remeber that that you opened your response to my objective falsifications in an abusive way.

Originally Posted By: alfredo capurso

...“All equal temperaments have precise beat curves, this is not an exclusive feature of Chas.
Purity is well defined as abscence of beats. You contradict yourself here.”...

What is the theoretical curve of 19 root of three on 12ths? And theoretical 12 root of two’s on octaves? You may well know about theoretical “abscence of beats”.

You know well that there are way more intervals in those two tunings beside the octaves or duodecimes, which all have a distinct beat curve.

Originally Posted By: alfredo capurso

...“Another wrong statement of your side.
Read my post from June, 9, 09:
"Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET).Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET)."...

Yes, I confirm, I can not read about “validity”. There, you were only banalizing ETs “stretch”, while you may well understand that Chas ± Δ gains a very precise ET. This, in my opinion, may be what you do not like.

I was talking of "solutions". And any solution implies validity.
Did you know that the delta is just a fraction of the pythagorean comma?
The pythagorean comma can of course be determined very exactly. And thus the distribution of the pythagorean comma between duodecimes and double octaves. But with the same precision the pythagorean comma can be distributed in any other tuning. This is not an exclusive feature of Chas.

Originally Posted By: alfredo capurso

My way is describing Chas Theory’s relevance, a way that could also be your way for what concerns 19 root of three. But you have more interest in acting as a destroyer, as a detractor and as a defamer.

And i was just falsifiying the relevance in an objective and provable way. This is a normal process of reviewing in science. Just because you don´t like the results, the reviewer is not a destroyer a detractor and a defamer. What you accuse me to be just because of objective reviewing, is exactly what you are yourself then.

Originally Posted By: alfredo capurso

How to tell you that I find you too arrogant, deceitful and contorted.

I can live with that fact, but such statements are inadequate here.

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/06/10 03:39 PM


Stopper,

Any added line on your part, as any other contortion of yours can only consolidate the relevance of Chas ET Temperament Theory.

I thank you and I wish you all the best,

a.c.

Jeff, you write: “Get real. I have been critical of your paper all along. I posted in an effort to educate you.”

Yes, I think you have indeed. I thank you and I wish you all the best,

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 02:28 AM

Originally Posted By: alfredo capurso


I thank you and I wish you all the best,



I thank you too and i hope you may benefit from our discourse for improvements on your paper, which is the sense of critical reviewing in it´s best case.


Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 03:41 AM


… “I thank you too and i hope you may benefit from our discourse for improvements on your paper, which is the sense of critical reviewing in it´s best case.”

CHECK * Best case formula:
1 educator + 1 aspiring salesman = Sense of critical review

Stopper, you are very generous in trying to improve Chas paper. Let me suggest you though to publish your own scientific paper on only-pure 12ths. Only after that you may offer in actual fact your "careful" know-how.

In general and out of my own experience, never confuse yourself with what you are not, in other words: let men of science talk about science and let academic Professors improve scientific papers.

All the best,

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 04:50 AM

Originally Posted By: alfredo capurso


Stopper, you are very generous in trying to improve Chas paper. Let me suggest you though to publish your own scientific paper on only-pure 12ths. Only after that you may offer in actual fact your "careful" know-how.



By the way i did a publication about the 19th root of three when i was invited as a contributor for the scientific symposium "Incontro a Bolzano: Scale e harmonie" in italy 1991. (And it has been reviewed by professors) So i am happy that you can accept my offer now.

I wish you good luck.

Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 05:29 AM

Please, when will you post other recordings of piano tuned by the 2 methods ?

I tried again the fifth inversion of the Alfredo s method, and I dont get what it provide, still thinking.

Anyway I finally had a piano with enough stretch in the medium range, which to my ears is not really the case on the Steinway S recording (at 5 bps the C E 10th or M6, that is too slow to my ears.

So I guess I did not use the same temperament method in the end.


To me, a good part of the lively sensation we have in a piano tuning is due to the acceleration of stretch .

It catch our ears and keep them alive. Then a too large span for tempering is not ideal (but between 12th and 2 octave sounds good to me).

I guess also that purity is good when it is really pure, if it is only sort of pure the effect is not there as much.
But purity is somehow at the opposite of tempering so I understand it may be impossible to really reconciliate we only can go for a set of compromizing as it have been said often there.

Well you can also use a predetermined curve as with a PT100 or some old generation EDT, and put that on the piano saying that justness is only a question of having that curve there and not the piano affair. It will work as well, the piano will be "tuned" and playeable. Our ears seemm to recognize very fast any kind of organisation as soon it is used in a consistent way all along the scale.

I understand that definition of ET is something more evolved those days than it was.

I'll accept any definition, the intention is to have an instrument playeable that can use its own harmony to some point.

What counts in the end is how it tones and how well one can play with the instrument, and other instruments as well.

All of your definitions will be accepted with time, no doubt about that, does not mean that the tuners will change the way they learn to tune (because it is difficult enough to learn to settle pins and manipulate the hammer) but it will or it have yet do something (as Cordier's ideas have also do something, and even Bill Bremmer's option)

Very comfortable to me to look at it from that distance !

Hopefully practically when I tune a piano I am in a known land, and I see generally no many ways to have it sounding right, we have a little discussion with the instrument and found a cordial agreement.

And that is for that I am paid, but probably more often to correct voicing or regulation behaviour (at last this knowledge is way less common, and the time window is not as tight as for the yearly tuning so work conditions are way better !).
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 05:51 AM


Stopper, this is what I wrote:

Stopper, you are very generous in trying to improve Chas paper. Let me suggest you though to publish your own scientific paper on only-pure 12ths. Only after that you may offer in actual fact your "careful" know-how.

In general and out of my own experience, never confuse yourself with what you are not, in other words: let men of science talk about science and let academic Professors improve scientific papers.

Now you write: "By the way i did a publication about the 19th root of three when i was invited as a contributor for the scientific symposium "Incontro a Bolzano: Scale e harmonie" in italy 1991. (And it has been reviewed by professors) So i am happy that you can accept my offer now."

Please, do not misunderstand. If you are happy with your publication, you can now write more about why and how tuning pure 12ths ET, possibly in your own Topic.

About the rest, this is my advice: never confuse yourself with what you are not, in other words: let men of science talk about science and let academic Professors improve scientific papers.

In simpler words, I suggest you to offer your professional expertise and nothing else.

Say you understand these simple words, I thank you and I too wish you good luck.

Say you do not understand these simple words, I thank you and I too wish you good luck.

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 06:52 AM

Hello Kamin,

it is not that easy for me, for the time being, to arrange all things and provide more recordings. But I will as soon as I can.

..."I tried again the fifth inversion of the Alfredo s method, and I dont get what it provide, still thinking."...

The way I set 5ths is relating two 5ths with two 4ths and A3-A4 octave (first 4 tuning steps, on centre strings).

..."To me, a good part of the lively sensation we have in a piano tuning is due to the acceleration of stretch."...

To me too.

..."It catch our ears and keep them alive. Then a too large span for tempering is not ideal (but between 12th and 2 octave sounds good to me)."

For concert tuning my temperament span (possibly) is 88 keys.

..."I guess also that purity is good when it is really pure, if it is only sort of pure the effect is not there as much.
But purity is somehow at the opposite of tempering so I understand it may be impossible to really reconciliate we only can go for a set of compromizing as it have been said often there."...

Chas approach and my tuning is pro-beats. I go for beats, just propensity and slow progressive beats for octaves, but still beats. More than "compromising" in the sense of "making the best of a bad job", I compromise in the sense that I draw the progressive beat curve for all intervals.

..."Our ears seemm to recognize very fast any kind of organisation as soon it is used in a consistent way all along the scale."...

I agree, our ear (in my opinion) also recognize an higher degree of harmonicity and resonance, especially on a comparative basis.

..."What counts in the end is how it tones and how well one can play with the instrument, and other instruments as well."...

I agree, too often!

..."All of your definitions will be accepted with time, no doubt about that, does not mean that the tuners will change the way they learn to tune (because it is difficult enough to learn to settle pins and manipulate the hammer)...

In my opinion, the ET zero-beating octave's theory has lost any reference for the tuner, listen to the frustration of many colleagues, to their silence embarrassement when they have to describe 4ths, 5ths, octaves, 12ths and 15ths tuning...would you believe that 12ths and 15ths can routinely invert? This shows you the average degree of confusion, and this is why Chas ET Theory is good news: it describes a scientifically correct model that can finally represent a reliable reference. You can aim at Chas in practice and gain what the Theory describes.

..."Hopefully practically when I tune a piano I am in a known land, and I see generally no many ways to have it sounding right, we have a little discussion with the instrument and found a cordial agreement."...

That is good.

..."And that is for that I am paid, but probably more often to correct voicing or regulation behaviour (at last this knowledge is way less common, and the time window is not as tight as for the yearly tuning so work conditions are way better !)."...

Yeah, correct. The whole thing is "sound production and control", a combination of all dynamics, keyboard's reliability and accuracy, action's timing, strings and structure dynamics. For me, a world of proportions and exactitude.

Best regards,

a.c.
.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 07:44 AM

Originally Posted By: alfredo capurso


In general and out of my own experience, never confuse yourself with what you are not, in other words: let men of science talk about science and let academic Professors improve scientific papers.

There is no law that reserves reviewing scientific theories to academic professors. You have choosen to publish your paper on a public internet forum and you can not expect that your theory will be accepted like the infallibility of the pope.


Originally Posted By: alfredo capurso

In simpler words, I suggest you to offer your professional expertise and nothing else.

That is, what i have done with my falsifications of parts of your paper. I have professional expertise in the field of tuning including tuning theory. I repeat myself again as you do often and in simple words: There is no law that reserves reviewing scientific theories to academic professors.

And don´t you see a contradiction in your point of view that reviewing should be reserved to professors, and for yourself as a non-professor you claim the right to create a new theory?

And remember, the hoax example proves what can pass professor reviewing.

As you are a friend of simple words, please answer me this simple question:

Do you agree at least with one of my falsifications?




Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 12:52 PM

I have noticed in the file info of the most recent version of your paper, that it has been created by Prof. Filippo Spagnolo.

May be he can give a statement here concerning my three falsifiction points?

Posted by: UnrightTooner

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/07/10 01:12 PM

Mr. Stopper:

You may be interested in part of a post on May 25, 2009 in this Topic where Alfredo sort of answered some questions about the publishing of his paper:

”Tooner,

You ask: "Was it checked by the math department?"

Well, what do you think?

"Did you have to defend the paper to a board of professors?"

I had to rewrite the article 3 times, to explain things that on the way had resulted obscure. It took me almost 2 years.

"Did anyone at the University understand it?"

Yes, Chas maths is not that difficult and I'll demonstrate that.

"Did they agree with it?"

They checked Chas maths without playing any other role. We'd better talk about how could anyone disagree, don't you think?”
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/08/10 05:25 AM

Hi ALfredo,

Because the ear very easely accept stretched octaves and hear them as more consonant at the piano (eventually it ask for it) ther is no problem, the zero beating theory is just that, a theory, I doubt tuners ever used it while I know what Yamaha tuning training is, there is even on the video of the inauguration of the new factory, a CFIIIS grand which is tuned without enough stretch and have a very boring harmony, while being "just", no doubt on that).

Yes as you state, when you use 6:3 you are in the effect of ih, but when you use 2:1 or 4:2 also, I tune octaves so they are sounding well, and avoid the use of m3d M6 as being useless in the end, I am more found of resonance, hence the use of octave (too much, but when tuning constantly the same models you know how your octaves may sound) the fifths, the 12th, double octave, and the FBI as a tool for evenness and a limiter (no scream).

Switching to a larger span and trading octave for 12th and 15th is very well an interesting option, that is easier to produce/listen (I usually check the 12ths so not to have it farther than pure, while I certainly agree it may be easely the case in the high treble and low bass, as the fiths that grows to 6:3 very easely)

I still did not get where lies the effect of the inversion of 5ths/4th, , what it gives practically, on other intervals.

ih is a tool, for the tuner, it is what gives us room for stretch. I am unsure your method is something else than a way to temper (in that case it can be called a new definition of tuning method, but as this is the piano which decide the pitch in Hz of each note, I am unsure it state for a new ET, it is only for the piano, in my view.

Did you try to apply it to an organ ?
is it feasible for a guitar ?
for a singer ?

In that sense, the one who divide the 12th or the one who divide the 5ths are to me more straightforward, as a standard pitch can be used to determine all the pitches so other instruments can be tuned that way.

Not to lower what you find and put words and formula on, but is not it more a practical method than a theory ?

It may be a cultural thing but you (as many defensing their point)seem to insert an unnecessary layer of "marketing" in your presentation, or may be it is emphasis, but anyway that makes me always more doubtful than convinced, by reflex (sorry I dont appreciate advertising that much!)

Sorry if I am wrong there.

Best regards.
Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/08/10 08:50 AM

Originally Posted By: UnrightTooner
Mr. Stopper:

You may be interested in part of a post on May 25, 2009 in this Topic where Alfredo sort of answered some questions about the publishing of his paper:

”Tooner,

You ask: "Was it checked by the math department?"

Well, what do you think?

"Did you have to defend the paper to a board of professors?"

I had to rewrite the article 3 times, to explain things that on the way had resulted obscure. It took me almost 2 years.

"Did anyone at the University understand it?"

Yes, Chas maths is not that difficult and I'll demonstrate that.

"Did they agree with it?"

They checked Chas maths without playing any other role. We'd better talk about how could anyone disagree, don't you think?”



Thank you Mr. Deutschle, i just missed that.

I will send a request for a review of my falsifications of several point of the Chas paper directly to Prof. Spagnolo.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/08/10 11:25 AM

Hi Kamin,

you kindly write:...“Switching to a larger span and trading octave for 12th and 15th is very well an interesting option, that is easier to produce/listen (I usually check the 12ths so not to have it farther than pure, while I certainly agree it may be easely the case in the high treble and low bass, as the fiths that grows to 6:3 very easely)”...

Yes, if I’m not doing pitch raising, on center strings I keep 3:1 12ths, even if I need to wait some time and play a little bit, so to get it stable, and 5ths get even wider than 3:2 (on center strings).

...“I still did not get where lies the effect of the inversion of 5ths/4th, , what it gives practically, on other intervals.”...

In my experience, 4ths and 5ths inversion is needed to S shape the octaves. If I were to talk on three dimensions, I’d say that octaves (pure or stretched) can stretch 3rds, 4ths and 5ths can wring the octaves. Quite similar to what we do when placing a new bass string. And if I do not invert those intervals, I can not get nor justify (gain) good 5ths, octaves, 10ths, 12ths ecc. in the high registers.

I think of 3rds, 4ths 5ths and octaves (and all larger intervals) as one, as a whole, so I set the temperament stretching A3-A4 and stretch 4ths and invert 5ths for gaining wringing octaves.

The rest in a while...a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
.

Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/09/10 06:58 PM


Hello Kamin, there was more you were nicely saying:

...”ih is a tool, for the tuner, it is what gives us room for stretch.”...

Me too, I was taught that we need to stretch octaves because of iH. When I had the opportunity to tune harpsichords for concerts, I realized that I was able to improve resonance and harmonicity by tuning Chas, same sequence, same beats progression, same octaves stretch, perfect superimposition of two (separate/indipendent) tunings. Conclusion:

1) I truly like pure sounding 5ths in the top register, Chas wide-progressive octaves and Chas 12ths Vs 15ths 88 keys even beating, all along the keyboard.
2) Despite iH (and different degrees of iH?), I can still go for Chas, therefore iH is not the driving factor for stretch.

On the theoretical side of it, I realized that 12 root of two ET compromised 3rds and 5ths, it compromised their commas (surely you know about commas), but the real “stretchers” are the octave (for 3rds) and 5th /4ths (for octaves).

Pure-octaves 12th root of two ET, both in theory and practice, produces too narrow 12ths. Am I wrong? So octaves needed to be theoretically stretched as they are stretched in practice. This is so clear to me now, after 25 years of practical observations, reasoning and computing.

Cordier’s 3:2 ET option does stretch octaves, so does 3:1 Stopper’s option. (Ah, by the way, there is still “square root of 9/8” available, anyone wanting to be first and name it?).

But the last obstacle was the theoretical “pure intervals” approach. A “zero-beating” interval does not exist in actual fact and, favoring - in theory or in practice - any particular interval, goes to the detriment of all the others. You well know, pure 5ths produce too wide 3rds and octaves, pure 12ths produce too wide 3rds, 10ths and 15ths, pure octaves produce too narrow 12ths, 19ths.

...“I am unsure your method is something else than a way to temper (in that case it can be called a new definition of tuning method, but as this is the piano which decide the pitch in Hz of each note, I am unsure it state for a new ET, it is only for the piano, in my view.”...

One curious fact. If you read this thread you will realize what some posters say: Chas Theory and model can only be referred to harmonic tones. When I’ve talked about Chas with one pipe-organ tech, he immediately warried about stretched octaves and the many registers that would suffer from that. He said: Chas may work on pianos or for inharmonic tones! Then I worked (30 hours) at a compareson between 2:1 ET and Chas on digital sounds, to prove that stretched octaves sound better even in this case. You can find the details and link posted on this Topic, September 02, 09.

What to say then? Chas resolves the age-old partials conflict, i.e. it finds the precise mathematical ratio that can interrelate frequencies and beats (beats deriving from partials combinations), so frequencies and beats can share the same scale ratio. Is Chas ET a method?

I kindly ask you: is 12th root of two a method?
Would you say that “pure fifths” and “pure 12ths” ETs are methods?
I do not think so, they try to be Theories. A method is a procedure, in fact Chas Pre-form is meant to be that, for gaining Chas Form.

And please notice, Chas Pre-form (for intervals stretching) uses pure 12ths (on center strings), together with pure 5ths (on center strings), for gaining mild stretched and progressive octaves and Chas 88 tones even beating.

Sorry if I needed to split my posts, for time reasons. The first half is my previous post, tomorrow I'll complete my reply.

Best regards, a.c.
.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/10/10 06:08 AM


Voilà...

...“Did you try to apply it to an organ ?
is it feasible for a guitar ?
for a singer ?”...

No, I have not tried Chas on pipe organs. Though I can still hear the many organs I have heard: 3rds, 4ths, 5ths and octaves are tuned chaotic, 12ths and 19ths too narrow, high registers are flat. Is that only me?

I play guitar since I was 10, but I've not done any research. My opinion? Guitar tuning and frets relations can be improved.

Singers (amongst my customers), so far, are very happy with Chas. Maybe this is not the point though, music has been, is been and will be played in any case (opinion).

...“In that sense, the one who divide the 12th or the one who divide the 5ths are to me more straightforward, as a standard pitch can be used to determine all the pitches so other instruments can be tuned that way.”...

By “standard pitch” do you mean “standard pure ratio”? I’m not sure I get the point correctly.

Anyway. What follows is not even maths, but logics (pre-maths):

say you need an X factor for drawing a scale. Say this X factor is an irrational number, like X.fnsutnflhujx... (uncountable). I ask: how do you divide it? How do you multiply it? My logical answer: this approach, in the best case, opens to approximations, since we can not split an irrational number.

Instead of that, Chas fixis a proportion for our X scale factor, the only proportion that can work with any kind of number: 1:1.

Various X scale factors have so far been seeked amongst "pure intervals ratio" and semitone’s size. Chas approaches the X scale factor from the beats side, i.e. with proportional differences relative to partials matchings. I wanted to draw (numerically) progressive intervals, progressive wide octaves and 12ths/15ths even beating, i.e. what I can aim at, with my practical Pre-form tuning. So I wrote:

Be X (irrational) a resulting difference (in beat terms)

I want X to be even for 12ths and 15ths in a geometrical progression (like 12 root of two is).

So, in Chas equation, delta (Δ) stays for X, and I could write:

(3 – Δ) = (4 + Δ).

To get the ET property was enough rooting partial 3 (–Δ) to 19 and partial 4 (+Δ) to 24, in the way partial 2 is rooted with 12.

This is Chas “uncomprehensible” maths:

(3 – Δ)^(1/19) = (4 + Δ)^(1/24) = 1.0594865443501...

...“Not to lower what you find and put words and formula on, but is not it more a practical method than a theory ?”...

I think Chas Pre-form as the practical method to gain Chas Form.

...“It may be a cultural thing but you (as many defensing their point)seem to insert an unnecessary layer of "marketing" in your presentation, or may be it is emphasis, but anyway that makes me always more doubtful than convinced, by reflex (sorry I dont appreciate advertising that much!)”...

Yes Kamin, I think you are right, and I’m quite conscious of my not-very-shareable style. I would ask you not to bother...and still believe (or not) that I’m not selling anything. I’m offering my research results for free.

...“Sorry if I am wrong there.”...

Eventually you can freely decide, and I like the way you approach knowledge in any case.

Best regards, a.c.
.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/10/10 08:14 AM

Thanks a lot Alfredo for your answers and explanations.

I believe I can get my way thru them, and understand how you do that.

What I am unsure is if there is a so large necessity to treat the piano as a large chord, I see that more as an instrument with registers, and I fell the life in the instrument as based on the relation between those registers.

I've find the center 6th and 10ths to be a tad slow on your recording, while once can certainly get used to that, I cant rally make up my mind on where does the general tonal and harmony impression comes from, but our ear (human) ask also for some "falsness" (read "stretch" , may be).

So to have impression of more evenness of tone power, (if related ?) or to gain contrast between medium and treble for instance.

Most good eared pianist probably dont expect their piano to ever be in tune (nor organists may be).

What count is the expressiveness, articulation and why not contrast.
Some pianist hear in their brain, others with their fingers, other with their ears as well - amateur pianists with a little level often are way more exigent for tone than professionals , who knows how to build tone from within the keyboard -

I wonder if the one that like the HT are more listening than thinking "music architecture and articulation", in any case it change (as noticed by PPAT) the way they play and improvise, as a piano with a huge resonance and crispness will be more attractive, as a piano with a very definite energy at the rebound of the hammer will show the way to nuances and touch differences..

Professional pianists are often more exigent (without knowing it) for touch than for tone.

From this point of view, any tuning approach can be good.

As long as touch (tone energy) and harmony (resonance of the instrument in its own spectra) are treated in some way.

I'll read more deeply and try to put in practical way what you have written here, thanks for taking the time.

Not yet talked with my Bros( I guess he is in Switzerland actually).

I suggest that if you come by to Paris I could organize something with the technicians of the CNSM if you like.

I also understood that you have been attacked by Bernhard about the validity of the formula to compute the temperament , those points need to be cleared if possible, if not it is too easy to use them and contest the validity of the approach at large. (Then I also believe that maths can be used to put a mistake where there is none, as have been demonstrated to me by some friends with some mathematic incongruence, maths are an art as such !!)

This is not really important, I trust myself as being able to recognize if some practical method is good and add something or if the cost is too high somewhere.

Probably, most often, any valuable method is optimum for some particular circumstances and less for others.

As tuners I believe that we suspect that always.

Best regards, and I'll give all possible feedback (BTW I'll do the same with pure twelve's as soon as I can have an usable method to tune aurally with the concept)

Have a nice skying afternoon !!







Posted by: Bernhard Stopper

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/10/10 08:38 AM

Originally Posted By: Kamin

I also state that you have been attacked by Bernhard about the validity of the formula to compute the temperament , those points need to be cleared if possible, if not it is too easy to use them and contest the validity of the approach at large. (Then I also believe that maths can be used to put a mistake where there is none, as have been demonstrated to me by some friends with some mathematic incongruence, maths are an art as such !!)

This is not really important, I trust myself as being able to recognize if some practical method is good and add something or if the cost is too high somewhere.


I don´t find this statement to be true, Isaac.
My critics goes against Alfredo´s attempt to evidentiate Chas over other approaches, not against the validity of the Chas formula.

I falsified the arguments Alfredo used in his paper to make Chas evident over other approaches.
I falsified the statement (section 4.3) in Chas paper that beat inversion occurs because of the s variable, and the statement (section 4.5) in Chas paper that 19th root of three scale develops towards 12th root of three.

Every ET approach has it´s own outcome and all are mathematically valid. I would not having it mentioned here, but Alfredo and you were both refering to the approach i prefer in this thread, so i want to say that what i claim for the apporach i am using, is a maximum of purity and i find it important enough to be noticed as an interesting discovery, as this was not thought to be possible before with the common understanding of ET of whatever size. Some may not like this outcome of purity, but many do.

None of the approaches has more evidence than others, they are a question of personal taste.





Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/10/10 08:54 AM

thank you for the clearing Bernhard, as I am not a fluent reader of maths (far from that !) I did understood that you posted questions on the validity of the computation or at last the way the formulas are expressed.

The evidence of one other the other is a philosophical and a matter of taste.

I stated yet that the way the Chas theory is presented is deserving it somehow (to me) probably as my crusade for nice unisons !!.

But I also understand very well that one can get a tad obsessive if it took a so long time to do the researches , to understand a way to fight the frustration of musicians and tuners, it finally take may be more importance than it should ... (and it is difficult in our trade, to come with something new !)

Best regards.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 01/11/10 07:06 AM


Stopper states:..."Every ET approach has it´s own outcome and all are mathematically valid."...

Quiete reply for completeness sake.

I must precise that "approach" is not the same as "model", and the word “approach” might have been used, there, instead of “model”.

Then, Stopper’s statement may be either wrong (best case), or restrictive (chance), or purposive misleading.

Any geometric progression, maths wise, is equally valid.

http://en.wikipedia.org/wiki/Geometric_progression

This is true, in fact they are all geometric progressions, anyone could be a model. But if a 12 semitones scale model is needed, not all ETs geometric progressions - 12ths root of two, 3/2 root of 7 etc.) will work the same, and what makes the difference is the theoretical approach, it could be said “the theoretical assumptions” together with the actual target, i.e. our 12 semitones scale.

In Chas case, what makes the difference is the approach to the 12 semitones scale's ratio (as described in previous posts), together with the approach to partial sounds and actual partials matchings (Research report - sections 2.0 and 3.0 - Italian language translated into Oxford English by Liz Poore, Lecturer at the Universities of Torin and Milan, Italy).

..."I would not having it mentioned here, but Alfredo and you were both refering to the approach i prefer in this thread,"...

Nothing wrong with this (referring to and evaluating other models), none of us meant to be damaging.

What is Stopper’s intention (?), when he writes (Topic: Hysterical tunings):

"Dear Bill,...we met 2 years ago at the PTG convention in Anaheim...As i attended your class and listened to your tuning, i have to say that this was the sound character where i am usually phoned by my customers to pass by and do something about. Sincerely, Bernhard Stopper."

Then Stopper states..."so i want to say that what i claim for the apporach i am using, is a maximum of purity and i find it important enough to be noticed as an interesting discovery, as this was not thought to be possible before with the common understanding of ET of whatever size. Some may not like this outcome of purity, but many do."...

Yet, I do not understand why Stopper does not simply start his in-depth descriptive Topic, both for only-pure theory and practical aural tuning of 19th root of 3 ET.

Also, my logics does not help me to make the following addiction (+):

...Every ET approach has it´s own outcome and all are mathematically valid (+)

...what i claim...is a maximum of purity (+)

...i find it important enough to be noticed as an interesting discovery, as this was not thought to be possible before with the common understanding of ET of whatever size (+)

...None of the approaches has more evidence than others, they are a question of personal taste... = (?)

Quiete comment: About Stopper’s falsifications, I happen to have explained why I prefere Time to produce his/her effects, simply due to personal targets (opinion), manners-barrier and mistrust (on my part). So, I cannot even accept that role-play, Stopper asking about things that either he has already understood (I think he has the numbers - opinion), or he could ask to understand in a educated form. Not to mention the recent (my opinion) shameful, deceitful and revengeful attempt to squeeze Chas Theory into an easy-get-by gimmick.

Nevertheless, my human respect is solid.

Any other Colleague in need of more explainations may well post, I’ll be happy to contribute.

a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):

http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/14/10 02:59 PM

Jake, you write:

..."But can someone catch me up on what's happened since actual recordings of the Stopper and the CHAS became available?"...

Nothing really happened, as I say it takes time.

..."In that long thread, there was talk of Alfredo coming to the States to demonstrate an actual tuning, and people giving up on the discussion entirely,"...

The two things may go separatelly. Kent Swafford subsequently proposed October 2010, as this February a PTG seminar is taking place in Kansas City. And more than others, it was me who needed a rest from this Topic.

..."and a wish to have Bernhard Stopper in attendance."...

I wish Stopper had helped me to renew the whole ET horizon, though I cannot avoid being optimistic...you never know.

..."Not a fight over who has found the only valid temperament."...

May I ask you how old you are? That is wise of you.

..."A book, really, with chapters by each person and possibly critiques, and of course sound examples. A dvd with lectures and demonstrations?"...

All this is not easy nor direct, for many reasons... anyway Isaac Oleg is doing a lot and I hope we'll all manage to contribute, one way or another.

..."Or has everyone stopped talking?"...

I have not. Thank you and regards, a.c.

.
Posted by: Jake Jackson

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/14/10 04:06 PM

I didn't mean to extend this thread still longer! (Forgive me.) I look forward to learning more about, and hearing more of, these temperaments. I do hope that eventually something resembling an anthology of these temperaments is possible, whether at a meeting or in a book\journal issue.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/14/10 04:45 PM

..."I didn't mean to extend this thread still longer! (Forgive me.)"...

It was my intention to extend this thread, in any case, make it a very very very long thread, long like the many decades of frustration on tuners part for not being able to tune 12th root of two ET, very long like the age-old attraction for "pure" zero beating untuneable intervals.

..."I look forward to learning more about, and hearing more of, these temperaments. I do hope that eventually something resembling an anthology of these temperaments is possible, whether at a meeting or in a book\journal issue."...

Let's see, Bill Bremmer opened to the idea of an article about Chas on the PTG's Journal, perhaps he can conferm.

In any case, keep in touch and you'll find more and more material, and soon or later a video may be available (only with Isaac though, so I will not look so small!). As for the anthology, more frequently it comes...posthumous, but nowadays we have the web.

Regards, a.c.
Posted by: Phil D

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/15/10 09:08 AM

I'm eagerly following this thread, it's great to see an update to the theory of tuning that applies well to the modern piano in practice. Good work, Alfredo.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/15/10 04:18 PM

..."see an update to the theory of tuning that applies well to the modern piano in practice."...

This may well be how I should have started this Topic.

It is the story of few small numbers, all wanting to give in to one:+1.

Thanks for sharing. Are you a tuner?

Regards, a.c.

.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/16/10 01:32 AM

Hello ALfredo !

Little time those days, but I'll work on the translation of the basic article...


The recordings are in fact useful to compare the tuning with tempered intervals and the one with "pure" 12ths. Then ecah may choose depending it own inclinaison.


The reaction of some colleagues is the same than mine, till today.

Mr Stopper should give the method to tune it by ear, as your basis are in the neighbors of the same "stretch" to speak as tuners
!.
Then we can decide for a fight in 3 rounds (Ill manage the tickets !)


Yes our colleague is a tuner in Britany, studying tuning, he is lucky more informations availeable today !.


Its getting to quiet and polite there. could we have some action ? !!

For any further conversation with me call on the 08000-45-48-47-77 and note the code (12 Euros the 1 min call)
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 02/16/10 04:40 AM


Hello Isaac,

Let me tell our colleagues how nice it was meeting you in Paris. And right from the rendez-vous point, that peace of work with all those whatches piled up, suggesting the dissonance amongst those still minute hands and people fast moving around, running people witnessing still time, for once the other way around. Any idea why is that station called San Lazare?

It is very generous of you working at the translation of Chas article, I wish I could help and show you what is left of my studies...so much laughing with my sister every time I displayed my maccheroni Francais...

..."The recordings are in fact useful to compare the tuning with tempered intervals and the one with "pure" 12ths. Then ecah may choose depending it own inclinaison."...

About inclination and propensity, once I asked Stopper: if you tune pure 12ths, where do you think the piano will settle? He never answered. More recently he himself wrote in PW that pure 12ths tuning makes the tuning last longer...and I wonder if he was referring to the same concept of mine (for centre strings), i.e. you tune pure 12ths and you end up...gaining Chas. On this, I would again ask Kent Swafford (using Stopper's ETD) to record chromatic 12ths and 15ths right at the end of his own only-pure tuning, and also a week later, and two weeks later...it would be a good chance for evaluating the piano's "inclination", yes?

..."The reaction of some colleagues is the same than mine, till today."...

Do you mean no alchool, no smoke and Japanese food?

..."Mr Stopper should give the method to tune it by ear, as your basis are in the neighbors of the same "stretch" to speak as tuners !"...

Yes, I wonder why he has not provided. After all, he may well know that practical tunings is what tuners need to compare, so having a chance to achieve (aurally) what no ETD can ever gain.

..."Then we can decide for a fight in 3 rounds (Ill manage the tickets !)"...

Mmm...pure 3 again? Can we make it in 4 rounds? The 3rd round a little delta-shorter (there we need suspense), and the fourth little delta-longer (so we can finish in beating surplace).

..."Yes our colleague is a tuner in Britany, studying tuning, he is lucky more informations availeable today !"...

I agree, no more beats negation opens to a more relaxed approach, to the actual chance of enjoying beats...and turning our tuning lever with no fear.

..."Its getting to quiet and polite there. could we have some action ? !!"...

In a while I'll have to post again about Chas unique coherence, something may happen then.

..."For any further conversation with me call on the 08000-45-48-47-77 and note the code (12 Euros the 1 min call)"...

Wow, what did you record in that first minute?

Best regards, Alfredo.

.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 03/09/10 11:04 AM


Bill, you kindly write (Topic: My Piano in EBVT III):

...“Frankly, quite some time ago I gave up on following your threads. I cannot understand the math and I find your English barely understandable and therefore difficult to read.”...

Please, excuse my straight style, here in Sicily we say: the merciful doctor will turn banality into a bad disease.

For many reasons I'm sorry. I am still available for explaining Chas maths, but your difficulties may derive from lake of interest on your part.

You find my English barely understandable, and it must be true, though I tend to relate your difficulties to your approach more.

...“I am sorry but to this day, I do not know what "CHAS" means.”...

I'm sorry too, your statement tastes like an abstruse game to me.

...”I do get the idea that you tune in ET...”...

Correct.

...”and that you stretch the octaves in a particular way”...

Yes Bill, I stretch the octaves so that the octaves beat rate is progressive too, like RBIs, this opens to beat synergy. Actually, all intervals are stretched for gaining the most resonant and euphonious condition. Are you interested in synergy?

http://en.wikipedia.org/wiki/Synergy

And in Tensegrity?

http://en.wikipedia.org/wiki/Tensegrity

...”but that is about all I understand.”...

True? I do not know. Say it is true, as a teacher/examiner you could investigate more about tuning (O).

...“The same goes for the Stopper tuning.”...

This is a shame (O), since you have met Stopper, you could/should have gone deeper. And if I were you, I would take the chance to compare my tunings with colleagues, more than customers.

...”Whatever difference there may be between the way you stretch the octaves and the way Herr Stopper does seems to me to be quite small and virtually indistinguishable.”...

The same could be said about small differences in UTs, surely you would not like that, can it be a point? In actual tuning, the sum of many small differences can make a big difference and this, depending on the context, can be very relevant. I'd never teach approximation.

...”Yes, I liked the way Stopper's tuning sounded; it makes the piano sound crisp and clear. From what I have heard of your tunings, I certainly find nothing objectionable. However, when I played two recent examples that were posted, a CHAS tuning and a "standard" tuning, I frankly could not tell the difference.”...

Well, some do, some don't. In any case, you have had more Chas samples by now.

..."Both of you seem to feel that you have found the ultimate solution for tuning the piano.”...

This statement may be misleading. Is maybe that what you assumed? About Stopper's only-pure I'd let him say and, if you can, please make a distinction.

About me, I'm trying to share a new approach to the sound scale. Chas ET model is the end of the commas conflict, in a 12 semi-tonal scale and it opens to microtonal scales. Then, I suggest to always separate theory from tuning practice.

..."You want to prove somehow with math that it is valid and I have no argument with that but I personally cannot understand what the math I have seen tries to portray, so it is lost on me.”...

About music and maths interrelations I don't need to say, it is up to you wanting to understand or not, and you'd find plenty of literature. If I were you, I would not submit to simple fractions.

...”In any case, the difference in the way ET sounds, stretched very little, to moderately, to the most it could be only seems to yield very subtle nuances of difference.”...

Yes, so that you understand, subtle nuances similar to the one Serkin himself refers to.

...”Now, I did arrive at the conclusion about stretch in ET long ago as I have said. Stretching the temperament octave to a compromise between a 4:2 and a 6:3 octave, then causing an equal compromise between double octaves and octave and 5ths seemed to be ideal for me.”...

You should know, ET is not meant to be “ideal” for you only, it is meant to release a geometrical progression, ideal in that it gains “natural” proportions. I linked on this, but were you interested?

...”Many people, technicians and pianists alike expressed voluntarily how beautiful the octaves I tuned sounded.”...

No doubt about this.

...“As I had written to you long ago privately, I seemed to be able to turn the two problems in tuning, inharmonicity and the comma, against each other in a favorable way by using that approach. It reduced the "noise" inherent in tuning. It made the piano sound beautiful and clear.”...

I agree, so much so that I've tried to explain you why that can happen, but you seem to prefer “mindless” tuning, a kind of naivety that I cannot explain. Tuning EB 12ths and 15ths can somehow correct the temperament approximations. Any UT can be corrected by resorting to EB 12ths and 15ths, and Chas, nine months ago, could explain you why.

...”It turns out that this is the way most of the best tuners tune today whether they arrive at those results the way I did or not.”...

Most of the best tuners? Do you mean in USA? Have they written on this? Any name? You mentioned Steve Fairchild as your mentor, and I firmly believe he could profit from equal beating. Anyway, (O) if you want to overtake an “attempt” or “casual” level, you need to gain the maximum degree of consciousness.

...”Most technicians, pianists and music educators still believe firmly in ET as the best and/or only way to tune the piano.”...

When you understand what a natural geometric progression is, you'll also understand the “all the way” route for musicians, composers and technicians through centuries, and why we have been striving for a natural optimum.

...“No matter what is done, piano tuning is ultimately a compromise.”...

What you say is true in practice, due to pianos singularities. But for what concerns theory, today I cannot agree. The first 12th root of two ET was again a compromise, in the sense of “doing the best of a bad job”, since a pure 2:1 ratio was favored. Chas ET is an “Optimum”, and all partial ratios can now contribute proportionally.

Today for me, practical tuning is a trial of truth, the truth returned by beats, beats like rhythm, the same true rhythm that is played in music. Rhythmic synergy is what musicians go for, when playing together, synergy is what we grow up with, and what we can establish with beats when tempering.

...“Stretching ET to the point where the tempering of the 5ths is apparently hidden is one compromise, yes and it does yield its advantages and disadvantages. People can become accustomed to that sound and they can become fixated on that one sound being the one and only acceptable sound.”...

It should not be you talking about fixations. Actually, this is what maths are also meant for, to help you separate a personal fixation from a “fact”, a reality that can be objectively described and eventually shared.

...”Everyone already knows I have found another compromise.”...

Everyone? Don't say. Me? Yes. I think I told you that, like EBVT I, II, III, you may find many other compromises, I think as many as each single individual, should personal preferences be of any value. But again, if we had room for ever different/singular preferences, why promoting EBVT as the best compromise? Sorry, my logic does not help me.

...“I am in the minority, yes. Most people are skeptical about it. Some people reject it outright, some without ever hearing it. That does not hinder me because I have enough people who are interested to continue. Condemning what I do with ridicule and mockery however only invites the same in return. I recall the admonition, "If you can't say something nice, it is better to say nothing at all".”...

Have you stopped wanting to learn? I would not go for “something nice”, but relevant, possibly in scientific terms.

...“Therefore, I am not really interested in debating, analyzing, confirming or refuting which amount of stretch applied to ET is the ultimate solution. I already have my own idea about that but since I don't tune in ET, it is a moot point for me.”...

Bill, I can justify ignorance as a phenomenon, never as a choice.

If you ever really cared, you could find someone able to translate this quasi-English of mine into the most correct English. Then, if you preferred a practical demonstration, we could still try to arrange for that, then saving so many words. BTW, any news about that generous invitation? Any possible date?

Regards, a.c.

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 03/30/10 07:17 AM

Hello anybody .

Here are new samples :

Just recorded major and minor chords after a tuning on a vertical piano (Hoffmann 124 - 1985)

minor : http://www.box.net/shared/m03a2zgi9e

Major : http://www.box.net/shared/ndift3y4qd

Lately I've been also tuning very old pianos and forte using the same approach. I will post recordings , one of the advantages is a good stability, which is a real luxury on those instruments.
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 03/30/10 04:55 PM

Isaac,

I really like the balance of the CHAS ET. Open, but still together harmonically over the keyboard. I need to learn how to tune ET like this!
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 03/31/10 05:32 AM

Hi all

Here is that Chopin duo recorded on a 1843 "pianino" Pleyel (birdcage action, wood frame).


[url=http://www.box.net/shared/reu6ry9a77][/url]

The place was intended for movies so the acoustic is really dull.

What I noticed is that the justness was easy to catch for the cellist.



Last piece of the concert, and the piano have slighty moved (the bass strings wounded on brass !)

Interesting experience, I was not expecting the piano to stay playable in fact !.



Pianist : Aya akuyama, Cellist: Jerome Huille.

The piano is not really at its best level but that was the only one having the bottom C so the pianist choose it.

Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 03/31/10 07:11 AM

http://en.wikipedia.org/wiki/Tensegrity

I like that one indeed !

I have a recording of the Schott Viennese pianoforte 1833, wooden frame of course) that played in an opera, then supported a 100 km moving and was used on the next day without a complete tuning.

The piano is amazingly just.

So there is more than just numbers or a good ability with the tuning hammer .

I'll post the record.


Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/02/10 07:34 AM

Here are those links :

exerps of Beethoven sonatas piano violin.

The piano is a Scott forte dating 1833 - viennese action, leathered hammers, no iron frame of course.

What is amazing is how well it standed a moving in a little van on 100 km, no time for the tuning before the concert, only a few unisons have been tweaked (I was not there in fact)
It have been tuned twice the day before , raising from 225 to 230 Hz. (difficult to tune with those rectangular pins and T handle !)

http://www.box.net/shared/x5racjdiyf

http://www.box.net/shared/6f25atlga3


To me the Chas tuning showed there its coherence and good equilibrium. The violonist have no difficulties finding the pitch.

A little concert in the Germany house of the Cite Universitaire in Paris.


p.s I of course don't pretend the piano is perfectly tuned !
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/02/10 09:56 AM


Isaac, a few words, just to avoid advertising tones.

Your recordings sound to me very enjoyable and balanced, what a pleasure.

Then, I don't think it was Chas merit, but yours. Chas can only be a reference, for all the rest...Thank you.

Alfredo.
Posted by: Jake Jackson

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/02/10 10:33 AM

The second one, "audio track 5" is particularly good, despite the distance of the piano from the mic.

Which sonata is that?
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/03/10 02:55 AM

I dont wish to "pollute" the thread, but just get the "Cordier" pure 5th method used by my colleague .

He use an octave at 0.8 bps (4:2 test), and synchronim between the M6 and the 17th to "proove" the "pure" 5th. of course all RBI progressive.

synchronism used : M6 and the M3 ex A3 F#4 with C4 E4 (3half steps above the bass note of the M6)

I'll translate the temperament method if someone wish to try it, but only with those bases one can get there yet. because of the use of the parials match tests, as 4:2 and 3:2 the result, frequency wise, probably does not really relate exactly to the theory. But as I understand it he modified a little the original method, avoiding some of the tests proposed by Cordier.

1° accord de l'octave LA2-LA3 à 0,8 Bat/sec
2° accord de la quinte LA2-MI3 avec le synchronisme sixte-dixième
3° contrôle de la quarte ainsi formée LA2-RE3 qui bat en principe à 1,7 Bat/sec
4° accord de la quinte SOL2-RE3 idem qu'en 2°
5° contrôle de la sixte SOL2-MI3
6° accord de la tierce SIb2-RE3 en la faisant battre comme la sixte obtenue en 5°
7° accord quartes SO2-DO3 et DO3-FA3 respectivement à 1,5 et 2 Bat/sec avec , au passage , contrôle tierce DO3-MI3
8° accord de la quarte FA2-SIb2 à 1,3 Bat/sec
9° contrôle tierce FA2-LA2 et sixte FA2-RE3
10° contrôle de l'octave FA2-FA3 qui bat en principe à 0,6 Bat/sec
11° accord quartes SI2-MI3 et FA#2-SI2 respectivement à 1,9 et 1,4 Bat/sec
12° contrôle tierces FA#2-LA#(SIb)2 progressive par rapport à FA2-LA2 et SOL2-SI2
13° accord quarte SIb2-MIb3 1,8 Bat/sec avec contrôle , au passage , de tierce SI2-RE#3(MIb3) et contrôle sixte FA#2-RE#3 progressive par rapport à sixtes obtenues en 9° et en 5°
14° accord tierce LA2-DO#3 par progression de rapidité avec tierce SIb2-RE3 obtenue en 6° et/ou par synchronisme sixte-tierce (FA#2-RE#3 avec LA2-DO#3)
15° accord de LAb2 par progression tierce entre SOL2-SI2 et LA2-DO#3 et/ou par quarte SOL#(LAb)2-DO#3
16° contrôle synchronisme sixte-tierce entre LA2-FA3 et SI2-RE#(MIb)3

notes names : French terminology (mean an octave above to have the Anglo saxon usual names)
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/03/10 07:43 PM


Isaac,

thanks a lot for posting the Cordier's pure 5ths sequence.

...But as I understand it he modified a little the original method, avoiding some of the tests proposed by Cordier."...

Is this the modified version?

How do you expand this temperament?

To All: Happy (*) Easter
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/04/10 09:02 AM

Hi Alfredo !
I have to ask what differs from the original method proposed by Serge Cordier (I lost the book "Piano bien temperé et justesse orchestrale")

I believe that he did not mind the differnce between a 3:2 and a 6:4 5th, and proposed a synchronism between m3 and M3 within the 5ths , with at the same time synchronism between 10th and 17th. the 2 checks are not providing similar results.

For what I know the tuning is expanded with 10 ths and 17 ths progressiveness, with at the same time the 5th checked (I suppose).

I've find that tuning so strong that he hide the piano "natural harmony", to my ears. As I've always have been tuning "in the piano spectra" as many tuners that state that "the piano tells you" , I perceive a too strong construction as something that push the envelope, but at the same time take precedence on what I perceive as "natural harmony" indeed that may sound obscure, it is may be only due to the very strong habit of hearing tempered intervals, but it is.

SO I liked the Chas approach also because the tempering is saved (?) In fact it is even in Cordier despite what is said, a pure interval being something non existent at the piano.
BTW in classical harmony, the 5th is considered as a poor interval, too bland, and it is avoided in the chord progressions, as possible, inversions are used.



Happy **********, I am back to my chocolate eggs !
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/04/10 01:57 PM

Originally Posted By: Kamin

He use an octave at 0.8 bps (4:2 test), and synchronim between the M6 and the 17th to "proove" the "pure" 5th. of course all RBI progressive.

Isaac,

does he only use the test for 12ths - not the M6-M10 test?

'SIb2' is A#3, right? This is the note that is named slightly different from the solfege I learned as a kid at school (rusty, but it's there somewhere.. smile )
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/04/10 02:06 PM

Originally Posted By: Kamin

I've find that tuning [Cordier = pure fifths] so strong that he hide the piano "natural harmony", to my ears. As I've always have been tuning "in the piano spectra" as many tuners that state that "the piano tells you" , I perceive a too strong construction as something that push the envelope, but at the same time take precedence on what I perceive as "natural harmony" indeed that may sound obscure, it is may be only due to the very strong habit of hearing tempered intervals, but it is.

Yes, same thing for me. Equal-beating 12ths/15ths seems like my upper tolerance limit, I get harmonically uncomfortable with a wider tuning than that (as you know through discussions here).

Neither am I sure why this is, but I'm glad I'm not the only one smile

PS my pure 12ths tuning is a little over a week old now, and it's settling in... with unisons adjusted, most 12ths are not pure anymore, but closer to equal-beating 12ths/15ths! I'm going to check the 2 weeks old pure 12ths tuning at the conservatory on tuesday, and see if the same thing is happening there.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/04/10 04:02 PM

Originally Posted By: Kamin
I dont wish to "pollute" the thread, but just get the "Cordier" pure 5th method used by my colleague .

He use an octave at 0.8 bps (4:2 test), and synchronim between the M6 and the 17th to "proove" the "pure" 5th. of course all RBI progressive.

synchronism used : M6 and the M3 ex A3 F#4 with C4 E4 (3half steps above the bass note of the M6)

I'll translate the temperament method if someone wish to try it, but only with those bases one can get there yet. because of the use of the parials match tests, as 4:2 and 3:2 the result, frequency wise, probably does not really relate exactly to the theory. But as I understand it he modified a little the original method, avoiding some of the tests proposed by Cordier.

1° accord de l'octave LA2-LA3 à 0,8 Bat/sec
2° accord de la quinte LA2-MI3 avec le synchronisme sixte-dixième
3° contrôle de la quarte ainsi formée LA2-RE3 qui bat en principe à 1,7 Bat/sec
4° accord de la quinte SOL2-RE3 idem qu'en 2°
5° contrôle de la sixte SOL2-MI3
6° accord de la tierce SIb2-RE3 en la faisant battre comme la sixte obtenue en 5°
7° accord quartes SO2-DO3 et DO3-FA3 respectivement à 1,5 et 2 Bat/sec avec , au passage , contrôle tierce DO3-MI3
8° accord de la quarte FA2-SIb2 à 1,3 Bat/sec
9° contrôle tierce FA2-LA2 et sixte FA2-RE3
10° contrôle de l'octave FA2-FA3 qui bat en principe à 0,6 Bat/sec
11° accord quartes SI2-MI3 et FA#2-SI2 respectivement à 1,9 et 1,4 Bat/sec
12° contrôle tierces FA#2-LA#(SIb)2 progressive par rapport à FA2-LA2 et SOL2-SI2
13° accord quarte SIb2-MIb3 1,8 Bat/sec avec contrôle , au passage , de tierce SI2-RE#3(MIb3) et contrôle sixte FA#2-RE#3 progressive par rapport à sixtes obtenues en 9° et en 5°
14° accord tierce LA2-DO#3 par progression de rapidité avec tierce SIb2-RE3 obtenue en 6° et/ou par synchronisme sixte-tierce (FA#2-RE#3 avec LA2-DO#3)
15° accord de LAb2 par progression tierce entre SOL2-SI2 et LA2-DO#3 et/ou par quarte SOL#(LAb)2-DO#3
16° contrôle synchronisme sixte-tierce entre LA2-FA3 et SI2-RE#(MIb)3

notes names : French terminology (mean an octave above to have the Anglo saxon usual names)


Kamin, I can easily translate this into North American terminology although it will take a little time. I'll see how quickly I can do it. I remember we has discussed doing this two or three years ago.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/04/10 05:23 PM

Here is the English translation of the Cordier ET with pure 5ths sequence. I put the French in Italics and the ENglish in bold face for clarity.

Cordier Equal Temperament with Pure Fifths Sequence

1° accord de l'octave LA2-LA3 à 0,8 Bat/sec
1. Tune the A3-A4 octave to 0.8 beats per second wide.
2° accord de la quinte LA2-MI3 avec le synchronisme sixte-dixième
2.Tune the 5th A3-E4 pure using the M6-M10 check.
3° contrôle de la quarte ainsi formée LA2-RE3 qui bat en principe à 1,7 Bat/sec
3. Check the resultant A3-D4 4th which should beat at 1.7 beats per second wide.
4° accord de la quinte SOL2-RE3 idem qu'en 2°
4. Tune the G3-D4 5th pure (as in step 2).
5° contrôle de la sixte SOL2-MI3
5. Check the G3-E4 6th.
6° accord de la tierce SIb2-RE3 en la faisant battre comme la sixte obtenue en 5°
6. Tune the A#3-D4 M3 equal beating to the M6 in step 5.
7° accord quartes SO2-DO3 et DO3-FA3 respectivement à 1,5 et 2 Bat/sec avec , au passage , contrôle tierce DO3-MI3
7. Tune the G3-C4 and C4-F4 4ths at 1.5 and 2.0 beats per second respectively while checking the resultant C4-E4 M3.
8° accord de la quarte FA2-SIb2 à 1,3 Bat/sec
8. Tune the F3-A#3 4th at 1.3 beats per second wide.
9° contrôle tierce FA2-LA2 et sixte FA2-RE3
9. Check the F3-A3 M3 and the F3-D4 M6.
10° contrôle de l'octave FA2-FA3 qui bat en principe à 0,6 Bat/sec
10. Check the F3-F4 octave which should beat at 0.6 beats per second wide.
11° accord quartes SI2-MI3 et FA#2-SI2 respectivement à 1,9 et 1,4 Bat/sec
11. Tune the B3-E4 and F#3-B3 4ths respectively at 1.9 and 1.4 beats per second.
12° contrôle tierces FA#2-LA#(SIb)2 progressive par rapport à FA2-LA2 et SOL2-SI2
12. Check the progression of the F3-A3, F#3-A#3 and G3-B3 M3s.
13° accord quarte SIb2-MIb3 1,8 Bat/sec avec contrôle , au passage , de tierce SI2 RE#3(MIb3) et contrôle sixte FA#2-RE#3 progressive par rapport à sixtes obtenues en 9° et en 5°
13. Tune the A#3-D#4 4th at 1.8 beats per second wide while checking the resultant B3-D#4 M3 progression compared to the M6s formed in steps 5 and 9.
14° accord tierce LA2-DO#3 par progression de rapidité avec tierce SIb2-RE3 obtenue en 6° et/ou par synchronisme sixte-tierce (FA#2-RE#3 avec LA2-DO#3)
14. Tune the A3-C#4 M3 by progressive beat rate from the A#3-D4 M3 formed in step 6 or by M6-M3 comparison (F#2-D#4 M6 with A3-C#4 M3).
15° accord de LAb2 par progression tierce entre SOL2-SI2 et LA2-DO#3 et/ou par quarte SOL#(LAb)2-DO#3
15. Tune the G#2 by M3 beat rate progression between G3-B3 and A3-C#4 and/or by tuning the G#3-C#4 4th.
16° contrôle synchronisme sixte-tierce entre LA2-FA3 et SI2-RE#(MIb)3
16. Check the M3-M6 relationship of the B3-D#4 M3 and G#3-F4 M6 for quasi equal beating.
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/05/10 02:51 PM

Great work, Bill!
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 07:59 AM

Thanks so much Bill for that work, yes I recall I have send you the link to that Cordier discussion a few years ago.

I asked my friend how he expands the temperament, sor her is his answer (a little more work for you Bill , good that you learned French !)

Pour les prolongations vers octaves graves et aigues très simples , simplissime même :

Cibles ou targets : sortie vers aigus : do3-do4 1bat/sec , do4-do5 2bats/sec , do5-do6 4bats/ sur les très bons pianos , un poil plus (5-6) sur les pianos moins bons (là il faut de l'oreille musicale pour bien juger) , ensuite accélération exponentielle des bats dans la dernière octave .... Bien entendu le tout TRES progressif entre les cibles ! Je suis tellement habitué que je peux régler dans l'absolu une octave prise au hasard wink Of course , contrôle systématique des quintes qui se forment au fur et à mesure et elles doivent être toutes sans battement ! Sur les très bon pianos 12° obtenues automatiquement sans battement presque jusqu'en haut ; sur les autres pianos : bonnes jusque vers lab 60 après ça bat lentement entre plus ou moins 0,7 à 2 (ça dépend du piano), ensuite les bats se stabilisent (plafonnent). Entre fa3 et la3 je me sers des quartes (progressives , plus ou moins en les plafonnant suivant l'état de la quarte la2-re3 , plus ou moins lente , obtenue lors de la réalisation de la partition) en plus des octaves pour bien calibrer en combinant avec l'écoute des dixième jusque vers fa4 ...
Vers le grave utilisation :
1° des quartes jusque la1 env
2° des octaves dont il faut savoir apprécier la bonne lenteur (les quartes progressives et les quinte bien droites aident évidemment)
3° des quinte bien pures (du moins autant que le piano le permet à cet endroit)
4° écoute progressivité des sixtes et des tierces jusque vers le même La1 ; au delà 10° et 17° .

Au delà la1 quintes bien pures et octaves cassées écoutées avec autant de priorité l'une que l'autre.
Vers le grave 12° plus ou moins pures suivant qualité du piano . Pures sur les très bons , très très lentes sur les autres suivant état des cordes filées et qualité de la transition corde filées-cordes acier ... Là on rencontre souvent des discontinuités aux passages des cordes ... Preuve presque certaine de l'irrégularité de la progression de l'inh dans ce secteur ... :roll:

About the 12ths stability , he stated :

Il est absolument clair que les 12° suivant leur état initial pourront donner l'impression d'une stabilité incertaine ... Seulement la 12° semble avoir au piano une assez grande plasticité suivant le lieu de la tessiture ...


It seem clear that depending of their initial condition the 12 ths may give the impression of some instability. But semm to me that the twelve, at the piano have a somewhat large plasticity, depending of the placement in the scale...


SO for what I understand my colleague is tuning a real Cordier (because of the speed of doubles and tripes octaves. he may have changed the sequence and sequence checks, but those bps rates speak to me as the Cordier sign (more or well accepted depending of the instruments). A recording is promised ....

PS : above : C3-C4 = C3-C4 (not in French terminology where A0 is A-1)

PS What is interesting too , is that my colleague tune also standard ET, and he is organ player in churches, very used to the classical ET.

Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 08:06 AM

BTW he did not mention what is a "pure 5ths". When listening in the basses I suggest that there is a trade off between 3:2 and 6:4 ,

more probably the 3:2 is what considered the good relation as being the most audible beat most of the time.
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 09:35 AM

Kamin, that translation will take a little while longer. I won't have time for it today but I may tomorrow. The problem is with some jargon that I don't quite understand. I am not sure what is meant by the "stability" of octaves. In English, when we talk about stability, it means whether or not something stays in tune. Do I take "stability" in this sense to mean whether or not an octave is perceived as "pure" (beatless)?

Also, in the following phrase, the last word is incomprehensible to me:

"Preuve presque certaine de l'irrégularité de la progression de l'inh dans ce secteur ... :roll: "

What I get from it is this:

"Almost certain proof of irregularity in the progression of inharmonicity in this area ..." but then this "...:roll:" is incomprehensible to me. is he using the English word, "roll"? If so, what to the colon marks (:) on either side of it mean? If he is talking about a "roll" in the octaves, I think I can decipher it.

It seems to me that this French is not very well written. He seems to use two or three terms to describe the same thing sometimes. While that is OK, the whole thing seems to use a lot of jargon and abbreviations, some words left out which confuse me. This, of course is not unlike the way many people write in English. I, however have spent a great amount of time and effort trying to make the description I write as clear as possible and avoid the use of jargon. The abbreviations I use are necessary as they would be in French. As long as we all know what they mean, it is not a problem.

For example, a new reader many not understand "ETD", so I often write it out "Electronic Tuning Device (ETD)" first and use the abbreviation afterward. The system we use for naming the octaves was first advocated by Helmholtz. Dr. Sanderson used it for his first ETD and all of the other ETD manufacturers and software creators in the USA used the same one. I understand that in Europe, they think of "octave 1" or the "first octave" as beginning with the lowest note A, not the lowest C.

So, what does this phrase mean?:

"sur les autres pianos : bonnes jusque vers lab 60 "

I presume it to mean:

"on other pianos: good to about G#5 "

If so, you understand why I am a bit confused by the terminology and it will take me some time to sort it all out.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 09:44 AM

Hello, yes Bill I guess I should have cleared more the text.

I will answer to your questions. BTW "roll is some sign for a smiley , dont try to translate it !!

Best
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 09:48 AM

Originally Posted By: Bill Bremmer RPT
Kamin, that translation will take a little while longer. I won't have time for it today but I may tomorrow. The problem is with some jargon that I don't quite understand. I am not sure what is meant by the "stability" of octaves. In English, when we talk about stability, it means whether or not something stays in tune. Do I take "stability" in this sense to mean whether or not an octave is perceived as "pure" (beatless)?


STABILITY IS JUST THAT (stay in tune) USUALLY, BUT MAY BE IT CAN BE UNDESTOOD AS THERE FOR THE BEATS STOPPING TO PROGRESS AND "STABILIZE TO A GIVEN BEAT RATE" (the raising of beat rates slow then stop, hence the word)

Also, in the following phrase, the last word is incomprehensible to me:

"Preuve presque certaine de l'irrégularité de la progression de l'inh dans ce secteur ... :roll: "



What I get from it is this:

"Almost certain proof of irregularity in the progression of inharmonicity in this area ..." but then this "...:roll:" is incomprehensible to me. is he using the English word, "roll"? If so, what to the colon marks (:) on either side of it mean? If he is talking about a "roll" in the octaves, I think I can decipher it.

YOUR TRANSLATION IS PERFECT? THE ROLL IS OUT OF CONTEXT ! FORGET IT

It seems to me that this French is not very well written. He seems to use two or three terms to describe the same thing sometimes. While that is OK, the whole thing seems to use a lot of jargon and abbreviations, some words left out which confuse me. This, of course is not unlike the way many people write in English. I, however have spent a great amount of time and effort trying to make the description I write as clear as possible and avoid the use of jargon. The abbreviations I use are necessary as they would be in French. As long as we all know what they mean, it is not a problem.

For example, a new reader many not understand "ETD", so I often write it out "Electronic Tuning Device (ETD)" first and use the abbreviation afterward. The system we use for naming the octaves was first advocated by Helmholtz. Dr. Sanderson used it for his first ETD and all of the other ETD manufacturers and software creators in the USA used the same one. I understand that in Europe, they think of "octave 1" or the "first octave" as beginning with the lowest note A, not the lowest C.

So, what does this phrase mean?:

"sur les autres pianos : bonnes jusque vers lab 60 "

I presume it to mean:

"on other pianos: good to about G#5 "

YES, PERFECT

If so, you understand why I am a bit confused by the terminology and it will take me some time to sort it all out.


Sorry for the capitals, a fats way to have my comments clear.

Yes this is written in language as it is talked, I will have a look to translate tonight (I should have done it before).

Best regards

Isaac

Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 01:48 PM

Thanks, Kamin but I will be glad to do it. You answered what I needed to know. I think I can do a better job of turning the French into North American English. I would not presume to be able to translate very well what I have written into French but I can translate the French to English. It is all about idiomatic usage. I will have time for it tomorrow in the morning.
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 04:54 PM

Originally Posted By: Bill Bremmer RPT

[...]
I understand that in Europe, they think of "octave 1" or the "first octave" as beginning with the lowest note A, not the lowest C.


Bill,

we have yet another definition of the octaves, and because we've gotten our music terminology mostly from Germany, I think they might be using the same system (Maybe German forum members could give feedback on this?)

This is slightly off-topic, but just for curiosa.... smile These are our octaves:

C0-B0 = Subcontra octave
C1-B1= Contra octave
C2-B2 = Grand octave
C3-B3 = Small octave
C4-B4 = Octave 1
C5-B5 = Octave 2

... and so on. And, oh, The note B is still traditionally called H here, but that is slowly changing due to the musical influence from the US.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 05:43 PM


Thank you all for what you are adding.

More on octave designations:

http://www.music.vt.edu/musicdictionary/appendix/octaveregisters/octaveregisters.html

Regards, a.c.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 06:14 PM

Thanks Bill, I will ask for a recording on a grand Bechstein 2.20m , just bought by the pianist. I hope he will agree.
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 06:31 PM

Originally Posted By: alfredo capurso

Thank you all for what you are adding.

More on octave designations:

http://www.music.vt.edu/musicdictionary/appendix/octaveregisters/octaveregisters.html

Regards, a.c.


Great overview, Alfredo!
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/06/10 08:32 PM

Originally Posted By: alfredo capurso

Thank you all for what you are adding.

More on octave designations:

http://www.music.vt.edu/musicdictionary/appendix/octaveregisters/octaveregisters.html

Regards, a.c.


Thank you Alfredo and Patrick. Patrick, I have known about the verbal descriptions from my music education but I rarely think in those terms. The one I dislike the most is the one that identifies the notes from 1-88.

Alfredo, thank you for the other systems. I had no knowledge of these. I prefer the Helmholtz system that I described, of course and that is what the PTG Journal and the exams use, so I am used to that and prefer it.

However, knowing about other means of identification is useful and constructive, the same as knowing another language very thoroughly is. It is an expansion of the mind. When we do not limit ourselves to just one way of thinking, that we understand what other people say, in their own way, we become a more enlightened person.

That is why I did not choose to argue with you on the other thread when I gave my opinion. Yes, I do have my own opinion about what well temperament can provide. No, I do not choose to tune any pianos in the CHAS method because I feel that it is virtually the same concept that I would have if I were to tune a piano in ET but I choose not to do that.

This does not mean that I object to what you attempt to accomplish and any of the mathematical understanding of it. You should pursue it, by all means.

The Cordier method is yet another idea and is essentially off topic for this thread but it is noted by me as yet another variant of ET (in which I am not really interested). However, the translation of what is said is interesting to me. I have not yet commented on my opinion of the text but I do wish to provide an accurate translation of it in North American English so that anyone who is interested in it may form their own conclusions.
Posted by: Olek

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/07/10 02:57 AM

Originally Posted By: Bill Bremmer RPT
[
The one I dislike the most is the one that identifies the notes from 1-88.


That is how the piano makers talk to us about the notes, so to avoid any confusion. (when giving dimensions, for instance)
Posted by: Bill Bremmer RPT

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/07/10 02:54 PM

English translation of Cordier Octave Tuning

(The original French is in Italics and the English translation is in bold face for clarity)

Pour les prolongations vers octaves graves et aigues très simples , simplissime même :
To stretch the octaves in the the Bass and Treble is a very simple procedure, even extremely simple:


Cibles ou targets : sortie vers aigus : do3-do4 1bat/sec , do4-do5 2bats/sec , do5-do6 4bats/ sur les très bons pianos , un poil plus (5-6)
Objectives or Targets: Starting towards the Treble: C4-C5 1 beat per second, C6-C7 4 beats per second; on very good pianos, just a bit more, 5-6 beats per second.

sur les pianos moins bons (là il faut de l'oreille musicale pour bien juger) ,
On pianos of lesser quality, a very good ear for music is required.

ensuite accélération exponentielle des bats dans la dernière octave .... Bien entendu le tout TRES progressif entre les cibles !
Then, an exponential acceleration in the beats of single octaves in the High Treble...of course, everything must progress very evenly between the objectives.

Je suis tellement habitué que je peux régler dans l'absolu une octave prise au hasard wink Of course , contrôle systématique des quintes qui se forment au fur et à mesure et elles doivent être toutes sans battement !
I have so much experience that I can tune very accurately any octave taken at random (wink of course), check any of the 5ths that are formed as a result and they would all be found beatless!

Sur les très bon pianos 12° obtenues automatiquement sans battement presque jusqu'en haut ;
On very good pianos, 12ths (octave-5ths) are formed beatless automatically;

sur les autres pianos : bonnes jusque vers lab 60 après ça bat lentement entre plus ou moins 0,7 à 2 (ça dépend du piano),ensuite les bats se stabilisent (plafonnent).
On lesser quality pianos: good up to G#5, after that slowly progressing to between 0.7 and 2 beats per second (depending on the piano), then the beats stabilize (reach a maximum).

Entre fa3 et la3 je me sers des quartes (progressives , plus ou moins en les plafonnant suivant l'état de la quarte la2-re3 , plus ou moins lente , obtenue lors de la réalisation de la partition) en plus des octaves pour bien calibrer en combinant avec l'écoute des dixième jusque vers fa4 ...
Between F4 and A4, I reign in the beating of the 4ths (progressively but permitting only a certain maximum comparing them to the 4ths between F3 and A3, more or less slow, found when the division is created), [I do not know what is meant by the last six words but that is what it says] plus controlling the size (width) of the octave by also listening to the M10s up to about A4...

Vers le grave utilisation :
1° des quartes jusque la1 env
2° des octaves dont il faut savoir apprécier la bonne lenteur (les quartes progressives et les quinte bien droites aident évidemment)
3° des quinte bien pures (du moins autant que le piano le permet à cet endroit)
4° écoute progressivité des sixtes et des tierces jusque vers le même La1 ; au delà 10° et 17° .

Towards the Bass, I use:
1. 4ths down to about A0 [??? (that is what it says, probably means A2)]; beyond that, M10s and M17s.
2.Octaves with a nice, slow roll (progressive 4ths and very just [pure, beatless] 5ths help to justify the width of the octave)
3. Very pure 5ths (at least as far as the piano will permit them)
4. Listen to the progression of the M3s and M6s down to about A0 [again, that it is what it says, I believe he means A2], beyond that, use M10s and M17s


Au delà la1 quintes bien pures et octaves cassées écoutées avec autant de priorité l'une que l'autre.
Beyond A0 [again, I think he means A2] very pure 5ths and broken octaves [I don't know what he means by "broken" but that is what it says, perhaps "open" or slightly wide)] with as much priority [as possible] of one over the other [I presume he means to favor the 5ths over the octaves]

Vers le grave 12° plus ou moins pures suivant qualité du piano . Pures sur les très bons , très très lentes sur les autres suivant état des cordes filées et qualité de la transition corde filées-cordes acier ... Là on rencontre souvent des discontinuités aux passages des cordes ... Preuve presque certaine de l'irrégularité de la progression de l'inh dans ce secteur ... :roll:

Towards the Bass, 12ths (octave-5ths) [are to be] more or less pure according to the quality of the piano. Pure on the very good ones, very, very slow [narrow] on the others following the state [amount of inharmonicity?] of the wound strings and the quality of the transition between plain wire and wound strings [make a suitable compromise according to how good or bad the break is]...There [at the break] one often finds a large discontinuity [dilemma] at the change of [one] string [type to another]...Almost certain proof of irregularity in the progression of inharmonicity in this area [of the piano]... cursing

[Kamin]: About the 12ths stability , he stated :

Il est absolument clair que les 12° suivant leur état initial pourront donner l'impression d'une stabilité incertaine ... Seulement la 12° semble avoir au piano une assez grande plasticité suivant le lieu de la tessiture ...


It is absolutely clear that the 12ths (octave-5ths)following their initial state could give the impression of uncertain stability...Only the 12th seems to have a fairly large plasticity on the piano depending on the part of the scale. [I have no idea what the last paragraph means but that is a literal translation of what he said].

[Kamin]: [as he wrote it]It seem clear that depending of their initial condition the 12 ths may give the impression of some instability. But semm to me that the twelve, at the piano have a somewhat large plasticity, depending of the placement in the scale...


SO for what I understand my colleague is tuning a real Cordier (because of the speed of doubles and tripes octaves. he may have changed the sequence and sequence checks, but those bps rates speak to me as the Cordier sign (more or well accepted depending of the instruments). A recording is promised ....
Posted by: Jake Jackson

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/07/10 07:28 PM

Thank you for these translations.

Kamin is planning to recreate the tuning and record some pieces? Let's hope that the resulting tuning sounds good--it would be terrible if we found that the sound isn't good, after all of the attention Cordier has drawn and the time that you have given over to it.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 04/08/10 06:52 AM


Thank you very much, Bill, for your translation.

You wrote:... "The Cordier method is yet another idea and is essentially off topic for this thread but it is noted by me as yet another variant of ET (in which I am not really interested)."...

I'll write more about 1982 Cordier's ET variant in the Modern ETs thread.

Regards, a.c.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/03/10 01:32 PM


Hello.

Isaac, you wrote (April 06, 2010):

..."SO for what I understand my colleague is tuning a real Cordier (because of the speed of doubles and tripes octaves. he may have changed the sequence and sequence checks, but those bps rates speak to me as the Cordier sign (more or well accepted depending of the instruments). A recording is promised"....

It would be interesting to have the recording that was promised, also that colleague of ours may tell us whether he/she tunes three strings at the time or what. This could be relevant when having to count those beats.

Isaac (April 04, 2010)..."I've find that tuning [Cordier = pure fifths] so strong that he hide the piano "natural harmony", to my ears. As I've always have been tuning "in the piano spectra" as many tuners that state that "the piano tells you" , I perceive a too strong construction as something that push the envelope, but at the same time take precedence on what I perceive as "natural harmony" indeed that may sound obscure, it is may be only due to the very strong habit of hearing tempered intervals, but it is.

pppat (same day):..."Yes, same thing for me. Equal-beating 12ths/15ths seems like my upper tolerance limit, I get harmonically uncomfortable with a wider tuning than that (as you know through discussions here).

Neither am I sure why this is, but I'm glad I'm not the only one

PS my pure 12ths tuning is a little over a week old now, and it's settling in... with unisons adjusted, most 12ths are not pure anymore, but closer to equal-beating 12ths/15ths! I'm going to check the 2 weeks old pure 12ths tuning at the conservatory on tuesday, and see if the same thing is happening there."...

It would be nice to know from you, pppat, and about your tunings after some time was passed.

Regards, a.c.

CHAS Tuning MP3 (Granpianoman) on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/10 02:17 PM

Hello.

Bill,

You have written more on My EBVT thread. I shall reply here, with my personal opinions, on some issues that may result misleading.

You say:...”We'll wait a few days for all of you to actually prove that you can reliably tell the difference and describe what it is about the EBVT III that offends you as compared to ET.”...

You and GP are then comparing EBVT and ET. Actually, in my opinion, you may be comparing a modern quasi-ET to an historical quasi-ET.

In fact, EBVT temperament has very small cents deviations from ET electronic values, probably smaller than many first/wrong ET attempts and smaller than what the piano itself is bound to produce with its own settlings. This temperament is then expanded through the use of equal beating 12ths and 15ths, which can be two ET scale constants as described and proved by the most recent ET theory.

The ET you are comparing EBVT with, is an ETD variation that can only refer to 12th root of two, our historical model. This ETD variation is likely based on an inharmonicity model, therefore it is not directly profiting from Modern ET theories.

So I ask: What is the point of your test? To guess which is which? Or which sounds better?

Then, you ought to decide if you would like EBVT to be valid, in that it can be “confused” with ET, or if you are going for “which sounds better”.

In the first case, there would be no need to fight a war against ET and you may then acknowledge ET's evolution and modern ET achievements.

In the second case, you ought to define your audience target and consequently adjust the tunings and recordings qualities, as well as specifying which ET you want to establish the EBVT supremacy on.

You write:...”We who are enjoying the pursuit of what we believe to be state of the art innovation in tuning concepts do not need to read comments from people who have already dismissed any such ideas purely on pre-conceived notions or beliefs.”...

You talk about innovation, about pre-conceived notions, but you may quietly ask yourself: what is my notion of Modern ET's innovations?

...”To those who have been seeking perfection in modern piano tuning merely by manipulating the amount of stretch in the octaves but still insisting on ET only, may I suggest that you have been "looking for love in all the wrong places".”...

I do not know who “those” are, nor if they exist. Personally, I have been seeking resonance and harmoniousness by looking for “natural” proportions, in that nature can be our reference, our common ground, more than individual preferences. And together with octaves, Chas is offering a Rule for tuning 4ths, 5ths, 12ths, 15ths and nicely progressive RBIs.

...”The idea that tuning theory reached its pinnacle with ET was popular in the late 20th Century. ET was the ultimate. The more perfected it could be, the better the music would be.”...

Not only that, Bill. The more harmonious and resonant our tuning could be, the better we all – composers and pianists, listeners and tuners - would enjoy music.

...”So, now in 2010, the May issue of the PTG Journal has two important and pages-long articles that say essentially that we have not yet found any ultimate solution at all. We haven't found the ultimate design for a piano and we haven't found the ultimate tuning for the piano.”...

The ultimate tuning for the piano? That depends on what you are looking for. You can see, again with “Fluid piano”, that tastes and preferences may differ a lot. Though, when it comes to the semitonal scale we have longed-for, from Aristoxene to present days, you may well consider all the route, from pure fifths tuning to Chas ET.

...”If you want a real challenge, it will not be how to stretch so much or not stretch so much ET but how you can give each key signature its own definition and character yet still have the piano be able to perform all music the way that only ET is thought of as able to do.”...

You see, you talk about ET but maybe you have only experienced the historical ET and its ETD stretched variations. And to me it seems that your challenge is how much you can "magically" deviate from ET electronic values, “yet still have the piano be able to perform all music”. About cents deviations, isn't the piano "magic" enough?

...”I have done that with the EBVT III. Can you do better? If you can, I will be tuning your temperament and octave stretching style on the pianos I service as soon as you can write it out.”...

I think I've written my temperament out, both in theory and in tuning practice, but isn't you saying that you are not interested in ET? This is how I may be missing the sense of your comparison.

Regards, a.c.

CHAS Tuning MP3 - amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf


.
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/10 09:25 PM

Hi alfredo,

I think what Bill means is that EBVT III and modern versions of ET, although part of the thinking will coincide, still goes for different ideals.

You speak of making the piano as resonant as possible, where Bill wants it as musical as possible. You might argue that a piano sounding as resonant and balanced as possible is making it musical, but I'm not really sure about that.

EBVT III challenges harmony in a way that I really like. It gives movement to the progression of harmony. I'm not as concerned about whether this is the ultimate way of creating progression, I just say that it's there. And that thing - putting energy into harmonic movement - simply cannot be done using a symmetrical tuning.

So, I enjoy playing EBVT III. My studio at the conservatory have both grands in EBVT III, and it has become the first choice practice room among the students. Even when other grands are tuned more recently, it has stayed that way. They speak of music having a certain clarity and mood when played on these instruments.

I haven't told them that they are playing a different temperament.
Posted by: Jake Jackson

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/08/10 12:35 AM

Alfredo,

In the mp3 attached to your signature (the CHas Tuning.mp3 recorded on the Steinway S) the player often chooses the interval of the M6th, or at least bits of pieces that emphasize the M6. Nothing unusual there. Nice M6's.

You say earlier in this thread that in tuning, you see the use of "m3d M6 as being useless in the end." But I have to ask, in the final tuning, is there a specific check that you do on M6's? Narrow or wide in certain ranges? Beating equally with other intervals? They seem to have a specific color in CHas. Or does their sound arise only accidentally, in a manner of speaking, as a result of paying more attention to the M12's and M15's?
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/09/10 05:45 AM


Hello Jake, I'll reply asap.

Hello Patrick, you write:

...”I think what Bill means is that EBVT III and modern versions of ET, although part of the thinking will coincide, still goes for different ideals.”

Part of the thinking will coincide? To me, this sounds a bit vague, can you be more precise? From Bill I learn that he doesn't even want to know about modern versions of ET. Perhaps you are talking about your meanings.

Different ideals? Maybe, I talk about a "natural" ideal, Bill seems to talk about his own ideal.

I really wonder how Bill can still ignore modern versions of ET, and what follows is only one possible (rational) guess:

Bill (and not only him) may be convinced that ET is represented by the ETD stretched variations of 12th root of two (see PTG examination). Bill may then be interpreting some objective approximations, typical of ETD variations, as ET's limits. Then, Bill may think that in order to get color and emotions (what ETD misses?) we have to go through some “magic-out-of-tune” key-chords. Bill then refers his magic to those electronic cents deviations.

This may have convinced Bill that EBVT sounds nice in that EBVT is not ET, since ET (the ET he knows) goes to detriment of color and emotions. As a consequence, Bill is not acknowledging ET's evolution.

In my opinion, beyond any ideological position, Bill may click when he discovers that EBVT, compared to ETD values for ET, gives back a better approximation of Chas.

...”You speak of making the piano as resonant as possible, where Bill wants it as musical as possible.”...

Resonant, musical, harmonious, euphonious, colorful, motional, emotional....as a pro tuner I'd say “in tune”. A well tuned instrument opens to all the above. More than that, here I talk about a scientific Temperamental Theory, neither word-games nor magics.

...”You might argue that a piano sounding as resonant and balanced as possible is making it musical, but I'm not really sure about that.”...

What do you mean by “musical”?

...”EBVT III challenges harmony in a way that I really like.”...

That is good. I quite like it too, as an ordinary listener.

...”It gives movement to the progression of harmony.”...

Movement to the progression of harmony? What do you mean? And compared to what?

...”I'm not as concerned about whether this is the ultimate way of creating progression, I just say that it's there. And that thing - putting energy into harmonic movement - simply cannot be done using a symmetrical tuning.”...

Sorry, together with “progression of harmony”, what do you know about “symmetrical tuning”? I would like you to be a “symmetry” expert and I hope you are not improvising. And if I may suggest, try to be concerned about ultimate harmony, nature's harmony.

...”So, I enjoy playing EBVT III. My studio at the conservatory have both grands in EBVT III, and it has become the first choice practice room among the students. Even when other grands are tuned more recently, it has stayed that way. They speak of music having a certain clarity and mood when played on these instruments.”...

Clarity and mood, OK Patrick, but yesterday you wrote:...”EBVT III brings color to the keys. Harmonic motion, not clarity”.

Please, make up your mind, all this sounds like verbal contortions. And sorry, I'm not a student. Which is the tuning on the other grands?

You see Patrick, talking about tuning while mixing different levels of expertize is quite a non-sense, it can eventually be confusing and you can see in the EBVT thread what it is leading to. You are a pianist and a teacher, I am a piano technician. I do have an idea about piano conditions in schools and conservatories, and about common “in tune”standards, so I can understand your liking EBVT. Yet I would hopefully discuss the meaning of a handful of cents with Bill himself and/or experienced colleagues, while to you (teacher) I'd only say: would you like a demonstration of Chas tuning? Or: can I help you with Chas theory?

...”I haven't told them that they are playing a different temperament.”...

Is it that mysteries go along with magics? Today you may duly tell your students, that is a fairly close approximation of a modern ET Theory.

Nice reading about symmetries:

SYMMETRY AND COMPLEXITY
The Spirit and Beauty of Nonlinear Science
by Klaus Mainzer (University of Augsburg, Germany)
http://books.google.it/books?id=Gr3TEgy5...p;q&f=false

Regards, a.c.

CHAS Tuning MP3 - Amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

.
Posted by: DoelKees

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/09/10 02:18 PM

Originally Posted By: alfredo capurso


...”It gives movement to the progression of harmony.”...

Movement to the progression of harmony? What do you mean? And compared to what?

...”I'm not as concerned about whether this is the ultimate way of creating progression, I just say that it's there. And that thing - putting energy into harmonic movement - simply cannot be done using a symmetrical tuning.”...

Sorry, together with “progression of harmony”, what do you know about “symmetrical tuning”? I would like you to be a “symmetry” expert and I hope you are not improvising. And if I may suggest, try to be concerned about ultimate harmony, nature's harmony.



He means ET has the permutation group on the 12-tone set as symmetry, EBVT does not.
Therefore in a harmonic progression in ET all chords have the same quality,
but not in EBVT (or most well temperings). So if in a V-I cadence in C, GB is wider
than CE, the cadence is enhanced as we now also hear the resolution of a poor third
into a better third.

Interestingly the neutral harmonic progressions of ET are also present in meantone,
where we have a restricted symmetry group over a limited set of keys. During the
MT period there was no real harmony, so one could argue one of the reasons the well-temperaments became popular is that they also enhanced harmonic contrast.

Does your CHAS theory have anything to say about major thirds? These have always been
the major focus of unequal temperaments, and in my opinion make or break a temperament.

Kees
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/09/10 05:00 PM


Thank you, Kees, for helping with what Patrick may have meant.

Patrick is relating “ symmetrical tuning” with “putting energy into harmonic movement“.

You say:...”So if in a V-I cadence in C, GB is wider than CE, the cadence is enhanced as we now also hear the resolution of a poor third into a better third.”...

For example you say: G4-B4 is wider than C4-E4? And you mean EBVT? 12th root of two ET? Or?

Before I reply thoroughly, may I ask you whether you are a musician, and/or a theorist, a piano tuner? And/Or?

Regards, a.c.
Posted by: DoelKees

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/09/10 08:55 PM

Originally Posted By: alfredo capurso

Thank you, Kees, for helping with what Patrick may have meant.

Patrick is relating “ symmetrical tuning” with “putting energy into harmonic movement“.

You say:...”So if in a V-I cadence in C, GB is wider than CE, the cadence is enhanced as we now also hear the resolution of a poor third into a better third.”...

For example you say: G4-B4 is wider than C4-E4? And you mean EBVT? 12th root of two ET? Or?

Before I reply thoroughly, may I ask you whether you are a musician, and/or a theorist, a piano tuner? And/Or?

Regards, a.c.


Let me reword then.

If (if!) GB is wider than CE, in a V-I cadence in C,the cadence is enhanced.
GB is wider than CE in for example Werckmeister III, Lehman-Bach, EVBTIII,
but not in 1/4' meantone or ET (stretched or not).

I am all 3, but not a pro at piano practical tuning yet.

Kees
Posted by: pppat

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/09/10 09:44 PM

Originally Posted By: alfredo capurso
You see Patrick, talking about tuning while mixing different levels of expertize is quite a non-sense, it can eventually be confusing and you can see in the EBVT thread what it is leading to. You are a pianist and a teacher, I am a piano technician. I do have an idea about piano conditions in schools and conservatories, and about common “in tune”standards, so I can understand your liking EBVT. Yet I would hopefully discuss the meaning of a handful of cents with Bill himself and/or experienced colleagues, while to you (teacher) I'd only say: would you like a demonstration of Chas tuning? Or: can I help you with Chas theory?

No, I'm fine, thanks, I don't need that. Fact is, when you throw that kind of lines in my face, I couldn't be less interested.
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/10/10 07:37 AM


Hello Patrick,

I'm afraid you have misunderstood my words. With those lines I meant to say:

1- I find inadequate trying to fill up an expertize gap through posts
2- I do not find difficult to understand how you may enjoy one tuning or another
3- I would like to discuss about fine tuning with experienced colleagues
4- I'd be happy to give you, pro teacher, a demonstration of Chas tuning
5- I'll be happy to deepen with you any issue on Chas theory

I apologize for any other meaning you may have picked up, most probably due to my English (?).
Posted by: alfredo capurso

Re: CIRCULAR HARMONIC SYSTEM - CHAS - 05/10/10 08:10 AM


Hello Jake,

You say:...”In the mp3 attached to your signature (the CHas Tuning.mp3 recorded on the Steinway S) the player often chooses the interval of the M6th, or at least bits of pieces that emphasize the M6.”...

I should not think that was intentional. The player, Alessandro Petrolati, piano technician and lecturer at University of Macerata, after witnessing Chas tuning was kindly asked to play anything he wanted, possibly with a slow rhythm.

...”You say earlier in this thread that in tuning, you see the use of "m3d M6 as being useless in the end."...

Would you report my own words, so that I can contextualize them? Personally, I do not use those tests, but beat-coherence, i.e. the increment of the beat rates for chromatic intervals must make sense in proportional terms (a:b=b:c=c:d=etc.) and must be smooth.

...”But I have to ask, in the final tuning, is there a specific check that you do on M6's?”...

Generally, I check the progressiveness of M6's and other intervals on Pre-form tuning, on center strings, from about F2 to C6.

...”Narrow or wide in certain ranges? Beating equally with other intervals?”...

M6ths are never narrow, nor equally beating with other intervals. Chas form has only two equal beating intervals: 12ths (narrow) and 15ths (wide) all along the keyboard. All other intervals have their unique beat rates. If anything this, in my opinion, ends up adding color and character to each single key signature.

...”They seem to have a specific color in CHas. Or does their sound arise only accidentally, in a manner of speaking, as a result of paying more attention to the M12's and M15's?”...

Chas tuning is not an expansion of the temperament section, it is an ordered, inter-modular form (we could say a whole or a set), so nothing really can be accidental. In Chas form