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….

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.

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.

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.

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.”

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.

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.

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.

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.

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?

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

Gadzar, thanks for your outlook on Chas model V tuning. I'll contribute asap.

Regards, a.c.

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.

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.

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.

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

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!! 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.

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.

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.

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.

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.

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?

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.

Sorry for the typos. I changed 17ths to 15ths in the post. Thanks for pointing it out in time for me to edit it!

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.

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.

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?

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.

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!