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#1208192 - 05/29/09 04:31 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: ROMagister]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
Posts: 21431
Loc: Oakland
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.
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#1208202 - 05/29/09 04:47 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1708
Loc: Massachusetts
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.

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#1208767 - 05/30/09 06:04 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Roy123]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1208773 - 05/30/09 06:19 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
Posts: 21431
Loc: Oakland
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.
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Semipro Tech

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#1209059 - 05/31/09 06:07 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: BDB]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.)
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1209174 - 05/31/09 12:07 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
Roy123 Offline
1000 Post Club Member

Registered: 09/20/04
Posts: 1708
Loc: Massachusetts
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.

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#1209220 - 05/31/09 02:03 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Roy123]
alfredo capurso Offline
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Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1209281 - 05/31/09 03:59 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Kent Swafford Offline
Full Member

Registered: 06/06/07
Posts: 80
Loc: Kansas City
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.

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#1209798 - 06/01/09 01:55 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: ROMagister]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1209805 - 06/01/09 02:08 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
BDB Online   content
Yikes! 10000 Post Club Member

Registered: 06/07/03
Posts: 21431
Loc: Oakland
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.
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#1209817 - 06/01/09 02:32 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: BDB]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
This is like trying not to look at a train wreck.

Alfredo:

Can you state what you understand about iH?
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1210033 - 06/01/09 09:46 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Kent Swafford Offline
Full Member

Registered: 06/06/07
Posts: 80
Loc: Kansas City
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?

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#1210068 - 06/01/09 10:25 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Kent Swafford]
Bill Bremmer RPT Online   content
3000 Post Club Member

Registered: 08/21/02
Posts: 3195
Loc: Madison, WI USA
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.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1210181 - 06/02/09 06:35 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: BDB]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1210201 - 06/02/09 08:15 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bill Bremmer RPT]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1210224 - 06/02/09 09:51 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
Bill Bremmer RPT Online   content
3000 Post Club Member

Registered: 08/21/02
Posts: 3195
Loc: Madison, WI USA
5ths become wide on PTG Tuning Exam Master Tunings in the 6th octave.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1210239 - 06/02/09 10:25 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bill Bremmer RPT]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
Then I would think that the 12ths become wide, also.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1210612 - 06/02/09 09:01 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
Bill Bremmer RPT Online   content
3000 Post Club Member

Registered: 08/21/02
Posts: 3195
Loc: Madison, WI USA
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?
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1210835 - 06/03/09 06:57 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bill Bremmer RPT]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1210864 - 06/03/09 08:03 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1210960 - 06/03/09 11:04 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1211679 - 06/04/09 01:30 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1211692 - 06/04/09 02:00 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1211693 - 06/04/09 02:02 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.)
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1211711 - 06/04/09 02:35 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

Top
#1212089 - 06/05/09 07:05 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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."

_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

Top
#1212860 - 06/06/09 03:35 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1060
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1213205 - 06/07/09 08:21 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Robert Scott Offline
Full Member

Registered: 12/19/03
Posts: 283
Loc: Minnesota
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

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#1213393 - 06/07/09 03:54 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Robert Scott]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1213402 - 06/07/09 04:09 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4919
Loc: Bradford County, PA
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?
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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