Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum CIRCULAR HARMONIC SYSTEM - CHAS

Alfredo:

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

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

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

Bill, Tooner, all collegues,

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

Bill,

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

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

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

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

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

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

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

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

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

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

Tooner,

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

Well, what do you think?

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

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

"Did anyone at the University understand it?"

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

"Did they agree with it?"

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

Thanks, a.c.

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

Bill,

You say:

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

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

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

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

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

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

You end up saying:

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

When you started saying:

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

So, may I ask you for a wider explaination? a.c.

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

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

Roy:

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

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

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

Roy:

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

Alfredo:

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

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

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

Does this sound like what you are trying to say in your paper?

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

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

Roy123,

Thanks for joining in. You say:

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

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

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

Tooner,

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

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

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

Roy123,

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

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

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

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

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

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

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

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

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

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

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

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

Tooner,

I have to postpone your question, hope you do not mind. a.c.

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

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

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

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

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

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

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

Regards,

Bill:

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

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

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

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

Alfredo:

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

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

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

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

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

Roy:

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

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

Dear colleagues,

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

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

Bill,

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

Roy123, Tooner,

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

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

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

Then you read about the scale effects of s variable:

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

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

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

Equation (5) type:

(3–Δ)^(1/19) = (4 + (Δ*-9/8))^(1/24)

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

Substituting this Δ value:

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

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

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

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

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

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

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

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


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

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

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

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

0.0012725457675*8 = 0.01018036614

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

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

Roy123,

I would not like missing or evading any point.

Bill, Tooner,

I'll be with you asap, thank you. a.c.

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

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

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

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

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

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

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

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

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