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Alfredo

While others may disagree or come up with counter examples of ETs and UTs, these seem a reasonable set of premises to me.



To understand how this system works I'd like to look at the flowchart pdf but my application to the Canadian Association Of Piano Technicians Community Forum is still pending approval.



Yes, that what I thought and what Colin Pykett said other words. Optimal may not be the best word to use.



Clearly this is only possible in ET. Opinions may differ about what constitutes a "sound beating whole" on a particular piano, but yes for a model or an approach. You have to make choices in UT so self-ordering from "one root" without constraint is not possible.



Alfredo you are missing Colin Pykett's point here and elsewhere, and also my point in quoting him. He is talking about impure octaves in general, not your approach. One could say that you are attempting to update the concept of (equal) temperament on the piano as it is understood today, whereas he may have in mind all the horrors that impure octaves could bring to the organ.



And you are right.
And he is right,
And all is right as right as right can be
And all is right as right can be
All as right can be!

Hi Ian,

Thanks for your attentive replies.

..."You experimented, you stretched the octave, and found you tuned the most resonant and satisfying temperament when, as it so happened, E6 was as much below 1320 as A6 was above 1760. Your algorithm expressed these differences and gave you a scale factor of 1.059486544 instead of the conventional 1.059463094.

This resulted in:

E6: 1319.06482841531 = 1320 - 0.935171584
A6: 1760.93517158446 = 1760 + 0.935171584

and the whole constellation of relationships that give you with your inspiration."...

Let me add a few lines, not octaves only but all intervals, as mentioned above. The question regarded intervals beat-curves, beat rates and their coherent progressions as part of a whole* proportional geometry. Experimentations went through various steps of awareness. At one stage, coherent beat-curves could satisfy my ear indeed, but I could not notice the opposite-in sign delta. In fact, I was getting two different forms: the first, the most coherent, from the middle strings tuning, and there I was not looking for a scale constant; the second form I would get was after all unisons.

Then I understood that dynamic piano adjustments were to be dealt with and I accepted the necessity to anticipate the final form. In a way, I had to disappoint my ear and draw variable, more "salty" curves. Only then I could eventually achieve beat-rates coherence in consistent terms; only then I realized that two beat rates could well represent the one scale constant I was looking for. In fact, coherent beat-curves would result in opposite-in sign constant equal beating 12ths and 15ths.

(*) The first ET model's inspiration.

..."Now I also have a minor part of the answer to my questions about unequal temperament. The major part is skill."...

I'd say the major part is will, in order to acquire knowledge and achieve the required practical/physical skills.

..."It was a lucky coincidence that the scale factor I worked out from those inharmonicity numbers, 1.059506, was so similar to yours, 1.059486. That suggests to me that others may get the best resonances from a piano when they compensate for its harmonicity like you do."...

Yes, "others may get the best resonances from a piano when they" go for the most euphonious solution.

..."So I'd say what you have shown is that UT and WT octaves should be tuned with a similar stretch - what Bill also seems to be saying."...

Bill describes equal beating 12ths and 15ths as an easy, "mindless" and therefore convenient technique to expand the temperament octave. In my view, if I were to consider the "expansion" as a WT procedural issue, in general I'd say that pure 12ths (for middle strings) are even more "mindless". But each piano is unique and I'm for mindful tunings.

Would you like the sequence flowchart pdf. be e-mailed to you?

Regards, a.c.

Chas tuning mp3 - live recording on Fazioli
http://myfreefilehosting.com/f/07c3ca3905_6.32MB

CHAS Tuning mp3 - Amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

C.HA.S. The Harmonic Temperament, Chopin - Rondo in E-flat major, Op.16
http://www.youtube.com/watch?v=fWlTHnnN4cc

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. - Department of Mathematics, University of Palermo - 2009, Italy:
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

Article by Prof. Nicola Chiriano - published by P.RI.ST.EM (Progetto Ricerche Storiche E Metodologiche) - University "Bocconi" - Milano, 2010 - (Italian):
http://matematica.unibocconi.it/articoli/relazioni-armoniche-un-pianoforte

#1791859 - November 19, 2011 11:20 AM Re: CHAS PREPARATORY TUNING [Re: alfredo capurso]

Bill Bremmer RPT:

..."ET is now and always has been a theoretical model to which many people are drawn by its one-sided logic. It makes sense to many people to simply divide the 12 tone scale equally. However, the results were not what performing musicians wanted to hear in centuries past."...

Hi Bill,

I think people too use their ears. I'd say: no tuner could ever put the first ET theoretical model into practice. ET divides the whole scale logarithmically, but people are simply told it divides the 12 tone scale equally. However one-sided that logic is, today like in past centuries, quasi-ET tunings* result in what musicians want to hear.

(*) meaning "in practice", including my own tunings.

..."Helmholtz and Braide-White strongly advocated it as a solution. Isacoff recently described it as the "final solution" in his book. That was a very poor choice of words if you ask me!"...

I'm with you, 12 root of two would have made the "final unfeasible solution". But we don't know, perhaps Isacoff was unaware of that.

..."Throughout the 20th Century and now into the 21st, music education has made ET become the one and only frame of reference."...

But you understand that the musical world needs one reference.

..."The mere idea of "unequal" temperament seems unnatural and unwanted; not even to be considered."...

Well, perhaps not when few cents deviations here and there are not significant, to the point that not even tuners would detect them. Then the whole sense might get, what's the word, fishy?

..."Fortunately, when exposed to other possibilities, musicians often find something quite appealing to the re-introduction of Well Temperament to the modern piano."...

Yes, I'd only substitute Well Temperament with Good Tunings.

Regards, a.c.

#1792787 - November 20, 2011 09:12 PM Re: CHAS PREPARATORY TUNING [Re: alfredo capurso]

Bill Bremmer RPT:

..."Alfredo is welcome as anyone is. However, it seems to me that he is trying to convey some new discovery which he feels is the ultimate approach to tuning."...

Hi Bill,

Good for you. It took you a bit of time but you seem to have partially understood my efforts. Indeed, I'm trying to share the new ET model that perfectly fits our semitone division of the scale.

..."I well recall in my 1986 session at the Steinway factory with Bill Garlick (who had previously been a North Bennett Street School instructor, was the consultant to Dr. Al Sanderson and perhaps the most highly respected authority on tuning there could be), (also mentioned in one of Alfredo's earlier discussions), that he said, "There is nothing that anyone can do today with regard to tuning that someone else has not already done"."...

To get an idea of how narrow-minded that statement could be...depends on you. But you'll have to read and open your ears.

..."A temperament can be equal whether it is within an audibly narrow octave, an audibly beatless octave, an octave with a slight audible beat or an octave wide enough so that the fifths become beatless. From each to the next is a very small degree and each increment from an audibly beatless octave to slightly narrow or slightly stretched produces a nuance of effect, yet all tonalities will still have a character, one analogous to the other."...

To me, you sound "audibly" confused.

..."By definition and purpose, Equal Temperament (ET) has no tonal variation."...

Do you mean ET theory? I do not know who could teach you that, when I think about a model (the first ET) that nobody could apply and that, in our practice, has taken to very many ETD empirical variations. Or perhaps you mean in practice: Which "variation" are you missing?

..."Yet, Alfredo, while maintaining that ET is the only proper way to tune a piano, still maintains that there is a difference in each key and key signature."...

Please correct: I would never say that the way you tune pianos is not proper, you ought to remember, I like your tunings. What I don't like are some of your definitions, because to me, today, they sound repeated parrot fashion.

..."If it is because some people can recognize any note played on a piano without a reference and some people can recognize which key some music is in without a reference, one might be inclined to believe that ET does have distinctions."...

Yes, let's try to be inclined.

..."However, the same would be true if the piano is tuned in any Well Temperament (WT) or mis-tuned as often may be the case in Reverse Well or whether the piano is out of tune and needs tuning. There would be a limit to how far out of tune and off pitch the piano is, of course but anyone who has a good sense of pitch knows which note is being played and which key any particular chord may be in, regardless of any of the above variations."...

Sorry, I got lost. What are we talking about?

..."In all of the discussions between Alfredo and Bernhard Stopper whose amount of stretch in the octave results in a beatless octave-fifth (although Herr Stopper thinks of it as the other way around; the beatless octave-fifth results in a certain amount of stretch in the temperament octave), they seemed to be either trying to state the same idea or some very minute difference between what Alfredo suggests is optimal and what Herr Stopper suggests is optimal."...

Well, start focusing on just one or two issues: Equal Temperament has evolved. Today we have a model that we can actually tune.

..."Just how much different could a piano sound tuned by either Sr. Alfredo or Herr Stopper? Not much. Not much at all."...

That may depend on the listener, but in any case what is being shared is a new theoretical and practical reference.

..."Just how much different would a piano sound tuned by Alfredo by ear and one tuned by an optimized ETD program? Not much, if any at all."...

What makes you so sure?

..."So, the suggestion or implication that the Circular Harmonic System is the Holy Grail of tuning is far too overstated."...

Perhaps you pick up some superficial informations, you coin a new definition, like the one above, then you forget that you yourself coined it and it all turns into a real threat, what stops you from getting more knowledgeable.

..."Let's say for example that I said I had found a fabulous new cure for Hyperthermia. To cure a hyperthermic condition you will need to consume a vial of 236.6 milliliters of cryogenically treated monohydrogendioxide, assume a reclining posistion, apply a force of 3,600 meters per second of mixture of Dioxygen 23.2%, Dinitrogen 75.5%, Monocarbondioxide, 0.5% with added trace amounts of Dihydrogen, Argon, Neon, Helium, Krypton and Xenon for a period of 300,000 milliseconds.

I could then show pages of mathematics to show how the process works and pages of chemical symbols and proportions which few people could read or understand.
Or, I could say, "If you get all hot and sweaty, drink a glass of ice water, sit down and turn on a fan for 5 minutes"."...

Bill, the question would be: why do you get all hot and sweaty?

Regards, a.c.
.

Notice how Alfredo never addresses any objections regarding Chas, and never answers a specific technical question about it. Instead well founded objections (e.g. Bill's) are met with mockery whereas uninformed naive enthusiasm (e.g. Ian's) is met with praise.

This is typical behaviour of crackpots and con artists. Look up some interviews with Uri Geller (who claimed to have "real" magical powers) and you will see exactly the same behaviour.

Anybody can write 20 pages, call it a "paper" and put it on a web server. To get it published in a peer reviewed Journal is of course a different matter. This is why you can easily download papers with instructions on how to build a perpetuum mobile, but you can not find these kind of articles in peer reviewed scientific Journals. Same with this Chas "paper".

I still think (as was suggested by Bernhard Stopper well over a year ago) it is possible this Chas thing is an elaborate long running hoax and "Alfredo the tuner" exists only in the mind of some joker.

Has anyone here ever verified "Alfredo" really exists?

Kees

I am sure this could be asked of many a member here on this board including yourself. Interesting to note that while Alfredo posts under his real name you do not. Many technicians and other members of this board believe this is suspect.

Well over a year ago I was told by Isaac Oleg that Alfredo visited in his shop in France. I don’t believe that Isaac imagined the visit. I have also viewed photos of Alfredo and his children.

What a nasty insinuation, go wash your mouth with soap! Nonsense too, as I always sign my real name, and if you go to my website which is listed under my profile you can find out more about me than you probably care. Another option is to look at a list of PTG members, which has only one "Kees" as far as I know.

Kees

I know that I do not exist. I am just a figment of my imagination!

Come on Kees.

Theory and practice have become badly entangled in this thread. You and I both know the model will not stand up to rigorous mathematical scrutiny. That should not detract from Alfredo's tuning methodology or indeed from striving to translate his experience into a model.

From what I read in some of his posts, Isaac Oleg (Kamin) took Alfredo's approach seriously. When it came to comparing, so to speak, one fine wine with another, Isaac preferred French to Sicilian. Those posts may have preceded Alfredo's visit that Dan mentioned.

Alfredo seems reluctant to apply himself to UTs again. I'd be interested to know, from a modelling point of view, whether he has a point; if you want to derive a temperament from a single "root" are you inevitably in ET/Quasi ET territory? Alternatively, is it theoretically possible to derive UTs from one or more "roots" and constraints? Or must these always be entirely empirical matters?

I also wonder whether some temperaments, such as those Alfredo is aiming for, can have too much consonance, if that is the right way to describe how they sound, or whatever it was that led Isaac Oleg to say something was lacking.

As are 10,000 posts!

The CHAS system is very simple. As Bill has already pretty much established, it is almost identical to ET within his own mindless octaves paradigm. What Alfredo has done has encased it in a bit of esoteric mathematics and a bit of numerology which is valid but a little odd. However, in his analysis he found an important fact about the beat rates of the 4ths and 5ths, and has developed a way to incorporate this into his tuning. I do think the approach is very difficult, and on this forum we've not seen a satisfactory scheme in order to tune it. I've tried before from instructions posted in another thread, to great success. It took me four hours, but the resultant tuning was amazing.

(And had the damn cheek to not have English as his native language, or American as his native culture, and to not know how to use forums properly)

These attempts at character assassination are really petty. It's very difficult to engage with alfredo but that doesn't mean his ideas aren't valid.

To be clear: I am not going for character assassination, I just believe there is nothing to this Chas "theory".

I am not surprised if you spend 4 hours on a tuning the result will be good.

Kees

Hi.

Ian, here you find the review "QUADERNI DI RICERCA IN DIDATTICA" of G.R.I.M. N.19- PALERMO 2009

Chas research report is n. 5:

http://math.unipa.it/~grim/quaderno19.htm

There you read: "Every article published on the review is submitted referee from at least two members of editorial board." You'll also find the Editorial Board list of members.

Phil,

thank you for your words. Perhaps I don't know how to use forums properly* and for sure English is not my native language. Nevertheless the picture I'm getting is quite well defined and it is not difficult for me to recognize serious contributors.

I was asked for a tuning sequence and that was provided. Also Chas publication was called into question and I provided some links. Someone found that Chas maths had symbols that they could not understand, so I provided a simplified version. The same happened with Chas tuning recordings.

I agree, now it is petty and I cannot afford to go in circles, I cannot discuss absurdities and nonsense over and over again. Now I should prove that I exist... In any case, the validity of Chas materials and Chas threads contents is submitted to PW readers and colleagues.

(*):

#1792121 - November 19, 2011 07:33 PM
Re: CHAS PREPARATORY TUNING [Re: alfredo capurso]

DoelKees:

..."What I find disturbing is the refusal by Alfredo to provide a clear tuning sequence, despite being told by several of the most distinguished piano technicians in the USA and Europe, why the sequence he posted is too vague.

From this I conclude (no personal insult intended) that the whole Chas tuning is an illusion. There is no such thing. Alfredo just tunes ET like the best but that's all it is.

Added to this deception is the "mathematical" paper, which in my expert opinion as mathematician and tuning theory expert is what Wolfgang Pauli coined as "not even wrong". I call it crackpottery. You can find a forum on "not even wrong".

Reason I'm restating this and undoubtedly upsetting Alfredo again is that novice and aspiring tuners seem somehow drawn to this Chas stuff, and naively believe it must be better than the "conventional ET" that is usually taught (wow, so many equations!) and are thus led astray.

Hence I'd like clearly stated why Chas is not taken seriously by many (if not all) experts in the area.

Alfredo, you will get an apology from me if you can prove me wrong."...

Regards, a.c.

Chas tuning mp3 - live recording on Fazioli
http://myfreefilehosting.com/f/07c3ca3905_6.32MB

CHAS Tuning mp3 - Amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

C.HA.S. The Harmonic Temperament, Chopin - Rondo in E-flat major, Op.16
http://www.youtube.com/watch?v=fWlTHnnN4cc

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. - Department of Mathematics, University of Palermo - 2009, Italy:
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

Article by Professor Nicola Chiriano - published by P.RI.ST.EM (Progetto Ricerche Storiche E Metodologiche) - University "Bocconi" - Milano, 2010 - (Italian):
http://matematica.unibocconi.it/articoli/relazioni-armoniche-un-pianoforte

Alfredo

It was Kees who questioned whether or how your paper had been reviewed.

What concerns me about the mathematics is the definition and meaning of delta in your algorithm. As I said in a previous post, I am inclined to think the scale factor should be allowed to emanate from inharmonicity rather than equal deviations in frequency from just intonation.

My suggestion is to rewrite your algorithm like this:

(3 − Δ3)^(1/19) = (4 + Δ4)^(1/ 24)

and substitute (3 − Δ3) for (3 - Δ) and (4 + Δ4) for (4 + Δ) in sections 3.4 and 3.5.

Δ3 = Δ4 would be the special case you describe (S=1) but there would be no need for any S variable. Another special case could be 4*Δ3 = 3*Δ4.

Hi Ian.

#1796814 - 11/27/11 04:46 PM
Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: DoelKees]

Withindale:

"Come on Kees.

Theory and practice have become badly entangled in this thread."...

Really? Well, one of the original attempts here is to keep theory and practice separate.

..."You and I both know the model will not stand up to rigorous mathematical scrutiny."...

It will be relevant to exactly understand what you meant.

..."That should not detract from Alfredo's tuning methodology or indeed from striving to translate his experience into a model."...

We would not be here if the "striving" you mention hadn't already come to an end.

..."From what I read in some of his posts, Isaac Oleg (Kamin) took Alfredo's approach seriously. When it came to comparing, so to speak, one fine wine with another, Isaac preferred French to Sicilian. Those posts may have preceded Alfredo's visit that Dan mentioned."...

Perhaps you could read about Isaac's comments and preferences also at the Canadian Association of Piano Technicians Community Forum and here:

Accord tempéré en équation - CHAS
http://www.pianomajeur.net/forum/viewtopic.php?f=7&t=6210

..."I also wonder whether some temperaments, such as those Alfredo is aiming for, can have too much consonance, if that is the right way to describe how they sound, or whatever it was that led Isaac Oleg to say something was lacking."...

Perhaps you can explain "too much consonance" and also report what Isaac said that was lacking.

More recently you wrote:

..."What concerns me about the mathematics is the definition and meaning of delta in your algorithm. As I said in a previous post, I am inclined to think the scale factor should be allowed to emanate from inharmonicity rather than equal deviations in frequency from just intonation."...

Your premise needs to be corrected: Chas scale factor does not emanate from equal deviations in frequency; the scale factor emanates from the "s" variable.

You wrote:..."Δ3 = Δ4 would be the special case you describe (S=1) but there would be no need for any S variable. Another special case could be 4*Δ3 = 3*Δ4."...

As you say, (s = 1) is a special case. How would you cope with infinite special cases?

In section (3.3) you can read: "Equal temperament’s geometric progression, when clear of unjustified premises, suggested 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."

As I've mentioned, when we get to practice, "equal deviations in frequency from just intonation" as a rigid solution would not work, in that "tuning" is like drawing on a sheet of paper that is constantly shifting. For this reason we must be able an ready to tune different curves. This is what "s" represents.

Regards, a.c.

Hi Alfredo



This explains the difference between theory and practice very clearly. Practice does not depend on the way theory is written down on paper or whether it is written down at all.

I wrote from memory and I have not been able to find the post I had in mind. Perhaps it was not from Kamin as I thought. I am happy to add him to list of experienced tuners, like Phil, who really like the Chas tuning method.

Thank you for the links; unfortunately some of the sound samples they contain are no longer available.

When you tune an equal temperament as perfectly as you can, you are eliminating as many imperfections as possible. I used the word consonance to describe the results. Maybe it was not a good word to use. The more successful you are, the more you will emphasize any differences from the colour and character of other temperaments people hear and like. There is nothing you can do about this; it's their preference.

The equation I suggested is always true when the scale or stretch factor is constant across the octaves of interest. It could also be called s.

(3 − Δ3)^(1/19) = (4 + Δ4)^(1/ 24) = s * 2^(1/12)

This s is not the same as your s, but it covers the same infinite set of possibilities. [Note: other possibilities arise when s changes from note to note; Δ3, Δ4 will then have to change too or a pair of difference variables will have to come into the equations]

What would be really helpful to me, and maybe others, is a numerical example to illustrate sections 3.4 and 3.5.

PS This is what Kamin wrote and your reply:

..."BTW I managed to tune with a more closed tone than yours, giving all the energy to the attack and leaving the piano find its tone by itself for the rest, and I like it, it provide more tactile sensations to the pianist hence a larger dynamic palette. On a piano with not so rich tone due to old strings (I've done it yesterday with some at last 50 years old wire) that gives very pleasing results."...

I'm looking forward to hearing the tone you like the best, no doubt about superb results...Is it warmer now in Paris?

Hi Ian,

You wrote:..."This explains the difference between theory and practice very clearly. Practice does not depend on the way theory is written down on paper or whether it is written down at all."...

I'm not sure I understand what you mean, but whichever way I read your statement I get puzzled. Are you saying that theories, when it comes to practice, have no meaning? I try not to confuse, but "fuse" theory and practice. In my view, practice too can inspire theory and, if correct, the latter will improve the first.

..."Thank you for the links; unfortunately some of the sound samples they contain are no longer available."...

Thanks for reporting. I'm going to see what happened.

..."When you tune an equal temperament as perfectly as you can, you are eliminating as many imperfections as possible."...

Nop, it's not a question of imperfections alone, actually I like imperfections when I get their meaning.

..."I used the word consonance to describe the results. Maybe it was not a good word to use."...

The word is ok, do you hear too much consonance? If yes, what do you mean?

..."The more successful you are, the more you will emphasize any differences from the colour and character of other temperaments people hear and like. There is nothing you can do about this; it's their preference."...

We were not discussing people's preferences, nor color again.

..."The equation I suggested is always true when the scale or stretch factor is constant across the octaves of interest. It could also be called s.

(3 − Δ3)^(1/19) = (4 + Δ4)^(1/ 24) = s * 2^(1/12)

This s is not the same as your s, but it covers the same infinite set of possibilities."...

Have you checked (your) s ≤ 0 ?

..."[Note: other possibilities arise when s changes from note to note; Δ3, Δ4 will then have to change too or a pair of difference variables will have to come into the equations]"...

Hmmm…, do you prefer that way? If yes, why?

..."What would be really helpful to me, and maybe others, is a numerical example to illustrate sections 3.4 and 3.5."...

Numerical of which kind? Thanks for this feedback.

I see:..."PS This is what Kamin wrote...:

..."...more closed tone than yours, giving all the energy to the attack..."...

Yes, we were talking about attack Vs sustain. Isaac likes "all the energy to the attack", I prefer a beam.

Regards, a.c.

P.S.: ..."You and I both know the model will not stand up to rigorous mathematical scrutiny."...

It will be relevant to exactly understand what you meant.

Hi Alfredo

Theory and practice

You have explained how your experimentation led to your theory and, without question, your theory will have improved your practice. But you also imply you apply a more complex version of your theory than you have written down:

"As I've mentioned, when we get to practice, "equal deviations in frequency from just intonation" as a rigid solution would not work, in that "tuning" is like drawing on a sheet of paper that is constantly shifting. For this reason we must be able and ready to tune different curves."

In the paper the Chas Algorithm is stated in its simplest form in 3.1, and in 3.2 you say "the delta variable obtains two differences, in a 1:1 ratio, equal in value and opposite in
sign, one negative and one positive (0-19 negative and 0-24 positive)" . We see that in 4.0, Table 1; deviations of -/+ 0.93517 at XII (3-Δ) and XV (4+Δ).

It appears that you are talking about this specific scale of frequencies, but it is not explained where the differences in Table 2 come from or, because the rows are not identified, what they relate to. Nor is it explained how the beats and ratios in sections 3.4 and 3.5 relate to the numbers in in sections 4.0 and 4.1, or vice versa. This is makes it difficult for us to understand how to put the theory in practice. You know what it all means so you do not have the same problem.

So far I have been talking about s=1, but going back to section 3.3, you introduce integer and fractional values for s. Again it is very difficult to understand how these would be applied in practice. The significance of equations (5) and (6) and the two different values of delta is not clear. Without an explanation equations (5) and (6) seem to be in contradiction of each other.

All this is just an indication of why the paper, as it is, would not begin to stand up to the sort of scrutiny mathematicians such as Kees would undertake.

If we can see examples of specific values of delta and s resulting in specific beats and frequencies, we may begin to fully understand the theory.

I hope this answers your main questions, but it is getting rather late.

Hi Ian,

Thank you for your reply:

"Theory and practice

You have explained how your experimentation led to your theory and without question your theory will have improved your practice. But you also imply you apply a more complex version of your theory than you have written down.

"As I've mentioned, when we get to practice, "equal deviations in frequency from just intonation" as a rigid solution would not work, in that "tuning" is like drawing on a sheet of paper that is constantly shifting. For this reason we must be able and ready to tune different curves."

I see, actually I meant to say the opposite; in fact, at the end of that sentence I wrote "This is what "s" represents.", meaning that what I/we apply is not more complex than what is written down. We must be able and ready to tune different curves (in practice); and (in theory) This is what "s" represents.

..."In the paper the Chas Algorithm is stated in its simplest form in 3.1 and in 3.2 you say "the delta variable obtains two differences, in a 1:1 ratio, equal in value and opposite in sign, one negative and one positive (0-19 negative and 0-24 positive)" . We see that in 4.0, Table 1 deviations of -/+ 0.93517 at XII (3-Δ) and XV (4+Δ).

It appears that you are talking about this specific scale of frequencies, but it is not explained where the differences in Table 2 come from, or what they relate to (because the rows are not identified)."...

I hope this helps: the first line of differences in Table 2 (below) come from Table 1. If you notice, they are on the left in bold. All the other difference values come from chromatic octaves, 12ths and 15ths, i.e. differences related to partials 2, 3 and 4.

section 4.1 - Table 2
Partial 2 differences......Partial 3 differences.......Partial 4 differences
...0.0005312769............-0.002125389965.........0.002125389965

..."Nor is it explained how the beats and ratios in sections 3.4 and 3.5 relate to the numbers in in sections 4.0 and 4.1, or vice versa."...

In section 3.4 and 3.5 beats symmetries deriving from delta are visualized. (3–∆) and (4+∆) are also in section 4.0 - Table 1, on the left (red and green).

..."This makes it difficult for us to understand how to put the theory in practice. You know what it all means so you do not have the same problem."...

Yes, I understand that it is not straightforward, that's why I'm happy to help.

..."So far I have been talking about s=1, but going back to section 3.3, you introduce integer and fractional values for s. Again it is very difficult to understand how these would be applied in practice."...

As I've mentioned, integer and fractional values for s and s1 represent no more no less what we apply in practice.

..."The significance of equations (5) and (6) and the two different values of delta is not clear. Without an explanation equations (5) and (6) seem to be in contradiction of each other."...

Equations (5) and (6) were first pointed out by Roy123. So, I provided a case (s = -9/8) as a demonstration; you can find it here:

#1208057 - 05/29/09 01:19 PM
Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso

and I thought it was reasonable to improve the explanation in the paper along Roy123's advice:

#1208202 - 05/29/09 04:47 PM
Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]

Roy123:

..."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."

..."All this is just an indication of why the paper, as it is, would not begin to stand up to the sort of scrutiny mathematicians such as Kees would undertake."...

Well, Ian, we are both making a little, in my view positive effort and I hope we are making some progress. Even now, having acknowledged your points, I cannot imagine what sort of mathematicians Kees comments may belong to.

..."If we can see examples of specific values of delta and s resulting in specific beats and frequencies, we may begin to fully understand the theory."...

You can use the scale ratio and calculate specific frequencies; you can find specific beats in Chas sequence and flowchart.

Thank you very much for your valuable feedbacks. Let me know about any other question.

Regards, a.c.
.

Hi Alfredo

I'll have a look at the posts you mention, no point in repeating what has already be said.

Although I can't speak for Kees, my idea of a mathematical model is to represent what happens in practice with some accuracy.

What I'd be interested to know is how closely one of you tunings conforms to Table 1, or its equivalent for a particular piano. For instance, if your (4+Δ) was tuned to 1760.9 what might the actual values for the other notes be? Do you have some measurements of these frequencies (or cents) for any value of (4+Δ)?