To All, have a nice Sunday.

Jake wrote:..."I need a clarification here for the sentence "Stretch narrow fifths by subtracting a quantity (negative delta), and stretch wide octaves by adding the same quantity (positive delta).

Are you saying that:

1. In your CHas, M5's should be less narrow than they are in theoretical ET?

(1) - Considering both scale ratios, Chas M5's are less narrow than 12 root of two ET. What is most important is that M5's, in the Pre-tuning, must invert their beat rate progression.

2. That octaves should be wider than 2:1? (but not necessarily to 4:2)

(2) - Yes, Chas octaves are not 2:1, they are wide. I do not use 4:2 test, nor any other test of that kind.

3. That the M5's should be widened (from theoretical ET) to the same degree as the octaves are widened?

(3) - Nop. M5's and octaves have their own beat-curve. Chromatic M5's, with their beat rate progression, draw a spoon-shaped beat-curve. Octaves beat-curve is S shaped: in the midrange octaves are very little wide, and they get very slowly wider and wider towards the high and low registers.

4. Which do you tune first, or consider as the basis for the tuning--the octave or the M5?

(4) - First I tune the octave, A3-A4, but I consider five intervals as the basis of the tuning, five intervals that can be interrelated: one octave, two fourths and two fifths. Their beat-relation helps me to set the M5's inversion, the key to the whole beat-geometry.

5. Assuming that the answer to (3) is yes, is this widening of the octave a widening that starts at 2:1? In other words, if the M5 is stretched to +1 cent above the theoretical ET value, should the octave be stretched to +1 cent above the 2:1 partial match? (Or are you starting from 1:1?)

(5) - The answer to (3) is no; the widening of the A3-A4 octave starts from 2:1.

6. To ask (5) above in a slightly different way: Are these pitches set without first tuning the strings with a regard for inharmonicity? In other words, are you setting the M5 by listening only to the pitches of the fundamentals and then adding a stretch, or are you setting the M5 by a conventional method and then stretching slightly more?

(6) - No regard for inharmonicity whatsoever; I add stretch by listening and comparing beats that arise from small ratios, 2:1, 3:2, 5:4 etc.

7. What determines the degree to which the M5, and thus the octave are stretched? In other words, what gives you the number for Delta? (Is Delta the variable that accounts for the ih of the M5? If so, what partials or set of partials do you listen to, to determine Delta?)

(7) - Number (4) and (6) answer this question. It will not be one single interval, but the "relation" that you can establish between all intervals.

..."I'm sorry if I'm asking you to explain what you may have explained before, but the answers to these questions would help me to understand CHas. The discussion of CHas is spread over so many pages that finding many details is difficult.
(What confuses me, in part, is your speaking of a negative and a positive Delta, for both a positive and a negative Delta lead to a rise in pitch.)"...

Jake, I normally describe Delta only in order to explain Chas algorithm.

Your questions are welcome...let me know.

Regards, a.c.
.

Hi.

Ian wrote:..."To my mind your recordings are a perfect demonstration of your approach to ET. Some pianists will find your tuning to their taste and ideal for the way they want to play. Others, such as Kamin, would like to hear more contrast. That is why David, I think, and I are suggesting you might turn your attention to variations in temperament.

Although David might like to see a new model, I hope he would happy with a practical UT example inspired by CHAS; one that might appeal to Adolfo Barabino - http://www.youtube.com/watch?v=9zxNrQuxfNY. His homepage talks about a multitude of shades and tones, delicate details and subtle differentiations in expression.

Whether it proves a point or not does not really matter, but I am sure everyone would like to hear what you can do."...

Thank you, Ian.

In my view, there is a well defined difference between UT's and Chas ET: variable degrees of euphony (depending on the key and the type of chord) and non-structural form for the first; euphony of the whole proportional geometry for the latter. But I refer that to theory only.

When it comes to practice, that difference gets very subtle, perhaps it turns into an "arbitrary form" simulation for the first, and the attempt to the "reference form" for the latter, both forms resulting in a kind of interpretation. Again, when it comes to practice, I think we would all like to establish a steady euphonious dimension, but this cannot be the case in that the "backdrop" is always changing. In this sense, David and perhaps you too find one good reason, perhaps a reason that shouldn't even be discussed, for anticipating the changes. I'd say that, in a way, we both try to anticipate a possible scenery.

Please, keep in touch.

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

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

Alfredo,

Thanks for the detailed response. Just to be sure that I understand:

1. You say that you don't do 4:2 tests. But the octave is always above 2:1? (In other words, just to be sure we are using the same nomenclature, you make the width of the octave greater than would be achieved by matching the 2nd partial of the lower note in the octave to the 1st partial of the upper note of the octave? (You at one point speak of ratios, but the 2:1 notation denotes partial matching on the real strings to compensate for ih, not the doubling of the frequency.)

2. I'm still lost about Delta. In the context of setting the temperament, what determines the value of Delta? I may be thinking about it the wrong way. I understand Delta to be a variable. If it is a variable, and you do not do partial matching, what determines the actual number in pitching the octave and M5 when setting the temperament? The beat rate? Of the fundamentals? (I understand, I think, that the temperament is set beyond a single octave. Let's keep things to the first octave, for now, assuming that's possible.)

3. Are you creating an equal beating of partials over two octaves? If so, the partials and beat rates could be specified. (Are you using the beat rates for the fundamental in the lower octaves? Aren't fundamentals often at a low amplitude in the very lowest octaves?) Doing so would both bring us closer to understanding your tuning sequence and clarify the difference between CHas and a good ET. Perhaps you are creating an equal-beating ET?

Cheers.

More recordings coming soon?

And may I also vote yes for asking you to try your hand at a UT? I don't know why, or where it would lead, or what larger questions it would lead to. That, in part, is the point--to hear your tuning of, say, EBVT or another well temperament and read your impressions of the experience. I suggest EBVT only because it is equal beating and because I like it, and would, for simple listening pleasure, like to hear a recording of what you do with it.

Hi Jake.

...1. You say that you don't do 4:2 tests. But the octave is always above 2:1? (In other words, just to be sure we are using the same nomenclature, you make the width of the octave greater than would be achieved by matching the 2nd partial of the lower note in the octave to the 1st partial of the upper note of the octave? (You at one point speak of ratios, but the 2:1 notation denotes partial matching on the real strings to compensate for ih, not the doubling of the frequency.)

(1) - Yes, Chas octave is always above the usual theoretical 2:1 ratio, both in theory and practice, and I make the width of the octave in the way you have explained.

...2. I'm still lost about Delta. In the context of setting the temperament, what determines the value of Delta? I may be thinking about it the wrong way. I understand Delta to be a variable. If it is a variable, and you do not do partial matching, what determines the actual number in pitching the octave and M5 when setting the temperament? The beat rate? Of the fundamentals? (I understand, I think, that the temperament is set beyond a single octave. Let's keep things to the first octave, for now, assuming that's possible.)

(2) - Yes, when setting the temperament the beat rates determine the actual number in pitching the octave, two M5 and two fourths. Again, moving away from what would be achieved by matching these intervals relative partials. You are correct, the temperament is set beyond a single octave.

...3. Are you creating an equal beating of partials over two octaves? If so, the partials and beat rates could be specified. (Are you using the beat rates for the fundamental in the lower octaves? Aren't fundamentals often at a low amplitude in the very lowest octaves?) Doing so would both bring us closer to understanding your tuning sequence and clarify the difference between CHas and a good ET. Perhaps you are creating an equal-beating ET?

(3) - As you say, Chas theory and model describe an equal-beating ET. As for actual tuning (on a real piano), we can achieve that in one way only: by anticipating all the adjustments that possibly take place during our tuning. In other words, if we tried the direct tuning of equal beating 12ths and 15ths, we would not achieve Chas ET. For this reason, on a real piano, we have to be ready to manage the partials matching and beat-curves in more than one (theoretical) way; for that reason we need to consider also the tuning of pure fifths (Serge Cordier) and pure 12ths. For the very lowest octaves I use the beat rates relative to partials 3:2, 2:1, 5:1, 7:1, 10:1, depending on the overtones amplitude.

..."More recordings coming soon?"...

Yes but, please, leave everything aside and simply enjoy Chopin:

http://www.youtube.com/watch?v=fWlTHnnN4cc

..."And may I also vote yes for asking you to try your hand at a UT? I don't know why, or where it would lead, or what larger questions it would lead to. That, in part, is the point--to hear your tuning of, say, EBVT or another well temperament and read your impressions of the experience. I suggest EBVT only because it is equal beating and because I like it, and would, for simple listening pleasure, like to hear a recording of what you do with it."...

You see, my route went from a common non-structural tuning (where only intervals within one octave would find a strict rule) to the proportional beat-geometry of a coherent whole. So, I hope you understand that the tuning of a WT would not be a fresh experience as, instead, it still is for me when I go for Chas.

Regards, a.c.

P.S.: Can we moove technical Q/A in the Chas Pre.T. thread?

Hi Alfredo

I believe Jake was trying to be helpful, but I think what we are all suggesting is to explore, as you put it, other proportions of beat geometry within a coherent whole.

Taking fractals as an analogy you can have all sorts of repeating patterns within a structure. Some of these patterns may look the same but in some you will find subtle (or not so subtle) differences.

Fresh experience is exactly the right phrase!

Hi.

Bill, I'll have to reply little by little, I apologize for that.

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

..."I had not seen the above referenced sequence before but a few things about it it immediately come to mind." ... "Alfredo describes 4ths as "sharp" and 5ths as "flat" instead of "wide" or "narrow" respectively. This alone is a reason to dismiss the writing as poorly researched and written."...

You are right, a few things come to mind. The first version of my sequence, for example: it had "wide" and "narrow", but that didn't seem to be correct. Check in this thread (logged - p. 4):

#1204676 - May 23, 2009 06:54 PM
Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso

and following posts. My impression is that you over-load that issue. In any case, we have gone miles away from there, also thanks to Ernest Unrau RPT - PTG member, who kindly provided the complete sequence flow-chart (pdf), still available in the CANADIAN ASSOCIATION OF PIANO TECHNICIANS COMMUNITY FORUM - Tread: CHAS EQUATION FOR TUNING (you need to sign in).

Regards, a.c.

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

Bill Bremmer RPT:

..."It looks no different from the truncated sequence found in many tuning books that originated from the Braide-White book, Piano Tuning and Allied Arts. Many publications have taken that material and abbreviated or re-arranged it one way or another. It is still the same idea no matter how it is presented."...

Perhaps you are talking about the steps order, when the idea regards the intervals beat-curves and their interrelations. Do you know what I mean? I'm not sure, perhaps you were looking for sausages in a strawberry field.

A few things come to mind. This is your own idea of ET, non-ET and octaves:

#1204806 - May 23, 2009 11:30 PM
Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]

Bill Bremmer RPT:

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

Regards, a.c.

Alfredo

Your statement leads to two thoughts.

First, to show your equal temperament is the most appropriate way to tune a piano you need to demonstrate it is better than other temperaments with optimally stretched intervals.

Second, your model and temperaments could allow for different degrees of stretch between semitones. In just intonation, for example, E stretches 71 cents beyond D# while B stretches 119 cents beyond Bb. One might say a general version of the model should allow for the elasticity of the notes themselves, by an "s" vector or array perhaps.

Or perhaps an s matrix?

Kees

Originally Posted By: alfredo capurso
I believe the most appropriate way to tune all instruments is in a New Standard Equal Temperament With Optimally Stretched Intervals

Hi Ian. You wrote:

"Alfredo

Your statement leads to two thoughts.

First, to show your equal temperament is the most appropriate way to tune a piano you need to demonstrate it is better than other temperaments with optimally stretched intervals.

Second, your model and temperaments could allow for different degrees of stretch between semitones. In just intonation, for example, E stretches 71 cents beyond D# while B stretches 119 cents beyond Bb. One might say a general version of the model should allow for the elasticity of the notes themselves, by an "s" vector or array perhaps."...

First, I would need to know which "other temperaments with optimally stretched intervals" you are referring to.

Second, Chas algorithm calculates a figure that depends on the "s" parameter. One might already see that figure either as the scale incremental ratio or as the stretch value of a single note. Is this what you mean? Perhaps I don't understand when you say "elasticity of the notes themselves".

Thank you. a.c.
.

Alfredo

Optimally Stretched Intervals

The premises in section 3.0 of your paper appear to apply to both equal and unequal temperaments; pure intervals are not essential, differences translate into beats, a 12-semitone module is not sufficient, a double 12-semitone module is necessary.

So I imagine you could find optimally stretched intervals in both types of temperament. In saying this I have in mind have what Colin Pykett had to say in his article on impure octaves. I have quoted from it at some length at the end of the post.

The problem is that there is an indefinite number of such temperaments so you need some guiding principle(s) to focus your attention. Colin Pykett says "we have to develop the temperament after having first decided which intervals and in which keys give the pleasurable effects which are sought" and you have have your model.

I have to say I get totally lost in your paper after you introduce s at equation (5) but then you set s=1 so that we are in equal temperament. When you say the s value can determine the stretch value of a single note don't we move into the area of unequal temperaments?

What I meant were temperaments you could devise within the model by applying the algorithm to the stretch values of each and every note.

The question is how to choose values for s, s1 and so on. One starting point for selecting relative values could be a scale in just intonation (or any other set of musical tones you choose). I assume you would do this in some optimal way.

Elasticity of Notes

Let's suppose you connect each note to its neighbours with pieces of elastic. When stretched out you would expect to find the notes are equally spaced, but if the notes choose their own pieces of elastic they may be irregularly spaced.

In this way the elasticity of the notes determines the temperament and, by analogy, that could be the starting point for your pattern of s values.

In general, s values could vary across the chorale as well as between notes. Kees's s matrix?

How would those Faziolis have sounded if you had chosen other values of s?

Note: Colin Pykett's paper on Impure Octaves

Kees has pointed to this article before. Here are some of its comments on equal and unequal temperaments with impure octaves.

The major difference between the pure and impure octaves situation, as we have observed already, is that there is an indefinite number of both unequal and equal temperaments, unlike in the case of pure octaves in which there is only one Equal Temperament.

Therefore the obvious next step is to investigate unequal temperaments using impure octaves if only because these will possess key colour which the equal temperaments do not. Currently I have not gone this far. However the availability of impure octaves ought to give an additional degree of freedom when developing unequal temperaments which does not apply if the octaves are pure, consequently the unequal temperaments which can be developed would likely be original in the sense they had not been heard before.

There is an extra subjective dimension to the problem of devising unequal temperaments which does not apply to equal ones, and this relates again to the key colour issue. One cannot proceed until one has decided on the desired intonation of the 24 keys (12 major and 12 minor) in terms of the purities or otherwise of the intervals they contain. For instance, one might decide that certain keys must contain thirds as nearly pure as possible. Such constraints as these in the unequal temperaments using pure octaves with which we are familiar enable us to select which unequal temperament to use. However with impure octaves the problem is posed the other way round – instead of selecting a temperament which already exists because its intonation is pleasing in some way, we have to develop the temperament after having first decided which intervals and in which keys give the pleasurable effects which are sought.

Until we answer this question to our satisfaction we cannot proceed, but the answer will doubtless be different for different individuals. In this sense there cannot be a single unequal temperament using impure octaves which will please everybody, just as with the pure octaves case.

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

Bill Bremmer RPT:

..."The descriptions of how intervals and their checks should sound are too vague and thus open to interpretation. Quite a wide variety of results would be expected from following these instructions. The worst result, I am afraid would not be ET at all but you guessed it, Reverse Well. Show me a novice tuner who tries to tune a piano with those instructions and I'll show you a piano tuned in Reverse Well."...

Hi Bill. You were not being serious, were you? I don't know, you worry about novices; here I'm more concerned about masters and their teachings.

..."This does not mean I believe Alfredo tunes in RW. I have heard his recordings. They sound quite good. A very well executed ET, indeed. From what I could gather by reading what he has written, Alfredo advocates ET with a certain amount of stretch designed to incorporate inharmonicity to an optimum degree. That is fine, nothing wrong with that at all."...

It is not that really, please correct: Chas model has not been designed to incorporate inharmonicity; Chas has aimed at defining proportional beats for all intervals, once and for all, in theory and practice. Only then, one of the evidences is how Chas can deal with the incidence of iH.

..."However, that is what tuners have been doing in the USA now for over 30 years with far more clearly written instructions on how to do it."...

No doubt, I'm the first who thinks that ear-equipped tuners and musicians are sensible to euphony and resonance, and that proportions is what we like, iH or not iH but, talking about you, do you understand the relevance of a model? Are you able to make a difference between a sequence or a method, a technique, a model and a theory? When writing clear instructions, would you still say:…"...ET can exist with any conceivable amount of stretch or even within an octave which is deliberately narrowed. Stretching or narrowing an octave does not change any temperament,..."?

Hi Ian,

I'll reply asap. Thanks a lot.

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

Oh, my. I fear that we are still not using the same nomenclature: Setting an octave to 2:1 is NOT setting the octave to pure, theoretical ET, in which the fundamental of the upper note of the octave is twice the freq of the fundamental of the lower note. Instead, the figure "2:1" means setting the upper note's fundamental to the 2nd partial of the lower note. (On a perfect piano string, the freq of the lower note's 2nd partial would of course be the 2X the fundamental and thus the same freq as the upper note on a pure octave.

I understand that you stretch the octave a bit. My questions, then, are:

1. Do you set the first temperament octave so that the 1st partial of A4 (440) is pitched higher than the second partial of A3? (Or is the stretch instead just wider than a 2 to 1 ratio on the first partial, but more narrow than matching A3's 2nd partial to 440, which American nomenclature would describe as a 2:1 octave?)

2. When moving up to the next octave, above the conventional single octave temperament range, would you set the octave the same way, or would the stretch be higher?

I'm sorry to dwell on this point of nomenclature, but I fear that if we don't clear up this distinction between theoretical ratios and a 2:1 setting, no one will be able to accurately follow any tuning sequence for CHas, since the errors will be multiplied across the piano.

I look forward to hearing more recordings of CHas.

Hi.

Ian, thanks for your post.

Optimally Stretched Intervals

..."The premises in section 3.0 of your paper appear to apply to both equal and unequal temperaments; pure intervals are not essential, differences translate into beats, a 12-semitone module is not sufficient,."...

You say "...apply to both equal and unequal temperaments..."; one answer is:

"pure intervals are not essential" - all UT's and previous ET's have one pure interval as The reference;

"differences translate into beats" - meaning that by managing the beat-curves relative to 5:4, 4:3, 3:2, 2:1, 3:1 and 4:1, as in a system of levers, we are able to get to the One root;

"a 12-semitone module is not sufficient" - UT's ratios are referred to 12 semitones;

"a double 12-semitone module is necessary" - 24 semitones, i.e. 25 notes, this is the compass we need in order to inter-relate two (and all) octaves.

The above points summarize Chas approach and mention the limitations of UT's and other ET's "pure interval" approach.

..."So I imagine you could find optimally stretched intervals in both types of temperament."...

That is not automatic and may depend on your idea of optimality. Chas achieves proportional beats and a self-ordering form, as I've mentioned, a sound beating whole.

..."In saying this I have in mind have what Colin Pykett had to say in his article on impure octaves. I have quoted from it at some length at the end of the post."...

Thanks.

..."The problem is that there is an indefinite number of such temperaments so you need some guiding principle(s) to focus your attention. Colin Pykett says "we have to develop the temperament after having first decided which intervals and in which keys give the pleasurable effects which are sought" and you have have your model."...

Yes, I can see lots of UT's and lot's of WT's. I have focused my attention on the centuries old problem, how to combine prime numbers.

..."I have to say I get totally lost in your paper after you introduce s at equation (5) but then you set s=1 so that we are in equal temperament. When you say the s value can determine the stretch value of a single note don't we move into the area of unequal temperaments?"...

Conceptually, Chas opens to microtonality. In this sense, in force of a dynamic "s" parameter Chas is able to bridge old models and new approaches.

..."What I meant were temperaments you could devise within the model by applying the algorithm to the stretch values of each and every note."...

Yes, I get what you mean. Perhaps the area would be informatics, perhaps you want a program that allows designing of UT's. Have you heard of "Scala - ...tool for experimentation with musical tunings, such as just intonation scales, equal and historical temperaments, microtonal and macrotonal scales, and non-Western scales. It supports scale creation, editing..."?

http://www.huygens-fokker.org/scala/

"Elasticity of Notes"

..."Let's suppose you connect each note to its neighbours with pieces of elastic. When stretched out you would expect to find the notes are equally spaced, but if the notes choose their own pieces of elastic they may be irregularly spaced."...

ET notes are not equally spaced unless you convert the frequency values in cents.

..."In this way the elasticity of the notes determines the temperament and, by analogy, that could be the starting point for your pattern of s values."...

Yes, that happens with UT's and Chas algorithm does not exclude that, as a possible representation.

..."In general, s values could vary across the chorale as well as between notes. Kees's s matrix?"...

Thanks for mentioning the "chorale". I wanted to represent flows of beats and, as you say, "s" values can describe what could happen. Leave equation (5) aside, you perfectly got the meaning of "s".

..."How would those Faziolis have sounded if you had chosen other values of s?"...

I cannot say. In any case, "s" is also meant to visualize different beat-curves that we must be ready to tune, in order to anticipate the piano's adjustments.

..."Note: Colin Pykett's paper on Impure Octaves
Kees has pointed to this article before. Here are some of its comments on equal and unequal temperaments with impure octaves."...

The major difference between the pure and impure octaves situation, as we have observed already, is that there is an indefinite number of both unequal and equal temperaments, unlike in the case of pure octaves in which there is only one Equal Temperament.

In my view, when Pykett says "...indefinite number of both unequal and equal temperaments,..." he is following his own thesis; that I know, there are two other (edit: 12 TET) ET models, Cordier's and Stopper's, unless we count the numerous ETD variants.

Therefore the obvious next step is to investigate unequal temperaments using impure octaves if only because these will possess key colour which the equal temperaments do not. Currently I have not gone this far. However the availability of impure octaves ought to give an additional degree of freedom when developing unequal temperaments which does not apply if the octaves are pure, consequently the unequal temperaments which can be developed would likely be original in the sense they had not been heard before.

Here Pykett mentions the color issue, but let me quote:

http://www.pykett.org.uk/temperament_-_a_study_of_anachronism.htm#References

Yet impure octaves would without doubt have occurred if we accept the notion described earlier in which “good” Baroque temperaments arose largely through serendipity because of the tuning instability of the old stringed keyboard instruments.

So I ask: considering the same "tuning instability", what would the original color have been?

There is an extra subjective dimension to the problem of devising unequal temperaments which does not apply to equal ones, and this relates again to the key colour issue. One cannot proceed until one has decided on the desired intonation of the 24 keys (12 major and 12 minor) in terms of the purities or otherwise of the intervals they contain. For instance, one might decide that certain keys must contain thirds as nearly pure as possible. Such constraints as these in the unequal temperaments using pure octaves with which we are familiar enable us to select which unequal temperament to use. However with impure octaves the problem is posed the other way round – instead of selecting a temperament which already exists because its intonation is pleasing in some way, we have to develop the temperament after having first decided which intervals and in which keys give the pleasurable effects which are sought.

Pykett would still "slice" frequencies, as they have done in past centuries.

Until we answer this question to our satisfaction we cannot proceed, but the answer will doubtless be different for different individuals. In this sense there cannot be a single unequal temperament using impure octaves which will please everybody, just as with the pure octaves case.

You see, Pykett does not conceive a proportional sound whole, where partial 2, 3 and 5 are moulded in One. Let me quote (from the same page):

...pure octaves underpin the entire concept of temperament as it is understood today, and removing them will also remove the relative arithmetical simplicity of the subject.  If the octaves are no longer pure, the subject could easily become theoretically anarchic and entirely experiential.

Would you say that Chas is "anarchic"?

Any note on the keyboard could in principle take any frequency value, and the frequencies actually chosen would then arise solely through empiricism – trial and error
To prevent this unpleasing prospect developing, it will probably be necessary to impose a deterministic progression of octave sharpening across the keyboard.

That's done.

Each octave could contain a completely different temperament in principle, though in practice it will probably be better to regard the entire keyboard as a collection of notes upon which the notion of a distributed temperament is to be imposed.

Done.

... the results of any algorithm for distributing the note frequencies can be immediately assessed by ear.  And that, at the end of the day, is what it is all about.

That is in progress.

Regards, a.c.

Alfredo

Thank you for your extensive reply; I need to read it carefully.

Meanwhile I have been looking at your model and inharmonicity, and here are my notes:

The Introduction to Five Lectures on the Acoustics of the Piano says, "In real pianos, the resonance frequencies of the strings are not exactly harmonic. The frequency ratios are slightly larger than 1 : 2 : 3 : 4 . . . , more like 1 : 2.001 : 3.005 : 4.012 . . . ."

Allowing for inharmonicity a ratio 4 : 3 becomes 4.012 : 3.005. This equates to 1.335 and the scaling factor between semitones, the fifth root, is 1.059506.

These numbers are quite similar to those in the Chas scale (s=1) which are 1.334988 and 1.059487.

A simple coincidence?

Either way stretching out all octaves as you propose produces euphonius tunings.

Note: In the basic Chas model the delta variable Δ appears to affect the third and fourth partials in opposite directions:

(3 − Δ)^ (1/19) = (4 + Δ)^ (1/ 24)

I was trying to understand the physical significance of Δ when I thought of applying inharmonicity directly. The corresponding model is:

(4*(1+ΔiH4)/3*(1+ΔiH3))^(1/5)

ΔiH4 and ΔiH3 are inharmonicity variables.

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

Bill Bremmer RPT:

..."Furthermore, I have a problem with the title given to what is nothing more than Standard Equal Temperament with octaves (including the initial temperament octave) optimized for inharmonicity."...

Hi Bill. In my view, more than the title, you may have problems with what ET means, with optimized octaves and with inharmonicity. But, let's understand each other on the definition of the word "Standard". The word "Standard" used as an adverb means*:

In a normal, conventional, average or normative manner.
In a regular, unexceptional or frequent manner.
In an usual, common, general or habitual manner.
In a classical, classic or typical manner.
In a medium, adsorbent or median manner.
In a plain or trivial manner.
In a current or prevalent manner.
In a commonplace or familiar manner.
In a natural or classless manner.
Infrequently used adverbial inflection of the adjective standard.

Perhaps you refer to one or more of the above definitions. Perhaps you find Chas ET sounds "familiar" or "natural".

"Standard" as a noun also means*:

A basis for comparison;
A reference point against which other things can be evaluated;
The ideal in terms of which something can be judged;
That which is established by authority as a rule for the measure of quantity, extent, value, or quality;
That which is established as a rule or model by authority, custom, or general consent; criterion; test;

(*) http://www.websters-online-dictionary.org/definitions/standard

Now you may only need to sum 2 + 2 and reason upon 12 root of two ET, other ETD variants and Chas.

As I've mentioned, also the first ET was featuring a geometric set, all notes strung together in a logarithmic whole. Chas theory and practice explain why it failed.

Perhaps you still consider octaves on their own; you still ignore the bottom problem (prime numbers) and take iH as the reason for optimizing octaves.

Bill, are you with me?

Regards, a.c.
.

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

Bill Bremmer RPT:

..."."Circular Harmonic System" could be the description of any Well Temperament and Reverse Well for that matter. Any of them is "Circular" and any of them would be "Harmonic".

It reminds me of the labels that are put on agricultural produce in the USA and Europe. Produce which is grown with no manufactured chemicals or fertilizers is called "Organic" in the USA and "Biological" ("Bio" for short) in Europe. However, produce raised with manufactured chemicals and fertilizers are no less "organic" nor "biological" than those raised without them."...

Hi Bill.

I wonder what is bothering you. Is it a brief version of the essential meaning of something? Would you prefer to read "Fresh fruits or vegetable grown for the market with no material produced by involving changes in atoms or molecules" instead of "Organic"?

..."Finding a name that has not been commonly used before the general public and writing long papers with mathematics that the general public would not comprehend unfortunately does not amount to any kind of new discovery."...

I'm quite sure, someone from the PTG will be able to help you. Perhaps here and now you don't need to be concerned about the "general public" but yourself. Perhaps you are able to understand on your own why "Circular" and why a brief version:

#1241579 - August 01, 2009 06:07 PM
Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Gadzar] alfredo capurso

"I named Chas system “harmonic” because it deals with partials.

The word “circular” can paint Chas model’s soul, and it has conceptual, semantic, geometrical and numerical relevance. In a way the circularity concept, common to many cultures, substitues the usual “pure” founding concepts, those related to integer partial value’s incremental ratios.

Thinking “circular” has helped me to come over the theoretical dichotomy between “consonance” and “dissonance” (section 1.3), and suggested me to look at harmony in terms of consonance-within-dissonance. So conceptually, Chas model gains purity through a circular function, i.e. the strict relation between frequencies and beats. Beats themselves are determined by the interrelations amongst all intervals, and therefore amongst all theoretical partials, and little non-pure partials values can determine a pure-whole (section 2.0).

While “circular”, in its definition, can generally describe interrelation, this word may well evoke the continuous flowing of beats, what we should never try to stop (in theory as in practice) since on “beats-flowing” depend the natural dynamism of any frequency-whole. Chas model substitues the traditional theoretical concept of static “zero-beating” intervals – the product of any “pure”, integer ratio - with a dynamic and yet perfectly stable “beating-whole” (section 3.4). So again, the dichotomy between “static” and “dynamic” is overtaken by the concept of “stable-within-dynamic”.

In adopting the word “circular” I also meant to refer to what can be calculated in tables, as for ephemeris, and to what - based on beats - can geometrically be represented through circonferences (section 3.5), where a vector returns to the same point in a precise lap time. In fact “time”, perceptible as beat-rate, is the one Chas model’s root (section 1.2).

Also, “circular” as referred to the geometrical ideal where all points are equidistant from a centre, in the way the scale’s values relative to partial 3 and 4, in Chas model, are equidistant from there pure ratio, due to Chas delta in 1:1 proportion. From “delta” equidistance, with Chas algorithm, we could reach any difference/beating order to the nth decimal point, i.e. we could cut - precisely to infinite – any difference value."

Regards, a.c.
.

LOL. Does Chas also explain crop circles and UFO's?

Kees

Hi Alfredo

Aha! Now at last I understand.

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. I imagine the way you translate them into a tuning is a work of art.

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

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. 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.


Hi Kees

Shame on you! Have you no soul?

Sorry, I once stretched my soul by a factor
1.0384872894734862 instead of 1.0384872894735843
and it broke. So no chance of recruicruiting me
into the cult.

Kees

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!