Gadzar, you say: ...“I think iH has more to do with aural tuning than you say.”...

Ok, let’s see why you think so.

You write: ...“You don´t tune a concert grand the same way you tune a small spinet.”...

Actually, I do. Whether I tune a modern concert grand or a small Wurlitzer, an old clavicord or a cabinet or a harpsichord, I can go for the same beat-form, ET EB Chas form. And if I were to tune a pipe organ, I would try to do the same.

You say: ...“The difference is iH.”...

Yes, iH is a difference, but where? iH may differentiate the final frequencies values, but since I’m listening to beats, it does not make any difference at all.

You add: ...“No matter if you tune them aurally or using an ETD, iH is there and it must be taken into account.”...

When I’m tuning, what I take into account are beats.

You explain: ...“When tuning aurally you are listening to beats, not frequencies OK. But beats produced between inharmonical partials.”...

Ok, what’s the problem with “inharmonical partials”? They do not make me deaf. One by one, I go for the beat I want, and every time I recognize that it is not the beat I wanted I correct it.

...”So iH changes the actual frequencies in your aural tuning.”...

This is true, iH can change our tuning’s actual frequencies, but so what? When tuning, we have to order intervals beats.

You write: ...“I see a black spot on CHAS because there is no formula that contains iH data in an explicit way.”...

You see a black spot there, I see a blanc, mainly because iH data are not needed for aurally tuning Chas (nor for understanding Chas approach and maths), therefore it was not a matter of primary importance. Anyway, once we’ll be able to calculate the real incidence (the impact) of iH on Chas theoretical frequencies (in circle with strings scaling), it will not be difficult to formulate it.

For the time being, I’m happy to share a new outlook on tuning that finally makes theory applicable into practice, and get reed off theoretical wrong, damaging axiomes: the octave module, the 2:1 octave’s ratio and any “pure” partial’s incremental ratio. Would you please give me a good reason for theoretical zero-beating octaves (12th root of 2)? Or for “pure” (i.e. zero-beating) 12ths? Or “pure” 5ths or what’s so ever “pure” zero-beating ratios? Chas model’s theoretical key is “beating” in any dynamic-form, a form that can be said “ideal” when it can beat an yet remain perfectly stable.

You write: ...“By the way, I have read a number of times that fourths and fifths must progress.”...

Ok, but you red they must progress how?

You say: ...”I tune different types of octaves all along the scale favoring high partial octaves in the bass and low partial octaves in the treble, not that octaves exactly progress, but I don´t tune the same octaves all along the scale.”...

Chas octaves have a very very slow beating in the middle register, but are more and more beating when going towards bass and treble, always wide and always progressive.

...”But I confess that this is the first time I hear about tuning inverted fifths intentionally.”...

Good, you are one of the few that explicitly admit it.

You say: ...“So I don't know if my fifths become inverted at some spot in the treble.”...

Try to intentionally invert 5ths between A3 and A4.

You say:...”I tune equal beating 12ths and 15ths, so 12ths are narrow and 15ths are wide. But that doesn't mean 5ths are narrow, it will depend on how wide is the octave.”...

When in the treble you can check octaves, 10ths, 12ths, 15ths, 17ths and 19ths you do not need to check 5ths anymore. Check for progressive octaves (check with middle string only), tune middle string a bit higher, i.e. make your check-intervals a bit wider, so that when you join left and right strings you can get stable and constant 12ths and 15ths equal beating.

You write: ...”Another question: You named your system CIRCULAR HARMONIC but I don´t see why circular and why harmonic. Can you explain more about this?”...

Thanks, I had to give this answer to ROMagister too. 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 (as demonstrated with the kite analogy).

To appreciate the uniqueness of Chas model and have one more numerical proof of what I’m stating about circularity, let’s find together Chas 4th centre, i.e. the numerical middle distance between Chas scale’s values relative to intervals 12th and 15th:

- From Chas incremental ratio, we gain Chas scale’s values,

Chas incremental ratio = (3 - ∆)^1/19 = (4 + ∆)^1/24 = 1.0594865443501...

- subtract the scale’s value n. 19 (Chas 12th = 2.997874610034...) from the scale’s value n. 24 (Chas 15th = 4.002125389964...), and divide by 2.

The result is: 0.502125389964...

Chas 4th centre, i.e. the middle distance from Chas 12th and 15th shares Chas 15th’s decimal value. This explains also why Chas ratio, while resulting as well from a square root (octave ratio’s square root), has been described as deriving from a “circular” system.

Tooner, thanks for that story, good fun. Did you know that, once they left Euclide’s geometry behind, they went out together?

Regards, a.c.

"Tooner, thanks for that story, good fun. Did you know that, once they left Euclide’s geometry behind, they went out together?"

Yes, they went out together to the parking lot and settled it with their fists!

I doubt you do. With no iH, i.e. in a pipe organ, fifths must remain 2 cents narrow and will beat progressively faster as you go up the scale, you will never tune a piano this same way.

In a concert grand you can tune 6:3 octaves in the temperament section and they will sound good, if you do the same on a small spinet you get horrible soundig octaves. Why? Because partial 6 is much higher on a spinet than in a concert grand so you can not tune them in the same way, with the same stretching.

You say you only hear to beats and thus iH is mindless. I disagree. You tune adjusting beat rates but the way these beat rates progress is drastically affected by iH for some intervals more than for others. In order to get clean octaves you must tweak the beat rate progression of the other intervals:fifths, fourths, etc...

I am still confused by the inverted fifths. If we tune wide fifths then there is no equal beating possible between 12ths and 15ths. Because if the octave is wide and the fifth is inverted, i.e. wider than pure, then the 12th will be also wide and will beat at a slower rate than the 15th. Does CHAS tune equal beating 12ths-15ths?



I suppose ET EB Chas form means: Equal Temperament Equal Beating Chas form.

Equal Beating in what intervals?

Alfredo:

I have been trying to think of a single point of discussion that might be productive. I think your 5ths becoming wide between A3 and A4 may be one.

Can you show step by step how your CHAS ratio predicts wide 5ths between A3 and A4?

Gadzar, I'll soon answer to all you wrote in your post. Meanwhile, you say:

..."I suppose ET EB Chas form means: Equal Temperament Equal Beating Chas form.

Equal Beating in what intervals?...

Is this an attempt to drive me insane (with laughing)? ET = equal temperament, EB = equal beating, ever equal (and constant, i.e. non-progressive) 12ths (narrow) and 15ths (wide).

Tooner, you write: ..."I think your 5ths becoming wide between A3 and A4 may be one."...

Please, acknowledge that when I say 5ths invert I do not mean "5ths become wide", I mean that 5ths stop going progressively narrower and start going progressively less and less narrow, i.e. once 5ths invert (between A3 and A4), 5ths go progressively towards pure-crossing. Moreover, I talked of wide 5ths (untill you can ear 5ths) when tuning middle string only (check in my past posts), this to compensate string's, bridge's and sound-board's elasticity. In fact, when you unison left and right string, high notes (especially) will flatten a little bit, so wide 5ths (and generally wider check intervals) to counterbalance that flattening phenomenon.

Regards, a.c.

Mr. Capurso,

For me:

Pure fifth = beatless fifth with frequency of 3rd partial of lower note of fifth equal to frequency of 2nd partial of upper note of fifth.

Narrow fifth = beating fifth with frequency of 3rd partial of lower note greater than frequency of 2nd partial of upper note.

Wide fifth = beating fifth with frequency of 3rd partial of lower note of fifth being inferior to frequency of 2nd partial of upper note of fifth.

I am talking about 3:2 fifths type of course, not 6:4 fifths type.

I have understood that: inverted fifth = wide fifth.

But you say:



So, bellow inversion point, were fifths going progressively narrower? And at the inversion point they begin to stretch (go progressyvely towards pure)? Do they become pure at some point? Do they cross pure point and become wide at some other point? In the low bass, are they pure or even wide?


So, to put it in my own words:

You say that in E.T. the width of the fifths is no constant all along the scale but it progresses from pure, to narrow, to pure? And maybe even: it progresses from wide, to pure, to narrow, to pure, to wide? Is that it?

Note that I am not talking about beat rates, but about the width of the fifths. We can have constant width of the interval with progressing beat rates, because beat rates increase with frequencies of partials involved. But if the width is constant the beat rates can only progress in one direction, i.e. becoming faster all along the scale (or slower all along the scale), If width is constant there is no inversion point.

Gadzar thanks, I had prepared my answer to your second last post, hope it is still relevant.

To me saying: ...” Whether I tune a modern concert grand or a small Wurlitzer, an old clavicord or a cabinet or a harpsichord, I can go for the same beat-form, ET EB Chas form. And if I were to tune a pipe organ, I would try to do the same.”...

You answer: ...”I doubt you do. With no iH, i.e. in a pipe organ, fifths must remain 2 cents narrow”...

You say this because you are considering traditional ET pseudo-model.

...”and (fifths) will beat progressively faster as you go up the scale,”...

This is why, as a consequence, 12ths and 19ths are generally unbearable.

...”you will never tune a piano this same way.”...

Exactly, I’d never tune a piano nor a pipe organ that way. On both, I’d stretch octaves by inverting 5ths, and find ET EB Chas form.

You say: ...”In a concert grand you can tune 6:3 octaves in the temperament section and they will sound good, if you do the same on a small spinet you get horrible soundig octaves. Why? Because partial 6 is much higher on a spinet than in a concert grand so you can not tune them in the same way, with the same stretching.”...

In your way, iH can determine your octave. I do not let iH dictate me the amount of octave stretch in any section, as I’ve said I go for the beat I want, be it a concert grand or a spinet. If anything, in the temperament section, inverted 5ths determine octave stretch and vice-versa (sequence’s steps 1 – 4), i.e. by stretching the octave I can set up inverted 5ths.

You say:...”In order to get clean octaves you must tweak the beat rate progression of the other intervals:fifths, fourths, etc..."...

What do you mean with “clean octaves”? About having to “tweak the beat rate progression of the other intervals”, this happens becouse you do not invert 5ths. Try inverting 5ths as I suggest, then you can tell us.

You say: ...“I am still confused by the inverted fifths. If we tune wide fifths…”...

I hope you are not confused anymore. Inverting 5ths does not mean making 5ths wide, it means that 5ths go progressively towards pure-crossing, as I wrote in my last post.

Now you ask: ..."So, bellow inversion point, were fifths going progressively narrower?"...

Yes.

..."And at the inversion point they begin to stretch (go progressyvely towards pure)?"...

Yes.

..."Do they become pure at some point?"...

In Chas final tuning form, high 5ths will sound pure, although I think 5ths sound pure there because the beat-rate is to slow to get it in the sound duration. As I've said, to gain this final form, when tuning middle string you need to cross 5ths pure-point and make 5ths progressively wide.

...Do they cross pure point and become wide at some other point?...

Yes, when tuning middle strings. The point is between C5 and C6, but consider a very very slow progression.

..."In the low bass, are they pure or even wide?...

No, in low bass 5ths are narrow. I'll complete my answer with my next post, this i.point is shuting down.

Regards, a.c.

Gadzar, to complite answering your questions, you ask:

..."the width of the fifths is no constant all along the scale but it progresses from pure, to narrow, to pure?"...

In my final tuning, from bass to mid-range 5ths go progressively more and more narrow, between A3 and A4 5ths invert and progressively direct to pure. Notice that, when tuning high notes middle-string, it is necessary to cross 5ths pure-point (i.e. you need to stretch more than 3/2^(1/7) and therefore more than 19th root of 3) so that, after unisons and settlings, you'll be able to get Chas ET EB form.

You ask: ..."And maybe even: it progresses from wide, to pure, to narrow, to pure, to wide? Is that it?"...

No, 5ths invert only between A3 and A4. I'm going to replay to some other interesting post of our colleagues and of your's, thanks.

Regards, a.c.

Kent, on the 06/22/09 you kindly wrote:

...” it appears that talk of optimum stretch preferences seem to be giving way to talk of the optimum intervals on which to bass the underlying mathematical model of equal temperament.”...

So far, there were not many options for ET models, actually we could only choose amongst single “optimum intervals” pure ratios, more precisely 2:1, 3/2:1 or 3:1. Today, Chas model uses two intervals, the 12th relative to partial 3, and the 15th relative to partial 4. In this way, it is possible to proportion also partial 2 and partial 5, the latter being semitone n. 28, the 7th third. With any other ET mathematical pure ratio, where single partial’s ratio (2:1, 3/2:1, 3:1) determine the temperament, the other non-determining partials are bound to be damaged.

To Gadzar you wrote: ...”I also await the answer to the question you ask, that is, "How does Mr. Carpuso effectivelly tune CHAS?"...

This was my answer (in case you missed it): So far you could only read (and learn) that 4ths and 5ths can be tuned with “similar” beats/rate. I’m stating that octaves, 4ths and 5ths can - and need to - be progressive, like 3ds, 6ths, 10ths and so on. The first 4 steps of my sequence (like any other sequence could do) establishes “inverted 5ths” and wide octaves, the beat/rate is then described in “wider and narrower” terms because in aural tuning all intervals are related to each other, and it would be pratically of no use talking of very slow beat/rates. In fact, for the octaves, the only way to make them progressive, in my experience, is to calculate the time needed for the beat to rise, a question of very very small variations.

Kent, may I ask you: did you know that 5ths can invert in their being progressive? And that 4ths and octaves can be progressive too? Have you been able to see what 12ths you get at the end of your ETD tuning? Are 12ths pure, a little narrow or what? How are 12ths supposed to be?

You say: ...”Presently, I doubt that "CHAS" actually exists as a successful piano tuning system,"...

I do not really understand what you mean. You could believe that 2:1 ET ratio exists, today you say you believe that 3:1 ET exists, although it uses one more zero-beating, pure-ratio. Now, Chas model is featuring an ET ratio, ET like those above, where the difference regards the scale ratio’s attribute. In fact Chas ratio is not a “zero-beating” ratio, it is an ET EB ratio. Way do you doubt it exists? If anything, I’d ask: can “zero-beating” actually exist? On the theoretical and practical grounds, could you give me/us one good reason for adopting a zero-beating theoretical ratio?

...”but, though I am skeptical, I believe I am still open to CHAS.”...

Thanks for being still open to Chas. After 300 years of 2:1 ET ratio and the latest 3/2:1 and 3:1 pure ratio, I’m sure it will not take long to appreciate Chas ET EB ratio, coming at last from partials combination and from a comprehensive algorithm that can calculate those pure ratios as well. By the way, how clear (or obscure) did you find the “kite analogy” (06/04/09)?

Tooner, you kindly wrote (06/22/09): ...”Can all intervals be progressive on a piano that has iH? I am not so sure.”...

I’ve already stated precisely that Chas model describes 12ths and 15ths constant-equal-beating intervals, so 12ths (narrow) and 15ths (wide) are not progressive, they have the same, constant beat-rate all along. All the other intervals are progressive.

You say:...”There are many descriptions of tunings where fifths become wide in the high treble. Well, unless they also become fast in the bass, this would not be progressive.”...

In my opinion, the practical question is not if 5ths in the high treble are to be tuned wide, the practical point is how to manage 4ths, 5ths and octaves, starting from the temperament section. Right from the beginning we must determine the 4ths-5ths-octave’s relation, i.e. we must set up widening 4ths, directing-to-pure 5ths and a wide octave. This is way, to begin with, I tune A3 (from A4) and D4 and E4 so that I can temporarily evaluate together two 4ths, two 5ths and one octave (this sequence is not a must).

You write: ...” I asked a while back and it seems that the strings on a piano can only be set stabile within 0.3 cents.”...

As I’ve have said, string’s stability is the tuning must and it is the only real challenge, if a tuner can not set a string stable, he’ll get nowhere.

You say:...”One person’s perfect ET is another’s poison. Because of iH, and personal preference, ET is region not a location.”...

In my opininion, those conclusions are misleading. From what you say, it seems that nothing should ever go beyond iH (you were talking of “iH on everything”, remember?) or personal preference. I do not know what standards you are used to or referring to, but I can tell you that a truly good tuning will never be a poison for anybody. As for our practice, before surrendering to iH or talking about personal preferences, before saying that there is no reason for improving tuning theory and no way to improve tuning practice, I would take the chance to invert 5ths and go for their smoothest progression too. If I’m trying to share Chas ET EB model is because, in my professional experience, Chas is an absolutely precise ET location, inside the ET region. Way? Because Chas draws the precise ET form deriving from constant-equal-beating 12ths and 15ths, i.e. deriving from two opposite constants with the same one beat/rate all along.

May I ask you as well: could you give me/us one good reason for using an ET zero-beating (2:1, 3:1,…) theoretical ratio?

You write: ...“And I have already asked questions about how a piano is tuned to CHAS and cannot understand the answers.”...

Please, tell me if I can explain more about Chas tuning.

BDB, you wrote (06/22/09: ...” You guys are thinking about this too much. You tune a piano so that it sounds like it is equal tempered. That is all there is too it. Someone else can come along with a frequency counter and debate what it actually is, but as Duke Ellington put it, "If it sounds good, it is good."...

With this same approach to human progress, we would still light a fire by hands. Sure, also simplicity can be a life-key.

Tooner, the same day you wrote: ...”additional stretch can be given in an attempt to satisfy the well-documented human ear's desire to hear stretched octaves.”...

Not only, on my part there is also the desire to correct a wrong theoretical assumption, that you can base a tuning model on a zero-beating ratio, be it 2:1, 3/2:1 or 3:1.

You say: ...”The use of 12ths, not necessarily pure, is a great tool in controlling this additional stretch.”...

Even more, opposite-constant-equal-beating 12ths and 15ths, in practice, gives us a double errorless tool and a perfectly stable beating-whole (section 3.4).

Gadzar, you wrote (06/23/09): ...”Without iH all is easy, if we assume that

1 There is no iH
2 ET is defined as the division of the octave in 12 semitones equally tempered and
3 The octave's ratio is established to be 2:1

Then, the simple mathematical model of ET where

Semitone = 2^(1/12)

would perfectly do the work. Even if the “well-documented human ear's desire to hear stretched octaves” (by Tooner) is not satisfied.”...

Live human ear’s desire alone, all the above premises of yours are discretional, when not wrong:

1 If iH is relative to the mean, with different degrees iH is ever present.
2 Octave-based ET was an arbitrary and unlucky choice, the octave can not be cloned.
3 There was no logical reason for establishing 2:1 octaves.

You say: ...“With the presence of iH in pianos that model is no more applicable; we must tweak the frequencies calculated by this model in order to get acceptable tunings. That distorts what we understand by ET.”...

I look at it the other way around: 2:1 octaves lame ET model distorts the relation between ET and iH, in other words, we must tweak 2:1 ET’s frequencies firstly because those wrong values can not fit a combined-partials whole.

You add: ...”We lack a new mathematical model which includes iH and solves the problem of having several incompatible kinds of an interval. How can we tune an octave if there are 2:1, 4:2, 6:3, 8:4, 10:5, 12:6 octaves, and they are incompatible with each other? How can we tune a 5th if there are two distinct incompatible kinds of them?”...

The intervals incompatibility is firstly due to prime numbers 2, 3, 5 and their multiples. Theoretically, we needed to unite (mathematically) those “prime ratios” in one single ET ratio, what Chas model has finally done.

You say: ...“Models like semitone = 3^(1/19) or even more complicated, like the formulas used by Mr. Capurso, don't solve the problem because iH is not directly addressed.”...

The scale’s problems derive from non-combined partials. Firstly, we have to renounce “pure” ratios, then we can address iH. In fact, with Chas algorithm we could tweak frequencies the way we prefere, since “s” and “s1” variables work as a fine trimmer.

You say: ...“We need a new mathematical model where the octave's ratio is no more a constant but a variable value that will fit the piano's iH all along the scale.”...

If this was the problem, it would be solved. In fact, as I’ve said, Chas model’s algorithm allows you to variate the scale’s ratio as you like.

You ask for: ...“A new mathematical model where there will be only one type of each interval to tune, namely only one kind of octave, fifth, fourth, etc.”...

In Chas model each beat works for the others, each interval’s beat is solidly behind the beat of any other interval, due to a proportional “difference” ratio that involves all partials. Therefore, Chas intervals are of one precise mathematical type (the precise location). Also in practice, in addiction to usual check-intervals, as my constant-reference (the datum) I use just one type of beat (one beat/rate), 12ths (narrow) and 15ths (wide) constant-equal-beating all along. Traditional ET pseudo-model gave us a zero-beating constant that no tuner has ever been able to use, Chas ET EB model gives us two constants and the same one beat, so describing a tuning-form that we can always achieve into practice.

You say: ...“We have sometimes six contiguous unisons using the same size of wire and then the next unison has another size, here we have a jump in iH. The same happens when we get to the wound strings, and to the doublewound strings. Our new mathematical model can not just ignore this, because our ears do not.”...

Our ears can suffer jumps in iH and strings scaling, what a correct mathematical model can improve. Way? Because our expected frequencies (and our expected beats) affect strings scaling, and the latter affects iH.

Commenting Virgil Smith's concepts, you write: ...“So the “natural beat” concept can be applied to solve the problem of several kinds of intervals sounding simultaneously. It is then necessary to translate mathematically the ability of the human ear to combine all the partials into one sound and one pitch.”...

I can say that I use those “natural beats” for tuning Chas-form, “partial beats” for unisons. I think that listening to “partial beats” may expose you to iH. Chas model mathematically combines all partials into one natural ET semitone’s ratio, in the way our ear could do on larger intervals by using “natural beats”, with or without iH.

In my opinion though, one Smith’s line contains a partial information when, describing natural beats, he says:

...“ it combines, all the beats between the partials into one beat. The beat then comes from all the partials instead of one set of partials.
This beat can be tuned to the desired speed or eliminated completely.”...

Can we eliminate a beat completely? How long does it take the slowest beat to rise? If we do not hear any beat it is because we are in the partials leeways, where beats are going to define. Then, is it better counting beats or zero-beating? Shoud’nt we manage partials leeways? In my experience, that leeway allows me to define the beat’s rising and launch my tuning-form.

You say: ...“If we can translate to a mathematical formula this way of combining several “partial beats” into one “natural beat” there will be only one unique type of interval to tune. No more 2:1, 4:2, 6:3 octaves, nor 3:2, 6:4 fifths, nor 4:3, 8:6 fourths, but only one kind of octaves, fifths, etc.”...

Well, in Chas model each interval’s beat-curve is a function of all the others, i.e. partial beats (related to partial strings lengths) are expressed as a combined function. You will not find a zero-beating curve, nor an arbitrary “pure” ratio, you’ll find a purely dynamic form, so called “chorale” because deriving from all partials flows of beats combined. In Chas algorithm, delta variable determines an “ideal” beats-flowing, s1 and s variables open to the “real” infinite variety of flows.

You say: ...“In that respect I think Mr. Capurso's research although being a true effort to find a new model, missed the target by confusing iH with stretch.”...

I’m afraid I’ve to say that confusion took place when referring iH to wrong scale’s frequencies, those deriving from 2:1 octave’s ratio, a scale with a zero-beating-constant ratio that does not combine partials.

You say: ...“I think the solution to this problem is working on iH not stretching the width of the semitone in an arbitrary manner.”...

Somehow I agree, let’s combine partials in absolute terms and get the most natural (and logical) semitone’s ratio, then we can work on iH.

You say: ...“I think people like the late Dr. Albert Sanderson (Accutuner), Mr. Robert Scott (Tunelab), Mr. Dave Carpenter (Verituner), Mr. Dean Reyburn (Cybertuner), and Mr. Bernhard Stopper (Onlypure), who have developed the theory, algorithms, software and some of them even hardware of the most advanced ETDs available today, are aware of the lack of such a mathematical model.”...

I think so too.


Thanks amd regards, a.c.

I believe there is a wide world of tuning expertise in existence that is more sophisticated than you give it credit for.

I am sorry. I would really like to communicate meaningfully with you, but your language is impossible for me to follow.

You write:
"Today, Chas model uses two intervals, the 12th relative to partial 3, and the 15th relative to partial 4. In this way, it is possible to proportion also partial 2 and partial 5, the latter being semitone n. 28, the 7th third. With any other ET mathematical pure ratio, where single partial’s ratio (2:1, 3/2:1, 3:1) determine the temperament, the other non-determining partials are bound to be damaged."

I tune intervals. I don't understand your term, to "proportion partials."

You write:
"I’m stating that octaves, 4ths and 5ths can - and need to - be progressive, like 3ds, 6ths, 10ths and so on."

The definition of equal temperament is that temperament in which all intervals progress smoothly. You cannot possibly make the claim that this characteristic is unique to Chas.

I respectfully suggest that you need to fully study the existing literature on the subject of tuning equal temperament.

I repeat my suggestion that you make available recordings of your tuning.

I repeat my suggestion that you make available a coherent set of directions for executing your tuning.

Kent Swafford

Kent, you kindly say: ...” I believe there is a wide world of tuning expertise in existence that is more sophisticated than you give it credit for.”...

It would be interesting to know exactly what you are referring to. This would give depth to your convictions and may enlarge our horizons. If you wanted to tell me/us more about the world of tuning expertise and how it is more sophisticated, the time may be now.

You say: ...“I am sorry. I would really like to communicate meaningfully with you, but your language is impossible for me to follow.”...

Do not give up, on my part I’m trying to improve my language.

You say: ...“I tune intervals. I don't understand your term, to "proportion partials."...”...

Well, no problem. The history of temperaments is centred on the scale’s partials proportions. Briefly, the word “partial” generally refers to the vibrating string’s frequencies (section 1.1): the string vibrates at a foundamental frequency, say = 1, together with partial frequencies = 2, 3, 4, 5, and so on. Those integer numbers can determine an ET scale’s proportion, expressed by the semitone’s incremental ratio. In fact, traditional ET pseudo-model adopted the 2:1 partial proportion for dubbling the octave’s value.

“Chas model uses two intervals, the 12th relative to partial 3, and the 15th relative to partial 4. In this way, it is possible to proportion also partial 2 and partial 5…”... this is meant to explain how Chas theory proportionally combines – in its ratio - also scale’s partials 2 and 5. When in practice you tune intervals, you do proportion beats/rates and those are the direct product of partials proportional matching.

You write: ...“ The definition of equal temperament is that temperament in which all intervals progress smoothly. You cannot possibly make the claim that this characteristic is unique to Chas.”...

I’d never make that claim, since I would have no reason. In fact, Chas 12ths and 15ths do not progress at all, remember? In Chas model, 12ths (narrow) and 15ths (wide) are the two system’s constants, i.e. 12ths and 15ths have opposite, constant-equal-beating all along. So, if anything, Chas is unique in that it theoretically justifies octaves progression as the result of combined partials. Traditional ET pseudo-model fixes zero-beating 2:1 octaves, so our ET reference model leaves octaves out of any theoretical beat-progression, therefore it does not fit that definition either. Anyway, my goal is not ET’s definition, and what makes Chas model unique has been listed in this Topic’s first post.

You kindly say: ...“I respectfully suggest that you need to fully study the existing literature on the subject of tuning equal temperament.”...

Thank you for your suggestions. I can peacefully state that in equal temperament’s literature you will not be able to read about Chas tuning yet, and that is way I’m sharing Chas ET EB theory’s model and tuning practice here and now.

You say: ...“ I repeat my suggestion that you make available recordings of your tuning.”...

In a few days traditional ET and Chas ET EB digital comparison will be available. Meanwhile, in case you kindly wanted to contribute, I’ll list my latest questions:

1 - did you know that 5ths can invert in their being progressive?
2 - and that 4ths and octaves can be progressive too?
3 - have you been able to see what 12ths you get at the end of your ETD tuning? Are 12ths pure, a little narrow or what? How are 12ths supposed to be?
4 - way do you doubt Chas model exists?
5 - on the theoretical and practical grounds, could you give me/us one good reason for adopting a zero-beating theoretical ratio?
6 - how clear (or obscure) did you find the “kite analogy” (06/04/09)?

Regards, a.c.

Alfredo:

Consider this zero beating interval, 8:2 double octaves. In some parts of the scale (but not all) it will produce EB 12ths and 15ths. The test for the 8:2 double octaves is when the minor 6th above the lower note beats the same as the major 10th below the upper note. This will also produce octaves between 4:2 and 6:3. I have been striving for this on unwound strings as high as I can discern the beats. Wound strings are another matter…

Let me tell you what I think may be happening here.

In order for there to be a discussion of scientific/technical nature, the participants must share a common knowledge of the given subject along with a common vocabulary relating to that subject.

You and I simply do not have the required knowledge in common that is prerequisite to carrying on discussion on the subject of tuning theory.

For 30 years I have been reading the Piano Technicians Journal. It does not purport to be a scientific journal, but it is a technical Journal and it does contain the wealth of expertise that is the basis of my considerable understanding of piano tuning.

If you wish to participate in a discussion of piano tuning theory beyond your knowledge base, then I would suggest that you familiarize yourself with the Piano Technicians Journal, which is available on CD-ROM from the Piano Technicians Guild, www.ptg.org.

Take some advice, pending your completion of reading the Piano Technicians Journal. You simply, absolutely must drop your term "ET pseudo-model" in relation to the 12th root of two model of ET. Frankly, this term makes you look like a fraud and/or a clown to those who are familiar with the piano tuning theory to which I am accustomed.

When you are able to describe your ideas in terms of accepted piano tuning theory, or when you can communicate your ideas well enough for another tech to describe your ideas in terms of accepted piano tuning theory, then I look forward to continuing this discussion.

I hesitate to mention this; perhaps it will just open me to more of your insults; but I have published many articles on tuning theory in the Piano Technicians Journal, I have been an administrator of the PTG's tuning exam for some 25 years, and I am a past president of the Piano Technicians Guild. I state respectfully: I know what a partial is. <grin> You need to withdraw and learn the language of piano tuning theory. Once we have a common vocabulary, I suspect you would find me able to be of assistance to you in disseminating any good ideas you may have among the piano techs of the world.

This will be the last of my comments here. Farewell.

Kent, you kindly say: ...“In order for there to be a discussion of scientific/technical nature, the participants must share a common knowledge of the given subject along with a common vocabulary relating to that subject.”...

In this Topic, I do not see any “common knowledge” problem, nor a vocabulary problem. And if there were any, it could be easily solved by adding one more line.

You state: ...“You and I simply do not have the required knowledge in common that is prerequisite to carrying on discussion on the subject of tuning theory.”...

About knowledge in common, I do not understand what makes you so negative. In my professional experience, I have been able to debate tuning theory and practice with colleagues and pianists coming from all countries. What did we have in common? Our will to explain each other and to know from each other.

You say: ...“For 30 years I have been reading the Piano Technicians Journal. It does not purport to be a scientific journal, but it is a technical Journal and it does contain the wealth of expertise that is the basis of my considerable understanding of piano tuning.”...

I could then believe that your understanding of piano tuning, in terms of general knowledge, is not much different from mine.

You say:...“If you wish to participate in a discussion of piano tuning theory beyond your knowledge base, then I would suggest that you familiarize yourself with the Piano Technicians Journal, which is available on CD-ROM from the Piano Technicians Guild,www.ptg.org.”...

Kent, I’m sure you know that there is plenty of alternative literature available on temperament theory and tuning practice. Sure, I could familiarize with the Piano Technicians Journal, but this would not make our knowledge more common. In my opinion, the available literature on this subject, in any western language, is quite univocal.

You write:...“You simply, absolutely must drop your term "ET pseudo-model" in relation to the 12th root of two model of ET.”...

Here, I may think we do not have the same kind of knowledge, but it may not be a problem. I’m saying “pseudo-model” meaning that 12th root of two lacks in consistency, i.e. 12th root of two does not correspond tuning reality.

You say:...“Frankly, this term makes you look like a fraud and/or a clown to those who are familiar with the piano tuning theory to which I am accustomed.”...

I do not understand why you talk about frauds and clowns. In my opinion, many of us have long ago realized that 12th root of two, as a model, can not be put into practice, and I’m trying to explain why it is so. What 12th root of two should then be called, if model, proto-model, super-model or what, this may be an academical matter; my actual goal is sharing Chas approach, Chas theory’s model and its numerical evidencies. You see, in Chas article’s conclusions you can read: …“the numbers dispel all doubt concerning the simplicity and power of this long-awaited entity”.... so, since you can always check Chas numbers, you will never think in terms of clowns. Instead, you could now acknowledge that Chas scale’s ratio combines all partials, included partial 2, and derives from an algorithm that can proportion partial differencies, and therefore beats.

You say:...“When you are able to describe your ideas in terms of accepted piano tuning theory, or when you can communicate your ideas well enough for another tech to describe your ideas in terms of accepted piano tuning theory, then I look forward to continuing this discussion.”...

Well, I think it'll be up to you. I’m writing here about a new ET EB dynamic theory and, as you know, it may take some time for its terms to be accepted.

You write:..“I hesitate to mention this; perhaps it will just open me to more of your insults;”...

Maybe this is a problem of yours, you feel insulted but in this Topic nobody has insulted you.

...“but I have published many articles on tuning theory in the Piano Technicians Journal, I have been an administrator of the PTG's tuning exam for some 25 years, and I am a past president of the Piano Technicians Guild.”...

For what you have experienced, I’m sure you’ll soon appreciate the difference between traditional ET and Chas ET EB model.

You say: ...“You need to withdraw and learn the language of piano tuning theory. Once we have a common vocabulary, I suspect you would find me able to be of assistance to you in disseminating any good ideas you may have among the piano techs of the world.”...

If I had some vocabulary problem, I would like to find you able to assist me then, especially since you yourself think that vocabulary may help. Anyway, Chas model is not really a good idea only, it is a new ET theory that finally improves our traditional 12th root of two. The sooner you realize it, the sooner you’ll be ready to help. Thanks anyway.

Tooner, thanks for your post, I'll soon answer you.

Regards, a.c.

Dear colleagues, here

http://www.chas.it/ContentPartViewer.aspx?ID=confronto

you can listen to 24 digital tracks – it takes about 5 minutes - and compare four chords, formed with 12th root of two ET frequencies (odd tracks) and Chas frequencies (even tracks). Those are the notes that are played: A 3 - A 5 - B 5 - C# 6 - E 6.

This is the chords order:

From track 01 to 06 → A 3 - E 6
From track 07 to 12 → A 3 - A 5 - E 6
From track 13 to 18 → A 3 - A 5 - C# 6 - E 6
From track 19 to 24 → A 3 - A 5 - B 5 - C# 6 - E 6

Each chord is played using three kinds of wave (square, sawtooth, triangular). These are the notes and their frequencies:

Notes - 2:1 ET frequencies
A 3 - 220.00
A 5 - 880.00
B 5 - 987.76
C# 6 - 1108.73
E 6 - 1318.51

Notes - CHAS frequencies - offset in cents
A 3 - 220.00 - 0.0
A 5 - 880.46 - 0.9
B 5 - 988.33 - 1.0
C# 6 – 1109.41 - 1.1
E 6 - 1319.41 - 1.2

Those notes have been chosen to keep Chas frequencies approximation within +/- 0.2 cents, while 12th root of two ET frequencies could have no approximation (100 cents/semitones). For converting Chas frequencies in cents we have used Peterson’s software, available here:

http://www.petersontuners.com/index.cfm?category=15

For recording we have used: DIGIDESIGN, Mod.: PRO TOOLS HD 3 ACCEL
Interface: 192 I/O - Signal Generator: DIGIDESIGN
Software: PITCH DIGIDESIGN

This work has been done to aurally evaluate 12th root of two ET model and 2:1 theoretical octave ratio’s practical efficiency, when applied on digital sounds.

I need to precise that Chas model’s algorithm, while describing an ideal geometrical entity, can also provide infinite ET scale’s incremental ratio (section 3.3), in departure from any cultural, “euphonic” approach and any “pure” theoretical ratio (section 2.0).

In fact, Chas theory grows from the concept of dynamic and beating affinity in the sound-whole.

Regards, a.c.

Alfredo:

Yes, I hear a difference, but without a musical comparison, I would not be able to say which I prefer. For piano tuning there would need to be the same music played on the same piano tuned differently in order to determine a preference.

Dear Colleagues,

today, in my mail box, I found a message from Mr. Kent Swafford. It was sent on the 08/25/09 and, since it is about Chas, it may as well be discussed.

Here you'd have read the message but, as Tooner suggests, I'll ask for Mr. Swafford's permission.

Kent, can we publicly discuss your outlook on Chas theory?

Tooner, thanks for your feed-back, I'll answer you asap. Would you please tell me what you found "private" in Mr. Swafford's message?

Regards, a.c.

Alfredo:

The message that Mr. Swafford sent you seems to be a private message to you, and you alone. I think you should remove it from this Public Forum. You can do this in the near future by clicking the "edit" button on your post.

Kent,

you tell me that all of the matters Chas theory raises have been dealt with and settled over the years in alternative and superior ways.

On the one hand I could be quite glad because that “common vocabulary” is not a problem anymore and you could evaluate Chas theory’s matters. On the other hand I do not know how to take your latest understanding. Should I think that you were right, that there is a “common vocabulary” problem? Should I think that confusion is taking place? For both cases, I need to further precise a few points.

I’m not competing with theoretical or practical solutions that may represent alternative or superior ways for tuning pianos. In fact, I’m only trying to share what I think is a new theoretical approach to the sound scale, strictly related to my practical (aural) tuning experience.

Chas theory approaches the sound scale as a “beating-whole”. This is why in Chas algorithm (3 - ∆)^1/19 = (4 + ∆)^1/24 you find partials 3 and 4 together with ∆ (delta). In other words, Chas algorithm gains a basic scale incremental ratio = 1.0594865443501… with a delta beat-factor (section 3.1). So doing, Chas model can express an ET proportionate frequencies scale as the result of proportionate beats. Chas algorithm, in its basic form, describes then a precise geometrical entity (section 3.5) where 12ths (narrow) and 15ths (wide) are the equal-beating scale’s constants. This is why Chas is a precise ET EB theoretical model, what in my practical experience I consider the most euphonic and resonant semitonal sound-set. How Chas model combines partials is another matter and it is a mathematical evidence, as it is an evidence how, from Chas improved algorithm (3 – (∆*s1))^1/19 = (4 + (∆*s))^1/24 all ET conventional pure ratios can be gained.

If you can now evaluate the matters Chas theory raises and if you had heard of Chas model called in an other way, please let me/us know, we shall call it with the right name. Otherwise, try to take Chas model as it is, in its theory as in my practice, with no need to elbow.

Since my questions were serious, I need to go back there.

Question n. 1: did you know that 5ths can invert in their being progressive?

For the temperament, common teaching says: narrow and similar 5ths. In my experience, to get opposite and constant-equal-beating 12ths (narrow) and 15ths (wide), I have to stop 5ths from getting narrower in the temperament section, and I have to progressively drive 5ths towards no-beating. So, 5ths inversion = when tuning centre strings, invert 5ths in the temperament and let them go from progressive narrow to progressive wide.

Question n. 2: and that 4ths and octaves can be progressive too?

In my experience, to get progressive octaves I need to invert 5ths.

Question n. 3: have you been able to see what 12ths you get at the end of your ETD tuning? Are 12ths pure, a little narrow or what? How are 12ths supposed to be?

Answering this question is of general interest.

Question n. 4: on the theoretical and practical grounds, could you give me/us one good reason for adopting a zero-beating theoretical ratio?

I’ve also asked this question to Tooner and Gadzar, I’ve got no answer. What is a “zero-beating ET theoretical ratio”? It is a theoretical scale proportion that referes to any of those ET integer ratios: 2:1, 3/2:1, 3:1, 5:1, and so on. In my opinion, thinking in terms of “zero-beating ratio" is an abstraction that negates a dynamic reality.

Question n. 5: how clear (or obscure) did you find the “kite analogy” (06/04/09)?

Answering this last question may suggest to find a better way for sharing Chas model. Thanks.

Tooner,

hopefully in a short while I'll be able to add some recordings of Chas piano tuning. Remember though to separate what our musical taste can be - in my case Chas basic ET EB form - from a comprehensive model, what Chas can express with its "s" variable. In other words, I would not make it a question of personal preferencies. Thanks.

Regards, a.c.

Alfredo:

I don’t think you understood that Kent has said that he will not be replying to you anymore.

You have not shown a mutually exclusive relationship between zero beating tuning and equal beating tuning. So, the failure to prove the validity of one does not prove a validity of the other.

But here is something that you have not answered:

How does the Chas theory predict that fifths “invert”? The answer is that it does not predict this. It is an effect of inharmonicity when choosing a particular stretch style. Although you think that your theory is coherent, it is not because it is incomplete. It does not predict what actually happens.

Tooner,

You kindly say: “I don’t think you understood that Kent has said that he will not be replying to you anymore.”...

Thanks but yes, I understood what Kent wrote at the end of his last post, nevertheless Kent may change his mind and, rather than privately, we may discuss publicly about Chas, that is what I would prefere.

...“You have not shown a mutually exclusive relationship between zero beating tuning and equal beating tuning.”...

On the theoretical ground, I have not shown a “ mutually exclusive relationship between zero beating tuning and equal beating tuning” merely because I can prove the opposite, i.e. Chas ET EB theory can also comprehend any zero beating ratio (section 3.3 - THE S VARIABLE). Let’s see together one example:

From Chas algorithm (3 – (∆*s1))^1/19 = (4 + (∆*s))^1/24

If s1 = 1 and s = 0

we can find a delta value that makes our equation true:
Δ = 0.0033858462466

In fact:(3–(0.0033858462466*1))^(1/19) = (4+(0.0033858462466*0))^(1/24) =
= 1.059463094359 = 2^(1/12) = 12th root of 2 zero beating ratio.

As for 12th root of 2 octave’s zero beating ratio, Chas model does not exclude any theoretical ratio, zero beating or what ever. This is why, at the end of section 2.0 you can read: “In conceptual terms, the model is trans-cultural; it also responds to a new requirement on the contemporary music scene, by providing an algorithm which can give form to all kinds of microtonal sound structures.”

On the practical ground, in my experience, 12th root of 2 zero beating ratio is useless, 7th root of 3/2 and 19th root of 3 are unnecessarily extreme as a final tuning form, although not “sharp” enough when tuning mid-strings in the treble. In fact in my practice, when tuning centre strings in the treble I go beyond the 7th root of 3/2 pure fifths ratio and I stretch progressive wide fifths. As I’ve said, this is done to compensate sound-board, bridge and strings elasticity, so considering my tuning settling-down.

You say:...“So, the failure to prove the validity of one does not prove a validity of the other.”...

True, then the question may be: what makes Chas beating-whole’s ratio better than any zero beating ratio? Leave all previous reasonings aside, you find an answer reading Chas article’s section 4.5 - Sequence of quotients..., and Table 6: Comparison between quotients deriving from ratios 3:2, 5:4, 3:1 and 5:1. There you’ll be able to evaluate Chas validity only with the help of numbers.

...“But here is something that you have not answered: How does the Chas theory predict that fifths “invert”? The answer is that it does not predict this. It is an effect of inharmonicity when choosing a particular stretch style.”...

I'd put it in a slightly different way: when choosing a particular stretch style, we need to invert fifths. In other words, inverting fifths in the temperament is the technique I use for achieving Chas form.

12th root of 2 predicts zero beating octaves, 7th root of 3/2 and 19th root of 3 predict respectively zero beating fifths and zero beating 12ths, Chas model’s basic form predicts opposite and constant 12ths and 15ths equal beating. How to get to Chas basic form in practice is a different matter, and Chas algorithm, with its s variable, allows you to make use of all ratios you need.

...“Although you think that your theory is coherent, it is not because it is incomplete. It does not predict what actually happens.”

I would not fuse coherence with completeness and we can still discuss both concepts. Chas theory describes a beating-whole and it gains the scale’s frequency values with a delta (∆) beat-factor. I can not immagine anything more coherent, can you?

About prediction, fifths inversion and Chas theory’s completeness, maybe this can help. In section 3.4 - CHAS SET…and section 3.5 - EFFECT OF ±DELTA…, you can read that the Chas model opens up a module of 49 sound elements, in a semitonal order, from 0 to 48, whose scale ratio is (4 + Δ)^2. If 12th root of two zero beating symmetry encompasses 8 scale’s degrees (one octave), Chas model’s beating symmetry encompasses 29 degrees of our semitonal scale. Now, in section 4.3 (graph 5, 6) you can compare 12th root of two and Chas difference curves for ratios 4:3 and 3:2, and in section 4.8 (graph 10) you can see what happens: within Chas compass, the beat curve for ratio 3:2 inverts its progression. Using your words I ask you: is this an effect of inharmonicity? I do not think so but, as you'd say, I think it is the effect of a particular stretch style, in this case the (predicted ?) effect of Chas ET EB.

Thanks a lot and regards, a.c.

Alfredo:

Can you explain this in more detail:

“Now, in section 4.3 (graph 5, 6) you can compare 12th root of two and Chas difference curves for ratios 4:3 and 3:2, and in section 4.8 (graph 10) you can see what happens: within Chas compass, the beat curve for ratio 3:2 inverts its progression.”

You mention “ratios 4:3 and 3:2” but do not explicitly say what the ratios are of. Perhaps you mean partial matches? That is where beats come from, not ratios. Are you saying that using a semi-tone ratio of 1.0594865443501 will produce fifths that “invert”, in other words, beat faster and then slower when progressing up the scale?

Tooner,

...“You mention “ratios 4:3 and 3:2” but do not explicitly say what the ratios are of. Perhaps you mean partial matches? That is where beats come from, not ratios.”...

True. In section 4.3, at the bottom of graphs 5 and 6 you find the scale degrees that can be compared; as well as in section 4.8, were you also find Table 9 listing Graph 10 values.

...“Are you saying that using a semi-tone ratio of 1.0594865443501 will produce fifths that “invert”, in other words, beat faster and then slower when progressing up the scale?”

No, as I’ve said inverting fifths in the temperament is the technique I use for achieving the ET EB tuning form that Chas model describes. a.c.

Alfredo:

Then, your paper is incomplete as I said before. It does not predict the change in the beat rates of fifths that you find necessary for equal beating 12ths and 15ths on a real piano.

Tooner thanks, more often you help my mood.

Before you were talking about Chas theory being incomplete, now you are saying that my paper is incomplete. What happened in the meantime?

Could you tell me what can be average deducible from Graphs 5, 6 and 10? Could you get the partials-matching involving beats?

Maybe you know this:

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

I'd like to know your opinion sometime. a.c.

Alfredo:

Because your paper is the only place that Chas theory is explained, and the only topic in your paper is Chas theory, I see them as one thing: A problem with one is a problem with the other. But I understand that you would view them as separate things. So, where do you think the problem is between how your beat rates are when you tune and how your paper explains what the theory predicts beat rates should be?

I have pondered far too many times what graphs 5, 6 and 10 are indicating and still have no idea. I do not know what you mean by degrees. The only mention of degrees in music that I can think of is in the Psalms, which may have indicated ascending stair steps while singing. But your graphs supposedly indicate “differences”, which should mean that one value is subtracted from another, but I have no idea what the values are.

Sorry, I have very little interest in unequal temperaments. A disturbing reason I have read for tuning unequal temperaments is that hardly anyone can tune ET anyway, so it is far better to tune an UT, especially one that is designed to be easily tuned. It reminds me of excuses for “new morality” which is really just old sin. If you started a new Topic on the paper that you provided a link to, there is sure to be a great deal of response.

Tooner, thanks for your reply, I came back yesterday and finally I can answer you.

You kindly say:...“Because your paper is the only place that Chas theory is explained, and the only topic in your paper is Chas theory, I see them as one thing: A problem with one is a problem with the other. But I understand that you would view them as separate things.”...

It is good that you understand me when I view Chas theory and the Chas article as separate things, yet I do not understand what benefit you get from viewing at them as one thing. While Chas theory and maths are solid, the Chas article can be improved for sure, so I’d rather keep them separate.

You ask:...“So, where do you think the problem is between how your beat rates are when you tune and how your paper explains what the theory predicts beat rates should be?”...

I can only think that the problem you are pointing out may derive from the way you look at Chas model, at its scale and at the scale’s values (to know more about scientific modelling and systems:
http://en.wikipedia.org/wiki/Model_(abstract)

Once you refere to general modelling, you will not expect Chas model to predict what beat rates should be on pianos.

Chas model’s aim is to represent an ET scale of frequencies related to partials 3 and 4 differences, i.e. an ET scale related to the Chas system’s constants, the 12th and the 15th intervals.

Chas model describes a beating-whole where partials effects are finally combined. In other words, in Chas scale no interval is pure; all intervals have proportional differences from their pure partial value, in an intrinsic correlation between frequencies and differences arising from the infinite combinations of the scale’s elements (section 3.0). As a result, in theory as in practice Chas octaves are progressively stretched (section 4.2, graphs 3 and 4), and 12ths and 15ths have opposite equal beating (section 4.1, table 2, graph 2).

You write:...“I have pondered far too many times what graphs 5, 6 and 10 are indicating and still have no idea. I do not know what you mean by degrees. The only mention of degrees in music that I can think of is in the Psalms, which may have indicated ascending stair steps while singing.”...

Here you can get an idea:
http://en.wikipedia.org/wiki/Degree_(music)

You then say:...“But your graphs supposedly indicate “differences”, which should mean that one value is subtracted from another, but I have no idea what the values are.”...

You can find graph 6 differences values in table 4 (section 4.4, graph 7), in table 8 (section 4.7, graph 9) and in table 9 (section 4.8, graph 10). Sorry if it comes out a bit confusing.

About Professor Colin Pykett’s paper you write:...“Sorry, I have very little interest in unequal temperaments.”...

I should have been more precise, in part 4 – Impure octaves, these are the lines I found intriguing:

“...For example, the beat rate of any interval played depends on the octave in which the interval resides. In other words, a fifth played in the third octave will beat faster than if it is played in the second octave, but slower than if it were to be played in the fourth octave. With any temperament which uses pure octaves, the ratio of these beat frequencies has a simple numerical relationship to the octaves considered...”,

“...Why not ease this problem a little by making the octaves themselves adjustable as well?...”,

“...Currently I have yet to decide on a definite road map for the study though. Doing it with the degree of emphasis on arithmetic and theory which constitutes current work on temperament is almost certainly debarred. It is debarred because 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. 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.”...

Tooner, would you say that Chas model is anarchic? That Chas values arise from error? By the way, last saturday I could record Chas tuning on a small grand, now I only need to make an mp3 of it.

Regards, a.c.

About Colin Pykett’s paper - it seems to be heavily tilted to temperament and octaves in organs... There is only a passing explanation of inharmonicity's effect on piano tuning.

While "pure" octaves are the "norm", there have been plenty of piano tuners over the years adjusting the octave width to achieve a pleasing balance of sound across the range of the instrument. Even the pure fifth temperament as well as the pure 12th temperament stretches have been discussed. Bill Bremmer was one of the first I remember tempering his octaves to match the octave+5th in unequal temperaments.

Ron Koval
Chicagoland

Alfredo:

I am going to take a break from your Topic. The idea of using the degrees (or steps) of a scale (a major scale presumably, although since it is not specified, it could be any number of scales…) as a basis for analysis is just too foreign to me. And when I try to make sense of it by examining paragraph 4.3 I again read “The differences, divided by themselves,…”. ANYTHING EXCEPT ZERO DIVIDED BY ITSELF EQUALS ONE!!!!!

I am going to take a break….

RonTuner,

you kindly say:..."About Colin Pykett’s paper - it seems to be heavily tilted to temperament and octaves in organs..."

In one post (06/30/09) I wrote: "I’m still trying to explain why I can not agree with Mr. Deutschle when he says:….” The octave is tuned wider than theoretical due to iH.”

In fact what I’m saying (since I can prove it) is that, with or without iH, we need to stretch octaves. Why? Because also partial 2, with the other partials, through stretching can practically contribute to hold up a resonant beating-whole system. Negating the beat’s value (or relevance), we would never get to the Chas concept of a beating-whole."

One thing I find interesting is that Professor Colin Pykett is a pipe organ tuner and, as you can read, he admits octaves stretch.

Tooner, in one post I had already explained what "The differences, divided by themselves,…” means. Anyway, in section 4.5 you can always see what that means.

Thanks and regards, a.c.