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My own feeling is that my tolerance for narrow octaves is not as great as this calls for. When an octave is 2 Hz narrow in the center of the piano, the piano needs tuning.

Mr. Capurso, thanks for the additional explanation. As written, your paper is incorrect, or at least highly misleading, because Equations 5 and 6 show the same symbols for delta, S, and S1, and without some explanation, one would make the inevitable assumption that therefore the values of these three variables in both equations would be the same.

In order to make your paper read correctly, I suggest that you add some words to make your intent clear. You could say, for example. "In equation five, we will select values for S and S1, and calculate a new value for delta that makes the equality true. In Equation 6, we keep the same values of S and S1 and compute yet another value of delta that makes the equality true."

With such an explanation, I think your readers would have correctly interpreted your math--I certainly would have.

Roy 123,

thank you for having discounted your inflictions and above all for your suggestion that, in my opiniln also, can help our readers. Now I look forward to knowing about if and how you like Chas model.

ROMagister, BDB,

Chas octaves are not narrow. I'm sorry to have written "narrow" in stead of flat when referring to the note to be tuned. For istance, when in the sequence you read A4-A3 - narrow, I meant to say A3-flat, so A3-A4 is a wide interval.

ROMagister,

I'll have more time tomorrow to replay to your attentive post. Thank you.

Tooner,

you had already understood about Chas octaves, fifths, deltas and s, you devil. One day I'll tell you why you are not my enemy!

Bill,

I'm goin to work on the sequence and submit it to you. How do you like Chas inverting fifths? Did you find all those figures disgusting? a.c.

You say that if you start with a scale value of 1, the octave ratio will be 1,997134719564920 instead of 2, and two octaves will be 3,988547088091660 instead of 4. That is narrow.

Alfredo:

I am going to use analogies to try to describe the problems I see with your paper.

Lets say you get a call to tune a piano, and the customer says that their piano is a motorcycle. So you say your piano could not be a motorcycle, it is a piano. Motorcycles and pianos are not the same thing. So, the customer says that their piano says “Yamaha” on the fallboard and since Yamaha makes motorcycles then their piano is a motorcycle!

This is the problem with saying that equation 5 equals equation 6. Given certain values for the variables the terms can be equal to each other, but that does not make the equations equal to each other.

So you ask where the piano is so that you can go there and tune it and are told that the piano is in the front room, the room with the lovely drapes. From the customer’s point of view this is a perfectly good answer, but does not help you get from where you are to where the piano is.

Your paper seems to be written from your point of view and assuming that what you find to be desirable, everyone else will. By presenting your equations with variables on both sides it is very confusing as to what is being solved. For instance, if I was talking about the Pythagorean Theorem and said that the equation a^2 + b^2 = c^2 will give the length of the hypotenuse, it would be difficult to someone that did not already understand the equation to know that I mean that the Hypotenuse = (a^2 + b^2)^1/2. Your emphasis on delta is confusing. It is of no use in itself, but only as interim step in determining a ratio. It is proper to show and explain delta when showing how your equations are derived, but in the end the equation should be in the form of “ChasRatio = …..”

Then after talking to the customer more about where they live you find out that they live in Haiti. Ok, Haiti can be a nice place (I’ve been there), and it is interesting to think of different ways to get there. But besides not planning on going there, the directions from the customer are just too hard to follow because there are given from their point of view and not yours.

This is how I feel about your tuning from reading your sequence. I have tuned every widening octaves, but probably not to the point of wide fifths so low in the keyboard. It can be OK, but I don’t plan on tuning that way. But besides that, I can find nothing in your paper that goes from a fixed ratio to ever widening octaves. And as I continue to try to understand your paper I read the statement of “s=s/s1” (which can only be true if s1=1, but then what is the point?) and there is no explanation of what units s and s1 are in, nor how s and s1 are determined, nor why s/s1 must be a rational number. Since I am not interested in tuning as you do, the effort becomes too difficult to try to understand how you “get there” from your Chas ratio.

I will probably continue to read the posts to this Topic, and may or may post to it myself, but I doubt if I will put in the effort to really understand your paper. (I never cared for flowered drapes in a front room. I prefer lace.)

Tooner, if you read the explanation in my last post, you will see that with the use of different deltas, but the same values for S and S1, in Equations 5 and 6, that both equalities can be obtained, and, in fact, that the semitone ratio calculated for both is the same. I think much of the problem with Afredo's paper is the rather wordy, hyperbolic, and unclear (sorry, Alfredo) presentation. It is simply not written in a way that would be accepted by the scientific or mathematical community. The paper could have been much shorter, crisper, and more lucid.

Having said that, I've not ventured further to see if something is to be gained by using Alfredo's method.

Bill, you kindly say:

“Alfredo, while I can make no comment or judgment on the math, I can make a comment on the way the written sequence is described.”…

I take your's and all our colleagues math understanding to heart, so that we’ll be able to share Chas model in all its aspects.

Think about traditional ET ratio 12th root of 2.
Say you want all intervals to be ET progressive.
Say you want an equal beating on 12ths and 15ths.

You already know that theoretical 12th root of 2 would not satisfy your needs, since those 15ths are theoretically beatless. In fact, you know that the only way to have all intervals being progressive and equal beating 12ths and 15ths is to stretch your 12th root of 2 ratio. So now you are thinking in terms of (12th root of (2 + wide-stretch)).

Your experience tells you that P12’s (pure 12ths, ratio 19th root of3) would give you to wide octaves and 3ths, 10ths and so on, this is why you want 12ths a litle narrow, so you think at 19th root of (3 – stretch), while you want P15’s (pure double-octave, ratio 24th root of 4) to beat equally, say 24th root of (4 + stretch). So you conclude that, in order to have an equal beating on 12ths and 15ths, you can write:

19th root of (3 – stretch) must equal 24th root of (4 + the same stretch). This is Chas algorithm.

ROMagister,

I think you got the point in writing:

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

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

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

I'll answer your questions as soon as I can. Thank you

BDB,

in Chas article you'll find Chas octave ratio: 2.0005312...

The figures in my previous post came from an example, to show the effects of s variable.

Tooner,

I think your contributes are precious because you can think in abstract terms. Sorry for my style.

Roy123,

I think you have already been able to express your opinion about the article style, now if you like, you could help by considering the content. Your point and your suggestion have already solved a question.

Thank you, a.c.

Thanks for this most recent post, quoted below. It is a good explanation of your tuning; I can only wish that your previous writing was this clear.

However, I completely disagree that pure twelfth equal temperament is too wide. At least as tuned by the OnlyPure electronic tuning device, pure twelfth equal temperament yields beautiful, beautiful results.

Kent,

thanks for contributing. You tell me that with your ETD you get beautiful results, but I do not know what other ETD you are comparing it with. As you say, I do not even know what tuning curve I would then find in the piano, if 19th root of 3 or something else.

You see, I'm not promoting a precise tuning curve, having this to do with personal/cultural taste, I'm promoting an updated ET theory that is finally applicable in tuning practice.

Chas model discards traditional ET erroneous assumptions. When put into practice, Chas can help aural tuners dealing with iH and can correctly orientate to find the smoothest progression of RBI and SBI.

More precisely, Chas explains why and how 5ths invert, becoming less snd less narrow from the middle-high register goin up. Once you are aware of how your 12ths and 15ths are going, you would be able to fix your favorite tuning curve, while considering both iH and sound-board Vs strings adjustment.

ROMagister, you say:

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

Today, on well scaled pianos, inharmonicity is made quite even. With Chas correct standard frequency values we will improve the building and scaling of pianos and we'll better control iH. a.c.

I am not certain what that sentence means. In any case, a scale can be designed for other goals than inharmonicity, even now.

This is like trying not to look at a train wreck.

Alfredo:

Can you state what you understand about iH?

Of which erroneous assumptions do you speak? Usually, we speak of a mathematical model of equal temperament with no inharmonicity that we know very well doesn't exist on real pianos. Then we try to find the best fit of the model to the inharmonicity-laden piano in front of us. It isn't news that the model of equal temperament doesn't quite fit real pianos. If you have something to contribute, a way of better fitting equal temperament to real pianos, then we are all ears.

These two last posts of yours are providing good descriptions; please keep it up. Now, according to you, why do fifths invert going up the scale?

Kent, that first line that you quoted confused me. I thought it meant that he was writing about a non-ET and the "erroneous assumptions" that we may have that ET makes the piano sound best. I'm not bringing up that topic or argument here, mind you, it was just what I thought he meant and I was interested.

I would say, however that some of the points which have been raised will apply to a non-ET too, at least the way I prefer to tune a non-ET. It is interesting that theoretically, 5ths will increase in speed but the very last thing anyone wants to hear are "beating 5ths". Since what I normally tune is a mild Well Temperament in which some 5ths are beatless, others tempered a little less than in ET and some a little more than ET, I have long observed how 5ths actually widen when ascending the scale rather than maintain the same width as they do in the temperament/midrange. Regardless of whether they were tempered or not in the midrange, they all eventually become wide.

This will depend, of course on how aggressively or not the octaves are stretched. That kind of choice will make a difference in the part of the scale where 5ths do become wide. My observation has been that it is generally in the 6th octave. I also believe they become wide in the low bass.

I have copyed your posts from an Internet point. In those days I can not go in the web from home.

Today I'll work on my answers, meanwhle I thank you all. a.c.

It seems that fifths could not become wide going up the scale, unless 12ths become wide first, which does not happen with mindless octaves, Chas tuning, nor perfect 12ths. Since a 12th is made from a fifth and an octave, the only way to have a wide 5th and have a 12th that is pure or narrow, is to have an octave that is narrow while the fifth is wide. It seems that this could only happen if there is a tuning error.

5ths become wide on PTG Tuning Exam Master Tunings in the 6th octave.

Then I would think that the 12ths become wide, also.

I think they do. Anyway, I have a couple of master tuning records and when I get the time, I will post what I find on that but in a new thread. It doesn't belong here. In any case, the "mindless octaves" idea creates an exact compromise between the double octave and the 12th and I routinely see them invert themselves: the 12th becomes wider than the double octave but I still balance the two.

Then again, does that belong here? Alfredo seems to claim something unique as does Herr Stopper. How do either of them compare to what is considered a standard (a standard to which I freely admit I never adhere except for the purposes of the exam itself). It must be close to 20 years ago that I saw Steve Fairchild demonstrate that 5ths do become wide. He also said that 4ths become narrowed.

Whether they do or not, beating in either 4ths of 5ths cannot be heard beyond a certain point because the coincident partials are too high and too faint. Try it yourself: tune a 5th from G6 to D7. Tune it as wide or narrow as you like and you won't hear any beats. The coincident partials are in the 8th Octave. The beats for 4ths disappear on or about F5. If either is wildly wide or nefariously narrow, does it matter if you can't hear any beats? What does matter are beats that *can* be heard from the wider intervals such as double and triple octaves, 10ths, 12ths and 17ths.

As far as I am concerned, when you get to the top of the 7th octave, none of them matter at all any more, only a sense of pitch does but everything leading up to that must provide a foundation for stretching the top part of Octave 7 and C8 as much as I customarily do. The amount I stretch up there shocks may technicians (when they know the numbers) but I can assure you that many fine aural tuners go beyond that. I can at least justify what I do by making those highest pitches agree with notes in the midrange.

Just for consideration, C4 read at 0.0 on its fundamental will typically read 1.0-2.0 cents when read on C5, 3.0-6.0 when read on C6, 15-20 cents on C7 and 65-80 cents when read on C8. Wouldn't that be a reason to try to stretch the octaves to at least partially conform to the amount of inharmonicity there really is in piano strings?

BDB,

I meant to say that iH is, to a certain extent, predictable.

Tooner,

I would like to thank you again for what you had written:

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

I had never told you that I really think Chas is a gem. What a pure chance.

About iH, let me answer with a friend of ours words, from Topic “Tunelab 6:3 octaves”.

“Here’s iH in a nutshell. A piano sting with certain characteristics and at a certain tension should vibrate at a certain pitch, but doesn’t. The difference in cents that the fundamental frequency differs from the theoretical frequency is the string’s inharmonicity constant or inharmonicity coefficient. All of the string’s partials, the fundamental being the first partial, are sharp of their theoretical frequencies. The amount in cents that they are sharp is the square of the partial number times the iH constant.

Example: A string has an iH constant of 0.5. The first partial is 0.5 cents sharp of theoretical, the second 2.0 cents, the third 4.5 cents, the fourth 8.0 cents, and so on.

So you should see that the number of cents sharp that each partial is in relation to its neighbors is not linear, but logarithmic. Matching the 6th partial of one note to the 3rd partial of another will not make the 8th and 4th partials also align.

Now a graph of a piano’s iH approximates a “V”. But since the left hand scale on the graph is logarithmic, it actually is a curve. The iH constant will double around every 8 semi-tones or more. So in the treble, not only do the strings have a higher iH constant, but the iH constant increases more and more. The same thing occurs in the bass with the iH increasing toward the bass.”…

This is what you understand. The italian colleague Giovanni Bettin writes:

“Fino a tempi non lontani la disarmonicità ha rivestito importanza e rilevanza solo per corde poste in stato di tensione e vibranti. Esami più accurati e ricerche effettuate da parte di O. H. Schuck e R. W. Young (1943) e dallo statunitense H. Fletcher, hanno comprovato la sistematicità con la quale si producono gli spostamenti di frequenza dovuti alla disarmonicità, e hanno stabilito le formule in base alle quali calcolarli: diametro delle corda al quadrato (D), diviso; la lunghezza della corda alla quarta potenza (L) moltiplicato per la frequenza (F), il tutto moltiplicato per un fattore costante (K) che deriva dal modulo di elasticità (E) del materiale che costituisce la fonte sonora."

These are good examples of what I understand about how iH is understood. But maybe this was not your question’s target.

At one stage you wrote:

…”I had worried about this because I was thinking that if my fifths didn’t become wide, I wasn’t tuning “correctly”. But since this happens only in the very high treble, due to a greater slope of the iH curve, then fifths becoming wide is an inherent anomaly of some pianos, not the result of a tuning style.”

Would you tell us about your tuning style, especially regarding your 4ths, 5ths, 6ths, octaves, 12ths and 15ths?

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

Unluckily, I had to prove that Chas maths is errorless, otherwise I would have correct your understanding there and then: 5ths, from low notes, beat progressively faster (narrower), but then 5ths invert and beat as you say “progressively slower, become beatless, and then beat progressively faster but on the wide side of just intonation.” Please remember this as referred only to middle string tuning. Also I would like to understand more about you saying:

“Not too long ago I realized how the effects of iH are largely self-correcting on the theoretical beat rates of intervals.”

Kent, you ask:

“Of which erroneous assumptions do you speak?”

You find your answer in Chas article, section 3.0: “The chas model discards two unjustified assumptions: that the range of the scale module must be 12 semitones, and that the octave, the 12th semitone, must be double the first note.”

Also in Chas Topic you can still read: “About tuning and compromise - untill today we (aural tuners) could only think in terms of compromise becouse we had to get by with Equal Temperament and its unjustified premises, two unjustified assumptions that Chas model discards (section 3.0). Chas demontrates that the ratio 12th root of 2 is unsuitable, not only becouse of inharmonicity, but becouse it produces intervals incresingly narrow (12ths, 19ths and so on) together with intervals incresingly wide (10ths, 17ths ecc. - section 4.3 - graph 5). E.T. premises come out to be missleading.”

You say: “Usually, we speak of a mathematical model of equal temperament with no inharmonicity that we know very well doesn't exist on real pianos.”

Actually, I’m explaining why today we have good reasons to renew our old mathematical model of equal temperament with no inharmonicity, and adopt a mathematical model of equal temperament that can deal with inharmonicity, i.e. Chas theory’s mathematical model.

“If you have something to contribute, a way of better fitting equal temperament to real pianos, then we are all ears.”

Thank you so much for your interest and your encouragement. We know iH requires stretching, so why opposing an updated and reliable stretched-partials ET theory? This is what Chas can prove to be, despite banal prejudices and predictable distrustfulness.

I would like to ask you all:

1 - is it possible to have progressive M6’s (4th+ M3) without a correct ET progression of 4ths, meaning without a correct ET 4ths theoretical and practical progression? I would answer no. Actually, if we had theoretical stretched octaves we could, in fact today with Chas we can.

2 – if we can not get progressive M6’s, what happens to m3’s and how can we get progressive stretched octaves without progressive 6ths (4th+ M3)?

Maybe answering these question explains why we were in need of accuratelly theoretically stretched 4ths.

You ask: “Now, according to you, why do fifths invert going up the scale?”

Because if 5ths were not to invert, goin up the scale 5ths would unconveniently diverge from stretched octaves.

Bill,

You wrote:…“the "erroneous assumptions" that we may have that ET makes the piano sound best.”…

In a way I’m saying what you understood: traditional ET can not make a piano sound best. As I have said, traditional ET is a lame theory because of its two wrong theoretical basic assumptions. As you read in section 3.3, traditional ET is a particular case that we can still find included in Chas mathematical and geometrical entity (s = 0). As shown in section 4.5, traditional ET ratio is the only ratio that, as a tragic matter of fact, perfectly flattens octaves beat-curve.

To conclude, traditional ET manages to theoretically stretch only 3ths, 10ths and 17ths, when we were in need to theoretically and pratically stretch octaves.

You say: “It is interesting that theoretically, 5ths will increase in speed but the very last thing anyone wants to hear are "beating 5ths".”…

An again we would move on a debatable ground. Bill, I’m not talking about preferencies, I’m trying to share an "s" dynamical ET model.

Finally you say:” Regardless of whether they were tempered or not in the midrange, they all eventually become wide. This will depend, of course on how aggressively or not the octaves are stretched. That kind of choice will make a difference in the part of the scale where 5ths do become wide. My observation has been that it is generally in the 6th octave.”.

What you are saying seems to me very close to what I’m saying, talking about middle strings. In my experience, how much you need to stretch C8 will depend on how you have got to F7.

Thanks, a.c.

Bill:

I think it would be OK to continue to post about 12ths and 5ths on this Topic. I think “brain-storming” with similar subjects may help clarify what Alfredo is trying to say.

I admit that I cannot hear much more than how the 2:1 octaves beat after a certain point in the scale. But when evaluating the theory behind a tuning system, the only way I can do so objectively is by considering the beating of intervals, including those that cannot be heard. I don’t think the number of cents that any particular note is from a theoretical pitch means much in how a tuning is constructed nor how it sounds.

So thank you for confirming what I thought, but could not know from just listening: that if 5ths go wide, 12ths do also. Now the question this raises is whether a tuning that does not allow wide 12ths will make the high treble sound its best.