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Alfredo:

I have some time to reply to your posts now.

You wrote two things that seem to contradict each other:

”Ok, but you could also think a 12th as being made of 5th + 4th + 5th. What happens if they chromaticaly go: narrower + wider + narrower, narrower + wider + narrower, narrower + wider + narrower, untill 5ths in midrange invert so that 5th + 4th + 5th can go: less narrow + wider + less narrow? Can 12ths remain narrow-constant, and can 15ths remain EB wide-constant in this way? Let me pass you the answer: yes.”

”About 4ths and M6 you wrote:...“You can have 4ths beat slower, remain the same speed, or beat faster and still have M6s beat progressively faster.”...

I quite agree, but my challenge has been finding all intervals precise and univocal beats incremental curves, the only reality that would prove the sound set being perfectly coherent.”

You seem to say that the beat rate of all intervals must be progressive, but you also say that 5ths can first be narrower, but then later less narrower (and eventually wide).

There is an answer to such a phenomena, but it requires that all the effects of iH be taken into account.

You asked Mr. Scott for an example, but he did not seem to understand what you wanted to know. I think I understand what you want to know because I have been down this road recently. Much is said about iH, but very little is actually shown. I will work up some figures to give an example and post them within a few days.

Tooner, do not you think it is nice we can read again from Bernhard?

Anyway, you say: “You should have a very different perspective when you understand iH more.”

Konwing you are lovingly saying this, I need to tell you that what gave me a different perspective was a new approach to the scale frequencies values, it was looking for a sound scale’s beat-period’s ratio (or beat rate ratio, if you prefere), something had never been sought after, while I can quietly and friendly tell you that, what has been said so far about iH is common knowledge.

On my part, I prefer not thinking my self as being an iH expert mainly because I think that, the way tuning difficulties have been related to iH has been messed up by a bottom problem that has got nothing to do with iH, i.e. the 2:1 theoretical octave ratio. This is to say: iH exists but tuning difficulties do not derive only from iH.

You said: “…”However, if we look at the beat rates that a fixed ratio (ignoring iH) predicts, then the beat rates, (or at least the ratios between beat rates) can be used as a model for an aural tuning.”

Well, you can then be very happy since the Chas theory’s model is, for the first time, suppling us with a ratio between beat rates (12ths and 15ths), actually a ratio that you could modify by using “s” variable. I’ll be here just waiting for you (all) to realize it. My answers to your latest posts will follow in this topic post's order. Thank you.

Robert, you say: “I guess by "non-IH tones" you are talking about tones generated by an instrument like a pipe organ that does not have any inharmonicity.”

I thank you because your answers to my questions gives us a good chance to check the accuracy of our premises.

“...a pipe organ that does not have any inharmonicity…” Is what you are saying 100% true? Thinking theoretically, what about air pressure? Thinking in practice, have you heard organ’s 12ths and 19ths? Did you like them?

By "non-IH tones" I’m talking about tones generated by a digital instrument, i.e. tones resulting from a single sinusoid.

You say: “...And I guess by ET you mean the geometric sequence that exactly doubles every 12 steps...”

It is so when I mention “traditional ET pseudo-model”.

You say: ...“(although the term ET is also properly applied to a sequence that more than doubles, as in piano tuning with stretch).”

Correct, these are ET sequencies.

You say: “If that is what you mean, then A3=220 Hz exactly and A5=880 Hz exactly, assuming the goal was to tune the octaves beatless.”

So you are saying: ...“assuming the goal was to tune the octaves beatless” you would refer to traditional ET ratio 2:1.

We better stop for some reasoning, I hope you wont mind. For tuners, ET would also mean “progressive 3ths, 6ths, 10ths and so on”, what we can not get from traditional ET pseudo-model, while our-days ET means “progressive 3ths, 6ths, 10ths and so on + octave stretching”.

A few posts ago I highlighted a foundamental detail: unless you correctly and univocally stretch 4ths, you will not be able to obtain the correct progression of all scale’s intervals.

Why? Because 4th interval is the base component for 6ths (4th + M3), octaves (4th + 5th), 10th (4th + 5th + M3) and so on. As Tooner says, we may somehow stretch octaves and, by playing with leeways, obtain progressive M3’s and M6's, M10’s and so on but, those leeways will mess up our 4ths, 5ths and octaves progressions (with us thinking that this happens because of iH).

In other words, even if you can get progressive M3's, M10's and M17's, unless you order progressive 4ths correctly you will not get progressive 5ths nor progressive octaves.

What is normally thoght is that iH does not allow you to put traditional ET pseudo-model in practice, Tooner its self conferms that, saying …“The octave is tuned wider than theoretical due to iH…”.

What I’m stating is: even thinking to non-iH tones, traditional ET pseudo-model can not give you progressive intervals.

Why? Because basic ratio 2:1 is wrong. In fact, basic ratio 2:1 crushes all other intervals ratios, so that you could regularly only find a double ratio on any other clone-copied octave. We still call this a compromise, but really we should call it a ratio crushing. Traditional ET pseudo-model 2:1 ratio reads too much into the scale sound’s ratios.

Let’s go back to you saying: ““assuming the goal was to tune the octaves beatless”.

You may now understand that we have no reasons for tuning the octaves beatless, if not to refer to the ET first sequence, the one coming from a lame ET pseudo-model. You may also understand that, from Chas model’s natural interweaving of ratios 3:1 and 4:1 you get stretched octaves, before what iH could impose, i.e. with or without iH. So, my answer to question 1 is:

A3 = 219.94157505789133667677
A5 = 880.23376184808211425507

You then kindly say: “In the presence of inharmonicity, there is no single agreed-upon implementation of "two whole equal-tempered octaves i.e. 24 semi-tones"...”

I’m not surprised.

“...Some say that the octaves should be beatless 4:2 octaves. Some say they should be slightly wide of just. And they are both right.”

In my opinion, they are both – you choose the word – bewildered? Lost? Puzzled? Disconcerted? Mixed up?

And why can be so? Because they can not rely on a solid stretched octave theory.

“...ET encompasses any sequence of pitches where the ratio of consecutive pitches is the same. You can have more stretch or less stretch and still call it ET.”

True, so true that with Chas algorithm you can finally draw any kind of stretch. Was'nt the kite analogy clear enough?

“...If what you are asking is "what effect will IH have on the tuning of A3-A5", then the answer is generally that more IH will cause any implementation of ET to have more stretch than it would if there were less IH.”

You say: “…more iH…”, “…less iH…”. What I think is that, iH experts should calculate iH on the base of correct premises, something that up to now, on the basis of a lame pseudo-model, could not happen.

All this to say that in stead of unhappily submiting to iH, when we think in terms of euphonicity, what ever preference, we should take the chance to refer to a solid, reliable and comprehensive theory.

Bernhard,

how did you know that I play trumpet?

I'll answer your post in this topic's order, meanwhile I'd only kindly ask you to stop insinuating about the authenticity of Chas article. In case it was your favorite sport, check with your lawyers how far you can go. I understand that something is disturbing you very much, what I can tell you for the time being is that Chas model is meant to give, not to steal. So, please, calm down and stop celloing about.

Regards, a.c.

Alfredo,

You have posted your paper officially and i think you appreciate any comments if they are critical or not.

What is disturbing me, are the pathetic claims in your paper, which seem to be caused by incomplete research about prior art.

And no, i did not insinuate about the authenticity about your paper. I just wondered why nobody of the university staff has co-signed your paper. They usually do no not use such a pathetic language, what may be a reason for this.

Best Regards,

Bernhard Stopper

Clearly you have in mind a definition of the word "inharmonicity" that is different from what the rest of the world means by "inharmonicity". But in terms of the what the rest of the world means by that word, yes, it is 100% true that a pipe organ does not have any inharmonicity. It has harmonics. Those harmonics are true. They are locked to the fundamental. If air pressure changes, then the pitch of the pipe will change, but so will all its harmonics, and they will remain locked. That is the definition of zero inharmonicity. Whether or not 12ths and 19ths sound good does not change this fact.

Well, some digital instruments might generate a single sinusoid, but most of them generate a more complex waveform.

You certainly can and do get progressive intervals from the traditional no-stretch ET model, which is the model I was refering to because I was answering your question about non-IH tones. And when non-IH instruments, like pipe organs, are tuned, they most often are tuned this way.

Now when it comes to instruments that have inharmonicity, like the piano, nobody uses the traditional no-stretch ET model. So you are criticizing a model that nobody uses for the piano anyway.

Nonsense. Of course the intervals are progressive for non-IH tones. Just play any cheap electronic piano (I say "cheap" to make sure it does not simulate inharmonicity, which the expensive ones do). The beat rate of 3rds, 4ths, 5ths, etc. will be prefectly progressive. They will all increase as you go up the scale. (Unless you are inventing a new defintion for the word "progressive" too.)

I was not recommending it. I was just trying to guess what you meant by your "question #1".

I would say they are entitled to their own opinion.

As I said before, the IH of a piano string does not rest on any system of tuning, be it lame or otherwise. You say "IH experts" as if IH was some deep notion, accessible only to a few. It is an objective physical measurement that anyone can make with the proper equipment. They do not have to be an expert. By way of analogy, consider "length", which is another objective physical measurement that could be made of a piano string. Would you talk about "length experts"? No, anyone with a tape measure can measure the length. So anyone with an ETD can measure IH.

Robert Scott
Ypsilanti, Michigan

Alfredo:

I am going to show how to mathematically tune A3, A4 and A5 to 4:2 beatless octaves on a theoretical piano while accounting for iH, show what the octave ratios are, and what the beat speeds of the 2:1 partial matches are.

The model I am going to use is the one mentioned in Young’s paper: iH = 0.1 at C3 and doubles every 8 semi-tones.

The iH of other notes can be determined by multiplying 0.10000 times (2 ^ (1/8)) ^ (the number of semitones above C3). A3 is 9 semitones above C3. So the iH of A3 = 0.1 * (2 ^ (1/8)) ^ 9. Dong likewise with A4 and A5 we have:

iH of A3 = 0.218101
iH of A4 = 0.616884
iH of A5 = 1.74481

We know that the 1st partial of A4 is 440Hz, but the theoretical fundamental is 0.616884 cents lower. We will first determine the theoretical fundamental frequency so that we can then determine the theoretical partial frequencies by multiplying by the partial numbers. Finally, the theoretical partial frequencies are raised a number of cents (determined by multiplying the iH times the square of the partial number) to find the actual partial frequencies.

Using the equation: a = b * (2 ^ (n/1200)) where “a” is the actual frequency, “b” is the theoretical frequency and “n” is the number of cents, we can determine the theoretical frequencies from the actual frequencies and visa versa. So 440.000 / (2 ^ (0.616884/1200) = 439.843. In other words, the theoretical fundamental frequency of a string with a 1st partial of 440.000 Hz that has an iH of 0.616884 cents is 439.843. Multiplying this theoretical fundamental frequency by the partial number “2” gives us a theoretical 2nd partial frequency of 879.686 (2 * 439.843). The iH correction in cents for a partial’s theoretic frequency is the iH times the square of the partial number. So 0.616884 * (2 ^ 2) = 2.46753 cents. Again using the equation: a = b * (2 ^ (n/1200)), but this time to determine the actual frequency, we have the actual frequency of the second partial = 879.686 * (2 ^ (2.46753/1200) or 880.940 Hz. Doing the same for the 4th partial of A4 we have:

1st partial of A4 = 440.000
2nd partial of A4 = 880.940
4th partial of A4 = 1769.43

Next, let’s “tune” A3 for a beatless 4:2 partial match. This means that the frequency of the 4th partial of A3 is at the same frequency as the 2nd partial of A4. In this case the frequency is 880.940. I chose a beatless 4:2 octave because it results in the lower P4 interval in an octave beating at the same speed as the upper P5 interval in the same octave. (If you have questions about why this is so, please ask!!!) Since we know that the 4th partial of A3 must be 880.940 and the correction is the iH of A3 times the square of the partial number, we can say that the theoretical frequency of the 4th partial of A3 = 880.940 / (2 ^ ((0.218101 * (4 ^ 2)) / 1200)) or 879.166. So then the theoretical 4th partial divided by 4 equals the theoretical fundamental, 879.166 / 4 = 219.791 Applying the iH correction we get the actual 1st partial frequency of A3 = 219.791 * (2 ^ ((0.218101 * (1 ^ 2)) / 1200)) or 219.818. Working out the 2nd actual partial the same way gives us:

1st partial of A3 = 219.818
2nd partial of A3 = 439.803
4th partial of A3 = 880.940

And starting with the 2nd actual partial of A5 = 1769.43 we have:

1st partial of A5 = 882.043
2nd partial of A5 = 1769.43

So now we can see what the octave ratios are. The 1st partial of A5 divided by the 1st partial of A4 = 882.043 / 440.000, or 2.00464. But the 1st partial of A4 divided by the 1st partial of A3 = 440.000 / 219.818, or 2.00165. They are not the same. So if we wanted to have beatless 4:2 octaves we would need to have an increasing frequency ratio.

Now let’s look at the beat speeds of the 2:1 partial matches. The 1st partial of A5 minus the 2nd partial of A4 = 882.043 - 880.940, or 1.1 bps. But the 1st partial of A4 minus the 2nd partial of A3 = 440.000 - 439.803, or 0.2 bps. This is more than double per octave, not what would be expected.




Alfredo, I took the time to post this for a number of reasons. First, I think you are ready to look deeper into how iH affects frequencies, beat rates and their ratios. Everything that I showed can be derived from Young’s paper, that you referenced in your paper. That bothers me a little and leads to my second reason.

I have been able to learn a great deal from this Forum. Others have corrected errors when they find them. By doing so, it maintains the integrity of the vast knowledge that can be found in past posts. I feel an obligation to help correct errors, also. Your paper and posts are in error because of your misunderstanding of iH.

And finally, I have never seen anywhere where the math was actually demonstrated to show how to calculate the effects of iH. I thought I ought to do so.

So my point is that fixed frequency ratios and expected beat speeds are mutually exclusive. In fact, you can have only one interval with an expected beat speed progression. All others will vary dependant on the piano’s iH. That is why I said very early in this Topic:

”The theory of piano tuning fascinates me, but lately I am realizing its usefulness is limited in aural tuning. Aural tuning is all about compromises, compromises that can be heard. They don’t need to be theorized to be heard, just listened to and accepted. I am thinking that the theory is really only necessary for designing a mathematical model so that ETDs can make the compromises without actually “hearing” them. And then there is the final limit on accuracy imposed by the pinblock and rendering points. Not to mention what the next passing thunderstorm may do to a tuning!"

Robert, Tooner, thank you very much for your posts.

Bernhard,

You say: “It is usually highly recommendable to do research what others have said about the matter, before trumpeting out revolutionary ideas.”…

If not your style, I can at least share one non-revolutionary, recommendable and usual idea of yours.

...“Your claim that CHAS ET (equal temperament) is the first ET model that questions the the base of pure octaves for ET is wrong.”

You are missing the point, having developed my arguments I’m now more extensively explaining my logical, mathematical and practical reasons for sharing Chas theory, without really caring whether I’m the first or the last. For me, knowing I’m not alone on this path is only better. Anyway, before 1982, iH’s apparent need for stretched octaves questioned pure octaves first.

On the theoretical ground, what I’m proving is that, with octave’s 2:1 ratio, the sequence of difference quotients n/n+1 (section 4.5) cannot occur.

Why is this difference-quotients sequence important? Firstly, because Chas theorizes a dynamic sound-set and n/n+1 sequence helps to understand why beats have in Chas model the greater function. Secondly, because it proves that, the need for stretching octaves derives from a bottom question.

...“Serge Cordier published work about an equal temperament on pure fifths in 1982. (Accord bien temperé et justesse orchestrale, S. Cordier 1982). So it is his merit of having questioned octave based ET first and replaced it with a different theoretical model.”

Please adjust your throw, may Serge Cordier have recognised all its merits. Let me read his article, only then I’ll be able to value if he banally talked about one more pure ratio, like 19th root of 3, or like (why not?) square root of 9/8, or if he perfected what one would call a reliable (not just a different pure ratio) theoretical model.

If we look at frequencies from a dynamic point of view, the whole theoretical concept of pure ratios is useless, exactly like it would be useless – as you may know - trying to divide any lenth in two halfs without approximation. To zero beats is a theoretical absurd, it would be like zeroing a wave, or considering space as a plane. Instead you can theoretically set an equal beating, even if you may end up having to do with irrational ratios.

...”Chas (in the abscence of onharmonicity, where your s or s/s1 equals 1) is only one possible stretch point among let´s say millions of solutions between the 12th root of two (standard ET) and the 7th root of 1,5 (Cordier ET).”

I agree and I'm glad you recognize that. If you had red the “kite analogy” (in this topic), you would have understood how from Chas model’s algorithm also ratios 4th root of 5/4, 28th root of 5, 31st root of 6 can gush out. This is how, mathematically, Chas model proves to be coming from a comprehensive theory.

...“The solution you provide as a "discovery" of symmetry is rather an invention for me.”

Maybe you meant to say “fantasy”. Anyway, what you can see in Chas article (section 3.5 - Effect of delta on incremental scale ratios), is not fantasies, it is absolutelly correct and unique.

...“There are plenty of other "symmetric" solutions available.”

Good that you say that. As you know though, the partials that Chas model interwaves in its standard version, i.e. partials 3 and 4, just after foundamental tone and partial 2, are more often the most intense.

...”For example one could split the pythagorean comma in two equal parts and subtract one part from the twelve fifths and add the other part to the 7 octaves. But such symmetric methods must not end in symmetric beat patterns.”

You see, I decided to rely on beats. Once I was reasonably sure I could find again and again the one precise beat-set form, having dismissed all cultural influences, I traslated this dynamic set’s form into numbers.

...“The resulting numbers of CHAS in the beat tables don´t show a higher degree of symmtry than let´s say standard ET or Cordier ET to me. (See your table of different ET beat rates).”

Please mention precise numbered sections, otherwise I can not understand what you are referring to. Also remember that Chas has been conceived in dynamic terms, where symmetries of the beating-whole could continuously change.

...“I also disagree with your claim that ET based on pure duocimes (twelfths) does spoil symmetry.”

What I’ve said is: pure 12ths produce to wide M3’s, M6’s, octaves, 10ths and so on. Contextually, I’m stressing on Chas model’s different approach to the concept of “purity”: …“Purity no longer derives from a single combination or from a pure ratio…” (section 2.0).

...”The opposite is true: To verify, generate a beat table for a 19th root of three table (which you have left out in your paper).”

Don’t worry, you can put it in your paper.

...“You WILL find symmetries of highest degree in the resulting beat tables, which are not present in other ET solutions (including yours).”

I'm not competing for the highest degree of symmetry. Anyway, what you could conferm, reading section 4.5, is that Chas model standard version’s quotients values (s = 1) reach the 7th decimal point of the n/n+1 sequence, what no other pure ratio does.

...”And THAT is a discovery (discovered in 2004, as is wrote in an earlier post of this thread).”

What happened to your capital letters? I hope you are not living in a state of constant apprehension, you should know, time adjusts everything, also your claims.

...”From your writing about iH, it is obvious that you are not familiar with what an inharmonicity constant is.”

You are right, I’m more familiar with beats.

...”State of the art of available theoretical tuning models (which are in use for example in advanced electronic tuning devices) CAN handle with inharmonicity.”

For example, only-pure ETD Bernhard Stopper’s device?

...”You claims are once more wrong relating this fact, so you may not claim that CHAS is the first tuning model that can handle inharmonicity.”

It is important that you have understood how Chas model can deal with iH, I don't care being the first or what. Strangely enough though you are confusing Chas theory’s model with a device.

...”Finally your paper does appear to be an official paper of the university of ..., but i don´t find any names of representatives of the university.”

I don't know what you are talking about. Have you suddenly forgotten how to search? Look for: g.r.i.m. university of palermo. You find that heading on every single page of Chas model’s article.

...”That makes your paper very suspect to me.”

You are free to suspect as much as you like, as long as you do not spend words inconsiderately.

...”All your claims appear in those contexts as overblown and whithout reliable fundaments.”

When you state something you should also explain why, otherwise you sound like a defamer. Overblown or underblown, once again this is not the point. As for reliability, Chas model, as you may know, is numerically correct in absolute terms. This, together with its uniqueness have led to Chas model article’s publication.

Tooner, thanks again for your iH calculations.

following Bernhard post you say:

“I have to agree with most of what you posted. I cannot, of course, agree with what I do not know. I am still looking forward to your paper and hope that your “Super Symmetry” is more understandable than Alfredo’s “Synchronic Attractor”.”

I like it when you play the devil’s advocate, more than when you play the echo. Please, personalize all your statements, so that our contributing wont fall into triviality.

To know more about “attractors” you'd better ask a physicist.

Bernhard, you wrote:

...“As i wrote, you can verify it on your own by generating a 19th root of three beating table…”…”… a small example of the apparent symmetry in the 19th root of three ET whithout inharmonicity is...”…”...only in the case of 19th root of three this is true for all combinations of octaves fifths and fourths (with distances of octaves, fourths and fifths between them) etc.”…

As I could tell you, with 19th root of 3 you get a fast beating octave, as your own symmetry exercise’s figures conferm.

You also say:…” My model has perfect symmetry in abscence of inharmonicity, while CHAS or other ET solutions have not.”

If I were you I would start a new claims topic. There you could freely compare Chas theory’s model with your exercises figures. Meanwhile it may happen that you understand how Chas algorithm includes also 19th root of 3 ratio.

...“The point is, that the acoustic effects caused by the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.”

Untill you don’t explain on what basis you make your statements, I can only think your words as an act. Your experience should tell you that, a piano tuned using 19th root of 3 ratio sounds harsh. Anyway tell me please, on wich ratio do you think the piano will, after some playing or after some time, adjust?

Tooner,

You say: …“you also say that 5ths can first be narrower, but then later less narrower (and eventually wide). There is an answer to such a phenomena, but it requires that all the effects of iH be taken into account.”

You’d better figure out what happens.

Bernhard, you say:

..."What is disturbing me, are the pathetic claims in your paper, which seem to be caused by incomplete research about prior art."..."nobody of the university staff has co-signed your paper. They usually do no not use such a pathetic language, what may be a reason for this."

How strange, I've been listening to your claims. As for the rest, as I've said, I'm trying to share a new ET EB model that can deal with iH. Hope you do not mind if, despite my pathetic language, Chas theory's model can also include 19th root of 3.

Thanks a lot, a.c.

I have already voiced strong disagreement with this statement. What is your point in repeating it?

Are you suggesting that I and my customers and those who have heard my posted recordings do not know what a well-tuned piano should sound like?

When will you make some recordings available to attempt to back up your statements? I look forward to hearing the consonance in your tunings that will suddenly make mine sound "harsh".

You posted your tuning sequence here with the a number of directions reversed. Wouldn't it be appropriate (and face-saving) to post a corrected sequence in its entirety?

Mr. Swafford:

The harshness of a tuning is subjective. I find pure 12ths create a harsh sound in the tenor and bass, but make a very nice sound in the mid treble. In the high treble I prefer a little more “zing” than pure twelfths will provide. The recordings of Pure Sound and pure 12ths tunings that I have been able to get from the internet agree with what I hear when tuning aurally.

I don’t think this is a well-tuned piano verses a poorly-tuned piano judgment. It is a merely a personal preference.

My primary interest in all this is the mathematics. Unfortunately, there is only a mention of iH in discussions on tuning theory and not an integration of it.

Right. It is pointless to have a "debate" that consists of "this tuning is harsh" followed by "no, it isn't."

I had hoped to hear exactly what intervals or chords sound harsh in a pure 12th tuning and what can alleviate that harshness, according to those who hear such harshness. I can't hear such harshness so I haven't a clue where harshness might be coming from.

Actually, my hypothesis is that my execution of Stopper's implementation of a pure 12th tuning would not sound "harsh" to any listener. One shouldn't automatically assume that various tunings claiming to be pure 12th ET are all of the same quality. I have worked hard to learn to execute the Stopper tuning.

Stopper has made some rather bold claims; the difference is, however, that unlike Capurso, he has made it possible for us to duplicate his tuning and verify the claims.

I can't begin to follow all of this discussion or comment on it but I have finally had the time to look up a PTG Tuning Exam Master Tuning to show that 5ths become pure and then wide in an example of ET in which optimum stretch is used in the 5th and 6th octaves and conservative stretch in the 7th. So, for what it is worth:

If any two notes which form a 5th belong to the same octave and are read on the same partial and have the same numeric value, they would be theoretically equal tempered 5ths (2 cents narrow). We don't expect to see that at all and we don't in a master tuning. If the upper note of the 5th is 2 cents larger than the bottom note, the 5th is pure (beatless or just). If the upper note is more than 2 cents higher than the bottom note, the 5th is widened.

On a Yamaha C5 the C5-G5 5th reads: C5: 0.3, G5: 2.2. That is almost pure. The E5-B5 5th reads: E5: 0.6, B5: 3.0. That is 0.4 cents wide. The C6-G6 5th reads: C6: 3.6, G6: 5.1. That is 0.5 cents narrow. The E6-B6 5th reads: E6: 5.2, B6: 12.8. That is 5.6 cents wide. The C7-G7 5th reads: C7: 12.3, G7: 19.6. That is 5.3 cents wide. The E7-B7 reads: E7: 21.8, B7: 25.6. That is 1.8 cents wide.

Regarding 12ths: The Master tuning reads all pitches from C5 to B7 on the 1st partial. Therefore, 12ths can be analyzed similarly. The C5-C6 12th reads: C5: 0.3, G6: 5.1. That is 2.8 cents wide. The E5-G6 12th reads: E5: 0.6, G6: 5.5. That is 2.9 cents wide. Between the 6th and 7th octaves, with the 7th octave being tuned as 2:1 octaves with the 6th, an amount which the majority of technicians feel sounds "flat" and unacceptable, the 12s are still surprisingly wide. The C6-G7 12 reads: C6: 3.6, G7: 19.6. That is a whopping 14 cents wide. The E6-B7 12th reads: E6: 5.2, B7: 25.6. That is a mind blowing 18.1 cents wide!

By the way, I had nothing whatsoever to do with the Master Tuning I just quoted from. It was done late last year by 2 CTE's and a CTE Trainee from Chicago. Both of the CTEs have served more than 10 years as such. One of them is the Chicago Symphony Orchestra Concert Technician and the other is an instructor at the Chicago School of Piano Technology. I personally observed the CTE Trainee's qualifying exam and the results were superior indeed. He will no doubt be certified at the PTG Convention next month. Need I say more about the qualifications of the technicians who crafted this tuning record?

I can quote from other Master Tuning records and I can quote more examples than I have but all would reveal similar findings. Just consider now, that the above quotes are from what is considered to be the most standard form of ET and that the top octave is not stretched nearly as much as most people will do and certainly not as much as many other technicians, including myself do routinely.

So, to me, the issue is certainly not whether 5ths or 12ths become pure, then widened; they certainly do quite naturally. The ideas that there are out there for this to occur lower in the scale and the effects that has are what the issue is. I do what I do and I am pleased with the results and can document exactly how much I stretch octaves either numerically or in clear, readable text. From what I know, so can Bernhard Stopper.

Between what I do and what Bernhard Stopper does, although the two are aurally perceptive as different, there isn't really that much difference numerically or aurally as perceived by both casual and very educated listener alike. The difference amounts to a few cents here and there of manipulation of the temperament octave and midrange. If we take the "ET with pure 5ths" idea which I consider to be too extreme, there still is not all that much difference between a PTG Master Tuning and what would be required to produce those results in the midrange; a couple of cents worth at most.

If there were very much more of a difference in any of these, the results would inevitably be perceived as unacceptable by at least some and that "some" would be far too many to try to convince. There is simply a limit on how far one can go before it is too far and that limit has a very narrow range.

Therefore, it begs the question whenever any one of these new and improved ETs come out which claim to have the ultimate answer to universal beauty, "Just where is that sweet spot?" If we don't hit it this time but get it the next, will anyone really ever notice or care?

Bill:

Thank you very much for posting the figures from the master tunings.

You wrote: ” If any two notes which form a 5th belong to the same octave and are read on the same partial and have the same numeric value, they would be theoretically equal tempered 5ths (2 cents narrow). We don't expect to see that at all and we don't in a master tuning. If the upper note of the 5th is 2 cents larger than the bottom note, the 5th is pure (beatless or just). If the upper note is more than 2 cents higher than the bottom note, the 5th is widened.”

I respectfully disagree. The effects of iH are much, much greater on higher partials than lower partials, and iH is much, much more for higher notes than lower notes. So, the cents differences between theoretical frequencies and lower partials can be very, very different than the cents differences between theoretical frequencies and higher partials. I believe the only way to determine if a fifth is wide is to compare the frequency of the third partial of the lower note with the frequency of the second partial of the upper note. There is no short cut. This is not to say that fifths do not become wide, just that it needs to be demonstrated in another way.

I enjoyed your last paragraph very much:

” Therefore, it begs the question whenever any one of these new and improved ETs come out which claim to have the ultimate answer to universal beauty, "Just where is that sweet spot?" If we don't hit it this time but get it the next, will anyone really ever notice or care?”

Very excellent point! Who really notices or cares? Probably only the person that did the tuning. I sometimes think that many people could hear what I hear, but it is the caring that makes the difference. I have resigned myself to the burden of never being quite satisfied.

No. Re-think your assumptions please. The width of the fifths is determined by the 3:2 partial relationship; the width of fifths is not included in master tuning records. Sorry.

Well, of course I thought about this but consider this: if C3 and G3 both have the same number (which they would not), would that not mean that the 5th is 2 full cents narrow? From the same Master Tuning: C3: -3.7, G3: 0.7. That ;eaves 3 cents difference between the two. Subtract 2 cents that is already between C3 and G3 and it leaves a 5th that is 1 cent narrow. The theoretical 2 cent narrow 5th is widened by fully half the amount we expect theoretically to only 1 cent narrow.

Doesn't this apply to all the rest of it? While I agree with both Jeff and Kent that one would need to compare the actual partials, they are unknown and can't be determined without actually measuring them. Once the partials reach into the 8th octave and higher, that is impossible. They also cannot be heard, so it does not matter that 5ths and 12ths are wide.

Oops, I made a pos/neg error with the above which I can often do. The above 5th is actually 1 cent wide. Now, aurally, of course, it is still narrow but there is still 3 cents width between the two at the fundamental level. Between C6 and C7, C6: 3.6 and C7: 12.3, there is 8.7 cents width in the octave even though the octave sounds pure aurally. If C7 were any higher, there would be a beat, of course and that is the way most technicians would tune it. So, although 5ths and 12ths won't beat crazily sounding wide wherever we can hear them, I still conclude that they do, in fact become wide, even by an amount we can sometimes hear somewhere in the 6th octave on up.

I have often seen this happen when comparing the double octave and 12th electronically. Up to a certain point, the double octave is wider than the 12th but at some point they become approximately equal and then at another point, the 12th becomes wider than the double octave. If the 12th is wide, then the 5th must be wide too.

Bill:

First, if “C3: -3.7, G3: 0.7” is not a typo, then the algebraic difference is 4.2 cents because C3 is negative and G3 is positive. So, in your line of reasoning, this interval is 2.2 cents wide of just intonation. Would a master tuning have this interval beat wide of just intonation?

But let’s try a different sort of reasoning. Let’s say, just as an example, that C4 is +3.5 cents. This means that the octave is stretched 7.2 cents [3.5 - (-3.7) = 7.2], and again just as an example, this produces a beatless 4:2 octave. So how much is each semitone stretched due to iH? 7.2 / 12 = 0.6 cents. And how much is a fifth stretched due to iH? 0.6 x 7 = 4.2 cents. So we would expect the fifth to beat as if it were 2 cents narrow, because there is no “additional” stretch. The 4.2 cents is due to iH, which affects the 5th the same as it does the octave and all other intervals.

But even this kind of reasoning continues to have problems because the G3 – G4 octave would probably not be stretched 7.2 cents, but more. And there are even other considerations...

EDIT:

Sorry I cross posted. Glad you caught your own error. Still, there are other considerations. But if an interval beats wide, it is wide.

Robert, thank you.

In one previous posts of yours you wrote:

“I don't know what false assumptions you are attributing to ET. As far as I know, ET does not make any assumptions at all.”

Have I succeded to explain you that?

You say:

“Clearly you have in mind a definition of the word "inharmonicity" that is different from what the rest of the world means by "inharmonicity".

Have I talked about iH definition? No, I have and I am talking about the reference model on which basis iH’s effects have been calculated.

About the case of a pipe organ you wrote:

“It has harmonics. Those harmonics are true. They are locked to the fundamental. If air pressure changes, then the pitch of the pipe will change, but so will all its harmonics, and they will remain locked. That is the definition of zero inharmonicity.”

Please tell me, when you say “It has harmonics…..that is the definition of zero inharmonicity”, what are the harmonics values you are/they were referring to, when fixing zero iH?

“Whether or not 12ths and 19ths sound good does not change this fact.”

Whether or not 12ths and 19ths sound good, this may change our reasoning about traditional ET pseudo-model V iH.

“You certainly can and do get progressive intervals from the traditional no-stretch ET model,…”

Do you mean, on a pipe organ? By the way, are you an aural tuner?

…“Now when it comes to instruments that have inharmonicity, like the piano, nobody uses the traditional no-stretch ET model. So you are criticizing a model that nobody uses for the piano anyway.”…

I am criticizing traditional ET pseudo-model because it is a lame reference model, with or without iH.

When I wrote: “...What I’m stating is: even thinking to non-iH tones, traditional ET pseudo-model can not give you progressive intervals...

You answered back:

”Nonsense. Of course the intervals are progressive for non-IH tones. Just play any cheap electronic piano (I say "cheap" to make sure it does not simulate inharmonicity, which the expensive ones do). The beat rate of 3rds, 4ths, 5ths, etc. will be prefectly progressive. They will all increase as you go up the scale. (Unless you are inventing a new defintion for the word "progressive" too.)”

I think you are right, one of us is saying nonsense. In my opinion, if you were familiar with beats you could never say that. Give me a little time, and I’ll help you with precise reference brands and 4ths and 5ths reference intervals. In the meanwhile please, keep your definition of “progressive” but notice that RBI beat-rate progression can also be rough or smooth. A rough-hewn RBI progression will leave disorder amongst SBI, what some aural tuners may well know.

When you said:

“...Some say that the octaves should be beatless 4:2 octaves. Some say they should be slightly wide of just. And they are both right.”

I retourned you my opinion:

“In my opinion, they are both – you choose the word – bewildered? Lost? Puzzled? Disconcerted? Mixed up?”

You answered back saying:

”I would say they are entitled to their own opinion.”

Ok, generally speacking I quite agree, but I’m not writing about an opinion festival. Chas model’s interweaves prime numbers and relates scale’s frequencies on the basis of a new ET theory, a theory deriving from a dynamic approach to beats. In other words, Chas model proves how proportional beats can define an infinite number of scale’s ratios, quite the opposite of what has always been done.

You finally say:

"IH experts" as if IH was some deep notion, accessible only to a few. It is an objective physical measurement that anyone can make with the proper equipment. They do not have to be an expert. By way of analogy,…”

I am referring to those technicians able to make laboratory measurements and calculations with sophisticated technology. Masons still work with a tape (God bless them), land-surveyors don’t anymore. As you may notice, I’m treating a semitone ratio that, compared with traditional ET pseudo-model, differs 0.00002.., and produces an A5 octave difference of 0.2337..

Tooner, thanks for your commitment.

You say:

“I have been able to learn a great deal from this Forum. Others have corrected errors when they find them. By doing so, it maintains the integrity of the vast knowledge that can be found in past posts. I feel an obligation to help correct errors, also. Your paper and posts are in error because of your misunderstanding of iH.”

I feel a similar tipe of obligation, this is way I would ask you to be very responsible and utterly precise when you say “your paper is in error”, as I would also ask you to acknoledge that I do not ignore iH, I precisely think it should be calculated by using a correct reference model.

You kindly wrote:

“The iH of other notes can be determined by multiplying 0.10000 times (2 ^ (1/8)) ^ (the number of semitones above C3). A3 is 9 semitones above C3. So the iH of A3 = 0.1 * (2 ^ (1/8)) ^ 9.”…

This formula is the base of all your calculations. So I ask you: how wide are you calculating the semitone? What partial values would you expect with “zero inharmonicity”?

Certainly you will have red where our cello-expert writes: “…the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.” What do you think he meant, saying “…proper consideration of inharmonicity”?

Bill, Kent,

I will contribute to what you are saying asap, thank you.

Regards, a.c.

”I feel a similar tipe of obligation, this is way I would ask you to be very responsible and utterly precise when you say “your paper is in error”, as I would also ask you to acknoledge that I do not ignore iH, I precisely think it should be calculated by using a correct reference model."

You do not ignore iH in your paper. You mention it a number of times. That does not mean you are applying it correctly. You referenced Young’s paper, but seem to either not understand it, or believe that iH is different than described. Young’s paper describes how iH is calculated and how it affects the frequencies of partials. You have not used those calculations in your paper. I don’t know how I can be more - responsible and utterly precise when I say your paper is in error.

” You kindly wrote:

“The iH of other notes can be determined by multiplying 0.10000 times (2 ^ (1/8)) ^ (the number of semitones above C3). A3 is 9 semitones above C3. So the iH of A3 = 0.1 * (2 ^ (1/8)) ^ 9.”…

This formula is the base of all your calculations. So I ask you: how wide are you calculating the semitone? What partial values would you expect with “zero inharmonicity”?”

The width of the semitone has no effect on the iH of a particular note. The frequency of a note does not change a note's iH either, although the frequencies of the partials are affected by the both the fundamental pitch and the iH. The same calculations can be made with an iH of zero. The result will be partials that are whole number multiples of the base frequency.

Maybe you are asking about the width of semitones in regard to how I determined the width of the octaves in the example. If so, I did not calculate any semitones, only octaves. And I can think of a number of ways to calculate the semitones, but I would first have to decide on the width of more octaves so that there would be an accurate nonlinear interpolation.

[EDIT]: Ok, I think I know what you mean, now. Rather than considering the semitone as being an interval that is tuned, it should be considered as a note on the piano. The iH doubles every 8 notes in this example. iH is a function of piano scale design, not tuning.

” Certainly you will have red where our cello-expert writes: “…the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.” What do you think he meant, saying “…proper consideration of inharmonicity”?”

I am not certain what Mr. Stopper means. It may be similar to what I call “the largely self-correcting effects of iH on beat rates”. We should just be patient and wait for his paper.

Alfredo, have you tried looking at iH from a fresh perspective? You may feel that this Forum has treated you poorly (It does treat people poorly, sometimes) and are hesitant to accept what is written. Let me suggest that you re-read Young’s paper one paragraph at a time, not going on to the next until it is fully understood. Also, here is a link to the graphs of iH curves for a number of pianos:

http://www.goptools.com/

After opening the link, click on “The Scale Collection - Browse Original Scales of over 40 Pianos “

And by the way, I think very few tuners completely understand all the effects of iH. I am still working on the finer details. But I don't believe it is necessary to understand it to tune well.

Unfortunately, no.

You may not have explicitly talked about your definition of IH. But one must learn to walk before one learns to run. How can you make accurate statements about IH if you don't understand what it is at a more fundamental level?

A harmonic is a sinusoidal tone whose frequency is an exact whole-number multiple of a fundamental frequency. For example, in the case of a pipe organ, if the fundamental frequency is, say, 511.2 Hz, then the 2nd harmonic is 1022.4 Hz, the 3rd harmonic is 1533.6 Hz, etc. And we don't "fix" zero IH. We observe it. A pipe will behave like this whether we want it to or not.

Yes, that is one of the places where the traditional no-stretch ET model is actually used. It is not used for pianos.

Robert Scott
Ypsilanti, Michigan

Equal temperament is defined as that temperament in which the beat speeds of like intervals progress smoothly across the scale.

I believe it is true that pianos tend to be scaled in such a way that beat speeds taken from the mathematical model of equal temperament can be used in the mid-range of pianos with only relatively small modifications due to inharmonicity. The necessary modifications of beat speeds would tend to increase as one moves toward the top and bottom of the piano scales. No?

"Proper consideration of inharmonicity" would be modifying a piano tuning's beat speeds away from that of the mathematical model in order to best preserve the overall smoothness of beat rate progressions (and desired purity of the slow-beating intervals) despite the changing level of inharmonicity through the scale.

The perturbations of inharmonicity upon the mathematical model of equal temperament as applied to piano tuning are significant, and are probably the problem that underlies this discussion. This problem is endlessly fascinating and some brilliant people have devoted a great part of their lives to the subject. A great example of these "perturbations" is the fact that a standard feature in Dr. Al Sanderson's FAC tunings as calculated by the Accu-Tuner is narrow 4:2 octaves in the piano's treble. Dr. Sanderson considered those narrow 4:2 octaves absolutely necessary to provide the desired overall purity of the slow-beating intervals; obviously, many piano techs have agreed over time.

This is but one example.

There probably are still contributions to be made concerning "the proper consideration of inharmonicity", but my opinion is that these contributions should be made with great respect for the coherent effort and considerable intellect that has already been devoted to this subject. After all, many pianos already sound very well tuned.

Mr. Swafford:

I think the problem that is underlying this discussion is bruised egos. I am trying to be more careful.

You also mentioned:

”There probably are still contributions to be made concerning "the proper consideration of inharmonicity", but my opinion is that these contributions should be made with great respect for the coherent effort and considerable intellect that has already been devoted to this subject. After all, many pianos already sound very well tuned.”

Is this information readily available? This also makes me think about how pianos might have been tuned through the centuries, without the use of iH theory. I know the standard text that I learned to tune with is out-of-date, but it still worked. It is interesting that the octave tests that were used were appropriate even though they were not designed in "consideration of inharmonicity." Just musing...