Alfredo:

We may come to a mutual understanding yet!

You posted: “"However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally."...

And what happened? Could you tell me more?”

The graphs show what happened. Even though the incremental ratio was greater than 2^1/12, it was not enough to make even the 2:1 partial matches beat wide, let alone the 4:1 partial matches to even be just. Equal beating 12ths and 15th did not happen.

And you posted: “Now you say:..."The desired beatrate is determined, and then the frequencies are calculated. Finally, if wanted, the frequency ratios can be ascertained. But they are a byproduct, not used in the calculations."

This is what I would do in practice too, I'd tune Chas form and then I would ascertain the frequencies values and ratios.”

Now you really have my interest. How did you ascertain the frequency values and ratios? If it was by using the CHAS algorithm, then this shows that the algorithm is inadequate. We have already looked at the results.

As was shown above, the CHAS theoretical frequencies, when iH is applied, do not have the expected results. In fact, even without including iH there is a problem. The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate.

I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless. I believe the simulation is accurate and this is also similar to what I hear when tuning this way. On the average the beatrate is less than 1 bps.

Alfredo:

My apologies. I miss read ”This is what I would do in practice” as ”This is what I did in practice”.

But you still have my interest! So what would you then do with the frequencies and the ratios?

Tooner,

I hope you’ll still play the Devil’s Advocate, despite our mutual understanding.

We wrote:...“However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally....

And what happened? Could you tell me more?

The graphs show what happened. Even though the incremental ratio was greater than 2^1/12, it was not enough to make even the 2:1 partial matches beat wide, let alone the 4:1 partial matches to even be just. Equal beating 12ths and 15th did not happen.”...

Going back to
http://www.box.net/shared/rxb631v2yz

I could again confirm that your ET-EB simulation do make the 2:1 partial matches beat wide. True?

Then you say “Equal beating 12ths and 15th did not happen”, but you have managed to straighten 12ths and 15ths. In my opinion, to get EB 12ths and 15ths you/we may have to adjust iH.

You say:...“The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate. I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless.”...

What you are saying may well be possible. I can not be 1000 % sure about choromatic equal beating, although I trust my sense of rhythm.

...“I believe the simulation is accurate”...

I think it is accurate only to some extent. Not because of you though.

...“and this is also similar to what I hear when tuning this way.”...

You prove to be a very good Advocate, and surely you are a very good tuner.

...“On the average the beatrate is less than 1 bps.”...

Yes, I'd say between 1/2.5 and 1/3 bps.

...“But you still have my interest! So what would you then do with the frequencies and the ratios?”

I would pass you those values and invite you to point f).

Regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (granpianoman)
http://www.box.net/shared/od0d7506cv

Alfredo:

Have no fear. Even with a mutual understanding on the relationship of frequency ratios, beat rates and inharmonicity there is plenty that we can disagree on.

The next step is to discuss the nature of iH itself. But first, since you and I have different native tongues, allow me to summarize the mutual understanding:

Frequency ratios are useful in predicting beat rates of intervals made from harmonic tones, but are not useful in predicting beat rates of intervals made from inharmonic tones.

Before we continue on the nature of inharmonicity, perhaps you could acknowledge this mutual understanding, just to be sure.

Oh, and if you choose to give me the frequencies of a tuning that produce a certain set of beat rates, please include the inharmonicity so that all other beat rates can be calculated. If the iH is not included, then it would be best to send me the frequencies on very soft paper, so that I could find some use for them�

Tooner,

You let me know about your preferencies in terms of paper and that’s ok, I understand that as being pretty original for you.

Then you say:...“The next step is to discuss the nature of iH itself.”...

For me, the next step is going back to what you/we have already stated. In fact, lots has already been written about the nature of iH and, above all, I would like to pursue my actual aim, I would like to share Chas model as a variant of ET where the octave ratio is something a little bigger than two, in the way we are actually tuning.

You wrote (11/11/09):...“I am not trying to evade your other questions.”...

So, the questions I posted on the 10th and the 12th of November are, in my opinion, quite crucial and may well represent the central issues of this discussion, better than us having different native tongues.

You say:...“Frequency ratios are useful in predicting beat rates of intervals made from harmonic tones, but are not useful in predicting beat rates of intervals made from inharmonic tones. Before we continue on the nature of inharmonicity, perhaps you could acknowledge this mutual understanding, just to be sure.”...

Instead, I think you should explain what you meant when you wrote (06/04/09):..."Because if we take the beat rates (or at least the ratio between beat rates, including equal beating) that are predicted from a frequency ratio (such as 2^1/12) that does not take into account iH, and then tune a piano with iH using the beat rates we end up with a decent tuning, but a different frequency ratio, one that is non-linear. So on the one hand, the frequency ratio is wrong, but on the other, the beat rates are correct. And since when tuning aurally, we listen to beat rates, the model works even though it is incorrect."...

And what you mean when you write (11/13/09)...“The CHAS model predicts that the beatrates of the 12ths and 15ths more than double each octave, but how you hear your tuning has all these intervals beating at the same rate.”...

Now, if frequency ratios are not useful in predicting beat rates of intervals made from inharmonic tones, why do you raise the question for Chas model? And again I ask you: how could you use 12th root of two?

You say:...“I believe the truth lies in between. When tuning equal beating 12ths and 15ths the beatrates increase, but less than double each octave, until the high treble is reached when they slow down, and become beatless.”...

Although I do not know whether you are talking about 12th root of two or what you are referring your "truth" to, I can confirm that, in my tuning form, 12ths and 15ths are opposite equal beating all along the scale. But maybe you have one more model in mind.

I asked you: “Could you also tell me more about that well scaled studio upright? Was it a real piano? Which temperament was used and how? What standard did you smooth the curve by? Do you know the approximation degrees, just to have an idea?

And I would now add a very simple question too: do you realize that, while 12th root of two ratio is a compromise between 3ds and fifths, Chas ratio is a compromise between 3ds, 5ths and octaves?

You asked for a Chas tuning example. Have you heard Granpianoman's MP3 conversion? Then we can talk about iH's effects on small pianos.

Thanks and regards, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv

Alfredo:

Even though it was believed (and apparently still is by you) that octaves on a piano are tuned (aurally) by frequency ratios, this is not true. They are tuned by partial matches. So I just cannot contribute to a discussion about tuning with frequency ratios. I cannot pretend “the earth is flat”.

Ok, on 10 Nov you asked: ”Could you also tell me more about that well scaled studio upright? Was it a real piano? Which temperament was used and how? What standard did you smooth the curve by? Do you know the approximation degrees, just to have an idea?

The model piano is a Charles Walter upright. The length of A0 is 48 inches. A file was generously provided to me that included the frequencies of the partials of an actual tuning. Given the frequencies and the partial numbers, I was able to calculate the inharmonicity for each note, which is affected very little by the actual tuning. Unfortunately, the file included only one partial frequency for the top octave; it takes the frequencies of two partials to calculate iH. Nonlinear extrapolation was used to estimate these iH values. It really does not matter much, since only the first partial is usable in the top octave.

The resulting curve was “V” shaped on a logarithmic graph with the left arm shorter than the right. Also the left arm, being wound strings, was “squiggly”. So by taking the value of iH for note 1, note 88 and the value of the lowest iH with its note number; an idealized “V” could be constructed using logarithmic interpolation. But this would produce an uncharacteristic sharp point to the “V”. By using a computer subroutine, this sharp point was rounded off by using increasing fractions of the slope for the eight notes centered on the point of the “V”.

I am not sure how to explain the approximation degrees in terms that would be valuable. But the purpose of the simulation was to show the general effect of iH on beat rates, and the approximations made this clearer than raw values would. Any piano’s iH values would have shown the same general effect.

The temperament was as equal as I thought practical. I had to start with a slightly wide 15th, calculate what beat rate this would produce in a 12th, and adjust back and forth. When the 3rds and 6ths were progressive, I decided that this was good enough for the simulation.

And on 12 Nov you asked: ” We may as well consider one evidence: 12th root of 2 predicts zero-beating octaves, 7th root of 3/2 predicts zero-beating 5ths, 19th root of 3 predicts zero-beating 12ths. Then I may ask you: when including iH, is it of value to theorize a zero-beating ET incremental ratio? In other words, taking your latest analisys to extremes, can an infinitesimal degree of iH agree with ET theoretical zero-beating choromatic intervals?”

“Then I may ask you: when including iH, is it of value to theorize a zero-beating ET incremental ratio?” My answer is: Yes there is no value. (“Yes, we have no bananas!”) When tones are inharmonic, octaves cannot be zero-beating. If one set of coincident partials are at the same frequency, none of the others will be. At some level the octave always beats, although it may not be noticeable. This is the case regardless of the accuracy of the iH.

Now, I have defined a limit to what I am able to discuss and have answered your outstanding questions. I am going to stop here. I am hoping that you will make shorter posts and try to deal with single subjects.

Tooner, thanks for your answer.

You say:...“Even though it was believed (and apparently still is by you) that octaves on a piano are tuned (aurally) by frequency ratios, this is not true. They are tuned by partial matches.”...

I aurally tune octaves, as well as all other intervals, by beats, like all aural tuners. Beats result from partial matches, so I can not disagree with you.

You write:...“So I just cannot contribute to a discussion about tuning with frequency ratios. I cannot pretend “the earth is flat”.”...

I’m trying to share Chas ET-EB theory because it is an improved ET model, in fact it is the model that can finally compromise all intervals, octaves included, into a beating-whole. And I could tell you about the relevance of a general temperament/tuning theory (11/10/09): “...a correct and reliable temperament theory will address aural tuners towards a practicable and euphonic model...”

Thanks for telling me more about your simulation. From what you say, I understand that there might be a chance to reduce approximations. Anyway, you've done a great job.

You say:...“I am not sure how to explain the approximation degrees in terms that would be valuable. But the purpose of the simulation was to show the general effect of iH on beat rates, and the approximations made this clearer than raw values would. Any piano’s iH values would have shown the same general effect.”...

I agree.

...“The temperament was as equal as I thought practical. I had to start with a slightly wide 15th, calculate what beat rate this would produce in a 12th, and adjust back and forth. When the 3rds and 6ths were progressive, I decided that this was good enough for the simulation.”...

I hope one day we’ll be able to work directly on Chas tuning, that day we’ll also make sure that 4ths, 5ths and octaves are progressive.

I asked you: “...when including iH, is it of value to theorize a zero-beating ET incremental ratio? In other words, taking your latest analisys to extremes, can an infinitesimal degree of iH agree with ET theoretical zero-beating choromatic intervals?”

You answer: “Yes there is no value. (“Yes, we have no bananas!”) When tones are inharmonic, octaves cannot be zero-beating. If one set of coincident partials are at the same frequency, none of the others will be. At some level the octave always beats, although it may not be noticeable. This is the case regardless of the accuracy of the iH.”...

Then you may agree on one issue (I’m asking you): an infinitesimal degree of iH makes any zero-beating theory no value.

...“I am hoping that you will make shorter posts and try to deal with single subjects.”

I’ll shorten. T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv

Alfredo:

You posted: “I’m trying to share Chas ET-EB theory because it is an improved ET model, in fact it is the model that can finally compromise all intervals, octaves included, into a beating-whole. And I could tell you about the relevance of a general temperament/tuning theory (11/10/09): “...a correct and reliable temperament theory will address aural tuners towards a practicable and euphonic model...”

That’s odd. You used the word “compromise.” I remember you taking exception when I described tuning as being about compromises…

You also posted: “Then you may agree on one issue (I’m asking you): an infinitesimal degree of iH makes any zero-beating theory no value.”

Yes, and iH makes other theories invalid (of no value), also. But the word infinitesimal can mean immeasurably or incalculably small, so there is a point when iH can be so small that its effect is negligible. I don’t believe that it is this small on any string of any piano and maybe not on other string instruments either. (There is no doubt in my mind that iH effects the tuning of guitars.)

Tooner and all Colleagues,

Often we read about having to compromise in tuning. I will treasure a concept, recently expressed by an American colleague I hold in high esteem. He makes a distinction between a “compromise” - what for me means “to make the best of a bad job” - and a “superior compromise”, what for all of us may represent an “optimum”. Today I get the chance to briefly write on this, the aim being to show how 12th root of two ET model results as a compromise, while Chas EB variant of ET can represent a superior compromise.

Most of you will know about the commas conflict. Also our practical experience confirms that if we tuned a pure interval all along the scale, this would be to all other intervals detriment and here is where the bottom problem lays.

In tuning - iH tones or non-iH tones - three contiguous pure 3rds will produce a narrow octave; in the opposite way, pure 5ths produce a wide octave. We all can experience the conflict amongst octaves, 3rds and 5ths.

Then, to get a zero-beating octave we have stretched – in theory and in practice - wide 3rds and narrow 5ths. In fact, this is what 12th root of two was meant for: this theoretical model stretches 3rds (wide) and 5ths (narrow) so to get a theoretical 2:1 pure octave. The compromise is then made between two intervals, 3rds and 5ths (considering 4ths as mirror-like 5ths).

Now, say that the octave module is a shelter, and that three 3rds are the three vehicles we room under our shelter. If we wanted to room three longer vehicles, shouldn’t we lengthen our shelter first?

So, considering a single 3rd as the octave’s sub-module, the question may be: Since three contiguous 3rds make an octave and we need to stretch 3rds, shouldn’t we stretch the octave?

As I say, this is where the bottom problem lays, the conflict that theoretical 12th root of two does not resolve in its entirety. This pure-octave ET model uses one single root, i.e. the “root of two”, so making a compromise between 3rds and 5ths, but crushes our choromatic stretched 3rds in an arbitrary 2:1 zero-beating octave. This theoretical, unpracticable and arbitrary constant (zero-beating octaves) has left tuners without a fair scale’s ratio and, above all, without reference. How could we ever go back home without a reference?

Chas ET-EB model, by using the root of 3 and the root of 4, manages to stretch 3rds (wide) and 5ths (narrow) by stretching octaves. Actually, what really happens is that 3rds, 5ths, octaves and all intervals stretch each other in a multiple function. No interval and no ratio are hold dearest, all intervals “compromise” in their own favor and in favor of a sound beating-whole. Our tuning form’s reference can now be double: 12ths and 15ths opposite equal beating in what Chas can describe as the practicable, optimized ET.

T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv

Alfredo:

Very nicely written. Your thoughts came through in an orderly, understandable way. Thank you.

The basis of the 12ths root of two is the idea that this will produce a temperament with all intervals being the same width (have the same frequency ratio), and at the same time octaves that are beatless. This is only possible with harmonic tones.

Keeping the discussion to harmonic tones (for now), the 12th root of any number will produce a temperament with all intervals being the same width. However, unless the number is two, the octaves will beat.

Now to have 12ths beat narrowly and at the same time 15ths beat widely the number has to be larger than 2 but smaller than (3^(1/19))^12 or 2.0014269… The compromise that is given results in equal beating 12ths and 15ths when these intervals have a common note on the bottom.

But it is not clear why this compromise is necessary at all, let alone why a superior compromise results with this sort of equal beating. Not to mention how the roots of any other numbers are needed to calculate this compromise. In fact, they are not needed nor actually used although it could seem that way.

The argument could easily be made that the common note should be on the top, or the 15th should beat faster than the 12ths that has a common note on the bottom, but slower than the 12ths than has a common note on the top. Another argument could be made that if anything should be equal beating, it should be the single octaves beating the same as the 5ths. But then the question again arises as to why, which note should be common, or should they actually beat equally?

Things get difficult when trying to use the 12th root of any number to describe the tuning of inharmonic tones. But rather than go into the difficulties, let’s look at what actually happens when a piano, with inharmonic tones, is tuned.

Oddly enough the tendency when tuning beatless sounding octaves is that the effects of inharmonicity produce narrowly beating 12ths and widely beating 15ths throughout much of the scale. The tuning can be adjusted so that these intervals beat equally in any or all parts of the piano, or unequally in any or all parts of the piano.

This is the true value of these intervals. They are a tool that the tuner can use to make compromises that are more important than arbitrarily equal beating intervals. They can be used to make and define compromises between melodic, harmonic and musical priorities in the tuning.

Tooner, this is not very short, I must apologize.

...“The basis of the 12ths root of two is the idea that this will produce a temperament with all intervals being the same width (have the same frequency ratio),”...

Maybe you meant “semitones being the same width”. Yes. Like for any geometric progression’s term, the semitones do have the same incremental ratio, but if we were to make a staircase with the scale’s values, each frequency value could give us the step’s depth, or lenth, and each step would proportionally differ from the next one.

...“and at the same time octaves that are beatless.”...

Yes, and this is one of The Problems. A theoretical beatless octave is a wrong assumption, although 300 years ago it could be in line with the common approach to temperament theories.

You say:...“This is only possible with harmonic tones.”...

Not true. You say that also with iH tones, on single partial matchings, we may go for beatless octaves. In any case, this is only temporarly and apparently true, any beatless interval will end up beating in a beat-flow. This is not to be understood, this is to be acknowledged or, once you acknowledge it you may understand.

...“Keeping the discussion to harmonic tones (for now), the 12th root of any number will produce a temperament with all intervals being the same width. However, unless the number is two, the octaves will beat.”...

Not correct. Like any interval, octaves will beat anyway, since 12 root of two is only an abstract case. Also “purely harmonic tones” is abstract thinking, the "pure" attraction again, it is forcing a zero-iH concept into a model.

...“Now to have 12ths beat narrowly and at the same time 15ths beat widely the number has to be larger than 2 but smaller than ((3^(1/19))^12 or 2.0014269…”...

Correct.

...“The compromise that is given results in equal beating 12ths and 15ths when these intervals have a common note on the bottom.”...

If you approach the scale in terms of mirror-like order, you will not need to discriminate between top and bottom anymore.

...“But it is not clear why this compromise is necessary at all,”...

I wrote about this in my previous post. This compromise is necessary in that all intervals, with their stretch, can now contribute to the tonicity of the tuning form.

...“let alone why a superior compromise results with this sort of equal beating.”...

Opposite equal beating 12ths and 15ths results in a superior compromise for three reasons: firstly because it involves all intervals, wich are now beating intervals; secondly because the set gains stability by opposing a constant counter-beat, so all intervals compromise then for determining a perfectly stable, counter-balanced beating-whole; thirdly because the 15th encloses two octaves, what is needed to gain and ensure the intermodular quality. So, from one zero-beating octave block we progress to a two octaves beating matrix.

...“Not to mention how the roots of any other numbers are needed to calculate this compromise. In fact, they are not needed nor actually used although it could seem that way.”...

Please argue this last statement and be aware that you are getting into maths details, so before I answer please confirm you will not regret.

...“The argument could easily be made that the common note should be on the top,”...

No need. Anyway, show me please how you’d build a house starting from the roof, then I’ll follow you.

...“or the 15th should beat faster than the 12ths that has a common note on the bottom, but slower than the 12ths than has a common note on the top.”...

Ok, we both may be keen on break-dance, but this is not the place.

...“Another argument could be made that if anything should be equal beating, it should be the single octaves beating the same as the 5ths. But then the question again arises as to why, which note should be common, or should they actually beat equally?...

You try then: tune EB 5ths and octaves and then tell me how you like it. If you really like it, you can still refer to Chas algorithm:

((3/2) – Δ)^(1/7) = (2 + (Δ*s))^(1/12)

s = 1

Δ = 0.001178134272…

Scale ratio = 1.05951508823057…

...“Things get difficult when trying to use the 12th root of any number to describe the tuning of inharmonic tones.”...

Thinks get difficult only if or when you expect to find the theoretical frequencies values on iH tones. As for describing, Chas model is derived from a precise beats order and therefore can perfectly describe our actual tuning.

...“The tuning can be adjusted so that these intervals (12ths and 15ths) beat equally in any or all parts of the piano, or unequally in any or all parts of the piano. This is the true value of these intervals. They are a tool that the tuner can use to make compromises that are more important than arbitrarily equal beating intervals.”...

I hope you can better understand now the value of EB-ET and why it results in a superior compromise.

And do not worry, there will always be room for melodic, harmonic and musical priorities. Instead of calling it compromise, we'll call it knowledge.

T & R, a.c.

First recording (.rar) of CHAS tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 (Granpianoman)
http://www.box.net/shared/od0d7506cv