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I love all words. I love the word archaic in all it's connotations. How do you get these ideas on the likes and dislikes of another?
Again, you make inferences and then ask questions based on your inference or interpretation. What you are doing is questioning your own inferences. Those are questions nobody can answer. Only you can.
Your after the fact explanation of how you used archaic is not convincing because you coupled it with the words "and unreliable".
"Archaic" coupled with the words "and unreliable"? You rephrased your question but still coupled these two words.
I write in long and hopefully coherent thoughts. You take one sentence out of context and question your inference from it. Since when have tenths and sevemteenths, implying ther connection to the third, of course, automatically form a 2-1 octave? Yet that is what you infer. Are you not denying your own knowledge and intelligence in order to make the is outrageous assumption?
I just got thru saying that there are many options yet you attempt to pin me down to just one while you attempt to promote your one way. This is classic projection.
My advocacy of a complete and thorough familiarity with all the interactions within a chosen temperament range you challenge as somehow not helping students. Would you teach chess without advocacy of mastering all the interconnected moves?
Every statement you make is of this nature. My words are taken with no thought to earlier contexts. Is it your memory, your comprehension or, maybe your agenda is overriding your intelligence. Why else would you attempt to revive your own dead or dormant thread in this deceitful and devious way if it weren't to keep your pet project alive when all others have long ago lost interest and abandoned it?
My posts from other threads have been used many times now in this abusive way to resurrect your capital threads from twelve months of obscurity . I notice nobody else has responded to your attempts at resurrection. Only I feel the need to correct your glaring misapprehensions. Only because it is exclusively my posts that you are taking, literally, out of context. To put it plainly, taking my post from another thread (context) and using them in each case, to bring back "life" your own agenda. By the notable absence of other posters here, nobody is interested. Least of all me. Let these posts die peacefully. As of right now.
I have been very patient with you, even overlooked your abusive and puerile name calling.
If this happens again, I shall refer the matter directly to the moderators for some sort of action. It is patently an abuse of the forums. Not only what you do but the way in which you repeatedly do it.
Maybe I should be flattered by the attention but I find it quite creepy.
I am grateful for the experience in dealing with cybercreepies on line in preparation for if I ever meet them in real life. So far I've managed to avoid them.
If other aspects of your life include stalking like this, it certainly adds credence to the option on these forums to be and remain anonymous.
Now. Either present your own fresh approach to keep your threads alive or let them die peacefully before I lose my patience.
Amanda Reckonwith Concert & Recording tuner-tech, London, England. "in theory, practice and theory are the same thing. In practice, they're not." - Lawrence P. 'Yogi' Berra.
Back to what I would like to investigate, about which I wrote three days ago.
Originally Posted by alfredo capurso
Hi All,
For some time I have been thinking of asking for your help in order to make 'objective' something I can observe. Considering a string and the tone it produces, I observe that partials can be spaced, and intervals can be better related to each other, depending on the tuning-hammer technique.
In other words, what I believe is that the original scaling of a piano can be somehow re-adjusted, meaning that there is a leeway, some 'room for manoeuvre' (is this idiom correct?) we can use, in a way re-ruling the relations amongst partial sounds of different strings.
What makes the difference would be how I/we get to the 'spot', whether we get there from a lower or a higher, or a much higher pitch.
No doubt, other colleagues may experience this. Actually, if you know of any research on this, or available data, please let me know.
I do not have the necessary equipment. Would any of you (pro or non-pro) like to get involved in this experimentation?
What might be needed is:
- some tuning-hammer skill, enough to be able to get to the spot; - an ETD that indicates the spot; - a reliable device that can record and analyze individual partial frequencies.
Comments, questions, suggestions and corrections are welcome.
Regards, a.c. .
I got some help and hope to be able to share some progress.
Tune A5 to A4 coming from above and stopping when the octave first sounds beatless. Then measure the pitch A at A5. Tune A5 to A4 coming from below and stopping when the octave first sounds beatless. Then measure the pitch B at A5. What is the difference between pitch A and pitch B?
Tune A5 to A4 coming from above and stopping when the octave first sounds beatless. Then measure the pitch A at A5. Tune A5 to A4 coming from below and stopping when the octave first sounds beatless. Then measure the pitch B at A5. What is the difference between pitch A and pitch B?
Yes, Chris, that is one type of test we could do, although we might be exposed to a variable, i.e. "when the octave first sounds beatless". That is why I was mentioning a non-aural device.
In the above case, we would then compare two pitches, pitch A and pitch B.
A second test might focus onto the partials, measuring how the individual tone's partials end up being spaced.
In this second case, together with A4(-A5), we may consider one more 'bottom pitch', say F4(-A5), or a note further down for a wider interval, like D4 or F3, and perhaps test the 'above/below' combinations between three pitchs and their partials' configuration?
Please, consider that I am in the "I wonder.." zone.
Yes, I would like you to try putting it into words. I'm not sure I can change habits that have been developed over 40 years (and that seem to work reasonably well) but it is always good to expand one's base of understanding.
I am posting my reply here in order to keep some information together.
We were talking about "pin setting", Peter. Perhaps you have already found some info in this thread or elsewhere, in any case I will expand further and.. keep my word.
My experience started on very very old pianos, they were ready for refurbishing and then I could only deal with very loose pins. There I needed to rely on some forces that could be left onto the pin, and the feeling that the foot of the pin was in the ideal position.
The principle is different then, you do not move the pin to find the correct pitch, you actually pull the string to "set the pin". Reading/feeling the occurring forces while you are pulling the string, allows you to set the pin and, as a result, get the pitch you wanted.
The execution is fairly straightforward, in the best case - nice pin-pin-block-string rendering - it is one move only CW, one move CCW.
You can start pulling the pin where you find it, or loose the pin CCW and zero old tensions. Start pulling the string CW, feel the torsion and the bending taking place while you figure out where the spot is. That is all the information you need and you can now over-pull the string CW.
The latter move refers to a precise amount of forces that you will have read/felt - pin bending, torsion and rotation + string rendering - and that now you want to invert.
We may say that the above sequence is "tuning the pin", what comes next is "setting the pin-string system".
You go CCW, perhaps putting the hammer in a different angle, do not rush and consider that matter adjusts in time, release all the CW forces and "charge" or load the pin with CCW forces, enough to re-find the spot you heard before, making sure that the pin and the string are now pulling each other in opposite direction, making a tight-united system, where the pin is more willing to pull the string - check that a tiny CW force would raise the pitch - than ready to give.
As mentioned, make sure that about 3/10 points of force CW would sharpen the pitch, against about 7/10 points of CCW forces for lowering the pitch.
There must be a misunderstanding somewhere. I am not addressing basic notions, whole tone octave purity, or iH as it is understood. I was trying to reason on beat perception in order to understand whether Bernhard, while mentioning pure 12ths, is actually achieving close to pure 12ths.
This interest of mine comes from a recent conversation I had with Kent where, mentioning pure 12ths, he would suggest to call them "clean" 12ths.
And because Bernhard has mentioned "minimum_overall beating", I wonder how that beating sounds to him, if wide or narrow.
Perhaps some differences in our practice become significant, for instance tuning mid-strings first on a wide range, which allows the tuning of perfectly still/beat-less 12ths (as demonstrated recently at a convention in Canada) or unisons as you go, but beyond that, I wonder what is left there of an original (edit: theoretical) pure-interval strech scheme.
Apologies for some background noise. .
Originally Posted by alfredo capurso
for instance tuning mid-strings first on a wide range, which allows the tuning of perfectly still/beat-less 12ths (as demonstrated recently at a convention in Canada)
Dear Alfredo,
this is the goal with pure twelfths.
Originally Posted by alfredo capurso
And because Bernhard has mentioned "minimum_overall beating", I wonder how that beating sounds to him, if wide or narrow.
Minimal overall beating means clean to me. I use that description (minimum overall beating) to make clear that if speaking of a pure or clear interval on a piano, there are generally all but at LEAST one partial pair that have not the same frequency, but are tuned to a target, where they sound perfectly clean, still, beat-less, same as you possibly demonstrated successfully in Canada.
Originally Posted by alfredo capurso
Perhaps some differences in our practice become significant, for instance tuning mid-strings first on a wide range, which allows the tuning of perfectly still/beat-less 12ths (as demonstrated recently at a convention in Canada) or unisons as you go, but beyond that, I wonder what is left there of an original (edit: theoretical) pure-interval strech scheme.
I see a serious gap between what you do in practice (essentially a pure twelfths temperament) and what you have theoretically build around about that as Chas theory:
You tune pure twelfths on the middle string over a vast part of the piano (pure twelfths equal temperament).
On a second step you tune unisons and claim by pitch sagging or coupling of the tree strings the pitch drops exactly for the specific amount that after tuning the unisons, the resulting twelfths are exactly of Chas twelfths size.
This contradicts my understanding of the physical relativity principle.
- Think about the lower note when tuning the unison, it drops too, right? If the dropping is the same, is the resulting twelfth logically not of the same size as before? Are you aware that pitch dropping from coupling does not occur on every note? How can you expect then that the resulting twelfth will land exactly on the Chas size twelfth, with your own claimed precision of 0.01 cts over an octave, that is required to be classified as a Chas interval?
- Let´s assume ideally that exactly the required pitch drop occurs from whatever effect, when starting with unison tuning : What will happen when tuning the unisons downward from the temperament region where the unisons are already tuned (and thus do not drop anymore): must not the lower note stay exactly where it was to obtain a Chas size twelfth? Or let´s assume the upper note dropped only half the amount required to obtain a Chas size twelfth: Must not climb the lower note toward the upper note then?
- Let´s proceed with finishing unisons upwards from the temperament region in the treble on the first twelfth above the temperament twelfth (i.e. D3-A4), AFTER completing the unisons over the temperament twelfth. If the pitch dropped over all notes by unison tuning in the temperament twelfth, all twelfths of the middle string in the second twelfth section above the temperament are wider of pure of an amount of the pitch drop of the notes of the temperament twelfth. Must not the notes drop then two times the pitch drop of the temperament twelfth to obtain a Chas size twelfth? And must not the notes in the twelfth above the second twelfth above the temperament twelfth drop three times the amount of the drop in the middle region?
Frankly, to me your concept of obtaining a Chas tuning by tuning unisons from pure twelfths temperament on the middle string is wishful thinking. What you get after tuning unisons after a pure twelfth temperament on the middle strings instead is more or less a pure twelfths temperament (which as a result sound wise is not wrong at all .
If you want a Chas tuning, you may preferrably do something Bill Bremmer does with mindless octaves, what would probably represent more a Chas tuning than your actual approach.
Hi,
A few considerations based on Bernhard’s recent comments. Hope this can clarify some points, also for other colleagues, before it gets… confusing. I have moved this reply from Toni's thread for obvious reasons. The original thread is here:
I wrote…â€Perhaps some differences in our practice become significant, for instance tuning mid-strings first on a wide range, which allows the tuning of perfectly still/beat-less 12thsâ€â€¦
Bernhard, you replied: “..this is the goal with pure twelfths.†…
Now, the goal for you, correct me if I am wrong, seems to be “minimum overall beating†12ths all across the keyboard. My own goal, Bernhard, is to tune a beat-curve that may anticipate any pitch drop. We may have to tune pure 12ths or pure 5ths at some points, that - for me - depends on the individual piano.
I believe that our experience takes us onto different observations. Depending on the piano, I may decide to tune pure or close_to_pure 12ths in some ranges and compensate for a possible pitch sagging. Considering that, every time I need to tune pure 12ths on middle strings, the end result is what I like, i.e. 12ths with a minimum narrow beating. Later on you will understand how.
I wrote: ..†And because Bernhard has mentioned "minimum_overall beating", I wonder how that beating sounds to him, if wide or narrow.â€
You replied: ..†Minimal overall beating means clean to me.â€â€¦
Also to me, though “clean†simply does not mean “pure†(to me). And it is not clear whether you hear narrow-beating or wide-beating. Perhaps for you that does not really matter?
… â€I use that description (minimum overall beating) to make clear that if speaking of a pure or clear interval on a piano, there are generally all but at LEAST one partial pair that have not the same frequency, but are tuned to a target, where they sound perfectly clean, still, beat-less, same as you possibly demonstrated successfully in Canada.â€â€¦
For me “minimum overall beating†cannot be represented by a “pure†stretch scheme. Strange how you do not get this contradiction in terms, and when we say "clean" it sounds so vague, to the point that an "overall beating" might sound "clean" subjectively, without being objectively "minimum".
For me there is a difference that can be heard distinctly on centre strings, pure is pure, i.e. still, perfectly beat-less, something I demonstrate on centre strings; “clean†meaning “minimum overall beatingâ€, think of it, could also apply to the pure 5ths stretch scheme, in fact aren’t “clean†5ths nice as well? Aren't we all targeting "clean" octaves, clean 5ths and clean 12ths? Can they all be " pure" at once?
Hence my question, if there is “..at LEAST one partial pair that have not the same frequency..â€, why talking about pure intervals? Wasn’t it understood that any “pure†interval would go to the detriment of all the other intervals?
In my practice, very luckily 12ths will be as I like them after unisons - and other intervals will not sound sour (hopefully) – “clean†12ths, sure, on the narrow side, and clean 5ths too. If I demonstrate (and mention) pure 12ths on middle strings in a specific range, it is in order to show how we can use that check and have control over the tuning curve and settlings.
You wrote: … â€I see a serious gap between what you do in practice (essentially a pure twelfths temperament) and what you have theoretically build around about that as Chas theory:.â€...
Let’s see.
…†You tune pure twelfths on the middle string over a vast part of the piano (pure twelfths equal temperament).†…
The part of the piano is not really vast, and the tuning curve will be a bit steeper of some degree for all intervals, not only 12ths. In proximity of / in the range C5-C6 we may need to tune 12ths very “clean†or even pure, as I say, depending on the piano.
… “On a second step you tune unisons and claim by pitch sagging or coupling of the tree strings the pitch drops exactly for the specific amount that after tuning the unisons, the resulting twelfths are exactly of Chas twelfths size.†…
The end_tuning_beats for all intervals need to be anticipated, and of course this requires some experience. But it is handy: while we do unisons we have plenty of intervals that we can check and that actually tell us how the individual piano is reacting, as we go. Then we can still adjust the tuning curve and make sure that our target is always in the viewfinder.
What is weird for me, in these days, is the idea of tuning “clean†12ths with a spanner, for instance, and expect that - at the end of the tuning - all the intervals progress harmoniously. I would see no point in gaining one single “pure†interval out of “brute forceâ€, octaves, 12ths or whatever, for how important it is to interlace all intervals, as in a whole. You can surely tune with a spanner but the progressions, in my experience, will all be broken.
You mention “whole tone purityâ€, you go for “clean†12ths but you support a “pure†scheme. Isn’t this a bit odd?
… “Frankly, to me your concept of obtaining a Chas tuning by tuning unisons from pure twelfths temperament on the middle string is wishful thinking.†…
Yes, it would be, I agree, if things were as you describe them, but they are not. I thank you for being frank, this is how we can progress faster.
12ths can well serve as a reference when we expand the temperament, as much as many other intervals, I hope they will never get "unapproachable"
… “What you get after tuning unisons after a pure twelfth temperament on the middle strings instead is more or less a pure twelfths temperament (which as a result sound wise is not wrong at all .†…
Please note, I speak about 12ths that are really pure, when and where it is the case. In my experience, what we get at the end of our tuning does not depend on one single interval (how strange that I have to say this here), it will depend on how carefully we temper the first octave, on how carefully we expand the temperament, and on how carefully we check the “temporary†results, while tempering, while expanding and while we execute the unisons.
On Chas theory, as soon as I have some time.
Bernhard (and All) please do not hesitate to let me know if you would like me to expand further.
If you are referring to the so called wenreich effect, this can so easily be dealt with as, when and if it appears by simply tuning from completed unisons. 3-10-17 as you go becomes second nature, particularly when, as all pro tuners have to occasionally, you are in a noisy environment . You will soon spot anything and everything that is the least suspect usually in the tenth as you are comparing the 3rd and the 17th.
Not "clever" enough for you?
Twelfths will either look after themselves if the fifths are secure or, since you tell us that you, personally, can't tell whether they are wide or narrow, (you could if you used the external checks) what's this whole thing about if you aren't at some level projecting your own problems onto others?
Of course you do know that worrying a string will heat up the bearing points and also destabilise it.
Amanda Reckonwith Concert & Recording tuner-tech, London, England. "in theory, practice and theory are the same thing. In practice, they're not." - Lawrence P. 'Yogi' Berra.
You mention “whole tone purityâ€, you go for “clean†12ths but you support a “pure†scheme. Isn’t this a bit odd?
I find it odd to find this odd.
Originally Posted by DoelKees
Originally Posted by rXd
I had thought when this stuff first started that it was harmless crackpot stuff ..
In my opinion C.HA.S.^(R) theory is total nonsense, based on a lack of understanding of basic piano physics, in particular inharmonicity.
Kees
Not only inharmonicity. There seems to be a lack of understanding of the basics of interval beating in different ET sizes as well. ET´s without iH (s=0 as Alfredo says) progress with all intervals perfectly, no matter what ET-size is choosen (except if a pure interval was selected as base interval where this interval would be beatless throughout the scale.) I figured out already in a review of Alfredo´s paper when he came up with his theory some years ago here, that he did not understand this basic principle, as he claimed that only in Chas ET all intervals do progress. Seems he refused to understand these basics within the time that has gone since.
Last edited by Bernhard Stopper; 09/19/1708:48 AM.
If you are referring to the so called wenreich effect, this can so easily be dealt with as, when and if it appears by simply tuning from completed unisons. 3-10-17 as you go becomes second nature, particularly when, as all pro tuners have to occasionally, you are in a noisy environment . You will soon spot anything and everything that is the least suspect usually in the tenth as you are comparing the 3rd and the 17th.
Not "clever" enough for you?
Twelfths will either look after themselves if the fifths are secure or, since you tell us that you, personally, can't tell whether they are wide or narrow, (you could if you knew the external checks) what's this whole thing about if you aren't at some level projecting your own problems onto others?
Of course you do know that worrying a string will heat up the bearing points and also destabilise it.
Rxd, I do not have any problem with 12ths, accuracy, interval checks, resonance or else. Thanks for checking.
..."..since you tell us that you, personally, can't tell whether they are wide or narrow..."...
You misunderstood. In Toni's thread I was actually asking Bernhard if he could tell us anything about the "minimum overall beating" he was talking about, whether it would be heard as narrow beating or wide. We got no reply on that, he just said that there will be ..."..at LEAST one partial pair that have not the same frequency..".
Originally Posted by Bernhard Stopper
Originally Posted by alfredo capurso
You mention “whole tone purityâ€, you go for “clean†12ths but you support a “pure†scheme. Isn’t this a bit odd?
I find it odd to find this odd.
Originally Posted by DoelKees
Originally Posted by rXd
I had thought when this stuff first started that it was harmless crackpot stuff ..
In my opinion C.HA.S.^(R) theory is total nonsense, based on a lack of understanding of basic piano physics, in particular inharmonicity.
Kees
Not only inharmonicity. There seems to be a lack of understanding of the basics of interval beating in different ET sizes as well. ET´s without iH (s=0 as Alfredo says) progress with all intervals perfectly, no matter what ET-size is choosen. I figured out already in a review of Alfredo´s paper when he came up with his theory some years ago here, that he did not understand this basic principle, as he claimed that only in Chas ET all intervals do progress. Seems he refused to understand these basics within the time that has gone since.
I do not think I have ever stated that "..only in Chas ET all intervals do progress".
Oh, Bernhard, the basic Chas algorithm version is with s = 1
Time has passed, this is true, apparently you need some more.
Arguments, gentlemen. Otherwise, you could start your own thread and denigrate as much as you like.
Regards, a.c.
G.R.I.M. - Università di Palermo (2009) http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
Oh, Bernhard, the basic Chas algorithm version is with s = 1
typo, sorry.
Originally Posted by alfredo capurso
I do not think I have ever stated that "..only in Chas ET all intervals do progress".
Ok i see a need to precise this. You stated that in Chas ET intervals progress. In your tractat you presented some other ET then, where fifths or other intervals invert at some point. This is factually wrong with no iH or s=1
My guess is, that this misunderstanding (that intervals do invert in some ETs which are not Chas) leads you the interpretation, that after tuning pure twelfths on the middle string, they get progressively smaller after tuning the unisons.
Last edited by Bernhard Stopper; 09/19/1710:31 AM.
Oh, Bernhard, the basic Chas algorithm version is with s = 1
typo, sorry.
Originally Posted by alfredo capurso
I do not think I have ever stated that "..only in Chas ET all intervals do progress".
Ok i see a need to precise this. You stated that in Chas ET intervals progress. In your tractat you presented some other ET then, where fifths or other intervals invert at some point. This is factually wrong with no iH or s=1
My guess is, that this misunderstanding (that intervals do invert in some ETs which are not Chas) leads you the interpretation, that after tuning pure twelfths on the middle string, they get progressively smaller after tuning the unisons.
No other ET was discussed, if not marginally, in the Chas paper, I would then say that your guess is wrong.
G.R.I.M. - Università di Palermo (2009) http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
No other ET were discussed, if not marginally, in the Chas paper, I would then say that your guess is wrong.
I quote from your initial paper page 76 (http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf ):
"f in a semitonal logarithmic scale we wanted to fa vour partial 5, we would have to take value 5 and position 12+12+4 = ordinal 28, so the f ormula will be 5^(1/28) = 1.059164008 ...In this scale, as incremental ratio of degree 9 of the scale (element 14), we find the 5^(1/2) component of the gold section. In distances of octaves, (5*2)^(1/40), (10*2)^(1/52) etc. this ratio modifies towards 2^(1/12).
If in a different logarithmic scale, we wanted to f avour partial 3 we would have to take value 3 and position 12+7 = ordinal 19, so the form ula will be 3^(1/19) = 1.059526065 ... This ratio, too, in distances of octaves, (3*2)^( 1/31), (6*2)^(1/43) etc, modifies towards 2^(1/12). The formula 2^(1/12), at distances of octaves (posi tion+12) does not change: 4^(1/24) = 8^(1/36) = 16^(1/48) = 1.059463094.."
You mentioned pure twelfth ET explicitely, and claimed that this ratio modifies towards 2^(12) which is pure nonsense. The correct octave in pure twelfth is not (3*2)^( 1/31), (6*2)^(1/43) etc, but correctly 3^(1/19), 3^(12/19), 3^(24/19) etc.
My guess seems indeed right, just from your misinterpretation of pure twelfth temperament beat rates you falsely came to the conclusion that pure twelfths converges to smaller twelfths when finishing the unisons.
Frankly, I do not understand what is wrong there. But even if something were wrong, it would have nothing to do with my observations from tuning practice.
In fact, I consider theory and practice separately as much as I can, you may have well noticed that.
Your guess on unisons, again, is banal and simply wrong.
G.R.I.M. - Università di Palermo (2009) http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
Frankly, I do not understand what is wrong there. But even if something were wrong, it would have nothing to do with my observations from tuning practice.
If you don´t understand what is wrong there, i can´t help you further. There are serious scientists who noticed what your theory is (many thanks to professor Kees van den Doel).
I see, you were addressing the whole theory, not unisons... lol. . G.R.I.M. - Università di Palermo (2009) http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
The self-published article "HARMONIOUS PROPORTIONS IN A PIANOFORTE - THE C.HA.S.® TEMPERAMENT" with author listed as "Professor Nicola Chiriano", who is actually a high-school teacher if you Google him, starts off with this first sentence:
"In 1691 the German organist Andreas Werckmeister discovered an ingenious way of tuning instruments, the closest ever achieved to an equal temperament [1], that is to say, to a tone system where the distance between semitones (two successive notes in the chromatic scale) is constant."
which is nonsense. He then goes on to describe Werckmeister, but gets the number of narrow fifths (4, not 5) wrong, as well as the amount by which they are narrow (1/4 of a Pythagorean comma, not 1/4' of a syntonic comma).
After that it is stated that "Werckmeister’s scale was extremely successful because of J. S. Bach’s use of it in his “Well-tempered Clavichordâ€" which is also nonsense.
No other ET were discussed, if not marginally, in the Chas paper, I would then say that your guess is wrong.
I quote from your initial paper page 76 (http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf ):
Originally Posted by alfredo capurso
"f in a semitonal logarithmic scale we wanted to fa vour partial 5, we would have to take value 5 and position 12+12+4 = ordinal 28, so the f ormula will be 5^(1/28) = 1.059164008 ...In this scale, as incremental ratio of degree 9 of the scale (element 14), we find the 5^(1/2) component of the gold section. In distances of octaves, (5*2)^(1/40), (10*2)^(1/52) etc. this ratio modifies towards 2^(1/12).
If in a different logarithmic scale, we wanted to f avour partial 3 we would have to take value 3 and position 12+7 = ordinal 19, so the form ula will be 3^(1/19) = 1.059526065 ... This ratio, too, in distances of octaves, (3*2)^( 1/31), (6*2)^(1/43) etc, modifies towards 2^(1/12). The formula 2^(1/12), at distances of octaves (posi tion+12) does not change: 4^(1/24) = 8^(1/36) = 16^(1/48) = 1.059463094.."
You mentioned pure twelfth ET explicitely, and claimed that this ratio modifies towards 2^(12) which is pure nonsense. The correct octave in pure twelfth is not (3*2)^( 1/31), (6*2)^(1/43) etc, but correctly 3^(1/19), 3^(12/19), 3^(24/19) etc.
My guess seems indeed right, just from your misinterpretation of pure twelfth temperament beat rates you falsely came to the conclusion that pure twelfths converges to smaller twelfths when finishing the unisons.
I have to correct a typo in my statement:
Originally Posted by Bernhard Stopper
You mentioned pure twelfth ET explicitely, and claimed that this ratio modifies towards 2^(12) which is pure nonsense. The correct octave in pure twelfth is not (3*2)^( 1/31), (6*2)^(1/43) etc, but correctly 3^(1/19), 3^(12/19), 3^(24/19) etc.
should be: "You mentioned pure twelfth ET explicitely, and claimed that this ratio modifies towards 2^(12) which is pure nonsense. The correct octave in pure twelfth is not (3*2)^(1/31), (6*2)^(1/43) etc, but correctly 3^(12/19), 3^(24/19), 3^(36/19), etc."
So to sum up again: My guess is that exactly this wrong model about pure twelfth ET that you have in mind is what leads you to the wrong guess that a pure twelfth ET over the whole piano can shrink to a Chas ET just by tuning unisons, which is not possible (i second also what Amanda had to say about pitch sagging) if there is no pitch raise, as every twelfth would need to shrink by 1,23 cents (the difference between a pure twelfth ET twelfth and a Chas twelfth), this required shrink would even increase linearly with every stacked twelfth, as i was figuring out with my questions earlier. The truth for me is, that in your tuning practice you are selling something else (a pure 12th ET temperament) under a wrong label (Chas theory).
Last edited by Bernhard Stopper; 09/20/1703:22 AM.
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