Welcome to the Piano World Piano Forums Over 2 million posts about pianos, digital pianos, and all types of keyboard instruments
Join the World's Largest Community of Piano Lovers
(it's free)
It's Fun to Play the Piano ... Please Pass It On!

non-irrational pure interval consonances. I have not completed a mathematical proof out to infinity, but these relationships are well known in analytic number theory.

This reminds me somehow a DVD I have of a show on the Riemann hypothesis which was called "music of the primes" I think.

Looking forward to what you and Alfredo come up with next in this thread.

non-irrational pure interval consonances. I have not completed a mathematical proof out to infinity, but these relationships are well known in analytic number theory.

This reminds me somehow a DVD I have of a show on the Riemann hypothesis which was called "music of the primes" I think.

Looking forward to what you and Alfredo come up with next in this thread.

Kees

I've heard Terrance Tao (a Field's medalist) describe the distribution of primes in this manner, in reference to the complex-analytic proof of the prime number theorem. (It states that the number of primes less than or equal to x is roughly x / log x.) The idea is to define a function that is "noisy" on prime numbers only, that is, it spikes up when it touches a prime (or prime power) and is zero otherwise. Then, you smooth it out with a signal transform (akin to things like Fourier transform in signal processing), and then you can apply tools from analysis to determine its behaviour, which eventually leads to the x / log x estimate.

Originally Posted By: UnrightTooner

I am enjoying this discussion, too. Just remember that when inharmonicity is a factor, things change.

Definitely. I think that the program is really to explain where the theory of ET comes from, so that we have some justification for why some version of it is sought after in practise. (Unfortunately, what we want in theory isn't always what we get in real life!)

Tunewerk wrote:…"It's difficult to talk about abstract concepts on this message board where one cannot interject to clarify.."…

You are right, and it is very kind of you making an effort.

Originally Posted By: Alfredo There is no way we can sing three (or more) notes simultaneously, and obtain pure (perfectly consonant) intervals only. In other words, when more than two notes are played simultaneously, we end up having to face the same (fixed scale) problem: how do we (referring to any ensemble) make complex chords sound "in tune"?

…"This is possible. It just requires movable, unequal divisions in-between."…

I'll use your way, Tunewerk, and give a basic example in order to explain briefly what I was referring to:

Tonic voice, at 1;

I'm asked to sing the P5th, at F1*(3/2);

You are asked to sing a M3rd two octaves above the tonic, that is a M17th.

Using pure (consonance(?)) ratios, you can calculate different values for your M17th, for example:

F1*((3/2)^2)*((4/3)^2)*(5/4) = 5 or F1*(3/2)^4 = 5.0625 or F1*(5/4)^7 = 4.7684… or F1*(3/2)*(4/3)*(5/4)^4 = 4.8828…

If you calculated the P5th backwards, from your M17th = 5 you would not get 3/2. How would you move unequal division in-between? Which is the consonant M17th?

..."...the program is really to explain where the theory of ET comes from, so that we have some justification for why some version of it is sought after in practise."...

Good thing, thank you Coolkid70.

Regards, a.c.

P.S.: For those who are familiar: "THE RULE BEHIND THE OCCURRENCE OF PRIME NUMBERS" Matteo Arpe - Claudia La Chioma

Well, it's true, in these cases you sacrifice one major harmony for another. In the examples you gave, using harmonies built on 5, 4, 7 and 6 voices respectively, the last 3 differ from the first, in that you use sections of a curve instead of the ratios as points, with complementary parts* (which usually occur in natural chords).

If a choir did sing these strings of 5ths 4ths or 3rds, they'd have to temper their intervals to achieve a 5/1. However, the difference between that and the piano, is all harmonies would be built on rational interference between waves.. meaning at least one wave pair in ratio, from each dyad, would constructively interfere.

You may have [3/2]^4 = [81/16], but this is still a consonance, just a thinner one. At some point the ear decides the constructive alignment is too thin and it becomes a dissonance.

Without being confined to the irrational frequencies of the 12-tone [2/1] curve, there are no fixed compromises based on one curve, and every frequency can adjust and build to a new chord based on the best possible divisions. This is what I meant by movable and unequal.

That is not to say there would not be compromised choices, but all curves would be available to micro adjust by, and all compromises would result in some kind of rational interference, which by definition is always consonant.

PS - Alfredo, thank you for the primes paper. I'm going to take a look. I wasn't aware that the distribution pattern for the primes had been solved.

* Complementary intervals, as I call them, are a really interesting concept. Many intervals we have come to regard in music arose from subdividing larger intervals. For example, the m3rd [6/5] is the source of the M6th [5/3], from the point division of the octave [2/1]. This is literally the source of that interval. This brings to mind that any integer ratio [n/n] is consonant. The question is only the degree, which regards psychoacoustics instead.

The ratio [n+1]/n is the smallest way to define consonant intervals, but it isn't the only source. As a result of interval splitting, most chord consonance was built around these structures. Perfect consonance could happen through this. This is what I was alluding to through the concept of geometric structure. In most, if not all common chord combinations, compromises are unnecessary because of this unequal but symmetric division of tone.

I would argue that it was the mechanization of music on keyboard instruments that resulted thinking in terms of compromises bound to curves as opposed to naturally evolving and changing tone structures, consonant within themselves.

Edited by Tunewerk (03/23/1201:04 AM) Edit Reason: Footnote

P.S.: For those who are familiar: "THE RULE BEHIND THE OCCURRENCE OF PRIME NUMBERS" Matteo Arpe - Claudia La Chioma

pdf available in the web.

I skimmed through this paper, and I actually failed to see what the point was. It seems like the authors are just restating the Sieve of Eratosthenes using some modern notation. I didn't see anything obvious about the distribution of prime numbers. Could you maybe point out what I need to be looking for?

You'll be able to read about the authors point in section 1, page 2 and 3.

Hi Tunewerk,

Thanks for your attentive reply. On the first half:

..."Well, it's true, in these cases you sacrifice one major harmony for another. In the examples you gave, using harmonies built on 5, 4, 7 and 6 voices respectively, the last 3 differ from the first, in that you use sections of a curve instead of the ratios as points, with complementary parts* (which usually occur in natural chords). If a choir did sing these strings of 5ths 4ths or 3rds, they'd have to temper their intervals to achieve a 5/1."...

Ok, that's the point I wanted to "fix": in general, any ensemble (choir included) in order to sound harmonious must temper their intervals. In other words, the necessity to temper intervals derives from having to combine prime-numbers ratios, 2/1, 3/2, 5/4, etc (that's the centuries-old problem). Now we may have to reason on how all intervals can (should) be tempered within a steady, coherent geometry.

..."However, the difference between that and the piano, is all harmonies would be built on rational interference between waves.. meaning at least one wave pair in ratio, from each dyad, would constructively interfere."...

Perhaps I do not understand: Would they choose to play "at least" one pure ratio? You say "one wave pair in ratio, from each dyad, would constructively interfere", meaning "interfere" what on?

..."Without being confined to the irrational frequencies of the 12-tone [2/1] curve, there are no fixed compromises based on one curve, and every frequency can adjust and build to a new chord based on the best possible divisions. This is what I meant by movable and unequal. That is not to say there would not be compromised choices, but all curves would be available to micro adjust by,..."...

Indeed, what we understand from literature is that (1) non-fixed instruments have a higher degree of freedom in that they can always "adjust"; and we are told that, as a consequence, (2) non-fixed instruments can achieve a higher degree of consonance, and that (3) only fixed-tones instruments need to "compromise".

Point (1): Once we "fix" the reference, i.e. the "common" pitch (A4 or any other note), what would they "micro adjust" on? Would they favor one interval? Which one? Which "unequal" variant, amongst many, will represent THE "ad-just-ed" chord geometry?

Point (2): Do they sing/play along chord-by-new-chord consonance? Say yes, which is the "new chord" best possible division? (edit: would that be the next "new" compromise?)

Point (3): We have seen that also non-fixed ensemble need to compromise, we can see why. You say: ..."all compromises would result in some kind of rational interference, which by definition is always consonant."...

Please, could you give an example of one "kind of rational interference", so I'll be able to understand what you mean?

...* Complementary intervals, as I call them, are a really interesting concept."...

Very glad to, Alfredo. I'm interested in thinking through these things.

Originally Posted By: Alfredo

Ok, that's the point I wanted to "fix": in general, any ensemble (choir included) in order to sound harmonious must temper their intervals. In other words, the necessity to temper intervals derives from having to combine prime-numbers ratios, 2/1, 3/2, 5/4, etc (that's the centuries-old problem).

Well, where this is true in a static example.. can you name one piece in choir music, where a choir sings stacked M3rds, 4ths or 5ths to that degree? Even if it is the case, then they do not also try to sing the octave or M17th.. they already have a rational harmony for that chord. They simply choose one rational harmony over another.

The prime number ratios only need to be tempered when treating them as curves and not scalars. If prime number ratios are combined with their complementary ratios, within other intervals, then tempering is unnecessary.

Tempering occurs when you define any scale by one prime number ratio (or a composite, but those are just composed of primes). If you do not define a scale, then you do not have temperament problems. The age old problem is true, but I think a little short sighted.

Euler once said, "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."

I think this is similarly the level of difficulty in visualizing and understanding the temperament problem.. why it hasn't been resolved for centuries. The underlying structure of temperaments are based on the primes and the composites that contain them. The very aligning of wave patterns works the same as the Sieve of Eratosthenes. Very similar problems, in the deep underlying structure.

I was looking into Harry Partch's work the other day and I was reminded of an important term he created, called 'limit' - or upper bound to the complexity of harmony. This term is extremely accurate and useful because it's been applied through the ages but only recently was the concept realized.

Limit is essentially the same concept as in the Sieve of Eratosthenes, when to find all the factors of a term, you only have to define all the primes up to the square root of that number. The largest prime in this case, would be the limit.

The largest prime in the denominator of the harmony ratio roughly defines the resolution of the scale necessary to contain the limit.. or the structure of the unequal temperament necessary. In our music, with 12-TET, we are generally 5-limit. The 3rds contain the prime 5 in their harmony as well as their complementary 6ths. I wouldn't go so far to say that our scale is 7-limit because the tritone is more a figment of the scale which was chosen for its alignment to the 4th and 5th.. not for the ratios 7/5 or 10/7 (lesser and greater septimal tritone).

Harry Partch composed mostly off the 43-tone scale in 11-limit harmony. If you look at the above illustrations I posted, you can see the 43-tone scale.. however, I was only analyzing to 5-limit.

1) There is no single adjusted chord geometry. Or a single fixed pitch. One starts on pitch, but may not end on the same pitch. The natural form of music (before unnatural fixed scales) was a flexing, moving geometry through time. Chords grew out of one another, perhaps never repeating exact note twice. When the concept of fixed notes did not exist, there was only ratio.

2) I'm not sure what you're saying here, but the chord geometries simply adjust to the new desired position to get the effect wanted. Since all harmonies would be rational, it is only a matter of which rational harmony to choose from. Tempering is not necessary.

3) No. Non-fixed ensemble does not necessarily need to compromise. They only need to choose. Since all harmony can be rational, they only make musical choices, which then alter the note frequencies slightly.

An example of rational interference is any whole number ratio of base frequencies that produces an interval: 2/1, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, 11/10 - or - 3/1, 5/3, 7/5, 9/7, 11/9 - or - 4/1, 5/2, 7/4, 8/5, 10/7, 11/8 - or - 5/1, 7/3, 9/5, 11/7, etc..

Originally Posted By: Alfredo Ok, that's the point I wanted to "fix": in general, any ensemble (choir included) in order to sound harmonious must temper their intervals. In other words, the necessity to temper intervals derives from having to combine prime-numbers ratios, 2/1, 3/2, 5/4, etc (that's the centuries-old problem).

..."Well, where this is true in a static example.. can you name one piece in choir music, where a choir sings stacked M3rds, 4ths or 5ths to that degree?"...

You are right, perhaps it was a "static" example, but in order to make my point I intended to use simple numbers; I'd appreciate if you could show me which "unequal" variant, amongst many, would represent THE "ad-just-ed" chord geometry. As for singer examples, I'd try not to mix theory with general practice, because then it may really get subjective, depending on individual skills. Nevertheless, you (All) may judge and decide the degree to which an ensemble "stacks" intervals, we are all allowed to our own opinion.

..."Even if it is the case, then they do not also try to sing the octave or M17th.."...

Well, in my view either it is the case - so ensemble too need to temper and compromise - or it is not. On top of that, I do not understand that line.

..."...they already have a rational harmony for that chord."...

Take my example above, please would you put "rational harmony" and new chord "best possible division" in simple numbers?

..."They simply choose one rational harmony over another."...

Do you refer to some "rational harmony" which they would have agreed on? Hmmm...

..."The prime number ratios only need to be tempered when treating them as curves and not scalars. If prime number ratios are combined with their complementary ratios, within other intervals, then tempering is unnecessary."...

I'm a bit confused, are you thinking in terms of intervals "within other intervals"?

..."Tempering occurs when you define any scale by one prime number ratio (or a composite, but those are just composed of primes). If you do not define a scale, then you do not have temperament problems. The age old problem is true, but I think a little short sighted."...

I see. In my view, even before the definition of a scale, the problem originates from multiple intervals combinations; as I could show in my other post, tempering becomes an evident issue every time you play more than two notes simultaneously. And yes, the problem was short sighted, due to the age-old equality "pure ratio = consonance = in tune", plus the unexpanded 2:1 octave module and the odd aversion for irrational numbers.

..."Euler once said, "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."...

BTW, have you had a look at that paper?

..."I think this is similarly the level of difficulty in visualizing and understanding the temperament problem.. why it hasn't been resolved for centuries. The underlying structure of temperaments are based on the primes and the composites that contain them. The very aligning of wave patterns works the same as the Sieve of Eratosthenes. Very similar problems, in the deep underlying structure."...

Well, I do not think that understanding the temperament problem is that difficult, nor mysterious. In Chas paper (section 1.2) it required a few lines only.

..."...Harry Partch's work..."...

Yes, interesting work although related again to pure ratios. I preferred to go for interrelated beats, where pure ratios (1:1, 3:1, 4:1) can justify and comprehend proportional deviations.

..."1) There is no single adjusted chord geometry. Or a single fixed pitch. One starts on pitch, but may not end on the same pitch. The natural form of music (before unnatural fixed scales) was a flexing, moving geometry through time. Chords grew out of one another, perhaps never repeating exact note twice. When the concept of fixed notes did not exist, there was only ratio."...

Even if that was the case, they still needed to combine intervals, in theory and in practice.

..."2) I'm not sure what you're saying here, but the chord geometries simply adjust to the new desired position to get the effect wanted. Since all harmonies would be rational, it is only a matter of which rational harmony to choose from. Tempering is not necessary."...

Hmmm..., I was asking: which is the "new chord" best possible division?

..."3) No. Non-fixed ensemble does not necessarily need to compromise. They only need to choose. Since all harmony can be rational, they only make musical choices, which then alter the note frequencies slightly."...

To me, this might be a respectable idea of yours; admittedly I'm a bit confused about your outlook. Is "harmony = rational" your fundamental premise? What happens to the frequency when they slightly "alter" the note? Will that still represent a whole integer ratio?

..."An example of rational interference is any whole number ratio of base frequencies that produces an interval: 2/1, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, 11/10 - or - 3/1, 5/3, 7/5, 9/7, 11/9 - or - 4/1, 5/2, 7/4, 8/5, 10/7, 11/8 - or - 5/1, 7/3, 9/5, 11/7, etc.."...

I see. So, they agree on a single whole number ratio and forget about all other intervals?

Regards, a.c.

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. - Department of Mathematics, University of Palermo - 2009, Italy: http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

And perhaps we will be able to expand on what is arbitrary-assumption and more...

Have a nice Sunday,

a.c.

It is funny that something with "scientific" in the title so thoroughly abuses the notion of science. But I get it. April 1st, right? They do that in Italy too?

Yes, here too is April 1st... how funny! I was surprised too, not only for the title, also for what the author asserts. I could only skimm it but found strange how things are ...mixed in? And the author seems to be an advisor... and that institute seems to really exist. Hmmm...

I understand most of the sources--Pythagoras, the music of the spheres, etc, and the strange mistakes (thinking that Helmholtz believed in a perfect partial series, for one, and that he originated the idea).

And I'm glad to know that earth is a G.

But from whence comes the notion that "A sound wave, we know today, is an electromagnetic process involving the rapid assembly and disassembly of geometrical configurations of molecules. In modern physics, this kind of self-organizing process is known as a 'soliton'"?

At first I thought he was he talking about how the force bumps air molecules against each other, but I get lost in his mention of an "electromagnetic process."

Well, really I could not refrain... "...A=440 is an insane tuning...", "...The equal-tempered system is only an approximation of a rigorous well-tempered system whose details have yet to be fully elaborated."..., "...Our solar system functions very well with its proper tuning, which is uniquely coherent with C=256. This, therefore, is the only scientific tuning."... I thought I should share that with my PW friends!

But wait: some cited concepts and geometries have sense (on their own), also in relation to what can be observed in nature; it is wanting to prove the original conjecture that messes everything up. In the end though, what counts for Mr. Tennenbaum should count also for us… so, let's double check each other.

I look forward to being able to expand on "in tune" singing, "stacked intervals" and "best possible divisions".

"dialoguing with myself..... within a few cents accuracy probably wouldn't even be noticed by the general public (who will only notice whether the unisons sound "nice")! So the question is papably MOOT for MOST tuning situations.

But for concert pianists in performance--with good ears to boot--the question matters A LOT.

Also, for fanatics like me, who just can't STAND the piano sitting in my home being even slightly out of tune, and I don't just mean the unisons!, it's also a big deal.

That's why we fork out the big bucks (relatively speaking) for these tuning programs: they permit us to keep our babies sounding as good as they possibly can at ALL times!!!

hence my question about microphones!!"

- . - . - . -

I would like to thank you, JG, for that post of yours. When I read it I suddenly felt less lonely. Today, after much discussing in this Forum, I realize that many issues about tunings and "out of/in tune" are messed up by some "literature" that is based more on fashionable opinions than provable facts. And I realize that my own approach, like you suggest, may well be described as "fanatic".

Curiously enough, I have been able to share my "in tune" sense with all musicians I've met, but when it comes to theory, practice, ETD's, non-unified few-cent-science and colleagues... I understand that some colleagues are ready to swear they would sing "pure" 3rds, they do not need to temper; others would offer and argue different (pure?) intonations for choirs and bow instruments, though when they are asked to prove their arguments some of them prefer to, what's the word, vanish? Many others seem to prefer a "home key" to a "whole-home" tuning, and say that we all (ear equipped music lovers) are able to tell whether some notes Are or are Not in tune..."and I don't just mean the unisons!".

I too happen to dialog with myself..., I too think that's a big deal, in my opinion too your question matters a lot, I too would not stand a piano being slightly out of tune, actually not only a piano but a singer too, a violin too, a harp too... hence my questions...

And thanks for your thread, I'm getting to know more about ETD's approximations and the so called "true" ET tuning. I better understand why the fixed-intonation of a piano can turn into a nightmare.

To All,

Last night I melted on my armchair, listening to the Italian TV News... I woke up on Rachmaninoff's concert n. 2, really enjoyed it, hope You like it too:

Would you say that bows adjust their intonation all over, perhaps every now and then? I haven't told you, but I think they Do stack all intervals, possibly all "in tune" intervals they have been practicing for long long years.

Here is a list of statements of yours (from 03/20/12) that I found confusing:

- If a flexible string instrument were used, the geometries could change for each harmonic and melodic alignment to provide perfect consonance continuously with new geometries internally aligned, not preformed to the 2^x curve. Each time the chords changed, the frequency values would shift.

- If a choir did sing these strings of 5ths 4ths or 3rds, they'd have to temper their intervals to achieve a 5/1. However, the difference between that and the piano, is all harmonies would be built on rational interference between waves.. meaning at least one wave pair in ratio, from each dyad, would constructively interfere.

- Without being confined to the irrational frequencies of the 12-tone [2/1] curve, there are no fixed compromises based on one curve, and every frequency can adjust and build to a new chord based on the best possible divisions. This is what I meant by movable and unequal.

- Tempering occurs when you define any scale by one prime number ratio (or a composite, but those are just composed of primes). If you do not define a scale, then you do not have temperament problems.

- There is no single adjusted chord geometry. Or a single fixed pitch. One starts on pitch, but may not end on the same pitch. The natural form of music (before unnatural fixed scales) was a flexing, moving geometry through time. Chords grew out of one another, perhaps never repeating exact note twice. When the concept of fixed notes did not exist, there was only ratio.

- the chord geometries simply adjust to the new desired position to get the effect wanted. Since all harmonies would be rational, it is only a matter of which rational harmony to choose from. Tempering is not necessary.

- Non-fixed ensemble does not necessarily need to compromise. They only need to choose. Since all harmony can be rational, they only make musical choices, which then alter the note frequencies slightly.

I look forward to knowing your (and all colleagues) thoughts and comments.

I hope you are doing well, I am pleased to see you writing on that forum

as you are may be aware, I am deeply using your tuning approach and , lets say philosophy, since we meet a few sciecles ago.

Something I noticed about justness, if that what seem to add to the justness sensation, or impression, is the level at which the piano sound in phase with itself (and its environment).

Then, all depends of the instrument when we put it in front of an ET intention :

Some instruments will nicely use the space left by the iH and the deviations from straight "rule" to gently install themself in a self resonating system (as you state at 1/3 1/4 level equilibrium. Others, (very few in my experience) I've find, find that when the 12/15 equilibrium is attained, the octave begin to sound a little instable, less calm than with the plain 2:1 type of Japanese tuning for instance (which sound dull to my ears generally , so to say) . It may have to do with the level of iH and the voicing, but also for sure with the kind of tone reinforcement that the instrument and the room are able to accept (and certainly the room acoustics). Well I had comments from some pianists that the tuning was not possibly what they prefer for their instrument, so, as I like and the method and the result, I begin to use a little water in the vine, that mean, I keep the temperament method, but stick to a shorter octave for the end of the mediums (when going up) , then, I chase for the resonance in the beginning of the treble, and so I have yet a little of both worlds, less active fast intervals progression, but a "stretch" based on the acoustical answer coming from the lower note of the Chas ratio.

I believe I have get so much used to that state of resonance and equilibrium that I refine it when tuning the unisons, that mean,if I am not aware of the sympathetic resonance of the lower note 12-15 my unison is not good. (sometime I use the tonal pedal to help, but it works without it, and the tonal pedal tend to make the notes tuned a tad too high (for what reason ?)**

There is something that happens when the Chas tuning is installed, the resonance is globally way stronger in all regions of the instrument. It tend to lower the contrast between portions of scale, so making a hair of step under Chas in the mediums leaves that contrast more apparent , while keeping the singing and clear treble present (one of the most evident feature of Chas, melody is always audible) for instance when the musical phrase begin in the mediums then raise to the treble there is an added impression, which makes the musical phrase more lively, in my impression).

I have been using the "pre tensionning tuning" method since our meeting, and I wait to find something more lmogical or more efficient, I stated it lately, I could make huge PR in one pass because of the very good habit to feel how much the notes will lower during the tuning ; when used to anticipate a few cts lowering and being sure it have nothing to do with string rendering of tuning pin settling back, it gets easy to feel what happens in the instrument, acoustically speaking. It makes the tuner very confident (and the result appreciated !)

The "how far from Chas resonance am I" test (it is simply an energy test most probably) is invaluable for many situation as instantaneously know where a note is while tuning... Then it is also less tiring to have some "listening distance" from the fast intervals, chasing only for their progressiveness ans only a little for their real speed, those pesky intervals are playing the game nice generally and they fall gently in place one following the other, without me doing much effort in the direction of their speed, simply aligning the energy...

I would add something to my comment above, I suppose that , if we play with the unison opening (regulating the attack stabilization delay and the phase difference between doublets of strings at different times) we can take an instrument that originally ask for a dark romantic tone and push it in the Chas envelope nicely, to me I have been doing that constantly just because I appreciate to get the maximum energy coming from the rest of the instrument when a note is played. Some pianists (2 , up to this day, one with a Yamaha U and the other with a BOsendorfer 2.20) told me that they appreciate the more lively behavior of the instrument and that they are aware of the justness impression, but simply they find that slightly limiting for some reason they could not explain me, I had to chase by myself (all others tend to post me little mails saying that they never heard their instrument sounding so nicely and that they where not even aware it could sing as much !!) .

SO I looked for some reason, and I find that low IH instruments in a little dull environment seem to need close harmony in the medium range (may be only one octave and a half as A3 E5, that is the kind of zone I stay attentive to 2:4 resonance in that case). Probably, mostly the slightly faster thirds where too much noticed...

The only reason I didn't respond further, is I noticed you defending your CHAS idea in light of my comments. What I said is really not related to your CHAS concept and I don't intend to threaten your beliefs or understanding. If anything, I have come to believe we may view tuning similarly based on our independent experience.

I also don't believe I could say what I intended to say any clearer.

I essentially drew a difference between open ensemble and fixed string instruments. Temperament is a problem for fixed string instruments.

Chorus and non-fretted string ensembles can require musical situations that involve temperament, but that is by choice. Proving this is exhaustive. All rational frequency combinations result in some form of consonance. No matter what pattern, this is true. The degree of consonance is the only question.

Fixed-tone instruments introduce irrational (beating) relationships when attempting to equally space tones on a single prime curve. These irrational relationships require temperament to find a solution of best fit.

I had some energy this morning to add on these ideas for the public forum, and I remembered that what is essential to what I am saying is the concept of movable tones. Pure progressive chord structures require note frequencies that shift slightly to account for their different internal interconnected ratios, as well as the leading tones that connect them.

Melodic lines tend to go sharp or flat depending on their direction. Cellos and violins rise in their intonation when playing upwards along a melodic line. They fall in their intonation when descending.

Without the harmonic constraint of a fixed scale, melody tends to be greatly expanded and chords can occur in new locations, as dictated by the progression of the melodic line.

To analyze keyboard music on this system becomes problematic because many tones, the 2nd and tritone for example, are germane to the 12-TET scale. Music composed in a system of temperament is then dependent largely on that system.

I'm glad you are again taking part to this Forum and contributing with your feedbacks. Good news from your customers... I really think your success has to do with your own constant commitment, your passion and sincere desire for improvements. For some reasons, I feel more comfortable to reply to you privately. Be sure, I too look forward to meeting you again.

Un caro saluto,

Alfredo

- . - . - . -

Hi Tunewerk,

..."The only reason I didn't respond further, is I noticed you defending your CHAS idea in light of my comments. I don't intend to threaten your beliefs or understanding. If anything, I have come to believe we may view tuning similarly based on our independent experience."...

Thank you for your reply. I was ready to post mine... now I'd better read your latest adding and see if my reasoning can still make sense. I need to precise that my comment was not addressed to you, so I apologize for having been ambiguous.

I also need to say that Chas is basically a temperament model and it is meant to describe a new approach to the ordering of a sound scale; Chas scale is modeled on pure ratios (1:1, 3:1 and 4:1) and enables the representation of an inter-modular ruled tone-geometry as one of infinite (s) solutions. I hope you understand that nothing there needs to be defended.

..."I have come to believe we may view tuning similarly based on our independent experience."...

Hi Tunewerk,

I would not be surprised if we were to view tuning similarly and I look forward to being able to expand on piano tuning, both in theory and practice. In fact, even now I would feel ready to ask you:

- Do you think we could succeed in sharing tunings 100%, with a theoretical model that could only rule 12 semitones?

- How do you tune chromatic 4ths, 5ths, octaves, 12ths and 15ths? Do you follow any rule?

The latest issue: "Tempering for chorus and non-fretted string ensembles". On the theoretical side, I would still like to understand, either through numbers or out of logics or common sense, which is the "best possible division" you mentioned, when considering complex chords.

I have read your adding. On that, I'll post separately.

You wrote: ..."Temperament is a problem for fixed string instruments. Chorus and non-fretted string ensembles can require musical situations that involve temperament, but that is by choice."…

Please, help me "translate". Perhaps you mean: There exist musical situations where chorus and non-fretted ensembles have to temper; the difference is that they can choose how to temper, depending on desired effects?

If that is correct, we would agree on one (simplified) point: There exist musical situations where chorus and non-fretted ensembles have to temper. And I'd rather consider "choice" and "desired effects" as two other different issues, in the "harmony - in tune" chapter.

..."Proving this is exhaustive. All rational frequency combinations result in some form of consonance. No matter what pattern, this is true. The degree of consonance is the only question."...

Ok, let's say that is true. Now let's consider the "degree of consonance". I get the impression that you move from these premises:

(1) consonance = rational. No problem, this is within tradition.

(2) harmony = rational. Problem. If that is true, then:

(3) consonance = harmony

We know that, in case of complex chords, favoring one single rational (pure) frequency combination would go to the detriment of all other combinations, precisely in those terms of consonance = harmony you refer to in your premise. As a consequence, consonance cannot equal harmony. This evidence takes me directly to the question: what drives chorus and ensembles towards the highest degree of harmony, as you say, towards the "best possible division"? Can it be one single "rational" combination? And can we talk about a "choice" or will it be more a "must"?

..."Fixed-tone instruments introduce irrational relationships when attempting to equally space tones on a prime curve. These irrational relationships require temperament to find a solution of best fit."

Indeed, it is that concept, perhaps how we order factors, that I'm trying to double check. Adopting your words, this is what I can say (after some numbers):

Prime curves require temperament when dealing with complex chords. These prime curves introduce irrational relationships when attempting to space proportional tones and find a solution of best fit.

One consequential question: Would you (All) say that "rational = consonance" equals "in tune"?

..."...what is essential to what I am saying is the concept of movable tones. Pure progressive chord structures require note frequencies that shift slightly to account for their different internal interconnected ratios, as well as the leading tones that connect them."...

Please, help. I do not understand whether you refer "Pure" to the word "progressive" or to "chord structures". In case of complex chord structures, even if we are enabled to shift frequencies, there is no way to "interconnect internal ratios" and get them all "pure". This is to say that "Pure ...chord structures" would not make sense to me. As you wrote a few days ago,

..."If a choir did sing these strings of 5ths 4ths or 3rds, they'd have to temper their intervals to achieve a 5/1. However, the difference between that and the piano, is all harmonies would be built on rational interference between waves.. meaning at least one wave pair in ratio, from each dyad, would constructively interfere."...

And we would still need to prove that in order to build harmonies we need "rational interference".

Movable tones. Yes, singers, chorus and non-fretted tones are movable indeed but that, objectively, can only translate into movable temperament, and I doubt how convenient that can sound. I remember your reply to one of my points:

..."1) There is no single adjusted chord geometry. Or a single fixed pitch. One starts on pitch, but may not end on the same pitch. The natural form of music (before unnatural fixed scales) was a flexing, moving geometry through time. Chords grew out of one another, perhaps never repeating exact note twice. When the concept of fixed notes did not exist, there was only ratio."...

As you say, singers do not have a single fixed pitch, they might start on one pitch but may not end on the same pitch (it seems that the tendency is to go flat), but if that occurs it is considered a problem. Bows do have fixed pitch, 4 free strings at fixed pitch. That's why they would not go for movable temperaments.

..."Melodic lines tend to go sharp or flat depending on their direction. Cellos and violins rise in their intonation when playing upwards along a melodic line. They fall in their intonation when descending."...

I do not know if what you say is scientific, it is not what I hear. In any case that does not prove they are released from having to temper. Do you agree? In my view this last one is a different issue, namely "intonation".

..."Without the harmonic constraint of a fixed scale, melody tends to be greatly expanded and chords can occur in new locations, as dictated by the progression of the melodic line."...

Well, what I can say is that chorus, yes, if (for any reason) they were to change the original pitch, rising or falling, they would have to find new chord locations. But this is not to say "they do not need to temper". I hope you acknowledge that this is not a question of personal opinion nor believes, but numbers: non-fretted instruments have their fixed pitch too, that is what determines the constraints; assigning chords a "new location" would cause an obvious problem, in fact free strings would not sound in tune.

I hope we can share this (numerical and musical) notion: one single integer ratio, one single wave pair in ratio, although producing as you say "high density" consonance, cannot make for harmony. Before any fixed scale, "prime curves" deriving from one single fixed pitch are enough to determine melodies-within-harmony constraint, making temperament a must. Perhaps we should go to the source to find the first cause of constraint and the reason why temperament is required always, whether we sing or play; perhaps we would see that it is not "fixing" but "ordering" tone steps in a scale, two different issues. In fact fixing can simply be fixing, while ordering can result in a poor, out of tune tuning.

Last but not least, I do not see any reason why our musical ear should fear irrational relationships. Do you?

Alfredo, I respectfully bow out of further discussion.. I just don't have time to give the proper attention to the litany of questions you have detailed, as well as the parsing of semantics.

I will perhaps write on this subject further in my online journal when I have the time. I have just pointed to some mathematical facts here.

If you prefer beating relationships and want to call that harmony, great, more power to you.

I look forward to other substantive additions on these topics.

Thank you, Tunewerk, for those mathematical facts you have been able to point out. Let me know where I can find a link to your online journal, there too I would surely appreciate your further elaborations.

One thought on what has recently been said about "intonation", "having to temper" and "movable tones".

We tuners happen to deal with pianos, what we call a fixed-tone instrument, and the majority of pianists are not enabled to improve their piano's intonation, in fact tones are "fixed" there where we leave them. Yet, when I tune pianos I've learned how important it is thinking in terms of "movable tones", perhaps I should say ever "adjusting tones", in that tones tend to adjust in time, tones "move" regularly even during our tuning, when we modify the strings tension and the load onto the entire structure.

On the other hand, singers and bows, although allowed to "movable tones", are always enabled to master their intonation and their tones, exactly where they hear them "in tune". One "alternative" picture might then be this:

Pianos are fixed tone instruments, we have to deal with their "movable" tones.

Singers and non-fretted instrument musicians are allowed to master their intonation and enabled to play in tune... "fixed" tones.

The video linked below reminded me of when I studied violin, when I was practicing intonation, looking for the "best possible division", wanting to hear my scales "in tune" always. One day I opened the case and my small chinese toy raised… the finger board had come unstuck…

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. - Department of Mathematics, University of Palermo - 2009, Italy: http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

There is one cliché that may cause one more point of confusion; this, in my view, unfortunately allows for mystification and it may open to ignorance as a consequence. Briefly, the cliché I'm talking about was again reported last year by an Interlocutor of mine:

Int.: ..."Also, not all instruments regard Equal Temperament as their standard, so any intended improvement on Equal will be mostly irrelevant to players of those other instruments, and a good portion of the rest aren't even aware that there is anything to improve on."...

Me: Which other instrument are you referring to? In my view, all instrument players might be happy to adopt the most correct reference and the most performing ET frequency scale.

Int.: "What I am referring to is that the players of instruments such as brass and woodwinds automatically adjust their playing with the intention of achieving beat-free intervals against other players,...//.snip.//... Since "in tune" ideally means beat-free intervals, then calling beating intervals (albeit with a different compromise of beats and key color than 12th root of two ET) "in tune" or not will always be subjective.//.snip.//... Free intonating instruments (horns, reeds, strings) do not need to use an equal temperament and will always gravitate to Just Intonation or "Natural" tuning as the more consonant sound; to them the piano is an out of tune instrument to be tolerated rather than imitated."...

- . - . - . -

Focusing on "in tune" and "Beat-free intervals"... had I thought that a piano "is an out of tune instrument to be tolerated...", 30 years ago I'd have gone for gardening.

I'm sure All of you understand that, once more, I'm addressing to temperament and intonation, a simple word that has a precise meaning and yet it seems to be loosing sense day after day, also amongst... Piano Tuners!!

Now, those simple numbers I recently showed can well prove that chords "must" be tempered, thus including chorus and non-fretted instruments.

Today I'm pleased to be able to share a recording of Chas tuning with an orchestra. Two more concerts had just been played on that piano and I did not correct anything, as that was a competition and this final was performed all in one go. I'm offering the last fourth of the whole recording not casually, but hoping we can discuss more about whole fusion, stability, sound coherence, singing tones and above all...intonation.