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I have recently found a mathematically-sound "proof" that the twelve-tone musical scale is optimal. I am looking for a similar explanation proving that the diatonic scale is optimal in some sense.

Although the Five-limit tuning on Wikipedia gives some explanation, it does not "prove" optimality.

I realize that this is not an exact science and other scales exitst such as the pentatonic scale having a really long history. Still, I am convinced that the 9000 years of history behind the diatonic scale has some rational explanation. (Rational is an interesting choice of words in this context :) )

My motivation is to understand Why are the white and black keys on the piano placed the way they are? Optimality of the twelve-tone musical scale explains why we have (7+5) keys in an octave, optimality of the diatonic scale would explain why the white keys are chosen the way they are.

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4 Answers

Not a "proof" but a very interesting property that makes the diatonic scale unique. Summarizing from http://www.andrewduncan.ws/cmt/index.html :

Diatonic scale (and its complementary, pentatonic scale) has the highest "entropy" (in other words, "variety") among all possible 7-note (or 5-note) scales (there are 66 of them). Therefore, the diatonic scale is the most rich in content 7-note scale which makes it a fertile ground for melodic ideas.

Neither 5 or 7 have common factors with 12 therefore it's not possible to distribute notes uniformly as it is with 6. Distributing 6 notes gives us the whole-tone scale {C, D, E, F♯, G♯, A♯, C} which is highly regular, has no tonality and creates a blurred, indistinct effect and thus, not very "useful".

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+1 Indeed, very interesting! –  Ali Feb 1 '13 at 17:10
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Here is my own "etiology", not to be taken too seriously.

In short:

(1) The division to 12 semitones is advantageous since it gives a good approximation to the important interval of perfect fifth (3/2).

(2) Since we like the fifth so much, we want to be able to "surf the cycle of fifths" for as long as possible. One way is to demand that for each note in the scale, the fifth above it should also be in the scale. But as $(7,12) = 1$, this would result in a trivial scale. So instead, we want a scale of length $n$ such that $7n \pmod{12}$ is small, and that way we will "almost" be able to go up and down by a fifth. For $7n \pmod{12} = \pm 1$ the solutions are 5 (pentatonic) and 7 (heptatonic). We go with the latter, though the former is pretty popular as well.

(3) Given that we have 7 notes in the scale, how should they be spaced? If we want to have six of the possible fifth intervals realized, it turns out that it must be (up to transposition) the diatonic.

(4) Given that the scale is diatonic (i.e. the major scale up to transposition), the only "shifts" giving rise to the same triads I,IV,V are the major and minor.

This is a nice story, but I don't think it has much to do with reality. First of all, I'm pretty sure that a music historian could convince you that things didn't really develop this way. Second, in other cultures you don't get the diatonic scale. What would be more convincing is a theory that would explain the variety of scales that are found in world cultures. Perhaps it's out there, waiting for someone to mention it in an answer...

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+1 and thanks! Your explanation does show that the pentatonic and diatonic scales stand out. They both have a long history too. "What would be more convincing is a theory that would explain the variety of scales that are found in world cultures." Yes, I would love to see that study too. –  Ali Nov 13 '11 at 20:21
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One answer is given in my paper with John Clough, Musical scales and the generalized circle of fifths, Amer. Math. Monthly 93 (1986) 695–701, MR 88a:05019.

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Could you expand on that, please? I have read your paper but I need further, intuitive explanation on the Myhill's property and why it is a desirable property. –  Ali Nov 13 '11 at 20:23
    
I think Figure 6 (and the paragraph preceding) go some way to explaining why the white keys are where they are. With 12 keys, 7 of which are white, the white keys would be uniformly distributed if they were at positions 12/7, 24/7, 36/7, etc. But these aren't integers, so you round down, and voila! the white keys. If you round to the nearest integer, you still get the white keys, only now the scale starts on D. –  Gerry Myerson Nov 14 '11 at 0:19
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Note, the equal $n$-tone systems are uniform, whereas the diatonic scale has differing distance between adjacent notes. So the methodology of the paper you cited in your first link does not apply directly. In fact, we may simply use a just intonation scheme such as: C=1/1, D=9/8, E=5/4, F=4/3, G=3/2, A=5/3, and B=15/8. This gives the most pleasing major triads: I = CEG, IV = FAC, V = GBD. As for minor triads, iii = EGB, and vi = ACE are pure, but ii = DFA is really grating (which explains, some think, why ii most often occurs in first inversion, FAD, up to about the Baroque period!)

Anyway, as to optimality, what do you want to optimize? Perhaps a different choice of "D" leads to a better ii chord (consequently messing other chords up). On the other hand, if you take the approach Western music has taken for the past few hundred years, then you just take the 12-tone equal temperament as having the best approximations for each of the diatonic notes. Many musicians will admit that they often play out of equal-temperament depending on the key of the music -- major thirds can be softened by decreasing the pitch slightly. A more radical departure is the use of 7-limit or higher harmonic structure. Indeed, the "blue" 7th probably arose as a result of trying to find that pure 7/4 ratio.

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You should check out the music of Harry Partch for an exploration of 11-limit harmonies! It takes some getting used to, but it's well worth the listening! –  Shaun Ault Nov 10 '11 at 22:19
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"as to optimality, what do you want to optimize?" Well, if I could tell you what to optimize, I would do the math myself :) I am afraid there is no single criterion that could accommodate the variety of scales that are found in world cultures. Yuval Filmus' answer does show that the pentatonic and diatonic scales stand out if we insist on the ratios 2:1 and 3:2. –  Ali Nov 13 '11 at 20:26
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