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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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    Xenharmonic music is really interesting, by the way. Here's a [22-tone scale piece.](http://youtu.be/h9lL9j5F_iw)2015-08-23
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    @columbus8myhw that song is awesome! :D I must try out 22TET now in my own compositions2015-08-23
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    @StanShunpike There's a xenharmonic wiki, if you're interested. (Another name for it is "microtonal music," though that mainly only refers to scales with more than 12 per octave.)2015-08-23

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