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.