12 Semitones 十二律 again

I struggled with the 12 semitones in the previous post. It's a little easier to explain with music jargon and freshmen math.

Given any note, another node within the same octave that goes well with it is 1.5 times its frequency, in music it's called the perfect fifth. What I tried to say was if we begin with a root note and its perfect fifth, we'll naturally end up with 12 semitones.

It's self evident on a string instrument the pitch and length relationship is logarithmic, every semitone is a multiple of the previous note, the math is:
(ratio of note to the previous notes) raised to the (number of semitones)th power = 2 (frequency doubles to the same note in the next octave).

Now, let x be the number of semitones, and y be the order of the perfect fifth (in relation to the root), to get 1.5 times the frequency, we have this equation:

2 ^ (y/x) = 1.5

Stick this equation into Wolfram, we get:

y=0.585 x

The only reasonable (small) integer ratio that comes very close to this is 7/12 (=0.583). There! The perfect fifth is 7th note out of 12 semitones.

Since the octaves are circular, if we keep adding 7 semitones to get to the next perfect fifth in modulo 12 math, we'll get the sequence of 0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5.  Align 0 with C in western notation of C, C#, D, D#, E, F, F#, G, G#, A, A#, B. it maps into C, G, D, A, E, B,  F#, C#, G#, D#, A#, F -- Circle of Perfect Fifths (with the 5 pentatonic notes leading the way).

There are integer ratios like 10/17 (=0.588) that works well also. For the Chinese, 12 was probably a natural choice, not only 12 is smaller and less odd than 17, there was already 12 months to a year, 12 hours to a day, 12 地支 to count things, etc.