I agree very much with your post's thesis (you have to play AND think; thinking alone won't help. It's an experimental subject.) Just noticed one thing:
> For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament
It doesn't seems surprising to me. If you start from a pitch and go upwards in both octaves and perfect fifths (2:1 and 3:2, the two most fundamental intervals), the perfect fifth sequence will land on 11 distinct tones before (nearly) meeting the octave sequence. Mathematically, (3/2)^12 ≈ 2^7.
So 12 semitones works out nicely because you can follow perfect fifths out in any direction as far as you want and never go outside the set of semitones. And most of the small-ratio'd intervals can be represented with pairs of notes inside this set.
Interesting idea indeed. I need to think about it.
Edit after thinking: still, it doesn't explain the number 12 IMO. It could be 17 or something else. Probably, it's a long chain of coincidences at play: Western music settled on 7-note scales long time ago (long before equal temperament was invented), and we should start looking for explanations from here.
Another edit: one of the important coincidences is that number 12 makes possible the existence of diminished scale, which serves as a "universal glue" due to 2 tritones. (There's not enough space here to elaborate, but you probably know what I mean). And maybe tritone itself is one of factors leading to number 12.
If you start off from assuming that the Do-Sol (fifth, 3:2) harmony is a "pleasing" one, and also the Do-Mi (third, 5:4) one, you can create new "mostly pleasing" harmonies by for example taking the fifth of a fifth (9:4, which can be transposed an octave to get 9:8, which is Re or a second), and doing similar things (you can also do things like finding the note whose fifth is Do, which is 2:3, or 4:3, a fourth or Fa).
Repeat this process and you start getting a bunch of notes which fall on the 7-note scale. In the blog post the major seventh is listed as 17:9, but by this method you get a 16:9. Basically the same thing.
At this stage, you may notice that the notes are roughly equidistant, except for Mi-Fa and Ti-Do, which are at ~half the distance. This is the first hint of the 12-note scale. We could have stopped earlier in the notemaking process and had a 6-note or a 5-note scale or whatever, but it wouldn't be so equidistant.
Now pick each note, and build an octave from it. The new notes created will invariably be very close to existing notes, or very close to the midpoint between existing notes. This gets us the 12 note scale (5 midpoints + 7 notes, the aforementioned half-step notes don't have midpoints), if you choose a canonical note for each part. The number 12 just happens to be the number where simple harmonic ratios can get you a mostly-equidistant scale.
At this stage, different music systems do different things.
One kind of Chinese scale uses a 2:3 ratio and generates ratios involving these numbers that form a 12-note (roughly equidistant) division.
Indian music does something similar, though it instead generates a 22-note scale, where many of the 12-note scale notes have two forms. It is rare that a given piece of music will use both forms of the same note.
Western music goes ahead and invents the piano, realizes that the piano is hard to tune/transpose, and settles on the twelfth-root-of-two stuff so that transposing becomes dead easy.
A bit better modification of the argument: continue cycle of 5th. After 12 steps, you get (3/2)^12=129.7, which is really close to 128=2^7 (whole number of octaves). That's where 12 steps come from!
And from here, the natural idea follows: what if we take not exactly 3/2 for fifth, but value x such that x^12 is exactly equal to 128? This leads to equal temperament.
Yeah, that might be it! (Not sure that it's true historically though).
I think that there are several thing that brought us to 12. First it is very easy to divide, this is why we use 12 hours clocks and between 2 octave the ear is able to distinguish 1/12 of an octave as 2 different sounds, it might be possible to do better but it would be unpractical because it would be more difficult to found chords that sound good. Arabic music use quater tones (24 quater tones per octave).
> For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament
It doesn't seems surprising to me. If you start from a pitch and go upwards in both octaves and perfect fifths (2:1 and 3:2, the two most fundamental intervals), the perfect fifth sequence will land on 11 distinct tones before (nearly) meeting the octave sequence. Mathematically, (3/2)^12 ≈ 2^7.
So 12 semitones works out nicely because you can follow perfect fifths out in any direction as far as you want and never go outside the set of semitones. And most of the small-ratio'd intervals can be represented with pairs of notes inside this set.