Doesn't this just get you the octaves, which are by no means unique to the 12-tone scale?
> group properties of C12
Can you expand on this? "Group" in what sense? C12?
I'm genuinely intrigued!
In any case, I don't think I disagree per se. But from what I can tell, the 12 tone scale only "falls out" if we apply two constraints: (1) roughly 10 tones in an octave, (2) steps are roughly equidistant (I'm _not_ alluding to equal temperament tuning here). Besides those constraints, why does, say, 16:15 get a tone (m2) but not 7:6?
So the question becomes: are those constraints justified?
EDIT: Partially answered my own question by consulting this diagram [1]: 16:15 has a lower "prime limit" than 7:6.
[1] https://en.m.wikipedia.org/wiki/File:Equal_Temper_w_limits.s...