> A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2?
That one isn't especially difficult. Humans like the combination of fundamental tone and first harmonic (I'm not aware that there's a culture with a musical system that doesn't respect the octave). In the western culture, we found the similarity between those tones so notable that we classify them as harmonically indistinguishable; thus, A3 (220 Hz) is harmonically tied to A4 (440 Hz). Many cultures, our western predecessors included, also appreciated the 2nd harmonic (f3, or, since octaves are now identical, f1.5) as consonant with the fundamental.
Following on to that, they developed a pattern of notes at f(1.5^2), f(1.5^3) etc. Here it can be noted that the first 12 exponents of 1.5 are (renormalized to the [1:2] space as necessary): 1 , 1.5, 1.125, 1.6875, 1.265625, 1.8984375, 1.423828125, 1.06787109375, 1.601806640625, 1.20135498046875, 1.802032470703125, 1.3515243530273438 and 1.0136432647705078.
Note how infuriatingly close that last is to 1; on a violin string (~.328m) that would be a 4mm difference. Human societies have wrestled with that discrepancy as far back as the Pythagoreans. Musicians and composers in western society didn't come to a compromise for the issue until ~late 16th century, when they DEFINED a semitone as the 12th root of two. This adds a little bit of error to every interval besides the octave in favor of consistency across every key.