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Eugine_Nier comments on Harry Potter and the Methods of Rationality discussion thread, part 15, chapter 84 - Less Wrong Discussion
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Notes sound good if they're approximately simple rational multiples of each other. Hence you want your scale to contain multiples.
Since the simplest multiple is x2 we use that for the octave. As for why we break it up into 12 semitones, the reason is that 2^(7/12) is approximately 3/2 and as a bonus 2^(4/2) is a passable approximation to 5/4.
I'm referring to the name. What relation does it have to eight?
Eight notes: C D E F G A B C. (People used to not know how to count properly.* I think it comes from not having a clear concept of zero.)
* One can argue that this counting system is no worse than ours, but to do so, one would have to explain why ten octaves is seventy[one] notes.
Similarly, other musical intervals -- i.e., ratios between frequencies -- have names that are all arguably off by one. A "perfect fifth" is, e.g., from C to G. C,D,E,F,G: five notes. So a fifth plus a fifth is (not a tenth but) a ninth.