Twelve feels arbitrary. An octave is a doubling of frequency, and nothing about doubling obviously suggests twelve steps rather than ten or sixteen.
Is there a mathematical reason it landed on twelve, or is it purely convention?
Accepted Answer
There is a reason, and it comes from trying to fit the perfect fifth into the octave.
A fifth is close to the ratio 3:2, which is the most consonant interval after the octave. Stack fifths and you keep generating new pitches. The question is when you land back near where you started.
Twelve fifths is almost exactly seven octaves. Almost: the discrepancy is the Pythagorean comma, about a quarter of a semitone. Close enough that twelve is where the circle nearly closes, and no smaller number comes anywhere near.
Equal temperament then spreads that error evenly, making every fifth very slightly narrow so all keys are usable. Twelve is the smallest division that gives good fifths and thirds at once.
One correction to the usual telling: equal temperament is not a refinement that made everything better.
Before it, meantone and just tunings had some intervals purer than anything we hear now. Equal temperament made every key equally usable by making every key slightly out of tune.
It was a trade, not an improvement, and it was resisted for a long time by people who could hear what was being given up.
Worth adding that 19 and 31 divisions also work well and have been built, 31 giving noticeably better thirds. They exist, they sound good, and twelve won because instruments and notation standardised around it.
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