Why twelve notes?
Why does the keyboard repeat after twelve keys, not ten or fifteen? Climb by pure fifths and after twelve hops you land almost exactly where you started, seven octaves up. The tiny miss, and how tuning spreads it around, is why every key on your keyboard works.
Twelve hops around
C, G, D, A... fold each hop back into hearing range and after twelve fifths you're back at C, having visited every one of the twelve notes exactly once. That's the circle of fifths, and it's why the octave splits into twelve.
The almost: a built-in wobble
Twelve mathematically pure fifths overshoot seven octaves by about a quarter of a semitone: the Pythagorean comma. Hear the target C, then the overshoot, then both together. The throbbing you hear is the size of the miss.
The deal: equal temperament
Modern tuning spreads the miss evenly: each of the twelve fifths is tuned two cents narrow, and in exchange the circle closes and every key sounds equally good. Here's the same loop in C, then in F sharp. On an old pure-tuned keyboard, the second would have been badly out of tune.
Quiz
1 / 3Why twelve notes per octave?