[n4r9]

I always knew this but used to wonder why 12?, as I'm sure a lot of people did. It turns out of course to be a human choice, but the convenience of that choice can be codified mathematically.

On a piano you move up an octave by going up by 8 white keys, or 12 semitones (white and black keys). Going up by 4 semitones is called a "major third", which multiplies the frequency of the note by 5/4. If you do three major thirds you get an octave. However, notice that (5/4) multiplied by itself three times is 125/64 which is actually slightly less than 2.

In fact there is no way to tune a piano perfectly - there has to be a compromise in the intervals somewhere. The reason for this is exactly that no rational number (fraction) equals 2 when raised to an integer power.

[soulofmischief]

This is referred to as equal temperament, but one can also use just intonation. Each approach comes with tradeoffs and Western music mostly decided that equal temperament was worth it because instruments can play in any key without retuning.

Just intonation also suffers from harmonic issues when building certain chords, but the tradeoff is that there isn't "beating", or resonant pulsing due to frequency mismatches, since in equal temperament, the notes are slightly detuned in order to fit into the scale, as you've mentioned. Another benefit of just intonation is that it's been observed to be the instinctive intonation used by humans.

https://en.wikipedia.org/wiki/Just_intonation