Math in… Instrument Tuning

Pipes, blocks, and strings with lengths in whole number ratios like 2/1, 3/2, and 4/3 sound nice when vibrating together, so early instruments were designed based on these ratios.

It turns out that major scales on such instruments sound great in one key and worse in others! For instruments to sound equally good in all keys, the ratios between neighboring notes must all be the same.

Since the ratio between a note and its octave is 2/1 and music is typically written using 12 notes and their octaves, that neighboring note ratio is