Pythagorean Tuning

stacked fifths · Pythagorean comma · ancient ratios

12
on
3
Physics: Perfect fifth = 3/2. Stacking 12 fifths: (3/2)^12 = 129.746..., but 7 octaves = 2^7 = 128. Pythagorean comma = 531441/524288 ≈ 23.46 cents — the gap when spiraling fifths fail to close. Ancient Greek music used Pythagorean thirds (81/64 ≈ 408 cents vs just 5/4 = 386 cents, ~22 cents sharp). Meantone temperament (1523) flattens fifths by 1/4 syntonic comma each to make major thirds pure. Continued fraction convergents of log₂(3/2) = [1;1,2,2,3,1,5,...]: 1/2, 2/3, 5/7, 7/10, 12/17... → 12 is a convergent, explaining why 12-TET works well.