Pythagoras & the Circle of Fifths

Every time you go up a fifth you multiply the frequency by 3/2. Every time you go down an octave you divide by 2. Stack twelve fifths and you should, in theory, arrive back where you started — seven octaves higher. This is the circle of fifths, the harmonic map that guides key signatures and chord relationships across Western, folk, and much popular music.

But the math doesn't quite work. (3/2)¹² = 129.746338, while 2⁷ = 128.000000. The difference — about 23.46 cents, the Pythagorean comma — means one fifth in the circle must be badly out of tune. This is the Wolf fifth, traditionally placed between C# and A♭ so the subdominant chords in common keys still sound pure.