I've written a lot about the Pythagorean comma — that stubborn ~23 cent gap that emerges when you stack 12 pure fifths and discover they overshoot seven octaves by just a hair. The comma is why every tuning system is a compromise, and why there are so many of them. But reading about it is not the same as hearing it.
This page lets you hear it. Compare how a C-major scale sounds in Pythagorean tuning versus equal temperament versus just intonation. Play individual notes, stack intervals, hear the beats when ratios aren't quite pure. The math is below each button — but the point is the sound.
Stack 12 pure perfect fifths (ratio 3:2 each). The result should land on the same note 7 octaves up — but it doesn't. It's sharp by one Pythagorean comma: ≈ 23.46 cents. Play both and hear the difference.
Choose a tuning system, then play the note buttons below. Each system divides the octave differently — the differences are subtle on individual notes but audible on thirds and chords. Play a major third (C + E) in Pythagorean vs Just to hear the difference.
Cents are a logarithmic measure: 100 cents = one equal-temperament semitone, 1200 cents = one octave. Differences under ~5 cents are hard for most people to notice in isolation; over 10 cents the mistuning becomes apparent in chords.
| interval | ratio | 12-TET | Pyth. | Just | 19-TET |
|---|
The Pythagorean major third (81:64 ≈ 407.8 cents) is noticeably sharper than just (5:4 = 386.3 cents), which is why medieval music avoided thirds as consonances — in Pythagorean tuning they beat badly. The equal-tempered third splits the difference, but is pure to neither.
The Pythagorean comma isn't a flaw — it's a mathematical inevitability. No finite stack of perfect fifths will ever land exactly on a perfect octave, because 2n = (3/2)m has no integer solution. The comma is where the two most fundamental intervals in music become incommensurable.
Equal temperament resolves this by making every fifth slightly wrong — widening each by 1/12 of the comma ≈ 1.955 cents. Pure in nothing, usable in everything. Whether that trade is worth it is one of the oldest arguments in Western music.