The Pythagorean comma is 23.46 cents — about a quarter of a semitone. Too small to be a note, too large to ignore. For 2,500 years, every tuning system has been a different strategy for distributing this error. Here you can hear them.
Play two temperaments simultaneously on the same interval. When ratios differ, you hear beating — a wavering that reveals the comma hiding between them.
Start at C. Go up by fifths twelve times. In equal temperament, you return exactly to C. In Pythagorean tuning, you arrive 23.46 cents sharp. Press play to hear the walk — and the drift.
Every temperament is a compromise. Equal temperament distributes the error evenly: each fifth is 2 cents flat, each third is 14 cents sharp. Nothing is pure, but everything is tolerable. This is the deal Bach demonstrated and the twentieth century accepted.
Pythagorean tuning refuses to compromise on fifths. The cost is a single interval — the wolf fifth — that absorbs all 23.46 cents of error. Eleven intervals are pure; one is unplayable.
Just intonation refuses to compromise at all. Every interval from the root is a simple ratio. The cost is that you cannot change key. C major is paradise; F-sharp major is a foreign country with different laws.
There is no solution. Only choices.