The Comma & Meantone Temperament

The Pythagorean major third is 81:64 = 1.2656, but the perfect just major third is 5:4 = 1.25. The difference between them — the ratio 81:80, about 21.51 cents — is the syntonic comma. It's the gap between a pure, ringing third and the slightly sharp third produced by stacking perfect fifths.

In meantone temperament, each fifth is narrowed by a quarter of the syntonic comma. This sacrifice spreads the error of the thirds across four fifths, so that major thirds land exactly on the perfect 5:4 ratio. The result: gorgeous, resonant thirds in the home key — but fifths that are slightly flat, and a Wolf fifth that's even worse than Pythagoras's. Meantone dominated Renaissance and early Baroque music for exactly this reason: in the keys that mattered, it sounded better than anything before or since.

There is no single "correct" temperament. Each system distributes the inevitable imperfection differently — Pythagorean favors fifths, just favors a single key's ratios, meantone favors thirds, equal favors all keys equally. The choice of temperament is a musical and aesthetic decision, not just a mathematical one.