An interactive exploration of why every piano is slightly wrong — and why that's the point
When two notes vibrate in simple ratios, they sound consonant — locked together, beating as one. The simplest consonance after the octave is the perfect fifth: a ratio of 3 to 2. Click to hear it.
The difference is two cents — almost nothing. But stack that difference twelve times and it becomes a chasm.
If you stack twelve pure fifths — C to G to D to A and around — you should arrive back at C, seven octaves higher. You don't. You overshoot by about a quarter of a semitone. This gap has a name: the Pythagorean comma.
The circle doesn't close. Twelve perfect fifths equal 531,441 / 524,288 — not exactly 27. The difference is 23.46 cents. Small. Inescapable.
Every tuning system is a different answer to the same impossible question: how do you close a circle that doesn't want to close?
Keyboard: A–K keys map to one octave. Try playing a melody in each tuning.
In equal temperament, the circle closes. Every fifth is narrowed by the same invisible amount — 1.96 cents — so that twelve steps return exactly to the start. The imperfection is distributed. The system works.
Any system with competing goods that can't all be optimized simultaneously faces a temperament problem. The question is never whether there will be compromise — but how the compromise is distributed.
Equal temperament says: make all the wrongness agree.
Then every key becomes playable.