Twelve fifths and seven octaves should land on the same note. They do not. This gap is why every piano is slightly out of tune on purpose. Guess the size of the gap, then hear it.
A pure fifth has the frequency ratio 3:2. Climb 12 of them from C, then fold the result back down by octaves. A semitone is 100 cents. How many cents away from C do you land?
Each dot is one more pure fifth, placed on a clock face where one lap is one octave. After 12 steps the chain should come back to dot 0. Drag the slider in step 3 to squeeze the gap shut.
Played at C4 = 261.63 Hz, computed from A4 = 440 Hz. The beat rate is computed: 261.63 Hz times the frequency error. It is the speed of the wobble in the first button.
Set the slider to 0 to keep 11 fifths pure and dump the whole gap in one "wolf" fifth. Move it to 100 to shave the same sliver off every fifth. That is equal temperament, the tuning of a modern piano. Then press a major chord and listen.
Major chord from each root. The bar shows how close its third is to a pure 5:4 third (386.31 cents, computed from the ratio). Full green means pure, red means sour.
| Interval | Pure | 12 equal | Error |
|---|---|---|---|
| Fifth (3:2) | 700.00 | ||
| Major third (5:4) | 400.00 | ||
| Minor third (6:5) | 300.00 |
Equal temperament spreads the 23.46-cent gap evenly, so each fifth is only about 1.96 cents flat. That is nearly inaudible. The price is that thirds are a little sour in every key, instead of some keys sweet and one key awful. All figures in cents are computed from the frequency ratios as 1200 × log2(ratio).
Gap of (3/2)12/27 = 531441/524288 ≈ 1.01364, about 23.46 cents: Pythagorean comma. A4 = 440 Hz, standardised as ISO 16: A440. The semitone ratio of the twelfth root of 2 (≈ 1.05946): Equal temperament. Everything else on this page is computed from those constants, not measured.