No piano can be perfectly in tune

The circle that won't close

A pure fifth is two notes whose frequencies stand in the ratio 3 to 2. Shorten the string that plays C to two thirds of its length and you hear the pure G. When the ratio is exact, the overtones of the two sounds line up and the wobble disappears; if G is slightly low or high, two of the overtones beat against each other and you hear a slow pulse. Tuners who work by ear listen for the same thing: they count the beats, and on a piano they leave the fifth a slow, precise beat instead of removing it.

Start on C and climb fifth by fifth: G, D, A, E, B, F sharp and so on, until after twelve fifths you are back on a C, seven octaves higher. If every fifth is perfectly pure, that last C comes out too high by 23.46 hundredths of a semitone, almost a quarter of a semitone. Musicians call a hundredth of a semitone a cent, and this remainder the Pythagorean comma. It already appears in the music theory of ancient Greece, and the reason fits in one line of arithmetic: twelve fifths would land exactly on seven octaves only if 3¹² = 531,441 were equal to 2¹⁹ = 524,288. The first number is odd, the second even, so they never meet; the ratio between them is the comma itself.

On the spiral, each turn is an octave. So you can hear it, the top C is brought down seven octaves and played beside the starting C: the two beat 3.6 times a second.

Where the leftover hides

An instrument with fixed keys has twelve notes to the octave, so it has to force the circle shut. The history of tuning is the story of where the leftover was buried. Pick a tuning and listen to three intervals in the middle of the keyboard.

Each bar is the fifth that starts on the note written under it: up, wider than pure; down, narrower. Beats per second between the overtones that should coincide, for strings with exactly harmonic overtones.

Pythagorean. Eleven perfectly pure fifths, and the whole comma falls into the twelfth, between G sharp and E flat, 23.46 cents narrow. Most major thirds, such as C–E, come out 21.51 cents too wide, though, and beat harshly.

Meantone. Renaissance music wanted pure thirds, so every fifth was narrowed by 5.38 cents. Eleven narrow fifths leave an even bigger hole behind: the last fifth, between G sharp and E flat, comes out 35.68 cents wider than pure. It became known as the wolf fifth, for its harsh, howling sound; composers avoided the keys that ran through it.

Equal. The leftover is cut into twelve identical slices: every fifth is 1.955 cents narrow, and every key has the same structure. Equal temperament won ground gradually, and by the 19th century it had become the rule for pianos. The major third pays the price: it is 13.69 cents wider than pure and, on middle C, beats about ten times a second.

At a piano, today

You have everything you need to hear the compromise on a real instrument. A guitar works too: its frets are also laid out in equal temperament.

A choir or a string quartet has no fixed keys, so it can adjust its pitch from one chord to the next. A piano can't: twelve fixed notes to the octave settled from the start that a leftover would remain somewhere.