Pythagorean comma

A calculation error that nobody can fix

The article Equal, meantone, just: why your piano is slightly out of tune introduces the comma like this: you stack twelve pure fifths on top of each other from C, that is C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯ and on to the note B♯. After twelve fifths the last note should be a C again, seven octaves higher. It is not: it lies off by just under a quarter of a semitone, more exactly by 23.46 cents.

The reason is purely arithmetical. A pure fifth has the ratio 3:2, an octave 2:1. Twelve fifths and seven octaves lead to two numbers that cannot be equal, because a power of 3 is never a power of 2.

It follows, according to the article: a keyboard instrument with twelve notes per octave cannot have all intervals pure. Somewhere the error has to be put. The whole history of tuning systems is the history of the question of where exactly.

Where to put the comma?

The article names three answers:

• Just intonation: The intervals of a single key are fixed exactly by simple ratios (third 5:4, fifth 3:2). In this key everything sounds flawless. If you modulate, it becomes unbearable, individual intervals go wrong by more than 20 cents. For choirs singing a cappella it remains the normal case, because singers can adjust the pitch continually. See Just intonation.
• Meantone temperament: It makes the major thirds pure and shrinks every fifth a little for it. In the common keys that sounds warm. The price is a single, dreadfully out-of-tune fifth, the wolf fifth, usually between G♯ and E♭.
• Equal temperament: The comma is distributed completely evenly, every semitone is the twelfth root of two. No fifth and no third is pure, but all keys are equally good. The fifth lies 2 cents too low, which nobody hears, the major third 14 cents too high.

The number 2 fits the comma: twelve fifths, each reduced by just under 2 cents (exactly 1.955), together make up just the 23.46 cents by which the pure fifths were too wide.

The comma and the circle of fifths

The page Circle of fifths gives the consequence for the diagram: the wide acceptance of the circle only became possible through equal temperament. Only then do the twelve fifths close into a perfect circle. Without equal temperament twelve stacked pure fifths exceed seven octaves by the Pythagorean comma of about 23.46 cents.

It also says there what well temperament was meant for: with the Well-Tempered Clavier Bach wrote a prelude and a fugue in each of the 24 major and minor keys, to show that every key was playable on a single instrument. Well-tempered does not necessarily mean equal-tempered, but tunings in which all keys sound usable.

What you hear of it

The article describes the consequences in everyday life. If you tune a piano and bring every string exactly to zero cents, you have an equal-tempered instrument and a C major triad whose third sits 14 cents too high. The chord floats slightly, and that is normal: a built-in compromise, not a defect. String quartets, brass choirs and vocal ensembles sing thirds lower than the piano would have them, because the ear corrects towards freedom from beating. That is why an a cappella chord often sounds purer than the same chord on the grand piano. A tuner, too, measures against the equal-tempered grid, that is, against a compromise.

Common mistakes

• Confusing comma and cent: The comma is a difference between two notes. The cent is the unit in which it is measured.
• Taking the comma for a tuning error: It is no inaccuracy of the instrument but a property of the numbers 3 and 2.
• Counting twelve notes for twelve fifths: From C eleven fifths reach the note E♯, only the twelfth leads to B♯. The chain of fifths C–G–D–A–E–B–F♯–C♯–G♯–D♯–A♯–E♯ has twelve notes but only eleven fifths.