The Compromise in Every Piano

Take a note. Go up a perfect fifth, the interval at the start of Twinkle Twinkle, whose frequencies stand in a 3:2 ratio. Do it again. Keep going: twelve fifths in all.

By the arithmetic of the keyboard you should arrive back at the note you started on, seven octaves higher. C to G to D to A and round the circle to C.

You do not arrive. You overshoot, by about a quarter of a semitone: small, and far too large to ignore.

The reason is arithmetic that cannot be argued with. Twelve fifths multiply the frequency by (3/2)¹², which is about 129.7. Seven octaves multiply it by 2⁷, which is 128. Powers of three and powers of two never coincide, so the circle of fifths is not a circle.

The excess is called the Pythagorean comma. It cannot be removed, only relocated, and every tuning system in history is a decision about where to put it.

The medieval solution was to keep the fifths pure and let one of them absorb the whole error. That interval is unusable (it howls, and it is called the wolf) so you avoid the key it lives in. Fine, as long as music stays in a few keys.

Renaissance and Baroque practice made a different trade. Meantone temperament narrows the fifths slightly in order to obtain pure major thirds, because thirds had become the harmonic basis of the music and pure ones are conspicuously beautiful. The price is again a wolf, and a set of remote keys that sound genuinely bad.

The consequence is one modern listeners have lost. In an unequal temperament the intervals really do differ from key to key, so E flat major has a different colour from D major, not higher, different. Eighteenth-century writing about the character of keys is describing something measurable, and the characters vanish the moment every key is tuned identically.

Which is what equal temperament does. Divide the octave into twelve mathematically identical steps, each a ratio of the twelfth root of two, and spread the comma evenly across all of them. Nothing is pure and nothing is unusable. Every fifth is narrow by about two cents, which almost nobody hears. Every major third is wide by about fourteen cents, which many people do hear: an equal-tempered third beats audibly, and that beating is the sound of the compromise.

It won because music kept modulating. Once composers wanted every key available, an unusable one was intolerable, and the price was paid in the thirds. Bach's Well-Tempered Clavier is evidence of the argument in progress. 'well-tempered' most likely means a good unequal temperament in which all keys work and each retains a character, not our uniform one.

Two footnotes that change how you hear instruments.

Pianos are not tuned to exact octaves. Real strings have stiffness, so their harmonics run progressively sharp rather than landing at exact multiples. A tuner therefore stretches the tuning, sharpening the top and flattening the bass to match the instrument's actual harmonics. A mathematically perfect tuning sounds flat at the extremes, and this is one of the rare cases where the theoretical answer is straightforwardly wrong.

And almost nobody who can avoid equal temperament uses it. Singers, string players, trombonists and wind players adjust continuously, leaning thirds flat towards pure, raising leading notes, tuning chords by ear against each other. A string quartet does not play in twelve equal steps and never has.

Equal temperament is a fixed-pitch keyboard problem. Everyone else politely works around it, and when a piano enters, they conform to the piano.