Stack twelve pure musical fifths and you should land back on the same note, seven octaves up. But you overshoot by a small amount called the Pythagorean comma, about a quarter of a semitone. Musicians have wrestled with this tiny gap for over two thousand years.
Truth label: Established. Strong evidence and broad agreement among experts.
Go deeper
The exact number
The gap is the ratio 531441 to 524288, or about 23.46 cents. Euclid is the first known writer to record it.
How pianos cope
Modern equal temperament shrinks every fifth by about 2 cents. No fifth is perfectly pure, but every key sounds equally in tune.
Learn
- What a fifth is. A pure fifth is two notes whose frequencies are in the ratio 3 to 2, like C up to G. It is one of the sweetest sounds in music.
- Known in China too. Chinese scholars noticed the same problem. Around 50 BCE, Jing Fang extended the cycle of fifths to 53 notes to get closer to a perfect fit.
Test yourself
About how big is the Pythagorean comma?
A quarter of a semitone. It is about 23.46 cents, roughly a quarter of a semitone.
Sources
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