Pythagorean comma
Difference between twelve pure fifths and seven octaves (about 23.5 cents).
What it means
The Pythagorean comma is the small difference that arises when twelve pure fifths (frequency ratio 3:2) are stacked and compared with seven octaves (ratio 2:1). The notes do not quite coincide: after twelve fifths one lands slightly above the starting note in the seventh octave. The ratio is 531441:524288, which is about 23.5 cents, just under a quarter of a semitone.
Musical use
The comma explains why the circle of fifths does not close with purely tuned fifths. In Pythagorean tuning the difference is placed on a single fifth, the so-called wolf fifth, which sounds distinctly bad. Later temperaments such as meantone, well-tempered and equal temperament distribute the error over several or all fifths. The name goes back to Pythagoras, to whom the discovery of the simple numerical ratios of intervals is attributed.
It should be distinguished from the syntonic comma (81:80, about 21.5 cents), which denotes the difference between the Pythagorean and the pure major third.
Performance practice
You can experience the comma on a small scale: tune pure fifths one above the other on a string instrument and you will hear how far your starting note drifts at the end. This shows why keyboard instruments are tuned in temperament.