Pythagorean comma
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In music, when ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. If this pitch is then lowered precisely seven octaves, it will be discovered that the resulting pitch is 23.46 cents (a very small amount) higher than the initial pitch. This microtonal interval
is called a Pythagorean comma, named after the ancient mathematician and philosopher Pythagoras. It is sometimes called a ditonic comma.
Put more succinctly, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.
This interval has serious implications for the various tuning schemes of the chromatic scale, because in Western music, 12 perfect fifths and seven octaves are treated as the same interval. Equal temperament, today the most common tuning system used in the West, accomplished this by flattening each fifth by a twelfth of a Pythagorean comma (2 cents), thus giving perfect octaves.
Chinese mathematicians had been aware of the Pythagorean comma as early as 122 BC (its calculation is detailed in the Huainanzi), and circa 50 BC, Ching Fang discovered that if the cycle of perfect fifths were continued beyond 12 all the way to 53, the difference between this 53rd pitch and the starting pitch would be much smaller than the Pythagorean comma, which was later named Mercator's comma. (see: history of 53 equal temperament).
Other intervals of similar size are the syntonic comma, and Holdrian comma.
