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In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the microtonal interval defined as the difference between a Pythagorean apotome and a Pythagorean limma, e.g.
The resulting pitch is 23.46 cents (approximately a quarter of a semitone).
Circle of fifths and enharmonic change
The Pythagorean comma can also be thought of as the discrepancy between twelve justly tuned perfect fifths (ratio 3:2) (
play (help·info)) and seven octaves (ratio 2:1), e.g.
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In the following table of musical scales in the circle of fifths, the Pythagorean comma is visible as the small interval between e.g. F♯ and G♭.
The 6♭ and the 6♯ scales* are not identical - even though they are on the piano keyboard - but the ♭ scales are one Pythagorean comma lower. Disregarding this difference leads to enharmonic change.
* The 7♭ and 5♯, respectively 5♭ and 7♯ scales differ in the same way by one Pythagorean comma. Scales with seven accidentals are seldom used, because the enharmonic scales with five accidentals are treated as equivalent.
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, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma (approximately 2 cents), thus producing perfect octaves.
Another way to express this is that the harmonic fifth has a frequency (compared to the tonic) of 1.5 whereas the seventh semitone (based on 12 equal logarithmic divisions of an octave) is the seventh power of the twelfth root of 2 which is 1.4983... ie not quite the same (out by about 0.1%). Take this to the twelfth power (ie over 12 octaves) and you get the Pythagorean comma (about 1.4% difference).
History
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. This much smaller interval was later named Mercator's comma (see: history of 53 equal temperament).
See also
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