In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯ ( Play), or D♭ and C♯. It is equal to the frequency ratio 531441:524288, or approximately 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma which musical temperaments often refer to tempering is the Pythagorean comma.
The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning), or the difference between twelve just perfect fifths and seven octaves, or the difference between three Pythagorean ditones and one octave (this is the reason why the Pythagorean comma is also called ditonic comma).
The diminished second, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides therefore with the opposite of a Pythagorean comma, and can be viewed as a descending Pythagorean comma (e.g. from C♯ to D♭), equal to about −23.46 cents.
Read more about Pythagorean Comma: Derivation, Size, Circle of Fifths and Enharmonic Change, History
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