Pythagorean comma
Small interval between musical notes / From Wikipedia, the free encyclopedia
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In musical tuning, the Pythagorean comma (or ditonic comma[lower-alpha 1]), 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♯, or D♭ and C♯.[1] It is equal to the frequency ratio (1.5)12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73[2]). The comma that musical temperaments often "temper" is the Pythagorean comma.[3]
The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma[4] (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning); the difference between 12 just perfect fifths and seven octaves; or the difference between three Pythagorean ditones and one octave (this is why the Pythagorean comma is also called a 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.