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Encyclopedia > Pythagorean comma

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 For other uses, see Music (disambiguation). ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... In music, an octave (sometimes abbreviated 8ve) is the interval between one musical note and another with half or double its frequency. ... The cent is a logarithmic unit of measure used for musical intervals. ... Microtonal music is music using microtones â€” intervals of less than an equally spaced semitone, or as Charles Ives put it, the notes between the cracks of the piano. ... In music theory, the term interval describes the difference in pitch between two notes. ...

is called a Pythagorean comma, named after the ancient mathematician and philosopher Pythagoras. It is sometimes called a ditonic comma. Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 BC and 490 BC) was an Ionian (Greek) philosopher and founder of the religious movement called Pythagoreanism. ...

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. In music, there are two common meanings for tuning: Tuning practice, the act of tuning an instrument or voice. ... The chromatic scale is a scale with twelve pitches, each a semitone or half step apart. ... In music theory, the circle of fifths (or cycle of fifths) is an imaginary geometrical space that depicts relationships among the 12 equal-tempered pitch classes comprising the familiar chromatic scale. ... An equal temperament is a musical temperament -- that is, a system of tuning intended to approximate some form of just intonation -- in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ...

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). The Huainanzi (æ·®å—å­) is a Chinese classic from the 2nd century BC written under the patronage of the Han dynasty nobleman Liu An. ... Ching Fang (78-37BC) was a music theorist, most known for being the first to notice how closely a succession of 53 just fifths approximates 31 octaves. ... Nicholas (Nikolaus) Mercator (c. ... In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into fifty-three equally large steps. ...

Other intervals of similar size are the syntonic comma, and Holdrian comma. The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... The Holdrian comma, also called Holders comma or the Arabian comma, is a musical interval of 22. ...

Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... In music theory, a comma is a small or very small interval between two enharmonic notes tuned in different ways. ... The schisma, also spelled skhisma, is the ratio between a Pythagorean comma and a syntonic comma and equals 32805/32768, which is 1. ...

• Tonalsoft Encyclopaedia of Tuning Results from FactBites:

 Syntonic comma (306 words) The syntonic comma, also known as the comma of Didymus, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents. Pythagorean tuning tunes the fifths as exact 3:2s, but uses the relatively complex ratio of 81:64 for major thirds. In Quarter comma meantone, the major and minor tones are made equal to the square root of 5:4.
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