Syntonic comma

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In music theory, the syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents. Two notes that differ by this interval would sound different from each other even to untrained ears[2], but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is referred to as Didymus' because it is the amount by which Didymus corrected the Pythagorean major third[3] to a just major third (81:64 or 407.82 cents - 21.51 = 386.31 cents or 5:4).

Composer Ben Johnston uses a "−" as an accidental to indicate a note is lowered 21.51 cents, or a "+" to indicate a note is raised 21.51 cents.[1]

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The syntonic comma is the interval between a just perfect third (5:4) and a Pythagorean ditone (81:64). Another way of describing the syntonic comma, as a combination of more commonly encountered intervals, is the difference between four justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has its notes in the frequency ratio 3:2, which is equal to 701.955 cents, and four of them are equal to 2807.82 cents (81:16). A just major third has its notes in the frequency ratio 5:4, which is equal to 386.31 cents, and one of them plus two octaves is equal to 2786.31 cents (5:1 or 80:16). The difference between these is 21.51 cents (81:80), a syntonic comma.

The difference of 21.51 cents has contemporary significance because on a piano keyboard, four fifths is equal to two octaves plus a major third. Starting from a C, both combinations of intervals will end up at E. The fact that using justly tuned intervals yields two slightly different notes is one of the reasons compromises have to be made when deciding which system of musical tuning to use for an instrument. Pythagorean tuning tunes the fifths as exact 3:2s, but uses the relatively complex ratio of 81:64 for major thirds. Quarter-comma meantone, on the other hand, uses exact 5:4s for major thirds, but flattens each of the fifths by a quarter of a syntonic comma. Other systems use different compromises.

In just intonation, there are two kinds of major second, called major and minor tone. In 5-limit just intonation, they have a ratio of 9:8 and 10:9, and the ratio between them is the syntonic comma (81:80). Also, 27:16 ÷ 5:3 = 81:80[3].

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