Syntonic comma
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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 21.51 cents.
It 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. 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. The difference between these is 21.51, a syntonic comma.
This difference is significant 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 5-limit just intonation, the syntonic comma is the ratio between the major tone of 9:8 and the minor tone of 10:9 (and is therefore 81:80). In meantone temperaments, the major and minor tones are made equal. In Pythagorean tuning, the minor tone, is replaced by the major tone of 9:8. In quarter-comma meantone, the major and minor tones are made equal to the square root of 5:4.
Mathematically, by Størmer's theorem, 81/80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios.
Another frequently encountered comma is the Pythagorean comma.
