Syntonic comma

In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (around 21.51 cents). Two notes that differ by this interval would sound different from each other even to untrained ears, 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 a "comma of Didymus" because it is the amount by which Didymus corrected the Pythagorean major third (81:64, around 407.82 cents) to a just major third (5:4, around 386.31 cents).

The prime factors of the just interval 81/80 known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as 81/1 * 1/80 or (fully expanded and sorted by prime) 1/2 * 1/2 * 1/2 * 1/2 * 3/1 * 3/1 * 3/1 *3/1 * 1/5. All sequences are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below:

On a piano keyboard (typically tuned with 12-tone equal temperament) a stack of four fifths (700 * 4 = 2800 cents) is exactly equal to two octaves (1200 * 2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). Pythagorean tuning uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. Quarter-comma meantone uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments.

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