Comma (music)

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways.[1] Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves".[2] The word comma used without qualification refers to the syntonic comma,[3] which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Syntonic comma on C Play 
Pythagorean comma on C Play 

The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "an act of cutting".

Within the same tuning system, two enharmonically equivalent notes (such as G and A) may have a slightly different frequency, and the interval between them is a comma. For example, in extended scales produced with five-limit tuning an A tuned as a major third below C5 and a G tuned as two major thirds above C4 are not exactly the same note, as they would be in equal temperament. The interval between those notes, the diesis, is an easily audible comma (its size is more than 40% of a semitone).

Commas are often defined as the difference in size between two semitones.[citation needed] Each meantone temperament tuning system produces a 12-tone scale characterized by two different kinds of semitones (diatonic and chromatic), and hence by a comma of unique size. The same is true for Pythagorean tuning.

Lesser diesis defined in quarter-comma meantone as difference between semitones (m2 − A1), or interval between enharmonically equivalent notes (from C to D). The interval from C to D is narrower than in Pythagorean tuning (see below). Play 
Pythagorean comma (PC) defined in Pythagorean tuning as difference between semitones (A1 − m2), or interval between enharmonically equivalent notes (from D to C). The interval from C to D is wider than in quarter-comma meantone (see above).

In just intonation, more than two kinds of semitones may be produced. Thus, a single tuning system may be characterized by several different commas. For instance, a commonly used version of five-limit tuning produces a 12-tone scale with four kinds of semitones and four commas.

The size of commas is commonly expressed and compared in terms of cents11200 fractions of an octave on a logarithmic scale.


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