Comma (music) in the context of "Pythagorean comma"

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In Western music, intervals are most commonly differencing between notes of a diatonic scale. Intervals between successive notes of a scale are also known as scale steps. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C♯ and D♭. Intervals can be arbitrarily small, and even imperceptible to the human ear.

In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma, is a small comma type interval between two musical notes, equal to the frequency ratio ⁠81/80⁠ (= 1.0125) (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 also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third (⁠81/64⁠, around 407.82 cents) to a justly intoned major third (⁠5/4⁠, around 386.31 cents).

The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "a thing cut off", or "a hair", as in "off by just a hair".

In music, the ratio 225/224 is called the septimal kleisma (play). It is a minute comma type interval of approximately 7.7 cents. Factoring it into primes gives 2 3 5 7, which can be rewritten 2 (5/4) (9/7). That says that it is the amount that two major thirds of 5/4 and a septimal major third, or supermajor third, of 9/7 exceeds the octave.

The septimal kleisma can also be viewed as the difference between the diatonic semitone (16:15) and the septimal diatonic semitone (15:14).