Comma (music) - Commas in Different Contexts

Commas in Different Contexts

In the column labeled "Difference between semitones", m2 is the minor second (diatonic semitone), A1 is the augmented unison (chromatic semitone), and S1, S2, S3, S4 are semitones as defined here. In the columns labeled "Interval 1" and "Interval 2", all intervals are presumed to be tuned in just intonation. Notice that the Pythagorean comma (PC) and the syntonic comma (SC) are basic intervals which can be used as yardsticks to define some of the other commas. For instance, the difference between them is a small comma called schisma. A schisma is not audible in many contexts, as its size is narrower than the smallest noticeable difference between tones (which is around six cents).

Name of comma Alternative Name Definitions Size
Difference between
semitones		 Difference between
commas		 Difference between Cents Ratio
Interval 1 Interval 2
Schisma Skhisma A1 − m2
in 1/12-comma meantone		 1 PC − 1 SC 8 perfect fifths +
1 major third		 5 octaves 1.95 32805:32768
Septimal kleisma 2 major thirds +
1 septimal major third		 Octave 7.71 225:224
Kleisma 6 minor thirds Tritave (1 octave +
1 perfect fifth)		 8.11 15625:15552
Small undecimal comma 17.58 99:98
Diaschisma Diaskhisma m2 − A1
in 1/6-comma meantone,
S3 − S2
in 5-limit tuning		 2 SC − 1 PC 3 octaves 4 perfect fifths +
2 major thirds		 19.55 2048:2025
Syntonic comma (SC) Didymus' comma S2 − S1
in 5-limit tuning		 4 perfect fifths 2 octaves +
1 major third		 21.51 81:80
Major tone Minor tone
Pythagorean comma (PC) Ditonic comma A1 − m2
in Pythagorean tuning		 12 perfect fifths 7 octaves 23.46 531441:524288
Septimal comma Minor seventh Septimal minor seventh 27.26 64:63
Diesis Lesser diesis m2 − A1
in 1/4-comma meantone,
S3 − S1
in 5-limit tuning		 3 SC − 1 PC Octave 3 major thirds 41.06 128:125
Undecimal comma Undecimal quarter-tone Undecimal tritone Perfect fourth 53.27 33:32
Greater diesis m2 − A1
in 1/3-comma meantone,
S4 − S1
in 5-limit tuning		 4 SC − 1 PC 4 minor thirds Octave 62.57 648:625
Tridecimal comma Tridecimal third-tone Tridecimal tritone Perfect fourth 65.3 27:26

The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from freely using triads and chords, forcing them to write music with relatively simple texture. In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if you decrease by a syntonic comma (81:80) the frequency of E, C-E (a major third), and E-G (a minor third) become just. Namely, C-E is flattened to a justly intonated ratio of

and at the same time E-G is sharpened to the just ratio of

This brought to the creation of a new tuning system, known as quarter-comma meantone, which permitted the full development of music with complex texture, such as polyphonic music, or melodies with instrumental accompaniment. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the family belonging to the syntonic temperament continuum, including meantone temperaments.

Read more about this topic:  Comma (music)

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