# Syntonic comma

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Syntonic comma
Syntonic comma on C  Play .
Just perfect fifth on D  Play . The perfect fifth above D (A+) is a syntonic comma higher than the just major sixth (A).[1]
3-limit 9:8 major tone  Play .
5-limit 10:9 minor tone  Play .

In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma[2] is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents. Two notes that differ by this interval would sound different from each other even to untrained ears,[3] 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 Didymus' because it is the amount by which Didymus corrected the Pythagorean major third[4] to a just major third (81:64 or 407.82 cents - 21.51 = 386.31 cents or 5:4).

Composer Ben Johnston uses a "−" as an accidental to indicate a note is lowered 21.51 cents, or a "+" to indicate a note is raised 21.51 cents.[1]

## Relationships

The syntonic comma is the interval between a just major third (5:4) and a Pythagorean ditone (81:64). Another way of describing the syntonic comma, as a combination of more commonly encountered intervals, 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 (81:16). 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 (5:1 or 80:16). The difference between these is 21.51 cents (81:80), a syntonic comma. Equally, it can be described as the difference between three justly tuned perfect fourths (64/27 or 1494.13 cents), and a justly tuned minor third (6/5) an octave higher (12/5 or 1515.64 cents).

The difference of 21.51 cents has contemporary significance 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 just intonation, there are two kinds of major second, called major and minor tone. In 5-limit just intonation, they have a ratio of 9:8 and 10:9, and the ratio between them is the syntonic comma (81:80). Also, 27:16 ÷ 5:3 = 81:80.[4]

Mathematically, by Størmer's theorem, 81:80 is the closest superparticular ratio possible with regular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whose prime factors are limited to 2, 3, and 5. 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.

## Syntonic comma in the history of music

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 using triads and chords, forcing them for centuries 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

${81\over64} \cdot {80\over81} = {{1\cdot5}\over{4\cdot1}} = {5\over4}$

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

${32\over27} \cdot {81\over80} = {{2\cdot3}\over{1\cdot5}} = {6\over5}$

The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean wolf fifth. But the fifth C-G stays consonant, since only E has been flattened (C-E * E-G = 5/4 * 6/5 = 3/2), and can be used together with C-E to produce a C-major triad (C-E-G). These experiments eventually brought to the creation of a new tuning system, known as quarter-comma meantone, in which the number of major thirds was maximized, and most minor thirds were tuned to a ratio which was very close to the just 6:5. This result was obtained by flattening each fifth by a quarter of a syntonic comma, an amount which was considered negligible, and permitted the full development of music with complex texture, such as polyphonic music, or melody 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.

## References

1. ^ a b John Fonville. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.109, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106-137. and Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p.78. ISBN 9780252030987.
2. ^ Johnston B. (2006). "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0252030982.
3. ^ "Sol-Fa - The Key to Temperament", BBC.
4. ^ a b Llewelyn Southworth Lloyd (1937). Music and Sound, p.12. ISBN 0836951883.

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