Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways.^{[1]} The word comma used without qualification refers to the syntonic comma,^{[2]} which can be defined, for instance, as the difference between an F♯ tuned using the Dbased Pythagorean tuning system, and another F♯ tuned using the Dbased quartercomma meantone tuning system.
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 fivelimit tuning an A♭ tuned as a major third below C5 and a G♯ tuned as two major thirds above C4 is 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. Each meantone temperament tuning system produces a 12tone 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.

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 fivelimit tuning produces a 12tone scale with four kinds of semitones and four commas.
The size of commas is commonly expressed and compared in terms of cents – 1/1200 fractions of an octave on a logarithmic scale.
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 S_{1}, S_{2}, S_{3}, S_{4} 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 that 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 audible difference between tones (which is around six cents, also known as just noticeable difference, or JND).
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 1/12comma meantone  1 PC − 1 SC  8 perfect fifths + 1 major third  5 octaves  1.95  32805:32768 
Septimal kleisma  3 major thirds  1 Octave − 1 septimal comma  7.71  225:224  
Kleisma  6 minor thirds  Tritave (1 octave + 1 perfect fifth)  8.11  15625:15552  
Small undecimal comma^{[3]}  1 neutral second  1 Minor tone  17.40  100:99  
Diaschisma  Diaskhisma  m2 − A1 in 1/6comma meantone, S_{3} − S_{2} in 5limit tuning  2 SC − 1 PC  3 octaves  4 perfect fifths + 2 major thirds  19.55  2048:2025 
Syntonic comma (SC)  Didymus' comma  S_{2} − S_{1} in 5limit 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^{[4]}  Archytas' comma  Minor seventh  Septimal minor seventh  27.26  64:63  
Diesis  Lesser diesis  m2 − A1 in 1/4comma meantone, S_{3} − S_{1} in 5limit tuning  3 SC − 1 PC  Octave  3 major thirds  41.06  128:125 
Undecimal comma^{[5]}^{[6]}  Undecimal quartertone  Undecimal tritone  Perfect fourth  53.27  33:32  
Greater diesis  m2 − A1 in 1/3comma meantone, S_{4} − S_{1} in 5limit tuning  4 SC − 1 PC  4 minor thirds  Octave  62.57  648:625  
Tridecimal comma  Tridecimal thirdtone  Tridecimal tritone  Perfect fourth  65.34  27:26 
Many other commas have been enumerated and named by microtonalists^{[7]}
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, CE (a major third), and EG (a minor third) become just. Namely, CE is flattened to a justly intonated ratio of
and at the same time EG is sharpened to the just ratio of
This brought to the creation of a new tuning system, known as quartercomma 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.
Alternative definitions
In quartercomma meantone, and any kind of meantone temperament tuning system that tempers the fifth to a size smaller than 700 cents, the comma is a diminished second, which can be equivalently defined as the difference between:
 minor second and augmented unison (also known as diatonic and chromatic semitones), or
 major second and diminished third, or
 minor third and augmented second, or
 major third and diminished fourth, or
 perfect fourth and augmented third, or
 augmented fourth and diminished fifth, or
 perfect fifth and diminished sixth, or
 minor sixth and augmented fifth, or
 major sixth and diminished seventh, or
 minor seventh and augmented sixth, or
 major seventh and diminished octave.
In Pythagorean tuning, and any kind of meantone temperament tuning system that tempers the fifth to a size larger than 700 cents (such as 1/12comma meantone), the comma is the opposite of a diminished second, and therefore the opposite of the abovelisted differences. More exactly, in these tuning systems the diminished second is a descending inteval, while the comma is its ascending opposite. For instance, the Pythagorean comma (531441:524288, or about 23.5 cents) can be computed as the difference between a chromatic and a diatonic semitone, which is the opposite of a Pythagorean diminished second (524288:531441, or about 23.5 cents).
In each of the above mentioned tuning systems, the abovelisted differences have all the same size. For instance, in Pythagorean tuning they are all equal to the opposite of a Pythagorean comma, and in quartercomma meantone they are all equal to a diesis.
Notation
In the years 20002004, Marc Sabat and Wolfgang von Schweinitz worked together in Berlin to develop a method to exactly indicate pitches in staff notation. This method was called the extended HelmholtzEllis JI pitch notation.^{[8]} Sabat and Schweinitz take the "conventional" flats, naturals and sharps as a Pythagorean series of perfect fifths. Thus, a series of perfect fifths beginning with F proceeds C G D A E B F♯ and so on. The advantage for musicians is that conventional reading of the basic fourths and fifths remains familiar. Such an approach has also been advocated by Daniel James Wolf and by Joe Monzo, who refers to it by the acronym HEWM (HelmholtzEllisWolfMonzo).^{[9]} In the SabatSchweinitz design, syntonic commas are marked by arrows attached to the flat, natural or sharp sign, septimal commas using Giuseppe Tartini's symbol, and undecimal quartertones using the common practice quartertone signs (a single cross and backwards flat). For higher primes, additional signs have been designed. To facilitate quick estimation of pitches, cents indications may be added (downward deviations below and upward deviations above the respective accidental). The convention used is that the cents written refer to the tempered pitch implied by the flat, natural, or sharp sign and the note name. One of the great advantages of any such a notation is that it allows the natural harmonic series to be precisely notated. A complete legend and fonts for the notation (see samples) are open source and available from Plainsound Music Edition. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E F G A B.
Composer Ben Johnston uses a "−" as an accidental to indicate a note is lowered a syntonic comma, or a "+" to indicate a note is raised a syntonic comma.,^{[10]} however, Johnston's "basic scale" (the plain nominals A B C D E F G) is tuned to justintonation and thus already includes the syntonic comma. Thus a Pythagorean scale is C D E+ F G A+ B+, while a just scale is C D E F G A B.
Tempering of commas
Commas are frequently used in the description of musical temperaments, where they describe distinctions between musical intervals that are eliminated by that tuning system. A comma can be viewed as the distance between two musical intervals. When a given comma is tempered out in a tuning system, the ability to distinguish between those two intervals in that tuning is eliminated. For example, the difference between the diatonic semitone and chromatic semitone is called the diesis. The widely used 12tone equal temperament tempers out the diesis, and thus does not distinguish between the two different types of semitones. On the other hand, 19tone equal temperament does not temper out this comma, and thus it distinguishes between the two semitones.
Examples:
 12TET tempers out the diesis, as well as a variety of other commas.
 19TET tempers out the septimal diesis and syntonic comma, but does not temper out the diesis.
 22TET tempers out the septimal comma of Archytas, but does not temper out the septimal diesis or syntonic comma.
 31TET tempers out the syntonic comma, as well as the comma defined by the ratio (99:98), but does not temper out the diesis, septimal diesis, or septimal comma of Archytas.
Comma sequence
A comma sequence defines a musical temperament through a unique sequence of commas at increasing prime limits.^{[11]} The first comma of the comma sequence is in the qlimit, where q is the nth odd prime, and n is the number of generators. Subsequent commas are in prime limits, each one prime beyond the last.
Other intervals called commas
There are also several intervals called commas, which are not technically commas because they are not rational fractions like those above, but are irrational approximations of them. These include the Holdrian and Mercator's commas.
See also
References
 ↑ Waldo Selden Pratt (1922). Grove's Dictionary of Music and Musicians, Volume 1, p.568. John Alexander FullerMaitland, Sir George Grove, eds. Macmillan.
 ↑ Benson, Dave (2006). Music: A Mathematical Offering, p. 171. ISBN 0521853877.
 ↑ Haluška, Ján (2003). The Mathematical Theory of Tone Systems, p.xxvi. ISBN 0824747143.
 ↑ David Dunn, 2000. Harry Partch: an anthology of critical perspectives.
 ↑ Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", Harry Partch: An Anthology of Critical Perspectives , p.34. Dunn, David, ed. ISBN 9057550652. Difference between 11limit and 3limit intervals.
 ↑ Rasch, Rudolph (1988). "Farey Systems of Musical Intonation", Listening 2, p.40. Benitez, J.M. et al., eds. ISBN 3718648466. Source for 32:33 as difference between 11:16 & 2:3.
 ↑ List of commas, by prime limit in the Xenharmonic wiki
 ↑ see article "The Extended HelmholtzEllis JI Pitch Notation: eine Notationsmetode für dienatürlichen Intervalle" in "Mikrotöne und Mehr  Auf György Ligetis Hamburger Pfaden", ed. Manfred Stahnke, von Bockel Verlag, Hamburg 2005 ISBN 393269662X
 ↑ Tonalsoft Encyclopaedia article about 'HEWM' notation
 ↑ 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. 106137. 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
 ↑ Smith, G. W., "Comma Sequences", Xenharmony, retrieved 20120726.