View source for Comma (music)

{{Short description|Very small interval arising from discrepancies in tuning}}
{{about|the interval|the silent pause|Caesura}}
[[File:Syntonic comma on C.png|thumb|right|[[Syntonic comma]] on C[[File:Syntonic comma on C.mid]]]]
[[File:Pythagorean comma on C.png|thumb|right|[[Pythagorean comma]] on C [[File:Pythagorean comma on C.mid]]]]

In [[music theory]], a '''comma''' is a very small [[interval (music)|interval]], the difference resulting from [[Musical tuning|tuning]] one [[note (music)|note]] two different ways.<ref>{{cite dictionary |first=Waldo Selden |last=Pratt |year=1922 |title=comma |dictionary=[[Grove's Dictionary of Music and Musicians]] |volume=1 |page=568 |editor1-first=J.A. |editor1-last=Fuller Maitland |editor1-link=John Alexander Fuller Maitland |editor2-first=G. |editor2-last=Grove |editor2-link=George Grove |publisher=Macmillan |edition=print }}</ref> Traditionally, there are two most common commata; the [[syntonic comma]] (80:81), "the difference between a [[just intonation|just]] major 3rd and four just perfect 5ths less two octaves", and the [[Pythagorean comma]] (524288:531441, approximately 73:74), "the difference between twelve 5ths and seven octaves".<ref>{{cite dictionary |first=Clive |last=Greated |year=2001 |title=Comma |dictionary=[[Grove Music Online]] |doi=10.1093/gmo/9781561592630.article.06186 |edition=online }} (''see also'' [[Grove's Dictionary of Music and Musicians]]). </ref> The word ''comma'' used without qualification refers to the [[syntonic comma]],<ref>{{cite book |last=Benson |first=Dave |year=2006 |title=Music: A mathematical offering |page=171 |ISBN=0-521-85387-7 }}</ref> which can be defined, for instance, as the difference between an F{{Music|#}} tuned using the D-based [[Pythagorean tuning]] system, and another F{{Music|#}} tuned using the D-based [[quarter-comma meantone]] [[tuning system]]. Pitches separated by either comma are considered the same note because conventional notation does not distinguish Pythagorean intervals from 5-limit intervals. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in {{nobr|[[53 equal temperament|53 {{sc|TET}}]],}} the [[harmonic seventh]] B{{music|7}}{{music|b}} and A{{music|#}} are both approximated by the same interval although they are a [[septimal kleisma]] apart.

== Etymology ==
Translated in this context, "comma" means "a hair" as in "off by just a hair"{{Citation needed|date=December 2023}}. The word "comma" came via Latin from [[Greek language|Greek]] ''[[wikt:κόμμα#Ancient_Greek|{{math|κόμμα}}]]'', from earlier *{{math|κοπ-μα}}: "the result or effect of cutting".

== Description ==
Within the same tuning system, two [[enharmonically equivalent]] notes (such as G{{Music|#}} and A{{Music|b}}) may have a slightly different frequency, and the interval between them is a comma. For example, in [[Five-limit tuning#Extension of the twelve-tone scale|extended scales]] produced with [[five-limit tuning]] an A{{Music|b}} tuned as a [[major third]] below C<sub>5</sub> and a G{{Music|#}} tuned as two major thirds above C<sub>4</sub> 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|date=August 2021}} Almost all of the [[meantone temperament|meantone tuning]] systems produce two different characteristic semitones: A [[diatonic semitone|''diatonic'' semitone]] for [[half step|short steps]] in the scale, and [[chromatic semitone|''chromatic'' semitone]] between a natural note, N{{music|n}}, and its sharp, N{{music|#}}. Their ratio (or in [[cents (music)|cents]], their difference) is a ''comma''; its size is unique to each [[meantone tuning]]. The same is true for Pythagorean tuning.
{| align="center"
|-
| [[File:Lesser diesis (difference m2-A1).PNG|thumb|center|467 px|Lesser [[diesis]] defined in [[quarter-comma meantone]] as difference between semitones ({{nowrap|m2 − A1}}), or interval between [[enharmonic|enharmonically equivalent]] notes (from C{{Music|#}} to D{{Music|b}}). The interval from C to D is narrower than in Pythagorean tuning (see below).[[File:Enharmonic scale segment on C.mid]]]]

[[File:Pythagorean comma (difference A1-m2).PNG|thumb|center|492 px|[[Pythagorean comma]] (PC) defined in [[Pythagorean tuning]] as difference between semitones ({{nowrap|A1 − m2}}), or interval between enharmonically equivalent notes (from D{{Music|b}} to C{{Music|#}}). 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 [[Five-limit tuning#Size of intervals|commonly used version]] of five-limit tuning produces a 12-tone scale with four kinds of semitones and [[5-limit tuning#Commas|four commas]].

The size of commas is commonly expressed and compared in terms of [[Cent (music)|cents]] – {{frac|1|1200}} fractions of an [[octave]] on a [[logarithm]]ic scale.

== Commas in different contexts ==
[[File:Comma size comparison.png|thumb|400px|Comparison of the size of different commas, in [[cent (music)|cents]]. Equal-tempered semitone added for comparison. JND is the [[just-noticeable difference]] between tones.]]

In the column below labeled "Difference between [[semitone]]s", '''{{small|min}}2''' is the minor second (diatonic semitone), '''{{sup|aug}}1''' is the augmented unison (chromatic semitone), and '''S{{sub|1}}''', '''S{{sub|2}}''', '''S{{sub|3}}''', '''S{{sub|4}}''' are semitones as defined [[5-limit tuning#Size of intervals|here]]. In the columns labeled "[[Interval (music)|Interval]]&nbsp;1" and "Interval&nbsp;2", all intervals are presumed to be tuned in [[just intonation]]. Notice that the [[Pythagorean comma]] ({{mvar|κ}}{{sub|𝜋}}) and the [[syntonic comma]] ({{mvar|κ}}{{sub|S}}) 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).

:{| class="wikitable"
!rowspan=3|Name of comma
!rowspan=3|Alternative name
!colspan=4|Definitions
!colspan=2|Size
|-
!rowspan=2|Difference between<br>[[semitone]]s
!rowspan=2|Difference between<br>commas
!colspan=2|Difference between
!rowspan=2|[[Cent (music)|Cents]]
!rowspan=2|[[Interval ratio|Ratio]]
|-
! [[Interval (music)|Interval]]&nbsp;1 !! Interval&nbsp;2
|-
| [[schisma]] || skhisma ||align="center"| {{sup|aug}}1 − {{small|min}}2<br>in {{nobr|{{sfrac| 1 | 12 }} comma}} [[meantone temperament|meantone]] || align="center"| 1 {{mvar|κ}}{{sub|𝜋}} − 1 {{mvar|κ}}{{sub|S}} || 8&nbsp;perfect fifths +<br>1&nbsp;[[major third]] || 5&nbsp;octaves || align="right"| 1.95 ||align="center"| <math>\tfrac{\ 32805\ }{ 32768 }</math>
|-
| [[septimal kleisma]] || || || || 3&nbsp;major thirds || 1&nbsp;[[octave]] &minus;<br> 1&nbsp;[[septimal comma]] || align="right"| 7.71 ||align="center"| <math>\tfrac{\ 225\ }{ 224 }</math>
|-
| [[kleisma]] || || || || 6 minor thirds || 1&nbsp;[[octave]] +<br>1&nbsp;[[perfect fifth]]<br/>   ("[[tritave]]") || align="right"| 8.11 ||align="center"| <math>\tfrac{\ 15625\ }{ 15552 }</math>
|-
| small undecimal comma<ref>{{cite book |last=Haluška |first=Ján |year=2003 |title=The Mathematical Theory of Tone Systems |page={{mvar|xxvi}} |publisher=CRC Press |isbn=0-8247-4714-3 }}</ref> || || || || 1&nbsp;[[neutral second]] || 1&nbsp;[[minor tone]] ||align="right"| 17.40 ||align="center"| <math>\tfrac{\ 100\ }{ 99 }</math>
|-
| [[diaschisma]] || diaskhisma || align="center"| {{small|min}}2 − {{sup|aug}}1<br>in {{nobr|{{sfrac| 1 | 6 }} comma}} meantone,<br>S<sub>3</sub> − S<sub>2</sub><br>in [[5-limit tuning]] || align="center"| 2 {{mvar|κ}}{{sub|S}} − 1 {{mvar|κ}}{{sub|𝜋}} || 3&nbsp;octaves || 4&nbsp;perfect fifths +<br>2&nbsp;major thirds || align="right"| 19.55 ||align="center"| <math>\tfrac{\ 2048\ }{ 2025 }</math>
|-
|rowspan=2| [[syntonic comma]]<br/>{{center|({{mvar|κ}}{{sub|S}})}} || rowspan=2| Didymus' comma || align="center" rowspan=2|S<sub>2</sub> − S<sub>1</sub><br>in 5&nbsp;limit tuning ||rowspan=2| || 4&nbsp;perfect fifths || 2&nbsp;octaves +<br>1&nbsp;major third ||rowspan=2 align="right"| 21.51 ||rowspan=2 align="center"| <math>\tfrac{\ 81\ }{ 80 }</math>
|-
| [[major second|major tone]] || [[minor tone]]
|-
| [[53 equal temperament|53&nbsp;{{sc|tet}}]] comma<br/>{{center|({{mvar|κ}}{{sub|53}})}} || [[Holdrian comma]]<br/>1&nbsp;step<br/>{{center|(in [[53 equal temperament|53&nbsp;{{sc|tet}}]])}} ||align="center"| {{nobr|{{sfrac| 1 | 9 }} [[major second|major tone]]}}<br/>{{center|(in [[53 equal temperament|53&nbsp;{{sc|tet}}]])}} ||align="center"| {{nobr|{{sfrac| 1 | 8 }} [[minor tone]]}}<br/>{{center|(in [[53 equal temperament|53&nbsp;{{sc|tet}}]])}} || [[major second|major tone]]<br/>{{center|(in [[53 equal temperament|53&nbsp;{{sc|tet}}]])}} || [[minor tone]]<br/>{{center|(in [[53 equal temperament|53&nbsp;{{sc|tet}}]])}} ||align="right"| 22.64 ||align="center"| <math>\bigl( 2 \bigr)^\tfrac{\ 1\ }{ 53 }</math>
|-
| [[Pythagorean comma]]<br/>{{center|({{mvar|κ}}{{sub|𝜋}})}} || ditonic comma || align="center"| [[Pythagorean apotome|{{sup|aug}}1]] − [[Pythagorean limma|{{small|min}}2]]<br>(in [[Pythagorean tuning]]) || || 12&nbsp;perfect fifths || 7&nbsp;octaves || align="right"| 23.46 ||align="center"| <math>\tfrac{\ 531441\ }{ 524288 }</math>
|-
| [[septimal comma]]<ref name=Rasch/> || [[Archytas]]' comma<br/>{{center|({{mvar|κ}}{{sub|A}})}} || || || [[minor seventh]] || [[septimal minor seventh]] ||align="right"| 27.26 ||align="center"| <math>\tfrac{\ 64\ }{ 63 }</math>
|-
| [[diesis]] || lesser diesis<br>[[diminished second]] || align="center"| {{small|min}}2 − {{sup|aug}}1<br>in [[Quarter-comma meantone|{{nobr|{{sfrac| 1 | 4 }} comma}} meantone]],<br>S<sub>3</sub> − S<sub>1</sub><br>in 5&nbsp;limit tuning || align="center"| 3 {{mvar|κ}}{{sub|S}} − 1 {{mvar|κ}}{{sub|𝜋}} || octave || 3&nbsp;major thirds ||align="right"| 41.06 ||align="center"| <math>\tfrac{\ 128\ }{ 125 }</math>
|-
| undecimal comma<ref name=Rasch>{{cite book |last=Rasch |first=Rudolph |year=2000 |section=A word or two on the tunings of Harry Partch |title=Harry Partch: An anthology of critical perspectives |page=[https://books.google.com/books?id=kIKar6TykE4C&pg=PA34 34] |editor-last=Dunn |editor-first=David |isbn=90-5755-065-2}} — Describes difference between 11&nbsp;[[limit (music)|limit]] and 3&nbsp;limit intervals.</ref><ref>{{cite book |last=Rasch |first=Rudolph |year=1988 |section=Farey systems of musical intonation |title=Listening |volume=2 |page=40 |editor1-last=Benitez |editor1-first=J.M. |display-editors=etal |isbn=3-7186-4846-6 }} = Source for 32:33 as difference between 11:16 & 2:3&nbsp;.</ref> || Undecimal [[quarter-tone]] || || || [[undecimal tritone]] || [[perfect fourth]] || align="right"| 53.27 ||align="center"| <math>\tfrac{\ 33\ }{ 32 }</math>
|-
| [[greater diesis]] || || align="center"| {{small|min}}2 − {{sup|aug}}1<br>in {{nobr|{{sfrac| 1 | 3 }} comma}} meantone,<br>S<sub>4</sub> − S<sub>1</sub><br>in 5&nbsp;limit tuning || align="center"| 4 {{mvar|κ}}{{sub|S}} − 1 {{mvar|κ}}{{sub|𝜋}} || 4&nbsp;minor thirds || octave || align="right"| 62.57 ||align="center"| <math>\tfrac{\ 648\ }{ 625 }</math>
|-
| tridecimal comma || tridecimal third-tone || || || [[Tritone#Size in different tuning systems|tridecimal tritone]] || [[perfect fourth]] ||align="right"| 65.34 ||align="center"| <math>\tfrac{\ 27\ }{ 26 }</math>
|}

Many other commas have been enumerated and named by microtonalists.<ref>{{cite web |url=https://en.xen.wiki/w/Comma |title=List of commas, by prime limit |website=Xenharmonic wiki}}</ref>

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 [[consonance and dissonance|dissonant]], and this prevented musicians from freely using [[triad (music)|triad]]s and [[chord (music)|chord]]s, forcing them to write music with relatively simple [[texture (music)|texture]]. Musicians in late [[Middle Ages]] recognized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made [[consonance and dissonance|consonant]]. For instance, if you decrease the frequency of E by a syntonic comma (81:80), C–E (a major third) and E–G (a minor third) become just: C–E is flattened to a [[just intonation|just]] ratio of

:<math> \frac{\ 81\ }{ 64 } \cdot \frac{\ 80\ }{ 81 } = \frac{\ 1 \cdot 5\ }{ 4 \cdot 1 } = \frac{\ 5\ }{ 4 }</math>

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

:<math> \frac{ 32 }{\ 27\ } \cdot \frac{ 81 }{\ 80\ } = \frac{ 2 \cdot 3 }{\ 1 \cdot 5\ } = \frac{ 6 }{\ 5\ }</math>

This led to the creation of a new [[tuning system]], known as [[quarter-comma meantone]], which permitted the full development of music with complex [[Texture (music)|texture]], such as [[polyphony|polyphonic music]], or melodies with [[homophony|instrumental accompaniment]]. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths throughout the family of [[syntonic temperament]]s, including [[meantone temperament]]s.

===Alternative definitions===
In [[quarter-comma meantone]], and any kind of meantone temperament tuning system that tempers the fifth to a size smaller than 700&nbsp;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 [[semitone]]s), 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&nbsp;cents (such as {{nobr|{{sfrac| 1 | 12 }} comma}} meantone), the comma is the opposite of a diminished second, and therefore the opposite of the above-listed differences. More exactly, in these tuning systems the diminished second is a descending interval, while the comma is its ascending opposite. For instance, the Pythagorean comma (531441:524288, or about 23.5&nbsp;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&nbsp;cents).

In each of the above-mentioned tuning systems, the above-listed differences have all the same size. For instance, in [[Pythagorean tuning#Size of intervals|Pythagorean tuning]] they are all equal to the opposite of a [[Pythagorean comma]], and in [[Quarter-comma meantone#Size of intervals|quarter comma meantone]] they are all equal to a [[diesis]].

== Notation ==
In the years 2000–2004, [[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 Helmholtz-Ellis [[just intonation|JI]] pitch notation.<ref>see article "The Extended Helmholtz-Ellis 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|3-932696-62-X}}</ref> 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 {{nowrap|C G D A E B F{{music|#}}}} 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 (Helmholtz-Ellis-Wolf-Monzo).<ref>[http://tonalsoft.com/enc/h/hewm.aspx Tonalsoft Encyclopaedia article about 'HEWM' notation]</ref> In the Sabat-Schweinitz 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 [[flat (music)|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.{{full citation needed|date=December 2011}} Thus a Pythagorean scale is {{nowrap|C D E F G A B C}}, while a just scale is {{nowrap|C D E[[File:HE syntonic comma - natural down.png|9px]] F G A [[File:HE syntonic comma - natural down.png|9px]] B[[File:HE syntonic comma - natural down.png|9px]] C}}.

Composer [[Ben Johnston (composer)|Ben Johnston]] used a "−" as an accidental to indicate a note is lowered a syntonic comma, or a "+" to indicate a note is raised a syntonic comma;<ref name="Fonville">[[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 [[Ben Johnston (composer)|Johnston, Ben]] and [[Bob Gilmore|Gilmore, Bob]] (2006). "A Notation System for Extended Just Intonation" (2003), ''"Maximum clarity" and Other Writings on Music'', p. 78. {{ISBN|978-0-252-03098-7}}</ref> however, Johnston's "basic scale" (the plain nominals {{nowrap|A B C D E F G}}) is tuned to just-intonation and thus already includes the syntonic comma. Thus a Pythagorean scale is {{nowrap|C D E+ F G A+ B+ C}}, while a just scale is {{nowrap|C D E F G A B}}.

==Tempering of commas==
Commas are frequently used in the description of [[musical temperament]]s, 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 12&nbsp;tone [[equal temperament]] ''tempers out'' the diesis, and thus does not distinguish between the two different types of semitones. On the other hand, [[19 tone equal temperament]] does not temper out this comma, and thus it distinguishes between the two semitones.

Examples:
* {{nobr|[[12 equal temperament|12 {{sc|TET}}]]}} tempers out the diesis, as well as a variety of other commas.
* {{nobr|[[19 equal temperament|19 {{sc|TET}}]]}} tempers out the [[septimal diesis]] and [[syntonic comma]], but does not temper out the diesis.
* {{nobr|[[22 equal temperament|22 {{sc|TET}}]]}} tempers out the [[septimal comma]] of [[Archytas]], but does not temper out the septimal diesis or syntonic comma.
* {{nobr|[[31 equal temperament|31 {{sc|TET}}]]}} tempers out the syntonic comma, as well as the comma defined by the ratio {{sfrac| 99 | 98 }}, but does not temper out the diesis, septimal diesis, or septimal comma of Archytas.

The following table lists the number of steps used that represent various just intervals in various tuning systems. Zeros indicate that the interval is a comma (i.e. is tempered out) in that particular equal temperament. For example, the zero in the row labeled <math>\tfrac{ 128 }{ 125 }</math> and the column labeled <abbr title="12-tone equal divison of the octave">12 {{sc|T EDO}}</abbr> means that <math>\tfrac{ 128 }{ 125 }</math> is represented with zero steps in <abbr title="12-tone equal divison of the octave">12 {{sc|T EDO}}</abbr>, making it a comma in that temperament. Because commas are often thought of as the ratio between two different ways to tune the same note and <math>\tfrac{ 128 }{ 125 } = \tfrac{ 2 }{ 1 } \div \left(\tfrac{ 5 }{ 4 }\right)^3</math>, so the fact that this corresponds to zero steps in <abbr title="12-tone equal divison of the octave">12 {{sc|T EDO}}</abbr> means that in <abbr title="12-tone equal divison of the octave">12 {{sc|T EDO}}</abbr>, [[Octave|<math>\tfrac{ 2 }{ 1 }</math>]] is represented with zero more steps than (i.e., with the same number of steps as) three copies of [[Major third|<math>\tfrac{ 5 }{ 4 }</math>]]. (Specifically, <abbr title="12-tone equal divison of the octave">12 {{sc|T EDO}}</abbr> represents <math>\tfrac{ 2 }{ 1 }</math> with twelve steps and each copy of <math>\tfrac{ 5 }{ 4 }</math> with four steps.) Analogously, the 1 in the same row in the column labeled <abbr title="19-tone equal divison of the octave">19 {{sc|T EDO}}</abbr> means that <abbr title="19-tone equal divison of the octave">19 {{sc|T EDO}}</abbr> represents <math>\tfrac{ 2 }{ 1 }</math> with one more step than three copies of <math>\tfrac{ 5 }{ 4 }</math>, the 2 in the same row in the column labeled <abbr title="41-tone equal divison of the octave">41 {{sc|T EDO}}</abbr> means that <abbr title="41-tone equal divison of the octave">41 {{sc|T EDO}}</abbr> represents <math>\tfrac{ 2 }{ 1 }</math> with two more steps than three copies of <math>\tfrac{ 5 }{ 4 }</math>, and so on. Analogously, the -1 in the same row in the column labeled <abbr title="5-tone equal divison of the octave">5 {{sc|T EDO}}</abbr> means that <abbr title="5-tone equal divison of the octave">5 {{sc|T EDO}}</abbr> represents <math>\tfrac{ 2 }{ 1 }</math> with one ''fewer'' step than three copies of <math>\tfrac{ 5 }{ 4 }</math>. Specifically, <abbr title="5-tone equal divison of the octave">5 {{sc|T EDO}}</abbr> represents <math>\tfrac{ 2 }{ 1 }</math> with five steps and each copy of <math>\tfrac{ 5 }{ 4 }</math> with two steps. Because <math>\tfrac{ 5 }{ 4 }</math> is slightly less than <math>2^\tfrac{ 1 }{ 3 }</math> but is represented in <abbr title="5-tone equal divison of the octave">5 {{sc|T EDO}}</abbr> by <math>2^\tfrac{ 2 }{ 5 }</math>, <abbr title="5-tone equal divison of the octave">5 {{sc|T EDO}}</abbr>'s representation of <math>\tfrac{ 5 }{ 4 }</math> is significantly sharper than the justly-intoned version, hence the -1. Thus, in <abbr title="5-tone equal divison of the octave">5 {{sc|T EDO}}</abbr>, going up by a factor of <math>\tfrac{ 128 }{ 125 }</math> (e.g., from the representation of <math>\tfrac{ 2 }{ 1 }</math>to the representation of <math>\left(\tfrac{ 5 }{ 4 }\right)^3</math>) would require one to go ''down'' by one scale step.

Each of the frequency ratios in the first column are linked to its respective article.

:{| class="wikitable mw-collapsible mw-collapsed" style="text-align:right;"
|-
!style="text-align:center;"| Interval<br/>{{small|(frequency ratio)}}
!style="text-align:right;"| [[5 equal temperament|5 {{sc|T EDO}}]]
!style="text-align:right;"| [[7 equal temperament|7 {{sc|T EDO}}]]
!style="text-align:right;"| [[12 equal temperament|12 {{sc|T EDO}}]]
!style="text-align:right;"| [[15 equal temperament|15 {{sc|T EDO}}]]
!style="text-align:right;"| [[19 equal temperament|19 {{sc|T EDO}}]]
!style="text-align:right;"| [[22 equal temperament|22 {{sc|T EDO}}]]
!style="text-align:right;"| [[31 equal temperament|31 {{sc|T EDO}}]]
!style="text-align:right;"| [[34 equal temperament|34 {{sc|T EDO}}]]
!style="text-align:right;"| [[41 equal temperament|41 {{sc|T EDO}}]]
!style="text-align:right;"| [[53 equal temperament|53 {{sc|T EDO}}]]
!style="text-align:right;"| [[72 equal temperament|72 {{sc|T EDO}}]]
|-
|style="text-align:center;"| [[octave| &nbsp;<math>\tfrac{\ 2\ }{ 1 }</math>&nbsp; ]]
| 5 || 7 || 12 || 15 || 19 || 22 || 31 || 34 || 41 || 53 || 72
|-
|style="text-align:center;"| [[major seventh| &nbsp;<math>\tfrac{\ 15\ }{ 8 }</math>&nbsp; ]]
| 5 || 6 || 11 || 14 || 17 || 20 || 28 || 31 || 37 || 48 || 65
|-
|style="text-align:center;"| [[minor seventh| &nbsp;<math>\tfrac{\ 9\ }{ 5 }</math>&nbsp; ]]
| 4 || 6 || 10 || 13 || 16 || 19 || 26 || 29 || 35 || 45 || 61
|-
|style="text-align:center;"| [[harmonic seventh| &nbsp;<math>\tfrac{\ 7\ }{ 4 }</math>&nbsp; ]]
| 4 || 6 || 10 || 12 || 15 || 18 || 25 || 28 || 33 || 43 || 58
|-
|style="text-align:center;"| [[major sixth| &nbsp;<math>\tfrac{\ 5\ }{ 3 }</math>&nbsp; ]]
| 4 || 5 || 9 || 11 || 14 || 16 || 23 || 25 || 30 || 39 || 53
|-
|style="text-align:center;"| [[minor sixth| &nbsp;<math>\tfrac{\ 8\ }{ 5 }</math>&nbsp; ]]
| 3 || 5 || 8 || 10 || 13 || 15 || 21 || 23 || 28 || 36 || 49
|-
|style="text-align:center;"| [[perfect fifth| &nbsp;<math>\tfrac{\ 3\ }{ 2 }</math>&nbsp; ]]
| 3 || 4 || 7 || 9 || 11 || 13 || 18 || 20 || 24 || 31 || 42
|-
|style="text-align:center;"| [[Euler's tritone| &nbsp;<math>\tfrac{\ 10\ }{ 7 }</math>&nbsp; ]]
| 3 || 3 || 6 || 8 || 10 || 11 || 16 || 17 || 21 || 27 || 37
|-
|style="text-align:center;"| [[diminished fifth| &nbsp;<math>\tfrac{\ 64\ }{ 45 }</math>&nbsp; ]]
| 2 || 4 || 6 || 7 || 10 || 11 || 16 || 17 || 21 || 27 || 37
|-
|style="text-align:center;"| [[augmented fourth| &nbsp;<math>\tfrac{\ 45\ }{ 32 }</math>&nbsp; ]]
| 3 || 3 || 6 || 8 || 9 || 11 || 15 || 17 || 20 || 26 || 35
|-
|style="text-align:center;"| [[Huygens' tritone| &nbsp;<math>\tfrac{\ 7\ }{ 5 }</math>&nbsp; ]]
| 2 || 4 || 6 || 7 || 9 || 11 || 15 || 17 || 20 || 26 || 35
|-
|style="text-align:center;"| [[perfect fourth| &nbsp;<math>\tfrac{\ 4\ }{ 3 }</math>&nbsp; ]]
| 2 || 3 || 5 || 6 || 8 || 9 || 13 || 14 || 17 || 22 || 30
|-
|style="text-align:center;"| [[septimal major third| &nbsp;<math>\tfrac{\ 9\ }{ 7 }</math>&nbsp; ]]
| 2 || 2 || 4 || 6 || 7 || 8 || 11 || 12 || 15 || 19 || 26
|-
|style="text-align:center;"| [[major third| &nbsp;<math>\tfrac{\ 5\ }{ 4 }</math>&nbsp; ]]
| 2 || 2 || 4 || 5 || 6 || 7 || 10 || 11 || 13 || 17 || 23
|-
|style="text-align:center;"| [[minor third| &nbsp;<math>\tfrac{\ 6\ }{ 5 }</math>&nbsp; ]]
| 1 || 2 || 3 || 4 || 5 || 6 || 8 || 9 || 11 || 14 || 19
|-
|style="text-align:center;"| [[septimal minor third| &nbsp;<math>\tfrac{\ 7\ }{ 6 }</math>&nbsp; ]]
| 1 || 2 || 3 || 3 || 4 || 5 || 7 || 8 || 9 || 12 || 16
|-
|style="text-align:center;"| [[septimal whole tone| &nbsp;<math>\tfrac{\ 8\ }{ 7 }</math>&nbsp; ]]
| 1 || 1 || 2 || 3 || 4 || 4 || 6 || 6 || 8 || 10 || 14
|-
|style="text-align:center;"| [[major tone| &nbsp;<math>\tfrac{\ 9\ }{ 8 }</math>&nbsp; ]]
| 1 || 1 || 2 || 3 || 3 || 4 || 5 || 6 || 7 || 9 || 12
|-
|style="text-align:center;"| [[minor tone| &nbsp;<math>\tfrac{\ 10\ }{ 9 }</math>&nbsp; ]]
| 1 || 1 || 2 || 2 || 3 || 3 || 5 || 5 || 6 || 8 || 11
|-
|style="text-align:center;"| [[major diatonic semitone| &nbsp;<math>\tfrac{\ 27\ }{ 25 }</math>&nbsp; ]]
| 0 || 1 || 1 || 2 || 2 || 3 || 3 || 4 || 5 || 6 || 8
|-
|style="text-align:center;"| [[septimal diatonic semitone| &nbsp;<math>\tfrac{\ 15\ }{ 14 }</math>&nbsp; ]]
| 1 || 0 || 1 || 2 || 2 || 2 || 3 || 3 || 4 || 5 || 7
|-
|style="text-align:center;"| [[just diatonic semitone| &nbsp;<math>\tfrac{\ 16\ }{ 15 }</math>&nbsp; ]]
| 0 || 1 || 1 || 1 || 2 || 2 || 3 || 3 || 4 || 5 || 7
|-
|style="text-align:center;"| [[septimal chromatic semitone| &nbsp;<math>\tfrac{\ 21\ }{ 20 }</math>&nbsp; ]]
| 0 || 1 || 1 || 1 || 1 || 2 || 2 || 3 || 3 || 4 || 5
|-
|style="text-align:center;"| [[just chromatic semitone| &nbsp;<math>\tfrac{\ 25\ }{ 24 }</math>&nbsp; ]]
| 1 || 0 || 1 || 1 || 1 || 1 || 2 || 2 || 2 || 3 || 4
|-
|style="text-align:center;"| [[septimal third tone| &nbsp;<math>\tfrac{\ 28\ }{ 27 }</math>&nbsp; ]]
| 0 || 1 || 1 || 0 || 1 || 1 || 2 || 2 || 2 || 3 || 4
|-
|style="text-align:center;"| [[greater diesis| &nbsp;<math>\tfrac{\ 648\ }{ 625 }</math>&nbsp; ]]
| −1 || 1 || 0 || 1 || 1 || 2 || 1 || 2 || 3 || 3 || 4
|-
|style="text-align:center;"| [[septimal quarter tone| &nbsp;<math>\tfrac{\ 36\ }{ 35 }</math>&nbsp; ]]
| 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 2 || 2 || 3
|-
|style="text-align:center;"| [[lesser diesis| &nbsp;<math>\tfrac{\ 128\ }{ 125 }</math>&nbsp; ]] 
| −1 || 1 || 0 || 0 || 1 || 1 || 1 || 1 || 2 || 2 || 3
|-
|style="text-align:center;"| [[septimal diesis| &nbsp;<math>\tfrac{\ 49\ }{ 48 }</math>&nbsp; ]]
| 0 || 1 || 1 || 0 || 0 || 1 || 1 || 2 || 1 || 2 || 2
|-
|style="text-align:center;"| [[jubilisma| &nbsp;<math>\tfrac{\ 50\ }{ 49 }</math>&nbsp; ]]
| 1 || −1 || 0 || 1 || 1 || 0 || 1 || 0 || 1 || 1 || 2
|-
|style="text-align:center;"| [[Archytas' comma| &nbsp;<math>\tfrac{\ 64\ }{ 63 }</math>&nbsp; ]]
| 0 || 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 || 2
|-
|style="text-align:center;"| [[Pythagorean comma| &nbsp;<math>\tfrac{\ 531441\ }{ 524288 }</math>&nbsp; ]]
| 1 || −1 || 0 || 3 || −1 || 2 || −1 || 2 || 1 || 1 || 0
|-
|style="text-align:center;"| [[syntonic comma| &nbsp;<math>\tfrac{\ 81\ }{ 80 }</math>&nbsp; ]]
| 0 || 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 || 1 || 1
|-
|style="text-align:center;"| [[diaschisma| &nbsp;<math>\tfrac{\ 2048\ }{ 2025 }</math>&nbsp; ]]
| −1 || 1 || 0 || -1 || 1 || 0 || 1 || 0 || 1 || 1 || 2
|-
|style="text-align:center;"| [[septimal semicomma| &nbsp;<math>\tfrac{\ 126\ }{ 125 }</math>&nbsp; ]]
| −1 || 1 || 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1 || 1
|-
|style="text-align:center;"| [[Orwell comma| &nbsp;<math>\tfrac{\ 1728\ }{ 1715 }</math>&nbsp; ]]
| 0 || −1 || −1 || 1 || 1 || 0 || 0 || −1 || 1 || 0 || 1
|-
|style="text-align:center;"| {{nobr|[[semicomma| &thinsp;<math>\tfrac{\ 2109375\ }{ 2097152 }</math>&thinsp; ]]}}
| 3 || −2 || 1 || 2 || −1 || 0 || 0 || 1 || −1 || 0 || −1
|-
|style="text-align:center;"| [[kleisma| &nbsp;<math>\tfrac{\ 15625\ }{ 15552 }</math>&nbsp; ]]
| 2 || −1 || 1 || 0 || 0 || −1 || 1 || 0 || −1 || 0 || 0
|-
|style="text-align:center;"| [[septimal kleisma| &nbsp;<math>\tfrac{\ 225\ }{ 224 }</math>&nbsp; ]]
| 1 || −1 || 0 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|style="text-align:center;"| [[schisma| &nbsp;<math>\tfrac{\ 32805\ }{ 32768 }</math>&nbsp; ]]
| 1 || −1 || 0 || 2 || −1 || 1 || −1 || 1 || 0 || 0 || −1
|-
|style="text-align:center;"| [[breedsma| &nbsp;<math>\tfrac{\ 2401\ }{ 2400 }</math>&nbsp; ]]
| −1 || 2 || 1 || -1 || −1 || 1 || 0 || 2 || 0 || 1 || 0
|-
|style="text-align:center;"| [[ragisma| &nbsp;<math>\tfrac{\ 4375\ }{ 4374 }</math>&nbsp; ]]
| −1 || 0 || −1 || 1 || 0 || 1 || −1 || 0 || 1 || 0 || 0
|}

The comma can also be considered to be the fractional interval that remains after a "full circle" of some repeated chosen interval; the repeated intervals are all the same size, in relative pitch, and all the tones produced are reduced or raised by whole octaves back to the octave surrounding the starting pitch. The Pythagorean comma, for instance, is the difference obtained, say, between A{{music|b}} and G{{music|#}} after a circle of twelve just fifths. A circle of three just major thirds, such as {{nobr|&thinsp;A{{music|b}} C E G{{music|#}} ,}} produces the ''small diesis'' {{sfrac| 128 | 125 }} (41.1&nbsp;[[musical cent|cent]]) between G{{sharp}} and A{{music|b}}. A circle of four just minor thirds, such as {{nobr|&thinsp;G{{music|#}} B D F A{{music|b}} ,}} produces an interval of {{sfrac| 648 | 625 }} between A{{music|b}} and G{{music|#}}, etc. An interesting property of temperaments is that this difference remains whatever the tuning of the intervals forming the circle.<ref>
{{cite book
 |first=Rudolf |last=Rasch
 |year=2002
 |section=Tuning and temperament
 |title=The Cambridge History of Western Music Theory
 |editor-first=Th. |editor-last=Christensen
 |publisher=Cambridge University Press
 |isbn=0-521-62371-5
 |page=201
}}
</ref>
In this sense, commas and similar minute intervals can never be completely tempered out, whatever the tuning.

===Comma sequence===
A comma sequence defines a [[musical temperament]] through a unique sequence of commas at increasing [[prime number|prime]] [[limit (music)|limits]].<ref>
{{cite web
 |last=Smith |first=G.W.
 |title=Comma sequences
 |series=Xenharmony
 |url=http://lumma.org/tuning/gws/commaseq.htm
 |access-date=2012-07-26  |df=dmy-all  |via=lumma.org
}}
</ref>
The first comma of the comma sequence is in the {{mvar|q}}-limit, where {{mvar|q}} is the {{mvar|n}}‑th odd prime (prime&nbsp;2 being ignored because it represents the octave) and {{mvar|n}} is the number of [[regular temperament|generators]]. Subsequent commas are in prime limits, each the next [[prime number|prime]] in sequence above 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 comma|Holdrian]] and Mercator's commas,<ref>
{{cite web
 |title=Mercator-comma / Mercator's comma
 |first=Joe |last=Monzo
 |website=tonalsoft.com
 |url=http://www.tonalsoft.com/enc/m/mercator-comma.aspx
}}
</ref>
and the pitch-to-pitch step size in {{nobr|[[53 equal temperament|53 {{sc|TET}}]]}}.

==See also==
* [[Interval (music)|List of musical intervals]]
* [[List of pitch intervals]]
*[[Semicomma]]

==References==
{{reflist|25em}}

{{Intervals|state=expanded}}

[[Category:Commas (music)| ]]
[[Category:Musical temperaments]]