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{{short description|Musical interval}}
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{{Redirect|Chromatic diesis|27/26|Comma (music)}}
{{multiple image
| direction = vertical
| header    = Syntonic comma (81:80) on C[[File:Syntonic comma on C.mid]]
| width     = 200
| image1    = Syntonic comma on C HE notation.png
| caption1  = Helmholtz-Ellis notation
| image2    = Syntonic comma on C.png
| caption2  = [[Ben Johnston notation|Ben Johnston's notation]]
}}
[[File:Just perfect fifth on D.png|thumb|right|Just perfect fifth on D[[File:Just perfect fifth on D.mid]] The perfect fifth above D (A+) is a syntonic comma higher than the (A{{sup|{{music|n}}}}) that is a [[just major sixth]] above C, assuming C and D are {{small|{{sfrac|9|8}}}} apart.<ref name=Fonville/>]]
[[File:Major second on C.svg|thumb|right|3-limit 9:8 [[major second|major tone]][[File:Major tone on C.mid]]]]
[[File:Minor tone on C.png|thumb|right|5-limit 10:9 [[major second|minor tone]][[File:Minor tone on C.mid]]]]

In [[music theory]], the '''syntonic comma''', also known as the '''[[chromatic]] diesis''', the '''Didymean comma''', the '''[[Ptolemy|Ptolemaic]] comma''', or the '''[[diatonic]] comma'''<ref name=Johnston/> is a small [[Comma (music)|comma]] type [[interval (music)|interval]] between two [[musical note]]s, equal to the frequency ratio {{small|{{sfrac|81|80}}}} (=&nbsp;1.0125) (around 21.51&nbsp;[[cent (music)|cent]]s). Two notes that differ by this interval would sound different from each other even to untrained ears,<ref name=BBC/> 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 the Musician|Didymus]] corrected the [[Pythagorean interval|Pythagorean]] [[major third]] ({{small|{{sfrac|81|64}}}}, around 407.82&nbsp;cents)<ref name=Lloyd/> to a [[just intonation|just]] / [[harmonic series (music)|harmonicly]] consonant [[major third]] ({{small|{{sfrac|5|4}}}}, around 386.31&nbsp;cents).

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

==Relationships==
The prime factors of the just interval {{small|{{sfrac|81|80}}}} known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as {{nobr|{{math|{{small|{{sfrac|81|1}} × {{sfrac|1|80}} }}}} &hairsp;}} or (fully expanded and sorted by prime) {{nobr|{{math|{{small| {{sfrac| 3 × 3  × 3  × 3 |&hairsp; 2 × 2 × 2 × 2 × 5 &hairsp;}} }}}} .}} All sequences of notes that produce that fraction are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below:
* The ratio of the two kinds of [[major second]] which occur in [[5-limit tuning]]: the greater [[whole tone|tone]] (9:8, about 203.91 cents, or {{nobr|C {{small|↗}} D}} in [[just intonation|just]] C&nbsp;major) and lesser tone (10:9, about 182.40 cents, or {{nobr|D {{small|↗}} E}}). Namely, {{nobr| 9:8 ÷ 10:9 {{=}} 81:80 ,}} or equivalently, sharpening by a comma promotes a lesser [[major second]] to a greater second {{nobr| {{math|{{small| {{sfrac|10|9}} × {{sfrac|81|80}} {{=}} {{sfrac|9|8}} }}}} .}}<ref name=Lloyd/>
* The difference in [[Interval (music)#size|size]] between a Pythagorean [[ditone]] ([[interval ratio|frequency ratio]] 81:64, or about 407.82 [[Cent (music)|cents]]) and a just major third (5:4, or about 386.31 cents). Namely, {{nobr|{{small|{{math| {{sfrac|81|64}} ÷ {{sfrac|5|4}} {{=}} {{sfrac|81|80}} }}}} .}}
* The difference between four [[just intonation|justly]] tuned [[perfect fifth]]s, and two [[octave]]s plus a justly tuned [[major third]]. A just perfect fifth has a size of 3:2 (about 701.96 cents), and four of them are equal to 81:16 (about 2807.82 cents). A just major third has a size of [[sesquiquartum|5:4]] (about 386.31 cents), and one of them plus two octaves (4:1 or exactly 2400 cents) is equal to 5:1 (about 2786.31 cents). The difference between these is the syntonic comma. Namely, {{nobr| 81:16 ÷ 5:1 {{=}} 81:80 .}}
* The difference between one octave plus a justly tuned [[minor third]] (12:5, about 1515.64 cents), and three justly tuned [[perfect fourth]]s (64:27, about 1494.13 cents). Namely, 12:5&nbsp;÷&nbsp;64:27 =&nbsp;81:80.
* The difference between a [[Pythagorean tuning|Pythagorean]] [[major sixth]] (27:16, about 905.87 cents) and a [[5-limit tuning#The justest ratios|justly tuned]] or "pure" [[major sixth]] (5:3, about 884.36 cents). Namely, 27:16&nbsp;÷&nbsp;5:3 =&nbsp;81:80.<ref name=Lloyd/>

On a [[piano]] keyboard (typically tuned with [[Equal temperament#Twelve-tone equal temperament|12-tone equal temperament]]) a stack of four fifths (700&nbsp;×&nbsp;4 = 2800 cents) is exactly equal to two octaves (1200&nbsp;×&nbsp;2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using [[Just intonation|justly tuned]] octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). [[Pythagorean tuning]] uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. [[Quarter-comma meantone]] uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why [[Equal temperament#Twelve-tone equal temperament|12-tone equal temperament]] is currently the preferred system for tuning most musical instruments{{clarify|date=July 2022}}.

Mathematically, by [[Størmer's theorem]], 81:80 is the closest [[Superparticular number|superparticular ratio]] possible with [[regular number]]s 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 factor]]s are limited to 2, 3, and&nbsp;5. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios.

==Syntonic comma in the history of music==
{{multiple image
| align     = right
| direction = vertical
| width     = 300
| image1    = Syntonic comma minor third Cuisenaire rods just.png
| caption1  = Syntonic comma (the mismatch at the top)
| image2    = Syntonic comma major third Cuisenaire rods ET.png
| caption2  = is tempered out in 12TET (bottom)
}}
{{multiple image
| align     = right
| direction = vertical
| width     = 300
| image1    = Syntonic comma major and minor tone Cuisenaire rods just.png
| image2    = Septimal and syntonic comma whole tones Cuisenaire rods ET.png
| footer    = Syntonic comma, such as between the 9/8 (203.91 approximate cents) and 10/9 (182.40 approximate cents) major and minor tones (top), is tempered out in 12TET, leaving one 200 cent tone (bottom).
}}
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 using [[Triad (music)|triad]]s and [[Chord (music)|chord]]s, forcing them for centuries to write music with relatively simple [[Texture (music)|texture]].

The syntonic tempering dates to [[Didymus the Musician]], whose tuning of the [[diatonic genus]] of the [[tetrachord]] replaced one 9:8 interval with a 10:9 interval ([[lesser tone]]), obtaining a just major third (5:4) and semitone (16:15). This was later revised by Ptolemy (swapping the two tones) in his "syntonic diatonic" scale (συντονόν διατονικός, ''syntonón diatonikós'', from συντονός + διάτονος). The term ''syntonón'' was based on [[Aristoxenus]], and may be translated as "tense" (conventionally "intense"), referring to tightened strings (hence sharper), in contrast to μαλακόν (''malakón'', from μαλακός), translated as "relaxed" (conventional "soft"), referring to looser strings (hence flatter or "softer").

This was rediscovered in the late [[Middle Ages]], where musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made [[Consonance and dissonance|consonant]]. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C-E (a major third), and E-G (a minor third) become just. Namely, C-E is narrowed to a [[just intonation|justly intonated]] ratio of

:<math> {81\over64} \cdot {80\over81} = {{1\cdot5}\over{4\cdot1}} = {5\over4}</math>

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

:<math> {32\over27} \cdot {81\over80} = {{2\cdot3}\over{1\cdot5}} = {6\over5}</math>

The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean [[Wolf interval|wolf fifth]]. But the fifth C-G stays consonant, since only E has been flattened (C-E&nbsp;×&nbsp;E-G =&nbsp;5/4&nbsp;×&nbsp;6/5 =&nbsp;3/2), and can be used together with C-E to produce a C-[[Major chord|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 narrowing 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 (music)|texture]], such as [[Polyphony|polyphonic music]], or melody 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 in an entire family of them. Namely, in the family belonging to the [[syntonic temperament]] continuum, including [[meantone temperament]]s.

==Comma pump==
[[File:Comma pump Benedetti.png|300px|thumb|Giovanni Benedetti's 1563 example of a comma "pump" or drift by a comma during a progression.<ref name="Historically"/> {{audio|Comma pump Benedetti.mid|Play}} Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones. {{audio|Comma pump Benedetti first last.mid|Play first and last chords}}]]

The syntonic comma arises in ''[[comma pump]]'' (''comma drift'') sequences such as C G D A E C, when each interval from one note to the next is played with certain specific intervals in [[just intonation]] tuning. If we use the [[frequency ratio]] 3/2 for the [[perfect fifth]]s (C-G and D-A), 3/4 for the descending [[perfect fourth]]s (G-D and A-E), and 4/5 for the descending [[major third]] (E-C), then the sequence of intervals from one note to the next in that sequence goes 3/2, 3/4, 3/2, 3/4, 4/5. These multiply together to give
::<math> {3\over2} \cdot {3\over4} \cdot {3\over2} \cdot {3\over4} \cdot {4\over5} = {81\over80}</math>
which is the syntonic comma (musical intervals stacked in this way are multiplied together). The "drift" is created by the combination of Pythagorean and 5-limit intervals in just intonation, and would not occur in Pythagorean tuning due to the use only of the Pythagorean major third (64/81) which would thus return the last step of the sequence to the original pitch.

So in that sequence, the second C is sharper than the first C by a syntonic comma {{audio|Comma pump on C.mid|Play}}. That sequence, or any [[transposition (music)|transposition]] of it, is known as the comma pump. If a line of music follows that sequence, and if each of the intervals between adjacent notes is justly tuned, then every time the sequence is followed, the pitch of the piece rises by a syntonic comma (about a fifth of a semitone).

Study of the comma pump dates back at least to the sixteenth century when the Italian scientist [[Giambattista Benedetti|Giovanni Battista Benedetti]] composed a piece of music to illustrate syntonic comma drift.<ref name="Historically"/>

Note that a descending perfect fourth (3/4) is the same as a descending [[octave]] (1/2) followed by an ascending perfect fifth (3/2). Namely, (3/4)&nbsp;=&nbsp;(1/2)&nbsp;×&nbsp;(3/2). Similarly, a descending major third (4/5) is the same as a descending octave (1/2) followed by an ascending [[minor sixth]] (8/5). Namely, (4/5)&nbsp;=&nbsp;(1/2)&nbsp;×&nbsp;(8/5). Therefore, the above-mentioned sequence is equivalent to:
::<math> {3\over2} \cdot {1\over2} \cdot {3\over2} \cdot {3\over2} \cdot {1\over2} \cdot {3\over2} \cdot {1\over2} \cdot {8\over5} = {81\over80}</math>
or, by grouping together similar intervals,
::<math> {3\over2} \cdot {3\over2} \cdot {3\over2} \cdot {3\over2} \cdot {8\over5} \cdot {1\over2} \cdot {1\over2} \cdot {1\over2} = {81\over80}</math>
This means that, if all intervals are justly tuned, a syntonic comma can be obtained with a stack of four perfect fifths plus one minor sixth, followed by three descending octaves (in other words, four '''P5''' plus one '''m6''' minus three '''P8''').

==Notation==
{{multiple image
| direction = vertical
| width     = 200
| image1    = Major chord on C.png
| caption1  = Just major chord on C in Ben Johnston's notation. {{audio|Major chord on C in just intonation.mid|Play}} Pythagorean major chord on C in Helmholtz-Ellis notation. {{audio|Pythagorean major chord on C.mid|Play}}
| image2    = Pythagorean major chord on C.png
| caption2  = Pythagorean major chord, Ben Johnston's notation.
| image3    = Just major chord on C HE notation.png
| caption3  = Just major chord, in Helmholtz-Ellis notation.
}}

[[Moritz Hauptmann]] developed a method of notation used by [[Hermann von Helmholtz]]. Based on Pythagorean tuning, subscript numbers are then added to indicate the number of syntonic commas to lower a note by. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E<sub>1</sub> F G A<sub>1</sub> B<sub>1</sub>. [[Carl Eitz]] developed a similar system used by [[J. Murray Barbour]]. Superscript positive and negative numbers are added, indicating the number of syntonic commas to raise or lower from Pythagorean tuning. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E<sup>−1</sup> F G A<sup>−1</sup> B<sup>−1</sup>.

In [[Helmholtz-Ellis notation]], a syntonic comma is indicated with up and down arrows added to the traditional accidentals. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is 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]].

Composer [[Ben Johnston (composer)|Ben Johnston]] uses a "−" as an accidental to indicate a note is lowered by a syntonic comma, or a "+" to indicate a note is raised by a syntonic comma.<ref name="Fonville"/> Thus a Pythagorean scale is C D E+ F G A+ B+, while the 5-limit Ptolemaic scale is C D E F G A B.

{| class="wikitable"
|-
!
! 5-limit just
! Pythagorean
|-
| HE
| 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 D E F G A B
|-
| Johnston
| C D E F G A B
| C D E+ F G A+ B+
|}

==See also==
*[[F+ (pitch)]]
*[[Holdrian comma]]
*[[Comma (music)]]
*[[Pythagorean comma]]

==References==
{{reflist|refs=

<ref name=BBC>{{citation |title=Sol-Fa – The Key to Temperament |publisher=[[BBC]] |website=bbc.co.uk |url=https://www.bbc.co.uk/dna/collective/A1339076 |archive-url=https://web.archive.org/web/20050208134843/http://www.bbc.co.uk/dna/collective/A1339076 |archive-date=2005-02-08 }} </ref>

<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>

<ref name="Historically">{{Citation| last1 =Wild| first1 =Jonathan| last2 =Schubert| first2 =Peter| date =Spring–Fall 2008| title =Historically Informed Retuning of Polyphonic Vocal Performance| journal =Journal of Interdisciplinary Music Studies| volume =2| issue =1&2| pages =121–139 [127]| url =http://www.musicstudies.org/JIMS2008/articles/Wild_JIMS_0821208.pdf| access-date =April 5, 2013| url-status =dead| archive-url =https://web.archive.org/web/20100911051454/http://www.musicstudies.org/JIMS2008/articles/Wild_JIMS_0821208.pdf| archive-date =September 11, 2010}}, art. #0821208.</ref>

<ref name=Johnston>{{citation |author-link=Ben Johnston (composer) |last=Johnston |first=B. |year=2006 |title="Maximum Clarity" and Other Writings on Music |editor-link=Bob Gilmore |editor-first=B. |editor-last=Gilmore |place=Urbana, IL |publisher=University of Illinois Press |isbn=0-252-03098-2 }}</ref>

<ref name=Lloyd>
{{citation
 |first=Llewelyn Southworth |last=Lloyd
 |year=1937
 |title=Music and Sound
 |page=12
 |publisher=Books for Libraries Press
 |isbn=0-8369-5188-3
}}

</ref>

}} <!-- end "refs=" -->

==External links==
*[http://music.indiana.edu/departments/offices/piano-technology/temperaments/syntonic-comma.shtml Indiana University School of Music: Piano Repair Shop: Harpsichord Tuning, Repair, and Temperaments: "What is the Syntonic Comma?"]
*[http://tonalsoft.com/enc/s/syntonic-comma.aspx Tonalsoft: "Syntonic-comma"]
*[https://www.youtube.com/watch?v=TYhPAbsIqA8 Explanation of comma drift]

{{Intervals|state=expanded}}

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[[Category:5-limit tuning and intervals]]
[[Category:Commas (music)]]
[[Category:Superparticular intervals|0081:0080]]