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{{Redirect|Chromatic diesis|27/26|Comma (music)}}
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| header    = Syntonic comma (81:80) on C {{audio|Syntonic comma on C.mid|Play}}.
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| image1    = Syntonic comma on C HE notation.png
| caption1  = Helmholtz-Ellis notation
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| caption2  = Ben Johnston's notation
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[[Image:Just perfect fifth on D.png|thumb|right|Just perfect fifth on D {{audio|Just perfect fifth on D.mid|Play}}. The perfect fifth above D (A+) is a syntonic comma higher than the [[just major sixth]] (A{{music|natural}}).<ref name="Fonville"/>]]
[[Image:Major second on C.svg|thumb|right|3-limit 9:8 [[major second|major tone]] {{audio|Major tone on C.mid|Play}}.]]
[[Image:Minor tone on C.png|thumb|right|5-limit 10:9 [[major second|minor tone]] {{audio|Minor tone on C.mid|Play}}.]]

In [[music theory]], the '''syntonic comma''', also known as the '''[[chromatic]] diesis''', the '''comma of [[Didymus the Musician|Didymus]]''', the '''[[Ptolemy|Ptolemaic]] comma''', or the '''[[diatonic]] comma'''<ref>Johnston B. (2006). "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.</ref> is a small [[Comma (music)|comma]] type [[interval (music)|interval]] between two musical [[note]]s, equal to the frequency ratio 81:80, or around 21.51 [[Cent (music)|cent]]s. Two notes that differ by this interval would sound different from each other even to untrained ears,<ref>[http://www.bbc.co.uk/dna/collective/A1339076 "Sol-Fa - The Key to Temperament"], ''BBC''.</ref> 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 a "comma of Didymus" because it is the amount by which Didymus corrected the [[Pythagorean interval|Pythagorean]] [[major third]]<ref name="Lloyd"/> to a [[just intonation|just]] major third (81:64 or 407.82 cents - 21.51 = 386.31 cents or 5:4).

==Relationships==
The syntonic comma can be defined as:
* 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, 81:64 ÷ 5:4 = 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 about 701.96 cents ([[sesquialterum|3:2]]), and four of them are equal to about 2807.82 cents (81:16). A just major third has a size of about 386.31 cents ([[sesquiquartum|5:4]]), and one of them plus two octaves (2400 cents, or 4:1) is equal to about 2786.31 cents (5:1). The difference between these is the syntonic comma. Namely, 81:16 ÷ 5:1 = 81:80.
* The difference between three justly tuned [[perfect fourth]]s (64:27 or 1494.13 cents), and one octave plus a justly tuned [[minor third]] (12:5 or 1515.64 cents). Namely, 64:27 ÷ 12:5 = 81:80.
* The difference between the two kinds of [[major second]] which occur in [[5-limit tuning]]: major [[whole tone|tone]] (9:8, or 203.91 cents) and minor tone (10:9 or 182.40 cents). Namely, 9:8 ÷ 10:9 = 81:80.<ref name="Lloyd" />
* The difference between a [[Pythagorean tuning|Pythagorean]] [[major sixth]] (27:16 or 905.87 cents) and a [[5-limit tuning#The justest ratios|justly tuned]] or "pure" [[major sixth]] (5:3 or 884.36 cents). Namely, 27:16 ÷ 5:3 = 81:80.<ref name="Lloyd">Llewelyn Southworth Lloyd (1937). ''Music and Sound'', p.12. ISBN 0-8369-5188-3.</ref>

On a [[piano]] keyboard (typically tuned with [[12-tone equal temperament]]) a stack of four fifths (700 * 4 = 2800 cents) is exactly equal to two octaves (1200 * 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 [[12-tone equal temperament]] is currently the preferred system for tuning most musical instruments.

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 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==
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| image1    = Syntonic comma minor third Cuisenaire rods just.png
| caption1  = Syntonic comma (top)

| image2    = Syntonic comma major third Cuisenaire rods ET.png
| caption2  = is tempered out in 12TET (bottom)
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| 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 203.91 and 182.40 cent major and minor tones (top), is tempered out in 12TET, leaving one 200 cent tone (bottom).
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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]]. In late [[Middle Ages]], musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made [[Consonance and dissonance|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 [[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 sharpened 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 * E-G = 5/4 * 6/5 = 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 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 (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==
[[Image: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">{{Citation| last    =Wild| first   =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| accessdate  =April 5, 2013}}, art. #0821208.</ref> {{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 [[just intonation]] tuning. If you 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 (you multiply ratios when you stack musical intervals like that).

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 you go around the sequence 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 [[Giovanni Benedetti (scientist)|Giovanni 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)=(1/2)*(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)=(1/2)*(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 21.51 cents, or a "+" to indicate a note is raised 21.51 cents.<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 Johnston, Ben and 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> 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.

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

==References==
{{reflist}}

==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"]

{{Intervals|state=expanded}}

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

[[uk:Кома (музика)#Дідімова кома]]