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| [[File:Syntonic comma on C.png|thumb|right|[[Syntonic comma]] on C {{audio|Syntonic comma on C.mid|Play}}.]]
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| [[File:Pythagorean comma on C.png|thumb|right|[[Pythagorean comma]] on C {{audio|Pythagorean comma on C.mid|Play}}.]]
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| In [[music theory]], a '''comma''' is a minute [[interval (music)|interval]], the difference resulting from [[Musical tuning|tuning]] one [[note (music)|note]] two different ways.<ref>Waldo Selden Pratt (1922). ''Grove's Dictionary of Music and Musicians, Volume 1'', p.568. John Alexander Fuller-Maitland, Sir George Grove, eds. Macmillan.</ref> The word "comma" used without qualification refers to the [[syntonic comma]],<ref>Benson, Dave (2006). ''Music: A Mathematical Offering'', p.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.
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| 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 [[5-limit tuning]] an A{{Music|b}} tuned as a [[major third]] below C5 and a G{{Music|#}} tuned as two major thirds above C4 will not be 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]]).
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| Commas are often defined as the difference in size between two [[semitone]]s. Each [[meantone temperament]] tuning system produces a [[Chromatic scale|12-tone 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.
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| | [[File:Lesser diesis (difference m2-A1).PNG|thumb|center|467 px|Lesser [[diesis]] defined in [[quarter-comma meantone]] as difference between semitones <br> (m2 − A1 ≈ 117.1 − 76.0 ≈ 41.1 cents), or interval between [[enharmonic|enharmonically equivalent]] notes <br> (from C{{Music|#}} to D{{Music|b}}). The interval from C to D is narrower than in Pythagorean tuning (see below). {{audio|Enharmonic scale segment on C.mid|Play}}]]
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| [[File:Pythagorean comma (difference A1-m2).PNG|thumb|center|492 px|[[Pythagorean comma]] (PC) defined in [[Pythagorean tuning]] as difference between semitones <br> (A1 − m2 ≈ 113.7 − 90.2 ≈ 23.5 cents), or interval between enharmonically equivalent notes <br> (from D{{Music|b}} to C{{Music|#}}). The interval from C to D is wider than in quarter-comma meantone (see above).]]
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| 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 5-limit tuning produces a 12-tone scale with four kinds of semitones and [[5-limit tuning#Commas|four commas]].
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| The size of commas is commonly expressed and compared in terms of [[Cent (music)|cents]] – 1/1200 fractions of an [[octave]] on a [[logarithm]]ic scale.
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| == Commas in different contexts == | |
| [[File:Comma size comparison.png|thumb|400px|Comparison of the size of different commas, in cents. Equal-tempered semitone added for comparison.]]
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| In the column labeled "Difference between [[semitone]]s", '''m2''' is the minor second (diatonic semitone), '''A1''' is the augmented unison (chromatic semitone), and '''S<sub>1</sub>''', '''S<sub>2</sub>''', '''S<sub>3</sub>''', '''S<sub>4</sub>''' are semitones as defined [[5-limit tuning#Size of intervals|here]]. In the columns labeled "[[Interval (music)|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 which 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 [[Just noticeable difference|smallest noticeable difference]] between tones (which is around six cents). | |
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| {| class="wikitable"
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| !rowspan=3|Name of comma
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| !rowspan=3|Alternative Name
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| !colspan=4|Definitions
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| !colspan=2|Size
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| !rowspan=2|Difference between<br>[[semitone]]s
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| !rowspan=2|Difference between<br>commas
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| !colspan=2|Difference between
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| !rowspan=2|[[Cent (music)|Cents]]
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| !rowspan=2|[[Interval ratio|Ratio]]
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| ! [[Interval (music)|Interval]] 1 !! Interval 2
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| | [[Schisma]] || Skhisma || align="center"| A1 − m2<br>in 1/12-comma [[meantone temperament|meantone]] || align="center"| 1 PC − 1 SC || 8 perfect fifths +<br>1 [[major third]] || 5 octaves || align="right"| 1.95 ||align="center"| 32805:32768
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| | [[Septimal kleisma]] || || || || 2 [[major third]]s +<br>1 [[septimal major third]] || Octave || align="right"| 7.71 ||align="center"| 225:224
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| | [[Kleisma]] || || || || 6 minor thirds || Tritave (1 [[octave]] +<br>1 [[perfect fifth]]) || align="right"| 8.11 ||align="center"| 15625:15552
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| | Small undecimal comma<ref>Haluška, Ján (2003). ''The Mathematical Theory of Tone Systems'', p.xxvi. ISBN 0-8247-4714-3.</ref> || || || || || || align="right"| 17.58 ||align="center"| 99:98
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| | [[Diaschisma]] || Diaskhisma || align="center"| m2 − A1<br>in 1/6-comma meantone,<br>S<sub>3</sub> − S<sub>2</sub><br>in [[5-limit tuning]] || align="center"| 2 SC − 1 PC || 3 octaves || 4 perfect fifths +<br>2 major thirds || align="right"| 19.55 ||align="center"| 2048:2025
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| | rowspan=2|[[Syntonic comma]] (SC) || rowspan=2|Didymus' comma || align="center" rowspan=2|S<sub>2</sub> − S<sub>1</sub><br>in 5-limit tuning || rowspan=2| || 4 perfect fifths || 2 octaves +<br>1 major third || rowspan=2 align="right"| 21.51 ||rowspan=2 align="center"| 81:80
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| | Major [[Major second|tone]] || Minor tone
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| | [[Pythagorean comma]] (PC) || Ditonic comma || align="center"| [[Pythagorean apotome|A1]] − [[Pythagorean limma|m2]]<br>in [[Pythagorean tuning]] || || 12 perfect fifths || 7 octaves || align="right"| 23.46 ||align="center"| 531441:524288
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| | [[Septimal comma]]<ref>David Dunn, 2000. [http://books.google.it/books?id=kIKar6TykE4C&pg=PA34&dq=undecimal+comma&hl=it&ei=W4huTOOxBYbK4gbI8aTeCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q=septimal%20comma&f=false Harry Partch: an anthology of critical perspectives].</ref> || Archytas' comma || || || [[Minor seventh]] || [[Septimal minor seventh]] || align="right"| 27.26 ||align="center"| 64:63
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| | [[Diesis]] || Lesser diesis || align="center"| m2 − A1<br>in [[Quarter-comma meantone|1/4-comma meantone]],<br>S<sub>3</sub> − S<sub>1</sub><br>in 5-limit tuning || align="center"| 3 SC − 1 PC || Octave || 3 major thirds || align="right"| 41.06 ||align="center"| 128:125
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| | Undecimal comma<ref name="Rasch">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 90-5755-065-2. Difference between 11-[[limit (music)|limit]] and 3-limit intervals.</ref><ref>Rasch, Rudolph (1988). "Farey Systems of Musical Intonation", ''Listening 2'', p.40. Benitez, J.M. et al., eds. ISBN 3-7186-4846-6. Source for 32:33 as difference between 11:16 & 2:3.</ref> || Undecimal quarter-tone || || || [[Tritone#Definition and nomenclature|Undecimal tritone]] || [[Perfect fourth]] || align="right"| 53.27 ||align="center"| 33:32
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| | Greater diesis || || align="center"| m2 − A1<br>in 1/3-comma meantone,<br>S<sub>4</sub> − S<sub>1</sub><br>in 5-limit tuning || align="center"| 4 SC − 1 PC || 4 minor thirds || Octave || align="right"| 62.57 ||align="center"| 648:625
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| | Tridecimal comma || Tridecimal third-tone || || || [[Tritone#Definition and nomenclature|Tridecimal tritone]] || [[Perfect fourth]] || align="right"| 65.34 ||align="center"| 27:26
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| |}
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| Many other commas have been enumerated and named by microtonalists<ref>[http://xenharmonic.wikispaces.com/comma List of commas, by prime limit] in the Xenharmonic wiki</ref>
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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 freely using [[Triad (music)|triad]]s and [[Chord (music)|chord]]s, forcing them 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
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| :<math> {81\over64} \cdot {80\over81} = {{1\cdot5}\over{4\cdot1}} = {5\over4}</math>
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| and at the same time E-G is sharpened to the just ratio of
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| :<math> {32\over27} \cdot {81\over80} = {{2\cdot3}\over{1\cdot5}} = {6\over5}</math>
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| This brought 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 in an entire family of them. Namely, in the family belonging to the [[syntonic temperament]] continuum, including [[meantone temperament]]s.
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| ===Alternative definitions===
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| In [[quarter-comma meantone]], and any kind of [[meantone temperament]] tuning system which 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:
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| * minor second and augmented unison (also known as diatonic and chromatic [[semitone]]s), or
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| * [[major second]] and [[diminished third]], or
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| * [[minor third]] and [[augmented second]], or
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| * [[major third]] and [[diminished fourth]], or
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| * [[perfect fourth]] and [[augmented third]], or
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| * [[augmented fourth]] and [[diminished fifth]], or
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| * [[perfect fifth]] and [[diminished sixth]], or
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| * [[minor sixth]] and [[augmented fifth]], or
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| * [[major sixth]] and [[diminished seventh]], or
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| * [[minor seventh]] and [[augmented sixth]], or
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| * [[major seventh]] and [[diminished octave]].
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| In Pythagorean tuning, and any kind of [[meantone temperament]] tuning system which tempers the fifth to a size larger than 700 cents (such as 1/12-comma meantone), the comma is the opposite of a diminshed second, and therefore the opposite of the above-listed 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).
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| 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]].
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| == Notation ==
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| 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 Notationsmethode für die natürlichen 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 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|date=December 2011}} Thus a Pythagorean scale is C D E F G A B, while a just 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]].
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| Composer [[Ben Johnston (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.<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 a just scale is C D E F G A B.
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| ==Tempering of commas==
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| 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-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.
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| Examples:
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| * [[12-TET]] tempers out the diesis, as well as a variety of other commas.
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| * [[19-TET]] tempers out the [[septimal diesis]] and [[syntonic comma]], but does not temper out the diesis.
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| * [[22-TET]] tempers out the [[septimal comma]] of [[Archytas]], but does not temper out the septimal diesis or syntonic comma.
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| * [[31-TET]] 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.
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| ===Comma sequence===
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| A '''comma sequence''' defines a [[musical temperament]] through a unique sequence of commas at increasing [[prime number|prime]] [[Limit (music)|limits]].<ref>{{citation|last=Smith|first=G. W.,|authorlink=Gene Ward Smith|url=http://lumma.org/tuning/gws/commaseq.htm|title=Comma Sequences|work=Xenharmony|accessdate=2012-07-26}}.</ref> The first comma of the comma sequence will be in the q-limit, where q is the nth odd prime, and n is the number of [[regular temperament|generators]]. Subsequent commas will be in prime limits each one prime beyond the last.
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| ==Other intervals called commas==
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| 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 [http://www.tonalsoft.com/enc/m/mercator-comma.aspx Mercator's commas].
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| ==See also==
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| *[[List of musical intervals]]
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| *[[Semicomma]]
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| ==Sources==
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| {{Reflist}}
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| {{Intervals|state=expanded}}
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| {{DEFAULTSORT:Comma (Music)}}
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| [[Category:Commas]]
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| [[Category:Musical temperaments]]
| |
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