Comma (music)
In
Etymology
Translated in this context, "comma" means "a hair" as in "off by just a hair"[citation needed]. The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα: "the result or effect of cutting". A more complete etymology is given in the article κόμμα (Ancient Greek) in the Wiktionary.
Description
Within the same tuning system, two
Commas are often defined as the difference in size between two semitones.[citation needed] Each meantone temperament tuning system produces a 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.
In
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 below labeled "Difference between
Name of comma Alternative name Definitions Size Difference between
semitonesDifference between
commasDifference between Cents Ratio Interval 1 Interval 2 schisma skhisma aug1 − min2
in 1 / 12 comma meantone1 κ𝜋 − 1 κS 8 perfect fifths +
1 major third5 octaves 1.95 septimal kleisma 3 major thirds 1 octave −
1 septimal comma7.71 kleisma 6 minor thirds 1 tritave")8.11 small undecimal comma[4] 1 neutral second1 minor tone17.40 diaschisma diaskhisma min2 − aug1
in 1 / 6 comma meantone,
S3 − S2
in5-limit tuning2 κS − 1 κ𝜋 3 octaves 4 perfect fifths +
2 major thirds19.55 syntonic comma (κS)Didymus' comma S2 − S1
in 5 limit tuning4 perfect fifths 2 octaves +
1 major third21.51 major tone minor tone53 TET comma (κ53)1 step (in 53 TET)1 / 9 major tone (in 53 TET)1 / 8 minor tone(in 53 TET)major tone (in 53 TET)minor tone(in 53 TET)22.64 Pythagorean comma (κ𝜋)ditonic comma ) 12 perfect fifths 7 octaves 23.46 septimal comma[5] Archytas' comma (κA)minor seventh septimal minor seventh27.26 diesis lesser diesis
diminished secondmin2 − aug1
in 1 / 4 comma meantone,
S3 − S1
in 5 limit tuning3 κS − 1 κ𝜋 octave 3 major thirds 41.06 undecimal comma[5][6] Undecimal quarter-toneundecimal tritoneperfect fourth 53.27 greater diesismin2 − aug1
in 1 / 3 comma meantone,
S4 − S1
in 5 limit tuning4 κS − 1 κ𝜋 4 minor thirds octave 62.57 tridecimal comma tridecimal third-tone tridecimal tritone perfect fourth 65.34
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. Musicians in late Middle Ages recognized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made 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 by a just ratio of
and at the same time E–G is sharpened to the just ratio of
This led to the creation of a new
Alternative definitions
In quarter-comma 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
- 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 / 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 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 above-listed differences have all the same size. For instance, in Pythagorean tuning they are all equal to the opposite of a Pythagorean comma, and in 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 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 (Helmholtz-Ellis-Wolf-Monzo).[9] 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 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] Thus a Pythagorean scale is C D E F G A B C, while a just scale is C D E F G A B C.
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 just-intonation and thus already includes the syntonic comma. Thus a Pythagorean scale is C D E+ F G A+ B+ C, while a just scale is C D E F G A B.
Tempering of commas
Commas are frequently used in the description of
Examples:
- 12 TET tempers out the diesis, as well as a variety of other commas.
- 19 TET tempers out the septimal diesis and syntonic comma, but does not temper out the diesis.
- 22 TET tempers out the septimal comma of Archytas, but does not temper out the septimal diesis or syntonic comma.
- 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.
The following table lists the number of steps used that correspond 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.[clarification needed] All of the frequency ratios in the first column are linked to their wikipedia article.
Interval
(frequency ratio)5 T EDO7 T EDO12 T EDO 19 T EDO 22 T EDO 31 T EDO 34 T EDO 41 T EDO 53 T EDO 72 T EDO 5 7 12 19 22 31 34 41 53 72 5 6 11 17 20 28 31 37 48 65 4 6 10 16 19 26 29 35 45 61 4 6 10 15 18 25 28 33 43 58 4 5 9 14 16 23 25 30 39 53 3 5 8 13 15 21 23 28 36 49 3 4 7 11 13 18 20 24 31 42 3 3 6 10 11 16 17 21 27 37 2 4 6 10 11 16 17 21 27 37 3 3 6 9 11 15 17 20 26 35 2 4 6 9 11 15 17 20 26 35 2 3 5 8 9 13 14 17 22 30 2 2 4 7 8 11 12 15 19 26 2 2 4 6 7 10 11 13 17 23 1 2 3 5 6 8 9 11 14 19 1 2 3 4 5 7 8 9 12 16 1 1 2 4 4 6 6 8 10 14 1 1 2 3 4 5 6 7 9 12 1 1 2 3 3 5 5 6 8 11 0 1 1 2 3 3 4 5 6 8 1 0 1 2 2 3 3 4 5 7 0 1 1 2 2 3 3 4 5 7 0 1 1 1 2 2 3 3 4 5 1 0 1 1 1 2 2 2 3 4 −1 1 0 1 2 1 2 3 3 4 0 1 1 1 1 2 2 2 3 4 0 0 0 1 1 1 1 2 2 3 −1 1 0 1 1 1 1 2 2 3 0 1 1 0 1 1 2 1 2 2 1 −1 0 1 0 1 0 1 1 2 0 0 0 1 0 1 0 1 1 2 1 −1 0 −1 2 −1 2 1 1 0 0 0 0 0 1 0 1 1 1 1 −1 1 0 1 0 1 0 1 1 2 −1 1 0 0 1 0 1 1 1 1 0 −1 −1 1 0 0 −1 1 0 1 3 −2 1 −1 0 0 1 −1 0 −1 2 −1 1 0 −1 1 0 −1 0 0 1 −1 0 0 0 0 0 0 0 0 1 −1 0 −1 1 −1 1 0 0 −1 −1 2 1 −1 1 0 2 0 1 0 −1 0 −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♭ and G♯ after a circle of twelve just fifths. A circle of three just major thirds, such as A♭ C E G♯ , produces the small diesis 128 / 125 (41.1
Comma sequence
A comma sequence defines a musical temperament through a unique sequence of commas at increasing prime limits.[12] The first comma of the comma sequence is in the q-limit, where q is the n‑th odd prime (prime 2 being ignored because it represents the octave) and n is the number of generators. Subsequent commas are in prime limits, each the next 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
See also
References
- Grove's Dictionary of Music and Musicians, Vol. 1, p. 568. John Alexander Fuller Maitland, Sir George Grove, eds. Macmillan.
- ISBN 0-521-85387-7.
- ISBN 0-8247-4714-3.
- ^ ISBN 90-5755-065-2. — Describes difference between 11 limitand 3 limit intervals.
- ISBN 3-7186-4846-6. = Source for 32:33 as difference between 11:16 & 2:3 .
- ^ "List of commas, by prime limit". Xenharmonic wiki.
- ISBN 3-932696-62-X
- ^ Tonalsoft Encyclopaedia article about 'HEWM' notation
- ISBN 978-0-252-03098-7
- ^
Rasch, Rudolf (2002). "Tuning and temperament". In Christensen, Th. (ed.). The Cambridge History of Western Music Theory. Cambridge University Press. p. 201. ISBN 0-521-62371-5.
- ^ Smith, G.W. "Comma sequences". Xenharmony. Retrieved 26 July 2012 – via lumma.org.
- ^ Monzo, Joe. "Mercator-comma / Mercator's comma". tonalsoft.com.