Comma (music)

Source: Wikipedia, the free encyclopedia.
Syntonic comma on C
Pythagorean comma on C

In

tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B7 and A 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]. 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

enharmonically equivalent notes (such as G and A) may have a slightly different frequency, and the interval between them is a comma. For example, in extended scales produced with five-limit tuning an A tuned as a major third below C5 and a G tuned as two major thirds above C4 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] 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.

enharmonically equivalent
notes (from C to D). The interval from C to D is narrower than in Pythagorean tuning (see below).
Pythagorean comma (PC) defined in Pythagorean tuning as difference between semitones (A1 − m2), or interval between enharmonically equivalent notes (from D to C). The interval from C to D is wider than in quarter-comma meantone (see above).

In

four commas
.

The size of commas is commonly expressed and compared in terms of cents11200 fractions of an octave on a logarithmic scale.

Commas in different contexts

Comparison of the size of different commas, in cents. Equal-tempered semitone added for comparison. JND is the just-noticeable difference between tones.

In the column below labeled "Difference between

here. In the columns labeled "Interval 1" and "Interval 2", all intervals are presumed to be tuned in just intonation. Notice that the Pythagorean comma (κ𝜋) and the syntonic comma (κS) 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).

Name of comma Alternative name Definitions Size
Difference between
semitones
Difference between
commas
Difference between Cents Ratio
Interval 1 Interval 2
schisma skhisma aug1 − min2
in  1 / 12 comma meantone
1 κ𝜋 − 1 κS 8 perfect fifths +
major third
5 octaves 1.95
septimal kleisma 3 major thirds octave
septimal comma
7.71
kleisma 6 minor thirds
tritave
")
8.11
small undecimal comma[4]
neutral second
minor tone
17.40
diaschisma diaskhisma min2 − aug1
in  1 / 6 comma meantone,
S3 − S2
in
5-limit tuning
2 κS − 1 κ𝜋 3 octaves 4 perfect fifths +
2 major thirds
19.55
syntonic comma
(κS)
Didymus' comma S2 − S1
in 5 limit tuning
4 perfect fifths 2 octaves +
1 major third
21.51
major tone
minor tone
53 TET comma
(κ53)
1 step
(in 53 TET)
 1 / 9 major tone
(in 53 TET)
 1 / 8
minor tone
)
major tone
(in 53 TET)
minor tone
)
22.64
Pythagorean comma
(κ𝜋)
ditonic comma ) 12 perfect fifths 7 octaves 23.46
septimal comma[5] Archytas' comma
(κA)
minor seventh
septimal minor seventh
27.26
diesis lesser diesis
diminished second
min2 − aug1
in  1 / 4 comma meantone,
S3 − S1
in 5 limit tuning
3 κS − 1 κ𝜋 octave 3 major thirds 41.06
undecimal comma[5][6] Undecimal
quarter-tone
undecimal tritone
perfect fourth 53.27
greater diesis
min2 − aug1
in  1 / 3 comma meantone,
S4 − S1
in 5 limit tuning
4 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

syntonic temperaments, including meantone temperaments
.

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:

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

19 tone equal temperament
does not temper out this comma, and thus it distinguishes between the two semitones.

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 EDO
7 T EDO
12 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 

cent) between G and A. A circle of four just minor thirds, such as  G B D F A , produces an interval of  648 / 625 between A and G, etc. An interesting property of temperaments is that this difference remains whatever the tuning of the intervals forming the circle.[11]
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 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

Holdrian and Mercator's commas,[13]
and the pitch-to-pitch step size in 53 TET.

See also

References

  1. Grove's Dictionary of Music and Musicians, Vol. 1, p. 568. John Alexander Fuller Maitland, Sir George Grove
    , eds. Macmillan.
  2. .
  3. .
  4. ^
    ISBN 90-5755-065-2. — Describes difference between 11 limit
    and 3 limit intervals.
  5. . = Source for 32:33 as difference between 11:16 & 2:3 .
  6. ^ "List of commas, by prime limit". Xenharmonic wiki.
  7. ^ Tonalsoft Encyclopaedia article about 'HEWM' notation
  8. ^ Rasch, Rudolf (2002). "Tuning and temperament". In Christensen, Th. (ed.). The Cambridge History of Western Music Theory. Cambridge University Press. p. 201. .
  9. ^ Smith, G.W. "Comma sequences". Xenharmony. Retrieved 26 July 2012 – via lumma.org.
  10. ^ Monzo, Joe. "Mercator-comma / Mercator's comma". tonalsoft.com.