Comma (music)#Comma sequence

Translated in this context, "comma" means "a hair" as in "off by just a hair"{{Citation needed|date=December 2023}}. The word "comma" came via Latin from Greek wikt:κόμμα#Ancient_Greek, from earlier *{{math|κοπ-μα}}: "the result or effect of cutting".

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 extended scales produced with five-limit tuning an A{{Music|b}} tuned as a major third below C₅ and a G{{Music|#}} 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|date=August 2021}} 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 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 commonly used version of five-limit tuning produces a 12-tone scale with four kinds of semitones and four commas.

The size of commas is commonly expressed and compared in terms of cents – {{frac|1|1200}} fractions of an octave on a logarithmic scale.

File:Comma size comparison.png. Equal-tempered semitone added for comparison. JND is the just-noticeable difference between tones.]]

In the column below labeled "Difference between semitones", {{small|min}}2 is the minor second (diatonic semitone), {{sup|aug}}1 is the augmented unison (chromatic semitone), and S{{sub|1}}, S{{sub|2}}, S{{sub|3}}, S{{sub|4}} are semitones as defined 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 ({{mvar|κ}}{{sub|𝜋}}) and the syntonic comma ({{mvar|κ}}{{sub|S}}) 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).

:

Many other commas have been enumerated and named by microtonalists.{{cite web |url=https://en.xen.wiki/w/Comma |title=List of commas, by prime limit |website=Xenharmonic wiki}}

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 to a just ratio of

:$\backslash frac\{\backslash \; 81\backslash \; \}\{\; 64\; \}\; \backslash cdot\; \backslash frac\{\backslash \; 80\backslash \; \}\{\; 81\; \}\; =\; \backslash frac\{\backslash \; 1\; \backslash cdot\; 5\backslash \; \}\{\; 4\; \backslash cdot\; 1\; \}\; =\; \backslash frac\{\backslash \; 5\backslash \; \}\{\; 4\; \}$

and at the same time E–G is sharpened to the just ratio of

:$\backslash frac\{\; 32\; \}\{\backslash \; 27\backslash \; \}\; \backslash cdot\; \backslash frac\{\; 81\; \}\{\backslash \; 80\backslash \; \}\; =\; \backslash frac\{\; 2\; \backslash cdot\; 3\; \}\{\backslash \; 1\; \backslash cdot\; 5\backslash \; \}\; =\; \backslash frac\{\; 6\; \}\{\backslash \; 5\backslash \; \}$

This led to the creation of a new tuning system, known as quarter-comma meantone, which permitted the full development of music with complex texture, such as polyphonic music, or melodies with instrumental accompaniment. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths throughout the family of syntonic temperaments, including meantone temperaments.

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
• augmented fourth and 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 {{nobr|{{sfrac| 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.

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.see article "The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmetode für dienatürlichen Intervalle" in Mikrotöne und mehr – Auf György Ligetis Hamburger Pfaden, ed. Manfred Stahnke, von Bockel Verlag, Hamburg 2005 {{ISBN|3-932696-62-X}} 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 {{nowrap|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).[http://tonalsoft.com/enc/h/hewm.aspx Tonalsoft Encyclopaedia article about 'HEWM' notation] 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|date=December 2011}} Thus a Pythagorean scale is {{nowrap|C D E F G A B C}}, while a just scale is {{nowrap|C D E9px F G A 9px B9px 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;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}} however, Johnston's "basic scale" (the plain nominals {{nowrap|A B C D E F G}}) is tuned to just-intonation and thus already includes the syntonic comma. Thus a Pythagorean scale is {{nowrap|C D E+ F G A+ B+ C}}, while a just scale is {{nowrap|C D E F G A B}}.

Commas are frequently used in the description of musical temperaments, 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.

Examples:

• {{nobr|12 equal temperament}} tempers out the diesis, as well as a variety of other commas.
• {{nobr|19 equal temperament}} tempers out the septimal diesis and syntonic comma, but does not temper out the diesis.
• {{nobr|22 equal temperament}} tempers out the septimal comma of Archytas, but does not temper out the septimal diesis or syntonic comma.
• {{nobr|31 equal temperament}} tempers out the syntonic comma, as well as the comma defined by the ratio {{sfrac| 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.{{clarify |date=January 2024 |reason=What do all the "−1" entries mean?}} All of the frequency ratios in the first column are linked to their wikipedia article.

:

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{{music|b}} and G{{music|#}} after a circle of twelve just fifths. A circle of three just major thirds, such as {{nobr| A{{music|b}} C E G{{music|#}} ,}} produces the small diesis {{sfrac| 128 | 125 }} (41.1 cent) between G{{sharp}} and A{{music|b}}. A circle of four just minor thirds, such as {{nobr| G{{music|#}} B D F A{{music|b}} ,}} produces an interval of {{sfrac| 648 | 625 }} between A{{music|b}} and G{{music|#}}, etc. An interesting property of temperaments is that this difference remains whatever the tuning of the intervals forming the circle.

{{cite book

|first=Rudolf |last=Rasch

|year=2002

|section=Tuning and temperament

|title=The Cambridge History of Western Music Theory

|editor-first=Th. |editor-last=Christensen

|publisher=Cambridge University Press

|isbn=0-521-62371-5

|page=201

}}

In this sense, commas and similar minute intervals can never be completely tempered out, whatever the tuning.

A comma sequence defines a musical temperament through a unique sequence of commas at increasing prime limits.

{{cite web

|last=Smith |first=G.W.

|title=Comma sequences

|series=Xenharmony

|url=http://lumma.org/tuning/gws/commaseq.htm

|access-date=2012-07-26 |df=dmy-all |via=lumma.org

}}

The first comma of the comma sequence is in the {{mvar|q}}-limit, where {{mvar|q}} is the {{mvar|n}}‑th odd prime (prime 2 being ignored because it represents the octave) and {{mvar|n}} is the number of generators. Subsequent commas are in prime limits, each the next prime in sequence above the last.

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,

{{cite web

|title=Mercator-comma / Mercator's comma

|first=Joe |last=Monzo

|website=tonalsoft.com

|url=http://www.tonalsoft.com/enc/m/mercator-comma.aspx

}}

and the pitch-to-pitch step size in {{nobr|53 equal temperament}}.

• List of musical intervals
• List of pitch intervals
• Semicomma

{{reflist|25em}}

{{Intervals|state=expanded}}

Category:Musical temperaments