Comma (music)

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In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways.^[1] Traditionally, there are two most common commata: the syntonic comma (81:80), the amount by which four just perfect fifths exceed two octaves plus a just major third; and the Pythagorean comma (531441:524288, approximately 74:73), the amount by which twelve just perfect fifths exceed seven octaves.^[2] The word comma used without qualification refers to the syntonic comma,^[3] which can be defined, for instance, as the difference between an FTemplate:Music tuned using the D-based Pythagorean tuning system, and another FTemplate:Music tuned using the D-based quarter-comma meantone tuning system. Pitches separated by either comma are considered the same note because conventional notation does not distinguish Pythagorean intervals from 5-limit intervals. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in Template:Nobr the harmonic seventh BTemplate:Music Template:Music and ATemplate:Music 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"Template:Citation needed. The word "comma" came via Latin from Greek [[wikt:κόμμα#Ancient_Greek|Template:Math]], from earlier *Template:Math: "the result or effect of cutting".

Description

Within the same tuning system, two enharmonically equivalent notes (such as GTemplate:Music and ATemplate:Music) 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 ATemplate:Music tuned as a major third below C₅ and a GTemplate: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.Template:Citation needed Almost all of the meantone tuning systems produce two different characteristic semitones: A diatonic semitone for short steps in the scale, and chromatic semitone between a natural note, NTemplate:Music, and its sharp, NTemplate:Music. Their ratio (or in cents, their difference) is a comma; its size is unique to each meantone tuning. 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 – Template:Frac fractions of an octave on a logarithmic scale.

Commas in different contexts

In the column below labeled "Difference between semitones", Template:Small2 is the minor second (diatonic semitone), Template:Sup1 is the augmented unison (chromatic semitone), and STemplate:Sub, STemplate:Sub, STemplate:Sub, STemplate:Sub 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 (Template:Mvar Template:Sub) and the syntonic comma (Template:Mvar Template:Sub) 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
		Difference between semitones	Difference between commas	Interval 1	Interval 2	Cents	Ratio
schisma	skhisma	Template:Sup1 − Template:Small2 in Template:Nobr meantone	1 Template:Mvar Template:Sub − 1 Template:Mvar Template:Sub	8 perfect fifths + 1 major third	5 octaves	1.95	$\frac{32805}{32768}$
septimal kleisma				3 major thirds	1 octave − 1 septimal comma	7.71	$\frac{225}{224}$
kleisma				6 minor thirds	1 octave + 1 perfect fifth ("tritave")	8.11	$\frac{15625}{15552}$
small undecimal comma^[4]				1 neutral second	1 minor tone	17.40	$\frac{100}{99}$
diaschisma	diaskhisma	Template:Small2 − Template:Sup1 in Template:Nobr meantone, S₃ − S₂ in 5-limit tuning	2 Template:Mvar Template:Sub − 1 Template:Mvar Template:Sub	3 octaves	4 perfect fifths + 2 major thirds	19.55	$\frac{2048}{2025}$
syntonic comma Template:Center	Didymus' comma	S₂ − S₁ in 5 limit tuning		4 perfect fifths	2 octaves + 1 major third	21.51	$\frac{81}{80}$
syntonic comma Template:Center	Didymus' comma	S₂ − S₁ in 5 limit tuning		major tone	minor tone	21.51	$\frac{81}{80}$
[[53 equal temperament\|53 Template:Sc]] comma Template:Center	Holdrian comma 1 step Template:Center	Template:Nobr Template:Center	Template:Nobr Template:Center	major tone Template:Center	minor tone Template:Center	22.64	$(2)^{\frac{1}{53}}$
Pythagorean comma Template:Center	ditonic comma	[[Pythagorean apotome\|Template:Sup1]] − [[Pythagorean limma\|Template:Small2]] (in Pythagorean tuning)		12 perfect fifths	7 octaves	23.46	$\frac{531441}{524288}$
septimal comma^[5]	Archytas' comma Template:Center			minor seventh	septimal minor seventh	27.26	$\frac{64}{63}$
diesis	lesser diesis diminished second	Template:Small2 − Template:Sup1 in [[Quarter-comma meantone\|Template:Nobr meantone]], S₃ − S₁ in 5 limit tuning	3 Template:Mvar Template:Sub − 1 Template:Mvar Template:Sub	octave	3 major thirds	41.06	$\frac{128}{125}$
undecimal comma^[5]^[6]	Undecimal quarter-tone			undecimal tritone	perfect fourth	53.27	$\frac{33}{32}$
greater diesis		Template:Small2 − Template:Sup1 in Template:Nobr meantone, S₄ − S₁ in 5 limit tuning	4 Template:Mvar Template:Sub − 1 Template:Mvar Template:Sub	4 minor thirds	octave	62.57	$\frac{648}{625}$
tridecimal comma	tridecimal third-tone			tridecimal tritone	perfect fourth	65.34	$\frac{27}{26}$

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

\frac{81}{64} \cdot \frac{80}{81} = \frac{1 \cdot 5}{4 \cdot 1} = \frac{5}{4}

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

\frac{32}{27} \cdot \frac{81}{80} = \frac{2 \cdot 3}{1 \cdot 5} = \frac{6}{5}

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.

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
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 Template:Nobr 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.

Tempering of commas

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:

Template:Nobr tempers out the diesis, as well as a variety of other commas.
Template:Nobr tempers out the septimal diesis and syntonic comma, but does not temper out the diesis.
Template:Nobr tempers out the septimal comma of Archytas, but does not temper out the septimal diesis or syntonic comma.
Template:Nobr tempers out the syntonic comma, as well as the comma defined by the ratio Template:Sfrac, but does not temper out the diesis, septimal diesis, or septimal comma of Archytas.

The following table lists the number of steps used that represent 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. For example, the zero in the row labeled $\frac{128}{125}$ and the column labeled 12 Template:Sc means that $\frac{128}{125}$ is represented with zero steps in 12 Template:Sc, making it a comma in that temperament. Because commas are often thought of as the ratio between two different ways to tune the same note and $\frac{128}{125} = \frac{2}{1} \div {(\frac{5}{4})}^{3}$ , so the fact that this corresponds to zero steps in 12 Template:Sc means that in 12 Template:Sc, $\frac{2}{1}$ is represented with zero more steps than (i.e., with the same number of steps as) three copies of $\frac{5}{4}$ . (Specifically, 12 Template:Sc represents $\frac{2}{1}$ with twelve steps and each copy of $\frac{5}{4}$ with four steps.) Analogously, the 1 in the same row in the column labeled 19 Template:Sc means that 19 Template:Sc represents $\frac{2}{1}$ with one more step than three copies of $\frac{5}{4}$ , the 2 in the same row in the column labeled 41 Template:Sc means that 41 Template:Sc represents $\frac{2}{1}$ with two more steps than three copies of $\frac{5}{4}$ , and so on. Analogously, the -1 in the same row in the column labeled 5 Template:Sc means that 5 Template:Sc represents $\frac{2}{1}$ with one fewer step than three copies of $\frac{5}{4}$ . Specifically, 5 Template:Sc represents $\frac{2}{1}$ with five steps and each copy of $\frac{5}{4}$ with two steps. Because $\frac{5}{4}$ is slightly less than $2^{\frac{1}{3}}$ but is represented in 5 Template:Sc by $2^{\frac{2}{5}}$ , 5 Template:Sc's representation of $\frac{5}{4}$ is significantly sharper than the justly-intoned version, hence the -1. Thus, in 5 Template:Sc, going up by a factor of $\frac{128}{125}$ (e.g., from the representation of $\frac{2}{1}$ to the representation of ${(\frac{5}{4})}^{3}$ ) would require one to go down by one scale step.

Each of the frequency ratios in the first column are linked to its respective article.

Interval Template:Small	[[5 equal temperament\|5 Template:Sc]]	[[7 equal temperament\|7 Template:Sc]]	[[12 equal temperament\|12 Template:Sc]]	[[15 equal temperament\|15 Template:Sc]]	[[19 equal temperament\|19 Template:Sc]]	[[22 equal temperament\|22 Template:Sc]]	[[31 equal temperament\|31 Template:Sc]]	[[34 equal temperament\|34 Template:Sc]]	[[41 equal temperament\|41 Template:Sc]]	[[53 equal temperament\|53 Template:Sc]]	[[72 equal temperament\|72 Template:Sc]]
$\frac{2}{1}$	5	7	12	15	19	22	31	34	41	53	72
$\frac{15}{8}$	5	6	11	14	17	20	28	31	37	48	65
$\frac{9}{5}$	4	6	10	13	16	19	26	29	35	45	61
$\frac{7}{4}$	4	6	10	12	15	18	25	28	33	43	58
$\frac{5}{3}$	4	5	9	11	14	16	23	25	30	39	53
$\frac{8}{5}$	3	5	8	10	13	15	21	23	28	36	49
$\frac{3}{2}$	3	4	7	9	11	13	18	20	24	31	42
$\frac{10}{7}$	3	3	6	8	10	11	16	17	21	27	37
$\frac{64}{45}$	2	4	6	7	10	11	16	17	21	27	37
$\frac{45}{32}$	3	3	6	8	9	11	15	17	20	26	35
$\frac{7}{5}$	2	4	6	7	9	11	15	17	20	26	35
$\frac{4}{3}$	2	3	5	6	8	9	13	14	17	22	30
$\frac{9}{7}$	2	2	4	6	7	8	11	12	15	19	26
$\frac{5}{4}$	2	2	4	5	6	7	10	11	13	17	23
$\frac{6}{5}$	1	2	3	4	5	6	8	9	11	14	19
$\frac{7}{6}$	1	2	3	3	4	5	7	8	9	12	16
$\frac{8}{7}$	1	1	2	3	4	4	6	6	8	10	14
$\frac{9}{8}$	1	1	2	3	3	4	5	6	7	9	12
$\frac{10}{9}$	1	1	2	2	3	3	5	5	6	8	11
$\frac{27}{25}$	0	1	1	2	2	3	3	4	5	6	8
$\frac{15}{14}$	1	0	1	2	2	2	3	3	4	5	7
$\frac{16}{15}$	0	1	1	1	2	2	3	3	4	5	7
$\frac{21}{20}$	0	1	1	1	1	2	2	3	3	4	5
$\frac{25}{24}$	1	0	1	1	1	1	2	2	2	3	4
$\frac{28}{27}$	0	1	1	0	1	1	2	2	2	3	4
$\frac{648}{625}$	−1	1	0	1	1	2	1	2	3	3	4
$\frac{36}{35}$	0	0	0	1	1	1	1	1	2	2	3
$\frac{128}{125}$	−1	1	0	0	1	1	1	1	2	2	3
$\frac{49}{48}$	0	1	1	0	0	1	1	2	1	2	2
$\frac{50}{49}$	1	−1	0	1	1	0	1	0	1	1	2
$\frac{64}{63}$	0	0	0	0	1	0	1	0	1	1	2
$\frac{531441}{524288}$	1	−1	0	3	−1	2	−1	2	1	1	0
$\frac{81}{80}$	0	0	0	1	0	1	0	1	1	1	1
$\frac{2048}{2025}$	−1	1	0	-1	1	0	1	0	1	1	2
$\frac{126}{125}$	−1	1	0	0	0	1	0	1	1	1	1
$\frac{1728}{1715}$	0	−1	−1	1	1	0	0	−1	1	0	1
Template:Nobr	3	−2	1	2	−1	0	0	1	−1	0	−1
$\frac{15625}{15552}$	2	−1	1	0	0	−1	1	0	−1	0	0
$\frac{225}{224}$	1	−1	0	1	0	0	0	0	0	0	0
$\frac{32805}{32768}$	1	−1	0	2	−1	1	−1	1	0	0	−1
$\frac{2401}{2400}$	−1	2	1	-1	−1	1	0	2	0	1	0
$\frac{4375}{4374}$	−1	0	−1	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 ATemplate:Music and GTemplate:Music after a circle of twelve just fifths. A circle of three just major thirds, such as Template:Nobr produces the small diesis Template:Sfrac (41.1 cent) between GTemplate:Sharp and ATemplate:Music. A circle of four just minor thirds, such as Template:Nobr produces an interval of Template:Sfrac between ATemplate:Music and GTemplate:Music, etc. An interesting property of temperaments is that this difference remains whatever the tuning of the intervals forming the circle.^[7] 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.^[8] The first comma of the comma sequence is in the Template:Mvar-limit, where Template:Mvar is the Template:Mvar‑th odd prime (prime 2 being ignored because it represents the octave) and Template:Mvar 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,^[9] and the pitch-to-pitch step size in Template:Nobr.

References

Template:Reflist

Template:Intervals

↑ Template:Cite dictionary
↑ Template:Cite dictionary (see also Grove's Dictionary of Music and Musicians).
↑ Template:Cite book
↑ Template:Cite book
↑ ^5.0 ^5.1 Template:Cite book — Describes difference between 11 limit and 3 limit intervals.
↑ Template:Cite book = Source for 32:33 as difference between 11:16 & 2:3 .
↑ Template:Cite book
↑ Template:Cite web
↑ Template:Cite web

[1] Template:Cite dictionary

[2] Template:Cite dictionary (see also Grove's Dictionary of Music and Musicians).

[3] Template:Cite book

[4] Template:Cite book

[Rasch-5] 5.0 ^5.1 Template:Cite book — Describes difference between 11 limit and 3 limit intervals.

[6] Template:Cite book = Source for 32:33 as difference between 11:16 & 2:3 .

[7] Template:Cite book

[8] Template:Cite web

[9] Template:Cite web

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Comma (music)

Contents

Etymology

Description

Commas in different contexts

Alternative definitions

Tempering of commas

Comma sequence

Other intervals called commas

See also

References

Navigation menu

Comma (music)

Etymology

Description

Commas in different contexts

Alternative definitions

Tempering of commas

Comma sequence

Other intervals called commas

See also

References

Navigation menu

Search