Comma (music)

Last updated January 30, 2024

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways.^[1] Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and 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 F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone 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, B ♭ 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.

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 – 1⁄1200 fractions of an octave on a logarithmic scale.

Commas in different contexts

In the column below labeled "Difference between semitones", min2 is the minor second (diatonic semitone), ^aug1 is the augmented unison (chromatic semitone), and S₁, S₂, S₃, S₄ 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 ( $κ$ _𝜋) and the syntonic comma ( $κ$ _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).

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	^aug1 − min2 in  1 / 12 comma meantone	1 $κ$ _𝜋 − 1 $κ$ _S	8 perfect fifths + 1 major third	5 octaves	1.95	${\tfrac {\ 32805\ }{32768}}$
septimal kleisma				3 major thirds	1 octave − 1 septimal comma	7.71	${\tfrac {\ 225\ }{224}}$
kleisma				6 minor thirds	1 octave + 1 perfect fifth ("tritave")	8.11	${\tfrac {\ 15625\ }{15552}}$
small undecimal comma^[4]				1 neutral second	1 minor tone	17.40	${\tfrac {\ 100\ }{99}}$
diaschisma	diaskhisma	min2 − ^aug1 in  1 / 6 comma meantone, S₃ − S₂ in 5-limit tuning	2 $κ$ _S − 1 $κ$ _𝜋	3 octaves	4 perfect fifths + 2 major thirds	19.55	${\tfrac {\ 2048\ }{2025}}$
syntonic comma ( $κ$ _S)	Didymus' comma	S₂ − S₁ in 5 limit tuning		4 perfect fifths	2 octaves + 1 major third	21.51	${\tfrac {\ 81\ }{80}}$
syntonic comma ( $κ$ _S)	Didymus' comma	S₂ − S₁ in 5 limit tuning		major tone	minor tone	21.51	${\tfrac {\ 81\ }{80}}$
53 TET comma ( $κ$ ₅₃)	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	${\bigl (}2{\bigr )}^{\tfrac {\ 1\ }{53}}$
Pythagorean comma ( $κ$ _𝜋)	ditonic comma	^aug1 − min2 (in Pythagorean tuning)		12 perfect fifths	7 octaves	23.46	${\tfrac {\ 531441\ }{524288}}$
septimal comma ^[5]	Archytas' comma ( $κ$ _A)			minor seventh	septimal minor seventh	27.26	${\tfrac {\ 64\ }{63}}$
diesis	lesser diesis diminished second	min2 − ^aug1 in  1 / 4 comma meantone, S₃ − S₁ in 5 limit tuning	3 $κ$ _S − 1 $κ$ _𝜋	octave	3 major thirds	41.06	${\tfrac {\ 128\ }{125}}$
undecimal comma^[5]^[6]	Undecimal quarter-tone			undecimal tritone	perfect fourth	53.27	${\tfrac {\ 33\ }{32}}$
greater diesis		min2 − ^aug1 in  1 / 3 comma meantone, S₄ − S₁ in 5 limit tuning	4 $κ$ _S − 1 $κ$ _𝜋	4 minor thirds	octave	62.57	${\tfrac {\ 648\ }{625}}$
tridecimal comma	tridecimal third-tone			tridecimal tritone	perfect fourth	65.34	${\tfrac {\ 27\ }{26}}$

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

{\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  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 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:

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
${\tfrac {\ 2\ }{1}}$	5	7	12	19	22	31	34	41	53	72
${\tfrac {\ 15\ }{8}}$	5	6	11	17	20	28	31	37	48	65
${\tfrac {\ 9\ }{5}}$	4	6	10	16	19	26	29	35	45	61
${\tfrac {\ 7\ }{4}}$	4	6	10	15	18	25	28	33	43	58
${\tfrac {\ 5\ }{3}}$	4	5	9	14	16	23	25	30	39	53
${\tfrac {\ 8\ }{5}}$	3	5	8	13	15	21	23	28	36	49
${\tfrac {\ 3\ }{2}}$	3	4	7	11	13	18	20	24	31	42
${\tfrac {\ 10\ }{7}}$	3	3	6	10	11	16	17	21	27	37
${\tfrac {\ 64\ }{45}}$	2	4	6	10	11	16	17	21	27	37
${\tfrac {\ 45\ }{32}}$	3	3	6	9	11	15	17	20	26	35
${\tfrac {\ 7\ }{5}}$	2	4	6	9	11	15	17	20	26	35
${\tfrac {\ 4\ }{3}}$	2	3	5	8	9	13	14	17	22	30
${\tfrac {\ 9\ }{7}}$	2	2	4	7	8	11	12	15	19	26
${\tfrac {\ 5\ }{4}}$	2	2	4	6	7	10	11	13	17	23
${\tfrac {\ 6\ }{5}}$	1	2	3	5	6	8	9	11	14	19
${\tfrac {\ 7\ }{6}}$	1	2	3	4	5	7	8	9	12	16
${\tfrac {\ 8\ }{7}}$	1	1	2	4	4	6	6	8	10	14
${\tfrac {\ 9\ }{8}}$	1	1	2	3	4	5	6	7	9	12
${\tfrac {\ 10\ }{9}}$	1	1	2	3	3	5	5	6	8	11
${\tfrac {\ 27\ }{25}}$	0	1	1	2	3	3	4	5	6	8
${\tfrac {\ 15\ }{14}}$	1	0	1	2	2	3	3	4	5	7
${\tfrac {\ 16\ }{15}}$	0	1	1	2	2	3	3	4	5	7
${\tfrac {\ 21\ }{20}}$	0	1	1	1	2	2	3	3	4	5
${\tfrac {\ 25\ }{24}}$	1	0	1	1	1	2	2	2	3	4
${\tfrac {\ 648\ }{625}}$	−1	1	0	1	2	1	2	3	3	4
${\tfrac {\ 28\ }{27}}$	0	1	1	1	1	2	2	2	3	4
${\tfrac {\ 36\ }{35}}$	0	0	0	1	1	1	1	2	2	3
${\tfrac {\ 128\ }{125}}$	−1	1	0	1	1	1	1	2	2	3
${\tfrac {\ 49\ }{48}}$	0	1	1	0	1	1	2	1	2	2
${\tfrac {\ 50\ }{49}}$	1	−1	0	1	0	1	0	1	1	2
${\tfrac {\ 64\ }{63}}$	0	0	0	1	0	1	0	1	1	2
${\tfrac {\ 531441\ }{524288}}$	1	−1	0	−1	2	−1	2	1	1	0
${\tfrac {\ 81\ }{80}}$	0	0	0	0	1	0	1	1	1	1
${\tfrac {\ 2048\ }{2025}}$	−1	1	0	1	0	1	0	1	1	2
${\tfrac {\ 126\ }{125}}$	−1	1	0	0	1	0	1	1	1	1
${\tfrac {\ 1728\ }{1715}}$	0	−1	−1	1	0	0	−1	1	0	1
${\tfrac {\ 2109375\ }{2097152}}$	3	−2	1	−1	0	0	1	−1	0	−1
${\tfrac {\ 15625\ }{15552}}$	2	−1	1	0	−1	1	0	−1	0	0
${\tfrac {\ 225\ }{224}}$	1	−1	0	0	0	0	0	0	0	0
${\tfrac {\ 32805\ }{32768}}$	1	−1	0	−1	1	−1	1	0	0	−1
${\tfrac {\ 2401\ }{2400}}$	−1	2	1	−1	1	0	2	0	1	0
${\tfrac {\ 4375\ }{4374}}$	−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 .

Related Research Articles

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.

<span class="mw-page-title-main">Meantone temperament</span> Musical tuning system

Meantone temperaments are musical temperaments, that is a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are temperaments in that the fifths are not pure.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. This relation extends to pitch classes, chords, and key signatures. The term derives from Latin enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος, from ἐν ('in') and ἁρμονία ('harmony').

In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125). Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.

In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯. It is equal to the frequency ratio (1.5)¹²⁄2⁷ = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often "temper" is the Pythagorean comma.

In classical music from Western culture, a diesis is either an accidental, or a very small musical interval, usually defined as the difference between an octave and three justly tuned major thirds, equal to 128:125 or about 41.06 cents. In 12-tone equal temperament three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short. For instance, an octave (2:1) spans from C to C', and three justly tuned major thirds (5:4) span from C to B♯. The difference between C-C' (2:1) and C-B♯ (125:64) is the diesis (128:125). Notice that this coincides with the interval between B♯ and C', also called a diminished second.

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.

In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.

A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.

Quarter-comma meantone, or 1⁄4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81 : 80), with respect to its just intonation used in Pythagorean tuning ; the result is 3/2 × ^1⁄4 = ⁴√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 2^1⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET or 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Each step represents a frequency ratio of ³¹√2, or 38.71 cents.

In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal temperament, with 31 equal temperament being a more or less optimal tuning for both the 5- and 7-limits. Meantone temperament represents a frequency ratio of approximately 5 by means of four fifths, so that the major third, for instance C–E, is obtained from two tones in succession. Septimal meantone represents the frequency ratio of 56 (7 × 2³) by ten fifths, so that the interval 7:4 is reached by five successive tones. Hence C–A♯, not C–B♭, represents a 7:4 interval in septimal meantone.

In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps. Each step represents a frequency ratio of ²²√2, or 54.55 cents.

In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 3¹/ 2¹) and the perfect fourth with ratio 4/3 (equivalent to 2²/ 3¹) are Pythagorean intervals.

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2⁻³·3¹·5¹ = 15/8.

References

↑ Waldo Selden Pratt (1922). Grove's Dictionary of Music and Musicians , Vol. 1, p. 568. John Alexander Fuller Maitland, Sir George Grove, eds. Macmillan.
↑ Clive Greated (2001). "Comma", Grove Music Online . doi : 10.1093/gmo/9781561592630.article.06186
↑ Benson, Dave (2006). Music: A Mathematical Offering, p. 171. ISBN 0-521-85387-7.
↑ Haluška, Ján (2003). The Mathematical Theory of Tone Systems. p. $xxvi$ . ISBN 0-8247-4714-3.
1 2 Rasch, Rudolph (2000). "A word or two on the tunings of Harry Partch". In Dunn, David (ed.). Harry Partch: An anthology of critical perspectives. p. 34. ISBN 90-5755-065-2. — Describes difference between 11 limit and 3 limit intervals.
↑ Rasch, Rudolph (1988). "Farey systems of musical intonation". In Benitez, J.M.; et al. (eds.). Listening. Vol. 2. p. 40. ISBN 3-7186-4846-6. = Source for 32:33 as difference between 11:16 & 2:3 .
↑ "List of commas, by prime limit". Xenharmonic wiki.
↑ 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
↑ Tonalsoft Encyclopaedia article about 'HEWM' notation
↑ 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
↑ 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.

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[1] Waldo Selden Pratt (1922). Grove's Dictionary of Music and Musicians , Vol. 1, p. 568. John Alexander Fuller Maitland, Sir George Grove, eds. Macmillan.

[2] Clive Greated (2001). "Comma", Grove Music Online . doi : 10.1093/gmo/9781561592630.article.06186

[3] Benson, Dave (2006). Music: A Mathematical Offering, p. 171. ISBN 0-521-85387-7.

[4] Haluška, Ján (2003). The Mathematical Theory of Tone Systems. p. $xxvi$ . ISBN 0-8247-4714-3.

[Rasch-5] 1 2 Rasch, Rudolph (2000). "A word or two on the tunings of Harry Partch". In Dunn, David (ed.). Harry Partch: An anthology of critical perspectives. p. 34. ISBN 90-5755-065-2. — Describes difference between 11 limit and 3 limit intervals.

[6] Rasch, Rudolph (1988). "Farey systems of musical intonation". In Benitez, J.M.; et al. (eds.). Listening. Vol. 2. p. 40. ISBN 3-7186-4846-6. = Source for 32:33 as difference between 11:16 & 2:3 .

[7] "List of commas, by prime limit". Xenharmonic wiki.

[8] 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

[9] Tonalsoft Encyclopaedia article about 'HEWM' notation

[Fonville-10] 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

[11] 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.

[12] Smith, G.W. "Comma sequences". Xenharmony. Retrieved 26 July 2012– via lumma.org.

[13] Monzo, Joe. "Mercator-comma / Mercator's comma". tonalsoft.com.

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