List of intervals in 5-limit just intonation

Last updated June 17, 2023

The intervals of 5-limit just intonation (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5. The fundamental intervals are the superparticular ratios 2/1 (the octave), 3/2 (the perfect fifth) and 5/4 (the major third). That is, the notes of the major triad are in the ratio 1:5/4:3/2 or 4:5:6.

List

Names	Ratio	Cents	12ET interval (in cents)	Definition	53ET interval (in Holdrian commas)	53ET interval (in cents)	Complement
unison	1/1	0.00	0		0	0	octave
syntonic comma	81/80	21.51	0	c or T − t	1	22.64	semi-diminished octave
diesis diminished second	128/125	41.06	0	D or S − x	2	45.28	augmented seventh
lesser chromatic semitone minor semitone augmented unison	25/24	70.67	100	x or t − S or T − L	3	67.92	diminished octave
Pythagorean minor second Pythagorean limma	256/243	90.22	100	Λ	4	90.57	Pythagorean major seventh
greater chromatic semitone wide augmented unison	135/128	92.18	100	X or T − S	4	90.57	narrow diminished octave
major semitone limma minor second	16/15	111.73	100	S	5	113.21	major seventh
large limma acute minor second	27/25	133.24	100	L or S + c or T − x	6	135.85	grave major seventh
grave tone grave major second	800/729	160.90	200	τ or Λ + x or t − c	7	158.49	acute minor seventh
minor tone lesser major second	10/9	182.40	200	t	8	181.13	minor seventh
major tone Pythagorean major second greater major second	9/8	203.91	200	T or t + c	9	203.77	Pythagorean minor seventh
diminished third	256/225	223.46	200	S + S	10	226.42	augmented sixth
semi-augmented second	125/108	253.08	300	t + x	11	249.06	semi-augmented sixth
augmented second	75/64	274.58	300	T + x	12	271.70	diminished seventh
Pythagorean minor third	32/27	294.13	300	T + Λ	13	294.34	Pythagorean major sixth
minor third	6/5	315.64	300	T + S	14	316.98	major sixth
acute minor third	243/200	333.18	300	T + L	15	339.62	grave major sixth
grave major third	100/81	364.81	400	T + τ	16	362.26	acute minor sixth
major third	5/4	386.31	400	T + t	17	384.91	minor sixth
Pythagorean major third	81/64	407.82	400	T + T	18	407.55	Pythagorean minor sixth
classic diminished fourth	32/25	427.37	400	T + S + S	19	430.19	classic augmented fifth
classic augmented third	125/96	456.99	500	T + t + x	20	452.83	classic diminished sixth
wide augmented third	675/512	478.49	500	T + t + X	21	475.47	narrow diminished sixth
perfect fourth	4/3	498.04	500	T + t + S	22	498.11	perfect fifth
acute fourth^[1]	27/20	519.55	500	T + t + L	23	520.75	grave fifth
classic augmented fourth	25/18	568.72	600	T + t + t	25	566.04	classic diminished fifth
augmented fourth	45/32	590.22	600	T + t + T	26	588.68	diminished fifth
diminished fifth	64/45	609.78	600	T + t + S + S	27	611.32	augmented fourth
classic diminished fifth	36/25	631.29	600	T + t + S + L	28	633.96	classic augmented fourth
grave fifth^[1]	40/27	680.45	700	T + t + S + t	30	679.25	acute fourth
perfect fifth	3/2	701.96	700	T + t + S + T	31	701.89	perfect fourth
narrow diminished sixth	1024/675	721.51	700	T + t + S + S + S	32	724.53	wide augmented third
classic diminished sixth	192/125	743.01	700	T + t + S + L + S	33	747.17	classic augmented third
classic augmented fifth	25/16	772.63	800	T + t + S + T + x	34	769.81	classic diminished fourth
Pythagorean minor sixth	128/81	792.18	800	T + t + S + T + Λ	35	792.45	Pythagorean major third
minor sixth	8/5	813.69	800	(T + t + S + T) + S	36	815.09	major third
acute minor sixth	81/50	835.19	800	(T + t + S + T) + L	37	837.74	grave major third
grave major sixth	400/243	862.85	900	(T + t + S + T) + τ	38	862.85	acute minor third
major sixth	5/3	884.36	900	(T + t + S + T) + t	39	883.02	minor third
Pythagorean major sixth	27/16	905.87	900	(T + t + S + T) + T	40	905.66	Pythagorean minor third
diminished seventh	128/75	925.42	900	(T + t + S + T) + S + S	41	928.30	augmented second
semi-augmented sixth^[1]	216/125	946.92	800	(T + t + S + T) + S + L	42	946.92	semi-augmented second
augmented sixth	225/128	976.54	1000	(T + t + S + T) + T + x	43	973.58	diminished third
lesser minor seventh Pythagorean minor seventh	16/9	996.09	1000	(T + t + S + T) + T + Λ	44	996.23	greater major second Pythagorean major second
greater minor seventh	9/5	1017.60	1000	(T + t + S + T) + T + S	45	1018.87	lesser major second
acute minor seventh	729/400	1039.10	1000	(T + t + S + T) + T + L	46	1041.51	grave major second
grave major seventh	50/27	1066.76	1100	(T + t + S + T) + T + τ	47	1064.15	acute minor second
major seventh	15/8	1088.27	1100	(T + t + S + T) + T + t	48	1086.79	minor second
narrow diminished octave	256/135	1107.82	1100	(T + t + S + T) + t + S + S	49	1109.43	wide augmented unison
Pythagorean major seventh	243/128	1109.78	1100	(T + t + S + T) + T + T	49	1109.43	Pythagorean minor second
diminished octave	48/25	1129.33	1100	(T + t + S + T) + T + S + S	50	1132.08	augmented unison
augmented seventh	125/64	1158.94	1200	(T + t + S + T) + T + t + x	51	1154.72	diminished second
semi-diminished octave	160/81	1178.49	1200	(T + t + S + T) + T + t + x + c	52	1177.36	syntonic comma
octave	2/1	1200.00	1200	(T + t + S + T) + (T + t + S)	53	1200.00	unison

(The Pythagorean minor second is found by adding 5 perfect fourths.)

The table below shows how these steps map to the first 31 scientific harmonics, transposed into a single octave.

Harmonic	Musical Name	Ratio	Cents	12ET Cents	53ET Commas	53ET Cents
1	unison	1/1	0.00	0	0	0.00
2	octave	2/1	1200.00	1200	53	1200.00
3	perfect fifth	3/2	701.96	700	31	701.89
5	major third	5/4	386.31	400	17	384.91
7	augmented sixth§	7/4	968.83	1000	43	973.58
9	major tone	9/8	203.91	200	9	203.77
11	major fourth	11/8	551.32	500 or 600	24	543.40
13	acute minor sixth§	13/8	840.53	800	37	837.74
15	major seventh	15/8	1088.27	1100	48	1086.79
17	limma§	17/16	104.96	100	5	113.21
19	Pythagorean minor third§	19/16	297.51	300	13	294.34
21	wide augmented third§	21/16	470.78	500	21	475.47
23	classic diminished fifth§	23/16	628.27	600	28	633.96
25	classic augmented fifth	25/16	772.63	800	34	769.81
27	Pythagorean major sixth	27/16	905.87	900	40	905.66
29	minor seventh§	29/16	1029.58	1000	45	1018.87
31	augmented seventh§	31/16	1145.04	1100	51	1154.72

§ These intervals also appear in the upper table, although with different ratios.

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.

In music, there are two common meanings for tuning:

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 temperament is a musical temperament, that is a tuning system, 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 the same way as Pythagorean tuning, as a stack of equal fifths, but it is a temperament 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 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. 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.

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. 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.

In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths span the same number of note letter names and staff positions, but consist of a different number of semitones.

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees (of a major scale are called major.

In Western classical music, a minor sixth is a musical interval encompassing six staff positions, and is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a minor sixth, as the note F lies eight semitones above A, and there are six staff positions from A to F. Diminished and augmented sixths span the same number of staff positions, but consist of a different number of semitones.

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. 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". The word comma used without qualification refers to the syntonic comma, 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.

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. Play (help·info) Each step represents a frequency ratio of 2^1⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.

In musical tuning theory, a Pythagorean interval is a musical interval with 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.

Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.

In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.

In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2, 1.5, and may be approximated by an equal tempered perfect fifth which is 2^7/12. If the A above middle C is 440 Hz, the perfect fifth above it would be E, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.

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.

<span class="mw-page-title-main">Lattice (music)</span>

In musical tuning, a lattice "is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio. The lattice can be two-, three-, or n-dimensional, with each dimension corresponding to a different prime-number partial [pitch class]." When listed in a spreadsheet a lattice may be referred to as a tuning table.

References

1 2 3 "Stichting Huygens-Fokker: List of intervals".

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[intervals-1] 1 2 3 "Stichting Huygens-Fokker: List of intervals".

[1]

List of intervals in 5-limit just intonation

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