List of intervals in 5-limit just intonation

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.

In all tunings, the major third is equivalent to two major seconds. However, because just intonation does not allow the irrational ratio of √5/2, two different frequency ratios are used: the major tone (9/8) and the minor tone (10/9).

The intervals within the diatonic scale are shown in the table below.

List

Names	Ratio	Cents	12ET interval (in cents)	Definition	53ET interval (in Holdrian commas)	53ET interval (in cents)	Arel-Ezgi-Uzdilek notation (a 53ET-based notation)	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.

See also

• List of musical intervals
• List of pitch intervals
• Pythagorean interval

References

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