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.

Names	Ratio	Cents	12ET Cents	Definition	53ET commas	53ET cents	Representation (Makam)	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 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
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	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	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
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
augmented sixth	225/128	976.54	1000	(T + t + S + T) + T + x	43	973.58		diminished third
Pythagorean minor seventh	16/9	996.09	1000	(T + t + S + T) + T + Λ	44	996.23		Pythagorean major second
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		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