List of superparticular intervals

This list of superparticular intervals ordered by prime limit. It reaches to the 101-limit and is complete up to the 17-limit.

Superparticular numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in just intonation and Harmonic Series music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio 21/20. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common commas are superparticular ratios.

The list below is ordered by harmonic limit, or the largest prime involved in the prime factorization. 36/35, for instance, is an interval of the 7-limit, as it factors to (2²*3²)/(5*7), while 37/36 would belong to the 37-limit.

Størmer's theorem guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8. A002071 -- OEIS gives the number of superparticular ratios in each prime limit, A145604 - OEIS shows the increment from limit to limit, and A117581 the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).

See also gallery of just intervals. Many of the names below come from here.

Ratio	Cents	Factorization	Monzo	Name(s)
2-limit (complete)
2/1	1200.000	2/1	[1⟩	octave, duple; after octave reduction: (perfect) unison, unity, perfect prime, tonic
3-limit (complete)
3/2	701.995	3/2	[-1 1⟩	perfect fifth, 3rd harmonic (octave reduced), diapente
4/3	498.045	22/3	[2 -1⟩	perfect fourth, 3rd subharmonic (octave reduced), diatessaron
9/8	203.910	32/23	[-3 2⟩	(Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)
5-limit (complete)
5/4	386.314	5/22	[-2 0 1⟩	(classic) (5-limit) major third, 5th harmonic (octave reduced)
6/5	315.641	(2*3)/5	[1 1 -1⟩	(classic) (5-limit) minor third
10/9	182.404	(2*5)/32	[1 -2 1⟩	classic (whole) tone, classic major second, minor whole tone
16/15	111.713	24/(3*5)	[4 -1 -1⟩	minor diatonic semitone, 15th subharmonic
25/24	70.672	52/(23*3)	[-3 -1 2⟩	chroma, (classic) chromatic semitone, Zarlinian semitone
81/80	21.506	(3/2)4/5	[-4 4 -1⟩	syntonic comma, Didymus comma
7-limit (complete)
7/6	266.871	7/(2*3)	[-1 -1 0 1⟩	(septimal) subminor third, septimal minor third, augmented second
8/7	231.174	23/7	[3 0 0 -1⟩	(septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic
15/14	119.443	(3*5)/(2*7)	[-1 1 1 -1⟩	septimal diatonic semitone
21/20	84.467	(3*7)/(22*5)	[-2 1 -1 1⟩	minor semitone, large septimal chromatic semitone
28/27	62.961	(22*7)/33	[2 -3 0 1⟩	septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone
36/35	48.770	(22*33)/(5*7)	[2 2 -1 -1⟩	septimal quarter tone, septimal diesis
49/48	35.697	72/(24*3)	[-4 -1 0 2⟩	large septimal diesis, slendro diesis, septimal 1/6-tone
50/49	34.976	2*(5/7)2	[1 0 2 -2⟩	septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma
64/63	27.264	26/(32*7)	[6 -2 0 -1⟩	septimal comma, Archytas' comma
126/125	13.795	(2*32*7)/53	[1 2 -3 1⟩	starling comma, septimal semicomma
225/224	7.7115	(3*5)2/(25*7)	[-5 2 2 -1⟩	marvel comma, septimal kleisma
2401/2400	0.72120	74/(25*3*52)	[-5 -1 -2 4⟩	breedsma
4375/4374	0.39576	(54*7)/(2*37)	[-1 -7 4 1⟩	ragisma
11-limit (complete)
11/10	165.004	11/(2*5)	[-1 0 -1 0 1⟩	(large) (undecimal) neutral second, 4/5-tone, Ptolemy's second
12/11	150.637	(22*3)/11	[2 1 0 0 -1⟩	(small) (undecimal) neutral second, 3/4-tone
22/21	80.537	(2*11)/(3*7)	[1 -1 0 -1 1⟩	undecimal minor semitone
33/32	53.273	(3*11)/25	[-5 1 0 0 1⟩	undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)
45/44	38.906	(3/2)2*(5/11)	[-2 2 1 0 -1⟩	1/5-tone
55/54	31.767	(5*11)/(2*33)	[-1 -3 1 0 1⟩	undecimal diasecundal comma, eleventyfive comma
56/55	31.194	(23*7)/(5*11)	[3 0 -1 1 -1⟩	undecimal tritonic comma, konbini comma
99/98	17.576	(3/7)2*(11/2)	[-1 2 0 -2 1⟩	small undecimal comma, mothwellsma
100/99	17.399	(2*5/3)2/11)	[2 -2 2 0 -1⟩	Ptolemy's comma, ptolemisma
121/120	14.376	112/(23*3*5)	[-3 -1 -1 0 2⟩	undecimal seconds comma, biyatisma
176/175	9.8646	(24*11)/(52*7)	[4 0 -2 -1 1⟩	valinorsma
243/242	7.1391	35/(2*112)	[-1 5 0 0 -2⟩	neutral third comma, rastma
385/384	4.5026	(5*7*11)/(27*3)	[-7 -1 1 1 1⟩	keenanisma
441/440	3.9302	(3*7)2/(23*5*11)	[-3 2 -1 2 -1⟩	Werckmeister's undecimal septenarian schisma, werckisma
540/539	3.2090	(2/7)2*33*5/11	[2 3 1 -2 -1⟩	Swets' comma, swetisma
3025/3024	0.57240	(5*11)2/(24*32*7)	[-4 -3 2 -1 2⟩	Lehmerisma
9801/9800	0.17665	(11/(5*7))2*34/23	[-3 4 -2 -2 2⟩	Gauss comma, kalisma
13-limit (complete)
13/12	138.573	13/(22*3)	[-2 -1 0 0 0 1⟩	tridecimal 2/3-tone
14/13	128.298	(2*7)/13	[1 0 0 1 0 -1⟩	2/3-tone, trienthird
26/25	67.900	(2*13)/52	[1 0 -2 0 0 1⟩	tridecimal 1/3-tone
27/26	65.337	33/(2*13)	[-1 3 0 0 0 -1⟩	tridecimal comma
40/39	43.831	(23*5)/(3*13)	[3 -1 1 0 0 -1⟩	tridecimal minor diesis
65/64	26.841	(5*13)/26	[-6 0 1 0 0 1⟩	wilsorma, 13th-partial chroma
66/65	26.432	(2*3*11)/(5*13)	[1 1 -1 0 1 -1⟩	winmeanma
78/77	22.339	(2*3*13)/(7*11)	[1 1 0 -1 -1 1⟩	negustma
91/90	19.130	(7*13)/(2*32*5)	[-1 -2 -1 1 0 1⟩	Biome comma, superleap comma
105/104	16.567	(3*5*7)/(23*13)	[-3 1 1 1 0 -1⟩	small tridecimal comma, animist comma
144/143	12.064	(22*3)2/(11*13)	[4 2 0 0 -1 -1⟩	grossma
169/168	10.274	132/(23*3*7)	[-3 -1 0 -1 0 2⟩	buzurgisma, dhanvantarisma
196/195	8.8554	(2*7)2/(3*5*13)	[2 -1 -1 2 0 -1⟩	mynucuma
325/324	5.3351	(52*13)/(22*34)	[-2 -4 2 0 0 1⟩	marveltwin comma
351/350	4.9393	(3/5)2*13/(2*7)	[-1 3 -2 -1 0 1⟩	ratwolfsma
352/351	4.9253	(25*11)/(32*13)	[5 -3 0 0 1 -1⟩	minthma
364/363	4.7627	(2/11)2*7*13/3	[2 -1 0 1 -2 1⟩	gentle comma
625/624	2.7722	(5/2)4/(3*13)	[-4 -1 4 0 0 -1⟩	tunbarsma
676/675	2.5629	(2*13/5)2/33	[2 -3 -2 0 0 2⟩	island comma
729/728	2.3764	(32/2)3/(7*13)	[-3 6 0 -1 0 -1⟩	squbema
1001/1000	1.7304	7*11*13/(2*5)3	[-3 0 -3 1 1 1⟩	sinbadma
1716/1715	1.0092	22*3*11*13/(5*73)	[2 1 -1 -3 1 1⟩	lummic comma
2080/2079	0.83252	25*5*13/(33*7*11)	[5 -3 1 -1 -1 1⟩	ibnsinma
4096/4095	0.42272	(26/3)2/(5*7*13)	[12 -2 -1 -1 0 -1⟩	tridecimal schisma, Sagittal schismina
4225/4224	0.40981	(5*13)2/(27*3*11)	[-7 -1 2 0 -1 2⟩	leprechaun comma
6656/6655	0.26012	(23/11)3*13/5	[9 0 -1 0 -3 1⟩	jacobin comma
10648/10647	0.16260	(2*11)3/((3*13)2*7)	[3 -2 0 -1 3 -2⟩	harmonisma
123201/123200	0.014052	(3/2)6*(13/5)2/(7*11)	[-6 6 -2 -1 -1 2⟩	chalmersia
17-limit (complete)
17/16	104.955	17/24	[-4 0 0 0 0 0 1⟩	17th harmonic (octave reduced)
18/17	98.955	(2*32)/17	[1 2 0 0 0 0 -1⟩	Arabic lute index finger
34/33	51.682	(2*17)/(3*11)	[1 -1 0 0 -1 0 1⟩	septendecimal 1/4-tone (greater)
35/34	50.184	(5*7)/(2*17)	[-1 0 1 1 0 0 -1⟩	septendecimal 1/4-tone (lesser)
51/50	34.283	(3*17)/(2*52)	[-1 1 -2 0 0 0 1⟩	septendecimal 1/6-tone (greater)
52/51	33.617	(22*13)/(3*17)	[2 -1 0 0 0 1 -1⟩	septendecimal 1/6-tone (lesser)
85/84	20.488	(5*17)/(22*3*7)	[-2 -1 1 -1 0 0 1⟩	septendecimal comma
120/119	14.487	(23*3*5)/(7*17)	[3 1 1 -1 0 0 -1⟩
136/135	12.777	(2/3)3*17/5	[3 -3 -1 0 0 0 1⟩
154/153	11.278	(2*7*11)/(32*17)	[1 -2 0 1 1 0 -1⟩
170/169	10.214	(2*5*17)/132	[1 0 1 0 0 -2 1⟩
221/220	7.8514	(13*17)/(22*5*11)	[-2 0 -1 0 -1 1 1⟩
256/255	6.7759	(28)/(3*5*17)	[8 -1 -1 0 0 0 -1⟩	255th subharmonic
273/272	6.3532	(3*7*13)/(24*17)	[-4 1 0 1 0 1 -1⟩
289/288	6.0008	(17/3)2/25	[-5 -2 0 0 0 0 2⟩
375/374	4.6228	(3*53)/(2*11*17)	[-1 1 3 0 -1 0 -1⟩
442/441	3.9213	(2*13*17)/(3*7)2	[1 -2 0 -2 0 1 1⟩
561/560	3.0887	(3*11*17)/(24*5*7)	[-4 1 -1 -1 1 0 1⟩
595/594	2.9121	(5*7*17)/(2*33*11)	[-1 -3 1 1 -1 0 1⟩
715/714	2.4230	(5*11*13)/(2*3*7*17)	[-1 -1 1 -1 1 1 -1⟩
833/832	2.0796	(72*17)/(26*13)	[-6 0 0 2 0 -1 1⟩
936/935	1.8506	(23*32*13)/(5*11*17)	[3 2 -1 0 -1 1 -1⟩
1089/1088	1.5905	(32*112)/(26*17)	[-6 2 0 0 2 0 -1⟩	twosquare comma
1156/1155	1.4983	(22*172)/(3*5*7*11)	[2 -1 -1 -1 -1 0 2⟩
1225/1224	1.4138	(52*72)/(23*32*17)	[-3 -2 2 2 0 0 -1⟩
1275/1274	1.3584	(3*52*17)/(2*72*13)	[-1 1 2 -2 0 -1 1⟩
1701/1700	1.0181	(35*7)/[(2*5)2*17]	[-2 5 -2 1 0 0 -1⟩
2058/2057	0.8414	(2*3*73)/(112*17)	[1 1 0 3 -2 0 -1⟩	xenisma
2431/2430	0.7123	(11*13*17)/(2*35*5)	[-1 -5 -1 0 1 1 1⟩
2500/2499	0.6926	(22*54)/(3*72*17)	[2 -1 4 -2 0 0 -1⟩
2601/2600	0.6657	(32*172)/(23*52*13)	[-3 2 -2 0 0 -1 2⟩
4914/4913	0.3523	(2*33*7*13)/(173)	[1 3 0 1 0 1 -3⟩
5832/5831	0.2969	(23*36)/(73*17)	[3 6 0 -3 0 0 -1⟩
12376/12375	0.1399	(23*7*13*17)/(32*53*11)	[3 -2 -3 1 -1 1 1⟩
14400/14399	0.1202	(26*32*52)/(7*112*17)	[6 2 2 -1 -2 0 -1⟩
28561/28560	0.0606	(134)/(24*3*5*7*17)	[-4 -1 -1 -1 0 4 -1⟩
31213/31212	0.0555	(74*13)/(22*33*172)	[-2 -3 0 4 0 1 -2⟩
37180/37179	0.0466	(22*5*11*132)/(37*17)	[2 -7 1 0 1 2 -1⟩
194481/194480	0.0089	(34*74)/(24*5*11*13*17)	[-4 4 -1 4 -1 -1 -1⟩	scintillisma
336141/336140	0.0052	(32*133*17)/(22*5*75)	[-2 2 -1 -5 0 3 1⟩
19-limit (incomplete)
19/18	93.603	19/(2*32)	[-1 -2 0 0 0 0 0 1⟩	undevicesimal semitone
20/19	88.801	(22*5)/19	[2 0 1 0 0 0 0 -1⟩	small undevicesimal semitone
39/38	44.970	(3*13)/(2*19)	[-1 1 0 0 0 1 0 -1⟩	undevicesimal 2/9-tone
57/56	30.642	(3*19)/(23*7)	[-3 1 0 -1 0 0 0 1⟩
76/75	22.931	(22*19)/(3*52)	[2 -1 -2 0 0 0 0 1⟩
77/76	22.631	(7*11)/(22*19)	[-2 0 0 1 1 0 0 -1⟩
96/95	18.128	(25*3)/(5*19)	[5 1 -1 0 0 0 0 -1⟩
133/132	13.066	(19*7)/(22*3*11)	[-2 -1 0 1 -1 0 0 1⟩
153/152	11.352	(32*17)/(23*19)	[-3 2 0 0 0 0 1 -1⟩
171/170	10.154	(32*19)/(2*5*17)	[-1 2 -1 0 0 0 -1 1⟩
190/189	9.1358	(2*5*19)/(33*7)	[1 -3 1 -1 0 0 0 1⟩
209/208	8.3033	(11*19)/(24*13)	[-4 0 0 0 1 -1 0 1⟩
210/209	8.2637	(2*3*5*7)/(11*19)	[1 1 1 1 -1 0 0 -1⟩
286/285	6.0639	(2*11*13)/(3*5*19)	[1 -1 -1 0 1 1 0 -1⟩
324/323	5.3516	(22*34)/(17*19)	[2 4 0 0 0 0 -1 -1⟩
343/342	5.0547	74/(2*33*19)	[-1 -2 0 3 0 0 0 -1⟩
361/360	4.8023	192/(23*32*5)	[-3 -2 -1 0 0 0 0 2⟩
400/399	4.3335	(24*52)/(3*7*19)	[4 -1 2 -1 0 0 0 -1⟩
456/455	3.8007	(23*3*19)/(5*7*13)	[3 1 -1 -1 0 -1 0 1⟩
476/475	3.6409	(22*7*17)/(52*19)	[2 0 -2 1 0 0 1 -1⟩
495/494	3.501	(32*5*11)/(2*13*19)	[-1 2 1 0 1 -1 0 -1⟩
513/512	3.378	(33*19)/29	[-9 3 0 0 0 0 0 1⟩	513th harmonic
969/968	1.7875	(3*17*19)/(23*112)	[-3 1 0 0 -2 0 1 1⟩
1216/1215	1.4243	(26*19)/(35*5)	[6 -5 -1 0 0 0 0 1⟩
1331/1330	1.3012	113/(2*5*7*19)	[-1 0 -1 -1 3 0 0 -1⟩
1445/1444	1.1985	5*(17/(2*19))2	[-2 0 1 0 0 0 2 -2⟩
1521/1520	1.1386	(3*13)2/(24*5*19)	[-4 2 -1 0 0 2 0 -1⟩
1540/1539	1.1245	(22*5*7*11)/(34*19)	[2 -4 1 1 1 0 0 -1⟩
1729/1728	1.0016	(7*13*19)/(26*33)	[-6 -3 0 1 0 1 0 1⟩
23-limit (incomplete)
23/22	76.956	23/(2*11)		greater vicesimotertial semitone
24/23	73.681	(23*3)/23		small vicesimotertial semitone
46/45	38.051	(2*23)/(32*5)
69/68	25.274	(3*23)/(22*17)
70/69	24.910	(2*5*7)/(3*23)
92/91	18.921	(22*23)/(7*13)
115/114	15.120	(5*23)/(2*3*19)
161/160	10.7865	(7*23)/(25*5)
162/161	10.720	(2*34)/(7*23)
208/207	8.343	(24*13)/(23*9)
576/575	3.008	(26*32)/(23*25)
29-limit (incomplete)
29/28	60.751	29/(22*7)
30/29	58.692	(2*3*5)/29
58/57	30.109	(2*29)/(3*19)
88/87	19.786	(23*11)/(3*29)
31-limit (incomplete)
31/30	56.767	31/(2*3*5)
32/31	54.964	25/31		31st subharmonic
63/62	27.700	(32*7)/(2*31)
93/92	18.716	(3*31)/(22*23)
37-limit (incomplete)
37/36	47.434	37/(22*32)
38/37	46.169	(2*19)/37
75/74	23.238	(3*52)/(2*37)
41-limit (incomplete)
41/40	42.749	41/(23*5)
42/41	41.719	(2*3*7)/41
82/81	21.242	(2*41)/34
43-limit (incomplete)
43/42	40.737	43/(2*3*7)
44/43	39.800	(22*11)/43
86/85	20.249	(2*43)/(5*17)
87/86	20.014	(3*29)/(2*43)
47-limit (incomplete)
47/46	37.232	47/(2*23)
48/47	36.448	(24*3)/47
94/93	18.516	(2*47)/(3*31)
95/94	18.320	(5*19)/(2*47)
53-limit (incomplete)
53/52	32.977	53/(22*13)
54/53	32.360	(2*33)/53
59-limit (incomplete)
59/58	29.594	59/(2*29)
60/59	29.097	(22*3*5)/59
61-limit (incomplete)
61/60	28.616	61/(22*3*5)
62/61	28.151	(2*31)/61
67-limit (incomplete)
67/66	26.034	67/(2*3*11)
68/67	25.648	(22*17)/67
71-limit (incomplete)
71/70	24.557	71/(2*5*7)
72/71	24.213	(23*32)/71
73-limit (incomplete)
73/72	23.879	73/(23*32)
74/73	23.555	(2*37)/73
79-limit (incomplete)
79/78	22.054	79/(2*3*13)
80/79	21.777	(24*5)/79
83-limit (incomplete)
83/82	20.985	83/(2*41)
84/83	20.734	(22*3*7)/83
89-limit (incomplete)
89/88	19.562	89/(23*11)
90/89	19.344	(2*32*5)/89
97-limit (incomplete)
97/96	17.940	97/(25*3)
98/97	17.756	(2*72)/97
101-limit (incomplete)
101/100	17.226	101/(22*52)
102/101	17.057	(2*3*17)/101