List of Superparticular Intervals

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 | 2 2 -1 -1 >, 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. OEIS A145604 gives the number of superparticular ratios in each prime 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	Monzo	Name(s)
2-limit
2/1	1200.000	| 1 >	(perfect) unison, unity, perfect prime, tonic, duple
3-limit
3/2	701.995	| -1 1 >	perfect fifth, 3rd harmonic (octave reduced), diapente
4/3	498.045	| 2 -1 >	perfect fourth, 3rd subharmonic (octave reduced), diatessaron
9/8	203.910	| -3 2 >	(Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)
5-limit
5/4	386.314	| -2 0 1 >	(classic) (5-limit) major third, 5th harmonic (octave reduced)
6/5	315.641	| 1 1 -1 >	(classic) (5-limit) minor third
10/9	182.404	| 1 -2 1 >	classic (whole) tone, classic major second, minor whole tone
16/15	111.713	| 4 -1 -1 >	minor diatonic semitone, 15th subharmonic
25/24	70.672	| -3 -1 2 >	chroma, (classic) chromatic semitone, Zarlinian semitone
81/80	21.506	| -4 4 -1 >	syntonic comma, Didymus comma
7-limit
7/6	266.871	| -1 -1 0 1 >	(septimal) subminor third, septimal minor third, augmented second
8/7	231.174	| 3 0 0 -1 >	(septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic
15/14	119.443	| -1 1 1 -1 >	septimal diatonic semitone
21/20	84.467	| -2 1 -1 1 >	minor semitone, large septimal chromatic semitone
28/27	62.961	| 2 -3 0 1 >	septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone
36/35	48.770	| 2 2 -1 -1 >	septimal quarter tone, septimal diesis
49/48	35.697	| -4 -1 0 2 >	large septimal diesis, slendro diesis, septimal 1/6-tone
50/49	34.976	| 1 0 2 -2 >	septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma
64/63	27.264	| 6 -2 0 -1 >	septimal comma, Archytas' comma
126/125	13.795	| 1 2 -3 1 >	starling comma, septimal semicomma
225/224	7.7115	| -5 2 2 -1 >	marvel comma, septimal kleisma
2401/2400	0.72120	| -5 -1 -2 4 >	breedsma
4375/4374	0.39576	| -1 -7 4 1 >	ragisma
11-limit
11/10	165.004	| -1 0 -1 0 1 >	(large) (undecimal) neutral second, 4/5-tone, Ptolemy's second
12/11	150.637	| 2 1 0 0 -1 >	(small) (undecimal) neutral second, 3/4-tone
22/21	80.537	| 1 -1 0 -1 1 >	undecimal minor semitone
33/32	53.273	| -5 1 0 0 1 >	undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)
45/44	38.906	| -2 2 1 0 -1 >	1/5-tone
55/54	31.767	| -1 -3 1 0 1 >	undecimal diasecundal comma, eleventyfive comma
56/55	31.194	| 3 0 -1 1 -1 >	undecimal tritonic comma, konbini comma
99/98	17.576	| -1 2 0 -2 1 >	small undecimal comma, mothwellsma
100/99	17.399	| 2 -2 2 0 -1 >	Ptolemy's comma, ptolemisma
121/120	14.376	| -3 -1 -1 0 2 >	undecimal seconds comma, biyatisma
176/175	9.8646	| 4 0 -2 -1 1 >	valinorsma
243/242	7.1391	| -1 5 0 0 -2 >	neutral third comma, rastma
385/384	4.5026	| -7 -1 1 1 1 >	keenanisma
441/440	3.9302	| -3 2 -1 2 -1 >	Werckmeister's undecimal septenarian schisma, werckisma
540/539	3.2090	| 2 3 1 -2 -1 >	Swets' comma, swetisma
3025/3024	0.57240	| -4 -3 2 -1 2 >	Lehmerisma
9801/9800	0.17665	| -3 4 -2 -2 2 >	Gauss comma, kalisma
13-limit
13/12	138.573	| -2 -1 0 0 0 1 >	tridecimal 2/3-tone
14/13	128.298	| 1 0 0 1 0 -1 >	2/3-tone, trienthird
26/25	67.900	| 1 0 -2 0 0 1 >	tridecimal 1/3-tone
27/26	65.337	| -1 3 0 0 0 -1 >	tridecimal comma
40/39	43.831	| 3 -1 1 0 0 -1 >	tridecimal minor diesis
65/64	26.841	| -6 0 1 0 0 1 >	wilsorma, 13th-partial chroma
66/65	26.432	| 1 1 -1 0 1 -1 >	winmeanma
78/77	22.339	| 1 1 0 -1 -1 1 >	negustma
91/90	19.130	| -1 -2 -1 1 0 1 >	Biome comma, superleap comma
105/104	16.567	| -3 1 1 1 0 -1 >	small tridecimal comma, animist comma
144/143	12.064	| 4 2 0 0 -1 -1 >	grossma
169/168	10.274	| -3 -1 0 -1 0 2 >	buzurgisma, dhanvantarisma
196/195	8.8554	| 2 -1 -1 2 0 -1 >	marveltwin comma
325/324	5.3351	| -2 -4 2 0 0 1 >
351/350	4.9393	| -1 3 -2 -1 0 1 >	ratwolfsma
352/351	4.9253	| 5 -3 0 0 1 -1 >	minthma
364/363	4.7627	| 2 -1 0 1 -2 1 >	gentle comma
625/624	2.7722	| -4 -1 4 0 0 -1 >	tunbarsma
676/675	2.5629	| 2 -3 -2 0 0 2 >	island comma
729/728	2.3764	| -3 6 0 -1 0 -1 >	squbema
1001/1000	1.7304	| -3 0 -3 1 1 1 >	sinbadma
1716/1715	1.0092	| 2 1 -1 -3 1 1 >	lummic comma
2080/2079	0.83252	| 5 -3 1 -1 -1 1 >	ibnsinma
4096/4095	0.42272	| 12 -2 -1 -1 0 -1 >	tridecimal schisma, Sagittal schismina
4225/4224	0.40981	| -7 -1 2 0 -1 2 >	leprechaun comma
6656/6655	0.26012	| 9 0 -1 0 -3 1 >	jacobin comma
10648/10647	0.16260	| 3 -2 0 -1 3 -2 >	harmonisma
123201/123200	0.014052	| -6 6 -2 -1 -1 2 >	chalmersia
17-limit
17/16	104.955	| -4 0 0 0 0 0 1 >	17th harmonic (octave reduced)
18/17	98.955	| 1 2 0 0 0 0 -1 >	Arabic lute index finger
34/33	51.682	| 1 -1 0 0 -1 0 1 >
35/34	50.184	| -1 0 1 1 0 0 -1 >	septendecimal 1/4-tone
51/50	34.283	| -1 1 -2 0 0 0 1 >	17th-partial chroma
52/51	33.617	| 2 -1 0 0 0 1 -1 >
85/84	20.488	| -2 -1 1 -1 0 0 1 >
120/119	14.487	| 3 1 1 -1 0 0 -1 >
136/135	12.777	| 3 -3 -1 0 0 0 1 >
154/153	11.278	| 1 -2 0 1 1 0 -1 >
170/169	10.214	| 1 0 1 0 0 -2 1 >
221/220	7.8514	| -2 0 -1 0 -1 1 1 >
256/255	6.7759	| 8 -1 -1 0 0 0 -1 >	255th subharmonic
273/272	6.3532	| -4 1 0 1 0 1 -1 >
289/288	6.0008	| -5 -2 0 0 0 0 2 >
375/374	4.6228	| -1 1 3 0 -1 0 -1 >
442/441	3.9213	| 1 -2 0 -2 0 1 1 >
561/560	3.0887	| -4 1 -1 -1 1 0 1 >
595/594	2.9121	| -1 -3 1 1 -1 0 1 >
715/714	2.4230	| -1 -1 1 -1 1 1 -1 >
833/832	2.0796	| -6 0 0 2 0 -1 1 >
936/935	1.8506	| 3 2 -1 0 -1 1 -1 >
1089/1088	1.5905	| -6 2 0 0 2 0 -1 >	twosquare comma
1156/1155	1.4983	| 2 -1 -1 -1 -1 0 2 >
1225/1224	1.4138	| -3 -2 2 2 0 0 -1 >
1275/1274	1.3584	| -1 1 2 -2 0 -1 1 >
1701/1700	1.0181	| -2 5 -2 1 0 0 -1 >
2058/2057	0.8414	| 1 1 0 3 -2 0 -1 >	xenisma
2431/2430	0.7123	| -1 -5 -1 0 1 1 1 >
2500/2499	0.6926	| 2 -1 4 -2 0 0 -1 >
2601/2600	0.6657	| -3 2 -2 0 0 -1 2 >
4914/4913	0.3523	| 1 3 0 1 0 1 -3 >
5832/5831	0.2969	| 3 6 0 -3 0 0 -1 >
12376/12375	0.1399	| 3 -2 -3 1 -1 1 1 >
14400/14399	0.1202	| 6 2 2 -1 -2 0 -1 >
28561/28560	0.0606	| -4 -1 -1 -1 0 4 -1 >
31213/31212	0.0555	| -2 -3 0 4 0 1 -2 >
37180/37179	0.0466	| 2 -7 1 0 1 2 -1 >
194481/194480	0.0089	| -4 4 -1 4 -1 -1 -1>	scintillisma
336141/336140	0.0052	| -2 2 -1 -5 0 3 1 >
19-limit (incomplete)
19/18	93.603	| -1 -2 0 0 0 0 0 1 >	undevicesimal semitone
20/19	88.801	| 2 0 1 0 0 0 0 -1 >	small undevicesimal semitone
39/38	44.970	| -1 1 0 0 0 1 0 -1 >
57/56	30.642	| -3 1 0 -1 0 0 0 1 >
76/75	22.931	| 2 -1 -2 0 0 0 0 1 >
77/76	22.631	| -2 0 0 1 1 0 0 -1 >
96/95	18.128	| 5 1 -1 0 0 0 0 -1 >
133/132	13.066	| -2 -1 0 1 -1 0 0 1 >
153/152	11.352	| -3 2 0 0 0 0 1 -1 >
171/170	10.154	| -1 2 -1 0 0 0 -1 1 >
190/189	9.1358	| 1 -3 1 -1 0 0 0 1 >
209/208	8.3033	| -4 0 0 0 1 -1 0 1 >
210/209	8.2637	| 1 1 1 1 -1 0 0 -1 >
286/285	6.0639	| 1 -1 -1 0 1 1 0 -1 >
324/323	5.3516	| 2 4 0 0 0 0 -1 -1 >
343/342	5.0547	| -1 -2 0 3 0 0 0 -1 >
361/360	4.8023	| -3 -2 -1 0 0 0 0 2 >
400/399	4.3335	| 4 -1 2 -1 0 0 0 -1 >
456/455	3.8007	| 3 1 -1 -1 0 -1 0 1 >
476/475	3.6409	| 2 0 -2 1 0 0 1 -1 >
495/494	3.501	| -1 2 1 0 1 -1 0 -1 >
513/512	3.378	| -9 3 0 0 0 0 0 1 >	513th harmonic
23-limit (incomplete)
23/22	76.956
24/23	73.681
46/45	38.051
69/68	25.274
70/69	24.910
92/91	18.921
115/114	15.120
161/160	10.7865
162/161	10.720
208/207	8.343
576/575	3.008
29-limit (incomplete)
29/28	60.751
30/29	58.692
58/57	30.109
88/87	19.786
31-limit (incomplete)
31/30	56.767
32/31	54.964		31st subharmonic
63/62	27.700
93/92	18.716
37-limit (incomplete)
37/36	47.434
38/37	46.169
75/74	23.238
41-limit (incomplete)
41/40	42.749
42/41	41.719
82/81	21.242
43-limit (incomplete)
43/42	40.737
44/43	39.800
86/85	20.249
87/86	20.014
47-limit (incomplete)
47/46	37.232
48/47	36.448
94/93	18.516
95/94	18.320
53-limit (incomplete)
53/52	32.977
54/53	32.360
59-limit (incomplete)
59/58	29.594
60/59	29.097
61-limit (incomplete)
61/60	28.616
62/61	28.151
67-limit (incomplete)
67/66	26.034
68/67	25.648
71-limit (incomplete)
71/70	24.557
72/71	24.213
73-limit (incomplete)
73/72	23.879
74/73	23.555
79-limit (incomplete)
79/78	22.054
80/79	21.777
83-limit (incomplete)
83/82	20.985
84/83	20.734
89-limit (incomplete)
89/88	19.562
90/89	19.344
97-limit (incomplete)
97/96	17.940
98/97	17.756
101-limit (incomplete)
101/100	17.226
102/101	17.057

Superparticular split intervals

Split intervals. Some edos temper out some of the resulting commas, and split commas themselves create smaller commas. There are also chained split commas, like 25/24 = 50/49 × 49/48 = (100/99 × 99/98) × 49/48, or 7/6 = 14/13 × 13/12 = 28/27 × 27/26 × 26/25 × 25/24.

3-limit

2/1 = 4/3 × 3/2

5-limit

3/2 = 6/5 × 5/4
5/4 = 10/9 × 9/8

7-limit

4/3 = 8/7 × 7/6
8/7 = 16/15 × 15/14
25/24 = 50/49 × 49/48

11-limit

6/5 = 12/11 × 11/10
11/10 = 22/21 × 21/20
28/27 = 56/55 × 55/54
50/49 = 100/99 × 99/98

13-limit

7/6 = 14/13 × 13/12
13/12 = 26/25 × 25/24
14/13 = 28/27 × 27/26
33/32 = 66/65 × 65/64
176/175 = 352/351 × 351/350

Split octave

2/1

3/2

4/3

5/4

6/5

7/6

8/7

9/8

10/9

11/10

12/11

13/12

14/13

15/14

16/15

-

-

21/20

22/21

-

25/24

26/25

27/26

28/27

-

-

-

-

-

-

-

49​/​48

50​/​49

-

-

55​/​54

56​/​55

-

-

-

-

-

-

-

-

9​9​/​9​8

1​0​0​/​9​9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

Edos by their first equalized superparticulars

3edo, 5edo: 4/3=5/4
7edo: 5/4=6/5
9edo, 12edo: 6/5=7/6
15edo, 19edo: 7/6=8/7
22edo: 8/7=9/8
26edo, 31edo: 9/8=10/9
41edo: 10/9=11/10
46edo, 53edo: 11/10=12/11
58edo: 12/11=13/12
72edo: 13/12=14/13
87edo: 14/13=15/14
94edo: 15/14=16/15
111edo: 16/15=17/16
130edo, 140edo: 17/16=18/17