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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. 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
Factorization
Monzo
Name(s)
2-limit (complete)
2/1
1200.000
2/1
| 1 >
(perfect) unison, unity, perfect prime, tonic, duple
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 >
marveltwin comma
325/324
5.3351
(52*13)/(22*34)
| -2 -4 2 0 0 1 >

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

| -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 (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 >

35/34
50.184
(5*7)/(2*17)
| -1 0 1 1 0 0 -1 >
septendecimal 1/4-tone
51/50
34.283
(3*17)/(2*52)
| -1 1 -2 0 0 0 1 >
17th-partial chroma
52/51
33.617
(22*13)/(3*17)
| 2 -1 0 0 0 1 -1 >

85/84
20.488
(5*17)/(22*3*7)
| -2 -1 1 -1 0 0 1 >

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 >

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 >

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
23-limit (incomplete)
23/22
76.956
23/(2*11)


24/23
73.681
(23*3)/23


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