List of superparticular intervals

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This list of superparticular intervals ordered by prime limit. It reaches to the 101-limit and is complete up to the 23-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 states 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: A002071 gives the number of superparticular ratios in each prime limit, OEIS: A145604 shows the increment from limit to limit, and OEIS: A117581 gives 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 the Scala website.

Ratio Cents Factorization Monzo Name(s) Meta[1]
2-limit (complete)
2/1 1200.000 2/1 [1 octave, duple; after octave reduction: (perfect) unison, unity, perfect prime, tonic
3-limit (complete)
3/2 701.955 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 S2
9/8 203.910 32/23 [-3 2 (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced) S3
5-limit (complete)
5/4 386.314 5/22 [-2 0 1 classic/just major third, 5th harmonic (octave reduced)
6/5 315.641 (2*3)/5 [1 1 -1 classic/just minor third
10/9 182.404 (2*5)/32 [1 -2 1 classic (whole) tone, classic major second, minor whole tone
16/15 111.731 24/(3*5) [4 -1 -1 classic/just diatonic semitone, 15th subharmonic S4
25/24 70.672 52/(23*3) [-3 -1 2 classic/just chromatic semitone, chroma, Zarlinian semitone S5
81/80 21.506 (3/2)4/5 [-4 4 -1 syntonic comma, Didymus comma S9
7-limit (complete)
7/6 266.871 7/(2*3) [-1 -1 0 1 (septimal) subminor third, septimal minor third
8/7 231.174 23/7 [3 0 0 -1 (septimal) supermajor second, septimal whole tone, 7th subharmonic
15/14 119.443 (3*5)/(2*7) [-1 1 1 -1 septimal major semitone, septimal diatonic semitone
21/20 84.467 (3*7)/(22*5) [-2 1 -1 1 septimal minor semitone, large septimal chroma
28/27 62.961 (22*7)/33 [2 -3 0 1 septimal 1/3-tone, small septimal chroma, (septimal) subminor second, septimal minor second, trienstonic comma
36/35 48.770 (22*32)/(5*7) [2 2 -1 -1 septimal 1/4-tone, septimal diesis S6
49/48 35.697 72/(24*3) [-4 -1 0 2 slendro diesis, large septimal diesis, large septimal 1/6-tone S7
50/49 34.976 2*(5/7)2 [1 0 2 -2 jubilisma, small septimal diesis, small septimal 1/6-tone, tritonic diesis, Erlich's decatonic comma
64/63 27.264 26/(32*7) [6 -2 0 -1 septimal comma, Archytas' comma S8
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 S15
2401/2400 0.72120 74/(25*3*52) [-5 -1 -2 4 breedsma S49
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, undecimal submajor second, Ptolemy's second
12/11 150.637 (22*3)/11 [2 1 0 0 -1 (small) undecimal neutral second
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 1/4-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 undecimal 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 mothwellsma, small undecimal comma
100/99 17.399 (2*5/3)2/11) [2 -2 2 0 -1 ptolemisma, Ptolemy's comma S10
121/120 14.376 112/(23*3*5) [-3 -1 -1 0 2 biyatisma, undecimal seconds comma S11
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 rastma, neutral thirds comma
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 werckisma, Werckmeister's undecimal septenarian schisma S21
540/539 3.2090 (2/7)2*33*5/11 [2 3 1 -2 -1 swetisma, Swets' comma
3025/3024 0.57240 (5*11)2/(24*32*7) [-4 -3 2 -1 2 lehmerisma S55
9801/9800 0.17665 (11/(5*7))2*34/23 [-3 4 -2 -2 2 kalisma, Gauss comma S99
13-limit (complete)
13/12 138.573 13/(22*3) [-2 -1 0 0 0 1 (large) tridecimal 2/3-tone, tridecimal neutral second
14/13 128.298 (2*7)/13 [1 0 0 1 0 -1 (small) tridecimal 2/3-tone, trienthird
26/25 67.900 (2*13)/52 [1 0 -2 0 0 1 (large) tridecimal 1/3-tone
27/26 65.337 33/(2*13) [-1 3 0 0 0 -1 (small) tridecimal 1/3-tone
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 animist comma, small tridecimal comma
144/143 12.064 (22*3)2/(11*13) [4 2 0 0 -1 -1 grossma S12
169/168 10.274 132/(23*3*7) [-3 -1 0 -1 0 2 buzurgisma, dhanvantarisma S13
196/195 8.8554 (2*7)2/(3*5*13) [2 -1 -1 2 0 -1 mynucuma S14
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 S25
676/675 2.5629 (2*13/5)2/33 [2 -3 -2 0 0 2 island comma S26
729/728 2.3764 (32/2)3/(7*13) [-3 6 0 -1 0 -1 squbema S27
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 schismina, tridecimal schisma S65
4225/4224 0.40981 (5*13)2/(27*3*11) [-7 -1 2 0 -1 2 leprechaun comma S66
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 S351
17-limit (complete)
17/16 104.955 17/24 [-4 0 0 0 0 0 1 large septendecimal semitone, 17th harmonic (octave reduced)
18/17 98.955 (2*32)/17 [1 2 0 0 0 0 -1 small septendecimal semitone, Arabic lute index finger
34/33 51.682 (2*17)/(3*11) [1 -1 0 0 -1 0 1 large septendecimal 1/4-tone
35/34 50.184 (5*7)/(2*17) [-1 0 1 1 0 0 -1 small septendecimal 1/4-tone
51/50 34.283 (3*17)/(2*52) [-1 1 -2 0 0 0 1 large septendecimal 1/6-tone
52/51 33.617 (22*13)/(3*17) [2 -1 0 0 0 1 -1 small septendecimal 1/6-tone
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 lynchisma
136/135 12.777 (2/3)3*17/5 [3 -3 -1 0 0 0 1 septendecimal major second comma
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 septendecimal kleisma, 255th subharmonic S16
273/272 6.3532 (3*7*13)/(24*17) [-4 1 0 1 0 1 -1 tannisma
289/288 6.0008 (17/3)2/25 [-5 -2 0 0 0 0 2 septendecimal 6-cent comma S17
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 dakotisma
715/714 2.4230 (5*11*13)/(2*3*7*17) [-1 -1 1 -1 1 1 -1 September comma, septembrisma, septendecimal bridge comma
833/832 2.0796 (72*17)/(26*13) [-6 0 0 2 0 -1 1 horizon comma
936/935 1.8506 (23*32*13)/(5*11*17) [3 2 -1 0 -1 1 -1 ainos comma, ainma
1089/1088 1.5905 (32*112)/(26*17) [-6 2 0 0 2 0 -1 twosquare comma S33
1156/1155 1.4983 (22*172)/(3*5*7*11) [2 -1 -1 -1 -1 0 2 septendecimal 1/4-tones comma S34
1225/1224 1.4138 (52*72)/(23*32*17) [-3 -2 2 2 0 0 -1 noellisma S35
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 palingenetic comma, palingenesis comma, palingenesma
2058/2057 0.84143 (2*3*73)/(112*17) [1 1 0 3 -2 0 -1 xenisma
2431/2430 0.71230 (11*13*17)/(2*35*5) [-1 -5 -1 0 1 1 1
2500/2499 0.69263 (22*54)/(3*72*17) [2 -1 4 -2 0 0 -1 S50
2601/2600 0.66573 (32*172)/(23*52*13) [-3 2 -2 0 0 -1 2 septendecimal 1/6-tones comma S51
4914/4913 0.35234 (2*33*7*13)/(173) [1 3 0 1 0 1 -3
5832/5831 0.29688 (23*36)/(73*17) [3 6 0 -3 0 0 -1 chlorisma
12376/12375 0.13989 (23*7*13*17)/(32*53*11) [3 -2 -3 1 -1 1 1 flashma
14400/14399 0.12023 (26*32*52)/(7*112*17) [6 2 2 -1 -2 0 -1 sparkisma S120
28561/28560 0.060616 (134)/(24*3*5*7*17) [-4 -1 -1 -1 0 4 -1 S169
31213/31212 0.055466 (74*13)/(22*33*172) [-2 -3 0 4 0 1 -2
37180/37179 0.046564 (22*5*11*132)/(37*17) [2 -7 1 0 1 2 -1
194481/194480 0.008902 (34*74)/(24*5*11*13*17) [-4 4 -1 4 -1 -1 -1 scintillisma S441
336141/336140 0.005150 (32*133*17)/(22*5*75) [-2 2 -1 -5 0 3 1
19-limit (complete)
19/18 93.603 19/(2*32) [-1 -2 0 0 0 0 0 1 large 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 hendrix comma
76/75 22.931 (22*19)/(3*52) [2 -1 -2 0 0 0 0 1 large undevicesimal 1/9-tone
77/76 22.631 (7*11)/(22*19) [-2 0 0 1 1 0 0 -1 small undevicesimal 1/9-tone
96/95 18.128 (25*3)/(5*19) [5 1 -1 0 0 0 0 -1 19th-partial chroma
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 ganassisma, Ganassi's comma
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 yama comma
210/209 8.2637 (2*3*5*7)/(11*19) [1 1 1 1 -1 0 0 -1 spleen comma
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 nusu comma S18
343/342 5.0547 73/(2*32*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 go comma S19
400/399 4.3335 (24*52)/(3*7*19) [4 -1 2 -1 0 0 0 -1 S20
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 3.5010 (32*5*11)/(2*13*19) [-1 2 1 0 1 -1 0 -1
513/512 3.3780 (33*19)/29 [-9 3 0 0 0 0 0 1 undevicesimal comma, undevicesimal schisma, Boethius' comma, 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 password comma, Eratosthenes' comma
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 aureusma
1521/1520 1.1386 (3*13)2/(24*5*19) [-4 2 -1 0 0 2 0 -1 pinkanberry S39
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 ramanujanisma
2376/2375 0.7288 (23*33*11)/(53*19) [3 3 -3 0 1 0 0 -1
2432/2431 0.7120 (27*19)/(11*13*17) [7 0 0 0 -1 -1 -1 1 Blumeyer comma
2926/2925 0.5918 (2*7*11*19)/(32*52*13) [1 -2 -2 1 1 -1 0 1
3136/3135 0.5521 (26*72)/(3*5*11*19) [6 -1 -1 2 -1 0 0 -1 S56
3250/3249 0.5328 (2*53*13)/(32*192) [1 -2 3 0 0 1 0 -2
4200/4199 0.4123 (23*3*52*7)/(13*17*19) [3 1 2 1 0 -1 -1 -1
5776/5775 0.2998 (24*192)/(3*52*7*11) [4 -1 -2 -1 -1 0 0 2 S76
5929/5928 0.2920 (72*112)/(23*3*13*19) [-3 -1 0 2 2 -1 0 -1 S77
5985/5984 0.2893 (32*5*7*19)/(25*11*17) [-5 2 1 1 -1 0 -1 1
6175/6174 0.2804 (52*13*19)/(2*32*73) [-1 -2 2 -3 0 1 0 1
6860/6859 0.2524 (22*5*73)/(193) [2 0 1 3 0 0 0 -3
10241/10240 0.1691 (72*11*19)/(211*5) [-11 0 -1 2 1 0 0 1
10830/10829 0.1599 (2*3*5*192)/(72*13*17) [1 1 1 -2 0 -1 -1 2
12636/12635 0.1370 (22*35*13)/(5*7*192) [2 5 -1 -1 0 1 0 -2
13377/13376 0.1294 (3*73*13)/(26*11*19) [-6 1 0 3 -1 1 0 -1
14080/14079 0.1230 (28*5*11)/(3*13*192) [8 -1 1 0 1 -1 0 -2
14365/14364 0.1205 (5*132*17)/(22*33*7*19) [-2 -3 1 -1 0 1 1 -1
23409/23408 0.07396 (34*172)/(24*7*11*19) [-4 4 0 -1 -1 0 1 -1 S153
27456/27455 0.06306 (26*3*11*17)/(5*172*19) [6 1 -1 0 1 0 -2 -1
28900/28899 0.05991 (22*52*172)/(32*132*19) [2 -2 2 0 0 -2 2 -1 S170
43681/43680 0.03963 (112*192)/(25*3*5*7*13) [-5 -1 -1 -1 2 -1 0 2 S209
89376/89375 0.01937 (25*3*72*19)/(54*11*13) [5 1 -4 2 -1 -1 0 1
104976/104975 0.01649 (24*38)/(52*13*17*19) [4 8 -2 0 0 0 -1 -1 -1 S324
165376/165375 0.01047 (29*17*19)/(33*53*72) [9 -3 -3 -2 0 0 1 1 decimillisma
228096/228095 0.007590 (28*34*11)/(5*74*19) [8 4 -1 -4 1 0 0 -1
601426/601425 0.002879 (2*72*17*192)/(37*52*11) [2 -7 -2 2 -1 0 1 2
633556/633555 0.002733 (22*7*113*17)/(33*5*13*192) [2 -3 -1 1 3 -1 1 -2
709632/709631 0.002440 (210*32*7*11)/(133*17*19) [10 2 0 1 1 -3 -1 -1
5909761/5909760 0.0002929 (112*132*172)/(28*35*5*19) [-8 -5 -1 0 2 2 2 -1 S2431
11859211/11859210 0.0001460 (7*13*194)/(2*34*5*114) [-1 -4 -1 1 -4 1 0 4
23-limit (complete)
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) vicesimotertial 1/5-tone
69/68 25.274 (3*23)/(22*17) large vicesimotertial 1/8-tone
70/69 24.910 (2*5*7)/(3*23) small vicesimotertial 1/8-tone
92/91 18.921 (22*23)/(7*13)
115/114 15.120 (5*23)/(2*3*19)
161/160 10.787 (7*23)/(25*5)
162/161 10.720 (2*34)/(7*23)
208/207 8.3433 (24*13)/(32*23)
231/230 7.5108 (3*7*11)/(2*5*23)
253/252 6.8564 (11*23)/((2*3)2*7)
276/275 6.2840 (22*3*23)/(52*11)
300/299 5.7804 ((2*5)2*3)/(13*23)
323/322 5.3682 (17*19)/(2*7*23)
391/390 4.4334 (17*23)/(2*3*5*13)
392/391 4.4221 (23*7*7)/(17*23)
460/459 3.7676 (22*5*23)/(33*17)
484/483 3.5806 (2*11)2/(3*7*23) S22
507/506 3.4180 (3*132)/(2*11*23)
529/528 3.2758 232/(24*3*11) S23
576/575 3.0082 (26*32)/(23*52) S24
736/735 2.3538 (25*23)/(3*5*72)
760/759 2.2794 (23*5*19)/(3*11*23)
875/874 1.9797 (53*7)/(2*19*23)
897/896 1.9311 (3*13*23)/(27*7)
1105/1104 1.5674 (5*13*17)/(24*3*23)
1197/1196 1.4469 (32*17*19)/(22*13*23)
1288/1287 1.3446 (23*7*23)/(32*11*13)
1496/1495 1.1576 (23*11*17)/(5*13*23)
1863/1862 0.92952 (34*23)/(2*72*19)
2024/2023 0.85556 (23*11*23)/(7*172)
2025/2024 0.85514 (34*52)/(23*11*23) S45
2185/2184 0.79251 (5*19*23)/(23*3*7*13)
2300/2299 0.75287 (22*52*23)/(112*19)
2646/2645 0.65441 (2*33*72)/(5*232)
2737/2736 0.63265 (7*17*23)/(24*32*19)
3060/3059 0.56586 (22*32*5*17)/(7*19*23)
3381/3380 0.51212 (3*72*23)/(22*5*132)
3520/3519 0.49190 (26*5*11)/(32*17*23)
3888/3887 0.44533 (24*35)/(132*23)
4693/4692 0.36893 (13*192)/(22*3*17*23)
4761/4760 0.36367 (32*232)/(23*5*7*17) S69
5083/5082 0.34063 (13*17*23)/(2*3*7*112)
7866/7865 0.22010 (2*32*19*23)/(5*112*13)
8281/8280 0.20907 (72*132)/(23*32*5*23) S91
8625/8624 0.20073 (3*53*23)/(24*72*11)
10626/10625 0.16293 (2*3*7*11*23)/(54*17)
11271/11270 0.15361 (3*13*172)/(2*5*72*23)
11662/11661 0.14846 (2*73*17)/(3*132*23)
12168/12167 0.14228 (23*32*132)/(233)
16929/16928 0.10227 (34*11*19)/(25*232)
19551/19550 0.088552 (3*73*19)/(2*52*17*23)
21505/21504 0.080506 (5*11*17*23)/(210*3*7)
21736/21735 0.079650 (23*11*13*19)/(33*5*7*23)
23276/23275 0.074380 (22*11*232)/(52*72*19)
25025/25024 0.069182 (52*7*11*13)/(26*17*23)
25921/25920 0.066790 (72*232)/(26*34*5) S161
43264/43263 0.040016 (28*132)/(32*11*19*23) S208
52326/52325 0.033086 (2*34*17*19)/(52*7*13*23)
71875/71874 0.024087 (55*23)/(2*33*113)
75141/75140 0.023040 (33*112*23)/(22*5*13*172)
76545/76544 0.022617 (37*5*7)/(28*13*23)
104329/104328 0.016594 (172*192)/(23*34*7*23) S323
122452/122451 0.014138 (22*113*23)/(3*74*17)
126225/126224 0.013716 (33*52*11*17)/(24*73*23)
152881/152880 0.011324 (172*232)/(24*3*5*72*13) S391
202125/202124 0.0085652 (3*53*72*11)/(22*133*23)
264385/264384 0.0065482 (5*112*19*23)/(26*35*17)
282625/282624 0.0061256 (53*7*17*19)/(212*3*23)
328510/328509 0.0052700 (2*5*7*13*192)/(33*233)
2023425/2023424 0.00085560 (32*52*17*232)/(213*13*19)
4096576/4096575 0.00042261 (26*112*232)/(34*52*7*172) S2024
5142501/5142500 0.00033665 (33*72*132*23)/(22*54*112*17)
29-limit (incomplete)
29/28 60.751 29/(22*7) Large vicesimononal 1/4 tone
30/29 58.692 (2*3*5)/29 Small vicesimononal 1/4 tone
58/57 30.109 (2*29)/(3*19)
88/87 19.786 (23*11)/(3*29)
116/115 14.989 (22*29)/(5*23)
117/116 14.860 (33*13)/(22*29)
145/144 11.981 (5*29)/(24*32)
175/174 9.9211 (52*7)/(2*3*29)
204/203 8.5073
232/231 7.4783
261/260 6.6458
290/289 5.9801
320/319 5.4186
378/377 4.5861
406/405 4.2694
494/493 3.5081
551/550 3.1448
552/551 3.1391
609/608 2.8451
638/637 2.7157
726/725 2.3863
31-limit (incomplete)
31/30 56.767 31/(2*3*5) large tricesimoprimal 1/4-tone
32/31 54.964 25/31 small tricesimoprimal 1/4-tone, 31st subharmonic
63/62 27.700 (32*7)/(2*31)
93/92 18.716 (3*31)/(22*23)
125/124 13.906 (53)/(22*31) twizzler
621/620 2.7901 (3³*23)/(2²*5*31) owowhatsthisma
3969/3968 0.43624 (34*72)/(27*31) yunzee comma S63
37-limit (incomplete)
37/36 47.434 37/(22*32) Large 37-limit quarter tone, 37th-partial chroma
38/37 46.169 (2*19)/37 Small 37-limit quarter tone
75/74 23.238 (3*52)/(2*37)
41-limit (incomplete)
41/40 42.749 41/(23*5) Large 41-limit fifth tone
42/41 41.719 (2*3*7)/41 Small 41-limit fifth tone
82/81 21.242 (2*41)/34 41st-partial chroma
43-limit (incomplete)
43/42 40.737 43/(2*3*7) Large 43-limit fifth tone
44/43 39.800 (22*11)/43 Small 43-limit fifth tone
86/85 20.249 (2*43)/(5*17)
87/86 20.014 (3*29)/(2*43)
129/128 13.473 (3*43)/27 43rd-partial chroma
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
7777/7776 0.223 7*11*101/(25*35)

Notes

  1. Denoted by s-expressions, where sk is defined as (k/(k - 1))/((k + 1)/k). See square superparticular for details.