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List of superparticular intervals

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This is a list of superparticular intervals ordered by prime limit. It reaches to the 101-limit and is complete up to the 23-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).

List of superparticular intervals

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, octave-reduced 3rd harmonic, diapente
4/3 498.045 22/3 [2 -1⟩ Perfect fourth, octave-reduced 3rd subharmonic, diatessaron S2
9/8 203.910 32/23 [-3 2⟩ (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, octave-reduced 9th harmonic or harmonic ninth S3
5-limit (complete)
5/4 386.314 5/22 [-2 0 1⟩ Classic(al)/just major third, octave-reduced 5th harmonic
6/5 315.641 (2*3)/5 [1 1 -1⟩ Classic(al)/just minor third
10/9 182.404 (2*5)/32 [1 -2 1⟩ Classic(al) (whole) tone, classic major second, minor whole tone
16/15 111.731 24/(3*5) [4 -1 -1⟩ Classic(al)/just diatonic semitone, 15th subharmonic S4
25/24 70.672 52/(23*3) [-3 -1 2⟩ Classic(al)/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, octave-reduced 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, tritonic diesis, small septimal diesis, small septimal 1/6-tone
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, octave-reduced 33rd harmonic
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⟩ Telepathma, eleventyfive comma, undecimal diasecundal 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, octave-reduced 17th harmonic
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, octave-reduced 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⟩ Semitonisma S17
375/374 4.6228 (3*53)/(2*11*17) [-1 1 3 0 -1 0 -1⟩ Ursulisma
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
833/832 2.0796 (72*17)/(26*13) [-6 0 0 2 0 -1 1⟩ Horizma, 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 (3*11)2/(26*17) [-6 2 0 0 2 0 -1⟩ Twosquare comma S33
1156/1155 1.4983 (2*17)2/(3*5*7*11) [2 -1 -1 -1 -1 0 2⟩ Quadrantonisma S34
1225/1224 1.4138 (5*7)2/(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
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 (2*52)2/(3*72*17) [2 -1 4 -2 0 0 -1⟩ Sperasma S50
2601/2600 0.66573 (3*17)2/(23*52*13) [-3 2 -2 0 0 -1 2⟩ Sextantonisma S51
4914/4913 0.35234 (2*33*7*13)/173 [1 3 0 1 0 1 -3⟩
5832/5831 0.29688 (2*32)3/(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 (23*3*5)2/(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 (3*7)4/(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 (2*32)2/(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 (22*5)2/(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)/(22*3)3 [-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 (23*7)2/(3*5*11*19) [6 -1 -1 2 -1 0 0 -1⟩ S56
3250/3249 0.5328 (2*53*13)/(3*19)2 [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 (22*19)2/(3*52*7*11) [4 -1 -2 -1 -1 0 0 2⟩ S76
5929/5928 0.2920 (7*11)2/(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 (32*17)2/(24*7*11*19) [-4 4 0 -1 -1 0 1 -1⟩ S153
27456/27455 0.06306 (26*3*11)/(5*172*19) [6 1 -1 0 1 0 -2 -1⟩
28900/28899 0.05991 (2*5*17)2/(32*132*19) [2 -2 2 0 0 -2 2 -1⟩ S170
43681/43680 0.03963 (11*19)2/(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 (2*32)4/(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) [1 -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 (11*13*17)2/(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) [-1 0 0 0 -1 0 0 0 1⟩ Greater vicesimotertial semitone
24/23 73.681 (23*3)/23 [3 1 0 0 0 0 0 0 -1⟩ Small vicesimotertial semitone
46/45 38.051 (2*23)/(32*5) [1 -2 -1 0 0 0 0 0 1⟩ Vicesimotertial 1/5-tone
69/68 25.274 (3*23)/(22*17) [-2 1 0 0 0 0 -1 0 1⟩ Large vicesimotertial 1/8-tone
70/69 24.910 (2*5*7)/(3*23) [1 -1 1 1 0 0 0 0 -1⟩ Small vicesimotertial 1/8-tone
92/91 18.921 (22*23)/(7*13) [2 0 0 -1 0 -1 0 0 1⟩
115/114 15.120 (5*23)/(2*3*19) [-1 -1 1 0 0 0 0 -1 1⟩
161/160 10.787 (7*23)/(25*5) [-5 0 -1 1 0 0 0 0 1⟩
162/161 10.720 (2*34)/(7*23) [1 4 0 -1 0 0 0 0 -1⟩
208/207 8.3433 (24*13)/(32*23) [4 -2 0 0 0 1 0 0 -1⟩
231/230 7.5108 (3*7*11)/(2*5*23) [-1 1 -1 1 1 0 0 0 -1⟩
253/252 6.8564 (11*23)/((2*3)2*7) [-2 -2 0 -1 1 0 0 0 1⟩
276/275 6.2840 (22*3*23)/(52*11) [2 1 -2 0 -1 0 0 0 1⟩
300/299 5.7804 ((2*5)2*3)/(13*23) [2 1 2 0 0 -1 0 0 -1⟩
323/322 5.3682 (17*19)/(2*7*23) [-1 0 0 -1 0 0 1 1 -1⟩
391/390 4.4334 (17*23)/(2*3*5*13) [-1 -1 -1 0 0 -1 1 0 1⟩
392/391 4.4221 (23*72)/(17*23) [3 0 0 2 0 0 -1 0 -1⟩
460/459 3.7676 (22*5*23)/(33*17) [2 -3 1 0 0 0 -1 0 1⟩
484/483 3.5806 (2*11)2/(3*7*23) [2 -1 0 -1 2 0 0 0 -1⟩ S22
507/506 3.4180 (3*132)/(2*11*23) [-1 1 0 0 -1 2 0 0 -1⟩
529/528 3.2758 232/(24*3*11) [-4 -1 0 0 -1 0 0 0 2⟩ S23
576/575 3.0082 (23*3)2/(23*52) [6 2 -2 0 0 0 0 0 -1⟩ S24
736/735 2.3538 (25*23)/(3*5*72) [5 -1 -1 -2 0 0 0 0 1⟩
760/759 2.2794 (23*5*19)/(3*11*23) [3 -1 1 0 -1 0 0 1 -1⟩
875/874 1.9797 (53*7)/(2*19*23) [-1 0 3 1 0 0 0 -1 -1⟩
897/896 1.9311 (3*13*23)/(27*7) [-7 1 0 -1 0 1 0 0 1⟩
1105/1104 1.5674 (5*13*17)/(24*3*23) [-4 -1 1 0 0 1 1 0 -1⟩
1197/1196 1.4469 (32*17*19)/(22*13*23) [-2 2 0 0 0 -1 1 1 -1⟩
1288/1287 1.3446 (23*7*23)/(32*11*13) [3 -2 0 1 -1 -1 0 0 1⟩
1496/1495 1.1576 (23*11*17)/(5*13*23) [3 0 -1 0 1 -1 1 0 -1⟩
1863/1862 0.92952 (34*23)/(2*72*19) [-1 4 0 -2 0 0 0 -1 1⟩
2024/2023 0.85556 (23*11*23)/(7*172) [3 0 0 -1 1 0 -2 0 1⟩
2025/2024 0.85514 (32*5)2/(23*11*23) [-3 4 2 0 -1 0 0 0 -1⟩ S45
2185/2184 0.79251 (5*19*23)/(23*3*7*13) [-3 -1 1 -1 0 -1 0 1 1⟩
2300/2299 0.75287 (22*52*23)/(112*19) [2 0 2 0 -2 0 0 -1 1⟩
2646/2645 0.65441 (2*33*72)/(5*232) [1 3 -1 2 0 0 0 0 -2⟩
2737/2736 0.63265 (7*17*23)/(24*32*19) [-4 -2 0 1 0 0 1 -1 1⟩
3060/3059 0.56586 (22*32*5*17)/(7*19*23) [2 2 1 -1 0 0 1 -1 -1⟩
3381/3380 0.51212 (3*72*23)/(22*5*132) [-2 1 -1 2 0 -2 0 0 1⟩
3520/3519 0.49190 (26*5*11)/(32*17*23) [6 -2 1 0 1 0 -1 0 -1⟩
3888/3887 0.44533 (24*35)/(132*23) [4 5 0 0 0 -2 0 0 -1⟩
4693/4692 0.36893 (13*192)/(22*3*17*23) [-2 -1 0 0 0 1 -1 2 -1⟩
4761/4760 0.36367 (3*23)2/(23*5*7*17) [-3 2 -1 -1 0 0 -1 0 2⟩ S69
5083/5082 0.34063 (13*17*23)/(2*3*7*112) [-1 -1 0 -1 -2 1 1 0 1⟩
7866/7865 0.22010 (2*32*19*23)/(5*112*13) [1 2 -1 0 -2 -1 0 1 1⟩
8281/8280 0.20907 (7*13)2/(23*32*5*23) [-3 -2 -1 2 0 2 0 0 -1⟩ S91
8625/8624 0.20073 (3*53*23)/(24*72*11) [-4 1 3 -2 -1 0 0 0 1⟩
10626/10625 0.16293 (2*3*7*11*23)/(54*17) [1 1 -4 1 1 0 -1 0 1⟩
11271/11270 0.15361 (3*13*172)/(2*5*72*23) [-1 1 -1 -2 0 1 2 0 -1⟩
11662/11661 0.14846 (2*73*17)/(3*132*23) [1 0 0 3 0 -2 1 0 -1⟩
12168/12167 0.14228 (23*32*132)/(233) [3 2 0 0 0 2 0 0 -3⟩
16929/16928 0.10227 (34*11*19)/(25*232) [-5 4 0 0 1 0 0 1 -2⟩
19551/19550 0.088552 (3*73*19)/(2*52*17*23) [-1 1 -2 3 0 0 -1 1 -1⟩
21505/21504 0.080506 (5*11*17*23)/(210*3*7) [-10 -1 1 -1 1 0 1 0 1⟩
21736/21735 0.079650 (23*11*13*19)/(33*5*7*23) [3 -3 -1 -1 1 1 0 1 -1⟩
23276/23275 0.074380 (22*11*232)/(52*72*19) [2 0 -2 -2 1 0 0 -1 2⟩
25025/25024 0.069182 (52*7*11*13)/(26*17*23) [-6 0 2 1 1 1 -1 0 -1⟩
25921/25920 0.066790 (7*23)2/(26*34*5) [-6 -4 -1 2 0 0 0 0 2⟩ S161
43264/43263 0.040016 (24*13)2/(32*11*19*23) [8 -2 0 0 -1 2 0 -1 -1⟩ S208
52326/52325 0.033086 (2*34*17*19)/(52*7*13*23) [1 4 -2 -1 0 -1 1 1 -1⟩
71875/71874 0.024087 (55*23)/(2*33*113) [-1 -3 5 0 -3 0 0 0 1⟩
75141/75140 0.023040 (33*112*23)/(22*5*13*172) [-2 3 -1 0 2 -1 -2 0 1⟩
76545/76544 0.022617 (37*5*7)/(28*13*23) [-8 7 1 1 0 -1 0 0 -1⟩
104329/104328 0.016594 (17*19)2/(23*34*7*23) [-3 -4 0 -1 -1 0 2 2 -1⟩ S323
122452/122451 0.014138 (22*113*23)/(3*74*17) [2 -1 0 -4 3 0 -1 0 1⟩
126225/126224 0.013716 (33*52*11*17)/(24*73*23) [-4 3 2 -3 1 0 1 0 -1⟩
152881/152880 0.011324 (17*23)2/(24*3*5*72*13) [-4 -1 -1 -2 0 -1 2 0 2⟩ S391
202125/202124 0.0085652 (3*53*72*11)/(22*133*23) [-2 1 3 2 1 -3 0 0 -1⟩
264385/264384 0.0065482 (5*112*19*23)/(26*35*17) [-6 -5 1 0 2 0 -1 1 1⟩
282625/282624 0.0061256 (53*7*17*19)/(212*3*23) [-12 -1 3 1 0 0 1 1 -1⟩
328510/328509 0.0052700 (2*5*7*13*192)/(3*23)3 [1 -3 1 1 0 1 0 0 -3⟩
2023425/2023424 0.00085560 (32*52*17*232)/(213*13*19) [-13 2 2 0 0 -1 1 -1 2⟩
4096576/4096575 0.00042261 (23*11*23)2/(34*52*7*172) [6 -4 -2 -1 2 0 -2 0 2⟩ S2024
5142501/5142500 0.00033665 (33*72*132*23)/(22*54*112*17) [-2 3 -4 2 -2 2 -1 0 1⟩
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 quartertone
32/31 54.964 25/31 Small tricesimoprimal quartertone, 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 quartertone, 37th-partial chroma
38/37 46.169 (2*19)/37 Small 37-limit quartertone
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)

See also

Notes

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

External links

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