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List of superparticular intervals

Revision as of 17:28, 18 July 2023 by Inthar (talk | contribs) (→List of superparticular intervals: Tabulated a few more.)

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⟩ Major naiadma
221/220 7.8514 (13×17)/(22×5×11) [-2 0 -1 0 -1 1 1⟩ Minor naiadma
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⟩ Seminaiadma
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⟩ Ainic comma, ainos
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⟩ Neuseisma 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 (7×19)/(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 2 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⟩ Worcester comma 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) 2.7.29 [-2 -1 1⟩ Large vicesimononal 1/4-tone
30/29 58.692 (2×3×5)/29 2.3.5.29 [1 1 1 -1⟩ Small vicesimononal 1/4-tone
58/57 30.109 (2×29)/(3×19) 2.3.19.29 [1 -1 -1 1⟩
88/87 19.786 (23×11)/(3×29) 2.3.11.29 [3 -1 1 -1⟩
116/115 14.989 (22×29)/(5×23) 2.5.23.29 [2 -1 -1 1⟩
117/116 14.860 (33×13)/(22×29) 2.3.13.29 [-2 3 1 -1⟩
145/144 11.981 (5×29)/(22×3)2 2.3.5.29 [-4 -2 1 1⟩ 29th-partial chroma
175/174 9.9211 (52×7)/(2×3×29) 2.3.5.7.29 [-1 -1 2 1 -1⟩
204/203 8.5073 (22×3×17)/(7×29) 2.3.7.17.29 [2 1 -1 1 -1⟩
232/231 7.4783 (23×29)/(3×7×11) 2.3.7.11.29 [3 -1 -1 -1 1⟩
261/260 6.6458 (32×29)/(22×5×13) 2.3.5.13.29 [-2 2 -1 -1 1⟩
290/289 5.9801 (2×5×29)/172 2.5.17.29 [1 1 -2 1⟩
320/319 5.4186 (26×5)/(11×29) 2.5.11.29 [6 1 -1 -1⟩
378/377 4.5861 (2×33×7)/(13×29) 2.3.7.13.29 [1 3 1 -1 -1⟩
406/405 4.2694 (2×7×29)/(34×5) 2.3.5.7.29 [1 -4 -1 1 1⟩
494/493 3.5081 (2×13×19)/(17×29) 2.13.17.19.29 [1 1 -1 1 -1⟩
551/550 3.1448 (19×29)/(2×52×11) 2.5.11.19.29 [-1 -2 -1 1 1⟩
552/551 3.1391 (23×3×23)/(19×29) 2.3.19.23.29 [3 1 -1 1 -1⟩ Marmosarubra
609/608 2.8451 (3×7×19)/(25×29) 2.3.7.19.29 [-5 1 1 1 -1⟩
638/637 2.7157 (2×11×29)/(72×13) 2.7.11.13.29 [1 -2 1 -1 1⟩
726/725 2.3863 (2×3×112)/(52×29) 2.3.5.11.29 [1 1 -2 2 -2⟩
783/782 2.2124 2.3.17.23.29 [-1 3 -1 -1 1⟩
784/783 2.2096 (22×7)2/(33×29) 2.3.7.29 [4 -3 2 -1⟩ S28
841/840 2.0598 292/(23×3×5×7) 2.3.5.7.29 [-3 -1 -1 -1 2⟩ S29
1015/1014 1.7065 2.3.5.7.13.29 [-1 -1 1 1 -2 1⟩
1045/1044 1.6575 2.3.5.11.19.29 [-2 -2 1 1 1 -1⟩
1276/1275 1.3573 2.3.5.11.17.29 [2 -1 -2 1 -1 1⟩
1450/1449 1.1944 2.3.5.7.23.29 [1 -2 2 -1 -1 1⟩
1596/1595 1.0851 2.3.5.7.11.19.29 [2 1 -1 1 -1 1 -1⟩
13456/13455 0.12866 2.3.5.13.23.29 [4 -2 -1 -1 -1 2⟩ S116
303601/303600 0.0057023 2.3.5.11.19.23.29 [-4 -1 -2 -1 2 -1 2⟩ S551
410670/410669 0.0042156 2.3.5.7.13.17.29 [1 5 1 -2 2 -2 -1⟩
418761/418760 0.0041342 2.3.5.7.17.19.23.29 [-3 2 -1 1 2 -2 1 -1⟩
613089/613088 0.0028238 2.3.7.17.23.29 [-5 6 -2 -1 -1 2⟩ S783
949026/949025 0.0018242 2.3.5.7.11.13.17.23.29 [1 1 -2 -1 -1 1 -1 3 -1⟩
1163800/1163799 0.0014876 2.3.5.7.11.13.23.29 [3 -2 2 -3 1 -1 2 -1⟩
1235169/1235168 0.0014016 2.3.11.13.17.23.29 [-5 5 -3 1 1 1 -1⟩
1243840/1243839 0.0013918 2.3.5.13.17.23.29 [6 -1 1 2 -1 1 -3⟩
1625625/1625624 0.0010650 2.3.5.7.11.13.17.29 [-3 2 4 -2 -1 -1 2 -1⟩ S1275
1852201/1852200 9.3469×10⁻⁴ 2.3.5.7.13.17.29 [-3 -3 -2 -3 1 3 1⟩
2697696/2697695 6.4175×10⁻⁴ 2.3.5.7.11.13.17.19.29 [5 2 -1 -3 -2 -1 1 1 1⟩
4004001/4004000 4.3238x10⁻⁴ 2.3.5.7.11.13.23.29 [-5 2 -3 -1 -1 -1 2 2⟩ S2001
4090625/4090624 2.5.7.11.17.19.29 [-8 5 1 1 1 -1 -2⟩
8268800/8268799 2.0937×10⁻⁴ 2.5.7.11.17.19.23.29 [10 2 -2 -1 1 1 -2 -1⟩
10556001/10556000 1.6400×10⁻⁴ 2.3.5.7.13.19.29 [-5 4 -3 -1 -1 4 -1⟩
18085705/18085704 9.5724×10⁻⁵ 2.3.5.7.11.13.17.23.29 [-3 -1 1 -3 1 -3 1 1 2⟩
96059601/96059600 1.8022×10⁻⁵ 2.3.5.7.11.13.29 [-4 8 -2 -2 4 -2 -1⟩ S9801
177182721/177182720 9.7709×10⁻⁶ 2.3.5.11.13.17.29 [-11 6 -1 -3 -1 2 2⟩ S13311
31-limit (incomplete)
31/30 56.767 31/(2×3×5) 2.3.5.31 [-1 -1 -1 1⟩ Large tricesimoprimal 1/4-tone
32/31 54.964 25/31 2.31 [5 -1⟩ Small tricesimoprimal 1/4-tone, octave-reduced 31st subharmonic
63/62 27.700 (32×7)/(2×31) 2.3.7.31 [-1 2 1 -1⟩
93/92 18.716 (3×31)/(22×23) 2.3.23.31 [-2 1 -1 1⟩
125/124 13.906 53/(22×31) 2.5.31 [-2 3 -1⟩ Twizzler
621/620 2.7901 (33×23)/(22×5×31) 2.3.5.23.31 [-2 3 -1 1 -1⟩ Owowhatsthisma
900/899 1.9247 (2×3×5)2/(29×31) 2.3.5.29.31 [2 2 2 -1 -1⟩ S30
961/960 1.8024 312/(26×3×5) 2.3.5.31 [-6 -1 -1 2⟩ S31
1024/1023 1.9247 210/(3×11×31) 2.3.11.31 [10 -1 -1 -1⟩ S32
3969/3968 0.43624 (32×7)2/(27×31) 2.3.7.31 [-7 4 2 -1⟩ Yunzee comma S63
37-limit (incomplete)
37/36 47.434 37/(22×32) 2.3.37 [-2 -2 1⟩ Large tricesimoseptimal 1/4-tone, 37th-partial chroma
38/37 46.169 (2×19)/37 2.19.37 [1 1 -1⟩ Small tricesimoseptimal 1/4-tone
75/74 23.238 (3×52)/(2×37) 2.3.5.37 [-1 1 2 -1⟩
1296/1295 1.3363 (2×3)4/(5×7×37) 2.3.5.7.37 [4 4 -1 -1 -1⟩ S36
1369/1368 1.2651 372/(23×32×19) 2.3.19.37 [-3 -2 -1 2⟩ S37
1444/1443 1.1993 (2×19)2/(3×13×37) 2.3.13.19.37 [2 -1 -1 2 -1⟩ S38
41-limit (incomplete)
41/40 42.749 41/(23×5) 2.5.41 [-3 -1 1⟩ Large quadracesimoprimal 1/5-tone
42/41 41.719 (2×3×7)/41 2.3.7.41 [1 1 1 -1⟩ Small quadracesimoprimal 1/5-tone
82/81 21.242 (2×41)/34 2.3.41 [1 -4 1⟩ 41st-partial chroma
1600/1599 1.0824 (23×5)2/(3×13×41) 2.3.5.13.41 [6 -1 2 -1 -1⟩ S40
1681/1680 1.0302 412/(24×3×5×7) 2.3.5.7.41 [-4 -1 -1 -1 2⟩ S41
43-limit (incomplete)
43/42 40.737 43/(2×3×7) 2.3.7.43 [-1 -1 -1 1⟩ Large quadracesimotertial 1/5-tone
44/43 39.800 (22×11)/43 2.11.43 [2 1 -1⟩ Small quadracesimotertial 1/5-tone
86/85 20.249 (2×43)/(5×17) 2.5.17.43 [1 -1 -1 1⟩
87/86 20.014 (3×29)/(2×43) 2.3.29.43 [-1 1 1 -1⟩
129/128 13.473 (3×43)/27 2.3.43 [-7 1 1⟩ 43rd-partial chroma
1764/1763 0.98170 (2×3×7)2/(41×43) 2.3.7.41.43 [2 2 2 -1 -1⟩ S42
1849/1848 0.93656 432/(23×3×7×11) 2.3.7.11.43 [-3 -1 -1 -1 2⟩ S43
1936/1935 0.98170 (22×11)2/(32×5×43) 2.3.5.11.43 [4 -2 -1 2 -1⟩ S44
47-limit (incomplete)
47/46 37.232 47/(2×23) 2.23.47 [-1 -1 1⟩
48/47 36.448 (24×3)/47 2.3.47 [4 1 -1⟩
94/93 18.516 (2×47)/(3×31) 2.3.31.47 [1 -1 -1 1⟩
95/94 18.320 (5×19)/(2×47) 2.5.19.47 [-1 1 1 -1⟩
2116/2115 0.81836 (2×23)2/(32×5×47) 2.3.5.23.47 [2 -2 -1 2 -1⟩ S46
2209/2208 0.78390 472/(25×3×5) 2.3.23.47 [-5 -1 -1 2⟩ S47
2304/2303 0.75157 (24×3)2/(72×47) 2.3.7.47 [8 2 -2 -1⟩ S48
53-limit (incomplete)
53/52 32.977 53/(22×13) 2.13.53 [-2 -1 1⟩
54/53 32.360 (2×33)/53 2.3.53 [1 3 -1⟩
2809/2808 0.61643 532/(23×33×13) 2.3.13.53 [-3 -3 -1 2⟩ S53
4081/4080 0.42427 (7×11×53)/(24×3×5×17) 2.3.5.7.11.17.53 [-4 -1 -1 1 1 -1 1⟩
59-limit (incomplete)
59/58 29.594 59/(2×29) 2.29.59 [-1 -1 1⟩
60/59 29.097 (22×3×5)/59 2.3.5.59 [2 1 1 -1⟩
3481/3480 0.49741 592/(23×3×5×29) 2.3.5.29.59 [-3 -1 -1 -1 2⟩ S59
61-limit (incomplete)
61/60 28.616 61/(22×3×5) 2.3.5.61 [-2 -1 -1 1⟩
62/61 28.151 (2×31)/61 2.31.61 [1 1 -1⟩
672/671 2.5782 (25×3×7)/(11×61) 2.3.7.11.61 [5 1 1 -1 -1⟩
1404/1403 1.2335 (22×33×13)/(23×61) 2.3.13.23.61 [2 3 1 -1 -1⟩
3721/3720 0.46532 612/(23×3×5×31) 2.3.5.31.61 [-3 -1 -1 -1 2⟩ S61
67-limit (incomplete)
67/66 26.034 67/(2×3×11) 2.3.11.67 [-1 -1 -1 1⟩
68/67 25.648 (22×17)/67 2.17.67 [2 1 -1⟩
71-limit (incomplete)
71/70 24.557 71/(2×5×7) 2.5.7.71 [-1 -1 -1 1⟩
72/71 24.213 (23×32)/71 2.3.71 [3 2 -1⟩
73-limit (incomplete)
73/72 23.879 73/(23×32) 2.3.73 [-3 -2 1⟩
74/73 23.555 (2×37)/73 2.37.73 [1 1 -1⟩
79-limit (incomplete)
79/78 22.054 79/(2×3×13) 2.3.13.79 [-1 -1 -1 1⟩
80/79 21.777 (24×5)/79 2.5.79 [4 1 -1⟩
83-limit (incomplete)
83/82 20.985 83/(2×41) 2.41.83 [-1 -1 1⟩
84/83 20.734 (22×3×7)/83 2.3.7.83 [2 1 1 -1⟩
89-limit (incomplete)
89/88 19.562 89/(23×11) 2.11.89 [-3 -1 1⟩ Sky comma
90/89 19.344 (2×32×5)/89 2.3.5.89 [1 2 1 -1⟩
97-limit (incomplete)
97/96 17.940 97/(25×3) 2.3.97 [-5 -1 1⟩
98/97 17.756 (2×72)/97 2.7.97 [1 2 -1⟩
101-limit (incomplete)
101/100 17.226 101/(2×5)2 2.5.101 [-2 -2 1⟩
102/101 17.057 (2×3×17)/101 2.3.17.101 [1 1 1 -1⟩
7777/7776 0.223 7×11×101/(2×3)5 2.3.7.11.101 [-5 -5 1 1 1⟩ Pulsar comma

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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