List of superparticular intervals

From Xenharmonic Wiki
Jump to: navigation, search

This list of superparticular intervals ordered by prime limit. It reaches to the 101-limit and is complete up to the 17-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 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. A002071 -- OEIS gives the number of superparticular ratios in each prime limit, A145604 - OEIS shows the increment from limit to 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> octave, duple; after octave reduction: (perfect) unison, unity, perfect prime, tonic
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> 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> 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> mynucuma
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
676/675 2.5629 (2*13/5)2/33 |2 -3 -2 0 0 2> island comma
729/728 2.3764 (32/2)3/(7*13) |-3 6 0 -1 0 -1> squbema
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> tridecimal schisma, Sagittal schismina
4225/4224 0.40981 (5*13)2/(27*3*11) |-7 -1 2 0 -1 2> leprechaun comma
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
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> septendecimal 1/4-tone (greater)
35/34 50.184 (5*7)/(2*17) |-1 0 1 1 0 0 -1> septendecimal 1/4-tone (lesser)
51/50 34.283 (3*17)/(2*52) |-1 1 -2 0 0 0 1> septendecimal 1/6-tone (greater)
52/51 33.617 (22*13)/(3*17) |2 -1 0 0 0 1 -1> septendecimal 1/6-tone (lesser)
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>
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> Tannisma
289/288 6.0008 (17/3)2/25 |-5 -2 0 0 0 0 2> septendecimal 6-cent comma
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> twosquare comma
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> flashma
14400/14399 0.1202 (26*32*52)/(7*112*17) |6 2 2 -1 -2 0 -1> sparkisma
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> undevicesimal 2/9-tone
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> 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>
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> go comma
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
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
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>
1521/1520 1.1386 (3*13)2/(24*5*19) |-4 2 -1 0 0 2 0 -1>
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>
23-limit (incomplete)
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)
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*52)
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