User:FloraC/Sequence of equal temperaments by error/Supplement

Below are the sequences of equal temperaments in various subgroups by TE absolute error (and TE relative error, in bold), searched up to 196608. These sequences are obtained with the Temperament Evaluator. For a method to iterate thru all GPVs, see Patent val/Properties.

GPVs starting with 0 are ignored as they would otherwise meddle with the results. GPVs starting with 1, 2, 3, and 4 may be regarded as trivial, and are not shown.

While looking at these sequences, an important consideration is whether a temperament tunes the relevant intervals monotonic. If a temperament is meant to be used in the p-prime-limit, it is reasonable to request that the order of size for the p-odd-limit should at least be kept. For this reason and for the sake of readability, each main sequence starts with the first temperament that passes the monotonicity test, with the entries before that point relegated to the exotemperaments databox below it. There are usually some more entries in the main sequence that fail the monotonicity test. These are noted with an apostrophe.

3-limit

5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, 79335, 111202, 190537, … (OEIS: A060528).

5-limit

5, 7, 12, 19, 31, 34, 46, 53, 118, 171, 289, 323, 388, 441, 559, 612, 1171, 1783, 2513, 3684, 4296, 12276, 16572, 20868, 25164, 46032, 48545, 52841, 73709, 78005, 151714, ….

7-limit

5, 7d*, 8d, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1407, 1547, 1578, 1718, 1848, 2019, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 51670, 63334, 70025, 81689, 84814, 103169, ….

11-limit

12, 15, 19e*, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 3395, 3737, 4007, 4501, 4843, 5585, 6079, 6691, 12276, 12770, 14618, 14960, 21309, 25046, 26894, 33315, 39664, 40006, 47933, 54624, 81860, 94630, 96478, 121524, 176148, ….

Exotemperaments

5c, 5, 7d, 8d, 9, 10, 12e

13-limit

15, 17c, 19e*, 19, 22*, 26, 27e, 29, 31f*, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, 11664, 13112, 14348, 14618, 20203, 22887, 27236, 33315, 40276, 41584, 46355, 63163, 65062, 66558, 69242, 73591, 81860, 87939, 96478, 161530, 184417, ….

Exotemperaments

5cf, 5f, 5, 5e, 7df, 7d, 7, 8d, 9, 10, 12, 12f, 14cf

17-limit

31, 38df, 41g*, 41, 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954, 1178, 1236, 1395, 1506, 2000, 2414, 2460, 3395, 4573, 5455, 5809, 5855, 6079, 7033, 8539, 10428, 11934, 14348, 16808, 20203, 27236, 41584, 51940, 58973, 60973, 76275, 79176, 82354, 88209, 102557, 128215, 129793, 155451, 161530, 169799, 181733, ….

Exotemperaments

5cf, 5f, 5, 5g, 5eg, 7dfg, 7dg, 7, 8d, 9, 10e, 10, 12, 12f, 14cf, 15, 15g, 16, 17cg, 19eg, 22, 26, 27eg, 29g

19-limit

34dh, 38df, 41g*, 41, 46*, 50, 53, 58h, 68, 72h*, 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h, 1178, 1578, 1889, 2000, 2460, 3178, 3395, 4349, 4501, 5498, 5527, 5809, 6079, 6961, 7527, 8269, 8539, 11934, 14348, 15854, 16066, 16808, 20203, 36269, 37011, 40770, 49039, 50434, 52434, 58973, 60973, 79176, 95984, 99162, 111407, 119365, 119946, 128215, 178338, ….

Exotemperaments

5cf, 5g, 5eg, 5egh, 7deefgh, 7dfgh, 7, 8d, 8dhh, 9g, 9, 10, 10h, 12e, 12, 12f, 14cf, 15, 15g, 17cgh, 19egh, 22, 22fh, 24, 26, 27eg, 29g, 31

23-limit

58hi, 62, 68*, 72hi*, 72i*, 80*, 87*, 94, 111, 121i, 130, 140, 152fg, 159, 183, 190g, 193, 217, 243e, 270, 282, 311, 373g, 422, 525, 566g, 581, 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, 1578, 1889, 2000, 2460, 3178, 4038, 4349, 5144, 5395, 5638, 5783, 5958, 6079, 6269, 7315, 8269, 8539, 10539, 11664, 14348, 16808, 28472, 28742, 29053, 32231, 36269, 38189, 44907, 45861, 49039, 58973, 75781, 81176, 95242, 102557, 119365, 140149, 148418, 178338, ….

Exotemperaments

5cfi, 5g, 5eg, 5egh, 7deefghi, 7dfghi, 7, 8d, 8di, 8dhhi, 9gi, 9i, 9, 10e, 10, 10h, 12e, 12, 12f, 12fi, 14cf, 15, 15g, 18ehi, 19eghi, 22i, 22fh, 24, 26i, 27egi, 29g, 31, 34dh, 38dfi, 39dfgi, 41g, 41, 41i, 43, 46, 50, 53

29-limit

99ef, 103h*, 113*, 118, 121i, 130, 140h*, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578, 1920, 2000, 2460, 2742, 3041, 3178, 3395i, 4038j, 4190g, 4349, 4501, 5395, 5783, 6079, 6893, 7315, 7361, 7527, 7847j, 8269, 8539, 9204, 9447, 9934, 10729, 11664j, 11934, 12245, 13112, 14066, 15230, 16808, 28742, 29053, 36269, 45861, 48945, 52704, 53077, 56513, 58392, 58973, 60973, 65322, 67242, 69512, 78716, 81176j, 83096, 87445, 92782, 102557, 119365, 148418, ….

Exotemperaments

5cfi, 5egj, 5eghj, 7deefghijj, 7dfghijj, 7, 8di, 8dhhijj, 9gij, 9, 10ej, 10, 10h, 12de, 12e, 12fij, 14cfjj, 15, 15g, 17cgh, 18ehij, 19eghi, 22i, 22h, 22fh, 26i, 29gj, 29g, 31j, 31, 34dh, 34dhj, 38dfij, 39dfgij, 41g, 41, 43, 46, 50, 53, 58hi, 62, 68j, 72hij, 72ij, 72, 77, 80, 87, 94j, 94

31-limit

99efk, 103h*, 121ik, 130j*, 130, 140k*, 140, 149, 152fgj, 159k*, 167hijk*, 181*, 183k*, 183, 190g, 217, 243ek*, 270, 311, 388, 422, 525, 566gj, 571, 624jk, 639hj, 643ijk, 653, 692ik, 718, 742i, 863efgjk, 882, 908, 954hj, 1205g, 1289, 1308, 1395, 1578, 1920, 2311, 2460j, 2460, 2684, 2742k, 2901, 3041, 3323, 3395ik, 3759, 4038j, 4349, 4501, 5395, 5455, 5809, 6079, 6219, 6893, 7315, 7361, 7527, 8539, 9447, 9934, 11353, 11664j, 12245, 13823, 14066k, 14348, 15112, 16808, 29053, 31920, 32231, 34691, 36269, 40078, 47309, 48514, 48945, 50434, 51499, 52704, 58973, 64368, 65322, 67242, 78716, 81176j, 83096, 92782, 109590, 119365, 131610, 148418, ….

Exotemperaments

5cfijjk, 5cfik, 5g, 5gj, 5egj, 5eghj, 7deefghijjk, 7dfghijjk, 7, 8d, 8di, 9gijk, 10ejk, 10h, 12de, 12e, 12fijk, 14cfjj, 15, 15k, 15gk, 17cghk, 18ehijk, 19eghikk, 19eghi, 19egh, 19, 22i, 22h, 22fh, 24, 26ik, 27egij, 29gjk, 31, 34dhk, 34dhjk, 38dfij, 39dfgijk, 41g, 41, 43, 46, 58hi, 58hik, 62, 68, 68j, 72hijk, 77, 80k, 87, 94

37-limit

152fgj, 159k*, 181*, 183k*, 183*, 190gl*, 198gl, 212ghj*, 212gh, 217*, 217l, 224, 239*, 243ekl*, 248h, 270l*, 270, 270i, 296, 311k*, 311, 388, 422l, 487, 525, 566gjl, 571, 597ik, 624jk, 643ijkl, 653, 692ik, 718, 742il, 814k, 814, 863efgjk, 882, 908, 954hjl, 1106, 1164fi, 1205gl, 1224, 1289, 1308, 1395, 1578l, 1600, 1696, 1821efk, 1920, 2460j, 2460, 2684, 2742kl, 2901, 3041, 3395ik, 3759, 3949i, 3992, 4190g, 4349, 4501, 6219, 6349, 6893, 7033, 7361, 7847jkl, 8269, 8893, 9281, 9934, 10652, 10729k, 11025, 11353, 12245, 13823, 14066k, 14348, 15854j, 16808, 20567, 23270, 25347, 26742, 31920, 32231, 34691, 40078, 47309, 48945, 52704, 53077, 57578, 58973, 65322, 67242, 81176j, 83096, 95524, 109590, 131610, 148418, ….

Exotemperaments

5cfijjkll, 5cfikll, 5cfik, 5, 5g, 5egj, 5eghj, 5eghjl, 6fhijl, 7deefghijjkll, 7dfghijjkll, 7dfghijjk, 7l, 8d, 8di, 9gijkl, 10ejk, 10h, 12del, 12el, 12fijk, 14cfjjl, 15, 15gk, 15gkl, 16j, 17cgk, 17cghk, 19eghikkl, 19eghil, 19egh, 19gh, 19g, 19, 22il, 22i, 22, 22h, 22fh, 24, 26ik, 27egij, 29gjk, 29gk, 31fghjk, 31l, 34dhk, 34dhjkl, 38dfijkkl, 38dfijl, 39dfgijkl, 43, 46l, 46, 50jk, 50k, 53, 53j, 58hik, 58hikl, 60el, 62, 65d, 68, 68j, 68jl, 72hijkll, 72hijk, 72ijk, 72, 77, 80k, 94, 99ek, 99efk, 103hl, 121ikl, 130j, 140k, 149

41-limit

224, 248h, 270*, 270i, 311k*, 311, 388, 422*, 422l, 525*, 540, 566gjl, 597ik, 624jk, 639hjm, 653, 692ikm, 718, 764, 764j, 814k, 814, 863efgjk, 908m, 954hjl, 1012, 1147i, 1164fim, 1178, 1205gl, 1289, 1385k, 1395, 1578l, 1696, 1817hi, 1920, 2159, 2270il, 2460jm, 2460m, 2460, 2554, 2581ij, 2703, 2742klm, 3041, 3084, 3337, 3395ik, 3759, 3992, 4349, 4501m, 5144, 5638, 5809, 6151, 6349, 7204, 7216, 7361, 7847jklm, 8893, 9717, 9934, 10729k, 11353, 11534, 12245, 12428, 13584, 14066k, 14348, 15854j, 16808, 17461, 19933, 20203, 20567, 23270, 25347, 27730, 28414hjk, 28742, 29053, 31920, 34691, 40078, 42052, 44268, 47309, 49256, 51176, 53077, 53612, 56354m, 57578, 58392, 58973, 65322, 81176j, 95524, 109590, 133431, 148418, 165226 ….

Exotemperaments

5cfijjkllm, 5cfikllm, 5cfikm, 5f, 5, 5g, 5egj, 5eghj, 5eghjl, 6fhijlm, 7dfghijjkllmmm, 7dfghijjkllm, 7dfghijjkm, 7l, 8d, 8di, 9gijklmm, 9gijkl, 10h, 12del, 12el, 12fijkm, 14cfjjlmm, 15, 15km, 15gkm, 15gklm, 16jm, 17cghk, 17cghkm, 18ehijkm, 19eghikklm, 19eghilm, 19, 22il, 22i, 22, 22h, 22fh, 24, 26ik, 27egij, 29gjk, 29gk, 31, 31l, 34dhjklm, 38dfijkklm, 38dfijlm, 39dfgijklmm, 41, 41i, 43, 43m, 46l, 46, 49fghjl, 50km, 50k, 53gk, 53k, 53, 53j, 58hik, 58hikl, 60ehl, 60el, 62, 72hijkllm, 72hijkm, 72, 77m, 80, 80k, 99ekm, 99efkm, 118fikl, 121iklm, 130j, 137, 140hk, 140k, 149, 159k, 166gl, 171, 181, 183k, 183, 188egijklm, 193, 193l, 198l, 198gl, 217, 217l

43-limit

236dghin, 239*, 243eklmn*, 248h*, 270*, 270i, 296, 311n*, 311, 373g, 388, 422*, 422l, 525*, 540, 566gjl, 597ik, 624jk, 639hjmn, 653, 711, 718, 764hj, 814k, 814, 863efgjk, 882, 908m, 954hjln, 1065, 1164fimn, 1178j, 1178, 1205gl, 1323, 1323k, 1357efhijlmn, 1385k, 1395, 1578kln, 1578ln, 1578l, 1600, 1696, 1817hi, 1920, 2159, 2270il, 2554n, 2581ijn, 2684m, 2684, 2703, 2742klm, 3041, 3084, 3337, 3395ik, 3759, 3949i, 3976j, 4132k, 4349, 4380, 4501m, 5395, 5638, 5809, 5927fk, 6349, 6597, 7033, 7204, 7216, 7361, 8893, 9717, 10729k, 11664jm, 12245, 12428, 12999, 13112, 14066kn, 14348, 16066i, 17461, 19933, 20203, 20567, 23270, 26742m, 26742, 28742, 29053, 32231, 34691, 36269, 37592, 37946, 40649, 44268, 47309, 52704m, 53612, 54997, 55407, 56354m, 58973, 73815, 81176jn, 83096, 95524, 109590, 135800g, 148418, 157865, 159070, 165226, ….

Exotemperaments

5cfijjkllmnn, 5cfikllmnn, 5cfikllm, 5cfikm, 5f, 5, 5g, 5eghj, 5eghjn, 5eghjln, 6fhijlm, 7dfghijjkllmmmn, 7dfghijjkllmn, 7dfghijjkmn, 7l, 8dn, 8din, 9gijkln, 10egjkm, 10ejkm, 10h, 12del, 12el, 12fijkmn, 14cfjjkklmmn, 14cfjjlmmn, 15mn, 15kmn, 15gkmn, 15gklmn, 17cghkn, 17cghkmn, 18ehijkm, 19eghikklmnn, 19g, 19, 22il, 22i, 22, 22h, 22hn, 22fhn, 24, 26iknn, 26ik, 27egj, 27egij, 29gjk, 29gk, 31l, 34dhkn, 34dhjkmn, 34dhjklmn, 38dfijkklmnn, 38dfijlmnn, 38dfijlm, 39dfgijkl, 39dfgijklmm, 41in, 43, 43m, 43mn, 46ln, 49fghjlnn, 50jkm, 50km, 50k, 53, 53j, 58hi, 58hik, 58hikl, 60ehln, 60eln, 62, 72hijkllmn, 72hijkmn, 77m, 80, 80k, 84, 87, 94, 99efkm, 99efkmn, 111jln, 113l, 113, 118fikl, 121iklm, 140kn, 140k, 140, 145jkl, 149n, 152fgjm, 171, 181, 183k, 183, 193, 198ln, 198gln, 217n, 217ln

47-limit

270i, 296*, 311n*, 311*, 311o*, 320ij, 354, 364i*, 383ij*, 388, 422*, 422l, 472, 487, 525*, 566gjl, 597ik, 639hjmno, 653, 711, 718, 764hj, 814k, 814, 863efgjk, 908mo, 954hjln, 1065, 1121, 1164fimn, 1178, 1289n, 1323k, 1357efhijlmn, 1385k, 1578kln, 1578ln, 1578l, 1646, 1696o, 1817h, 1817hi, 1920, 2072, 2132ehln, 2159, 2270il, 2554n, 2581ijno, , 2619l, 2684o, 2703, 2742klm, 3084o, 3224, 3337o, 3395ik, 3889, 3949i, 4038j, 4132ko, 4349, 4380, 4973, 5091, 4387ho, 5395, 5455l, 5809, 5927fk, 6151, 6219, 6349, 7204, 7216, 7315m, 7361, 7361o, 8269, 8539l, 8539, 8893, 9717, 10257, 10729k, 11664jm, 12245, 13823, 14124, 14348, 16066i, 17461o, 17883, 19381jm, 19849, 20567, 23270, 28742, 30631, 31243, 32137, 32231, 36269, 37592, 40649, 44268o, 45861, 47309, 52704m, 53077, 55407, 58973, 73815, 81176jno, 83096, 89023, 95524, 109590, 136173, 148418, 157865, 166431, 182687, 191580, ….

Exotemperaments

5cfijjkllmnno, 5cfikllmnno, 5cfikllmo, 5cfikmo, 5f, 5, 5g, 5eghj, 5eghjn, 5eghjln, 6fhijlmo, 7dfghijjkllmmmno, 7dfghijjkllmno, 7dfghijjkmno, 7l, 8dno, 8dino, 9gijklno, 9k, 9, 10egjkmo, 10ejkmo, 10h, 12delo, 12elo, 12fijkm, 12fijkmn, 14cfjjkklmmno, 14cfjjlmmno, 15gkmno, 15gklmno, 17cghkno, 17cghkmno, 19eghikklmnno, 19eghilmnno, 19eghmo, 19go, 19o, 22ilm, 22il, 22i, 22fhn, 22fhno, 24, 26iknn, 26ik, 29gjk, 29gk, 31l, 34dhkn, 34dhjklmn, 38dfijkklmnnoo, 38dfijlmnnoo, 39dfgijkl, 39dfgijklmm, 41in, 43, 43m, 43mn, 46lno, 49fghjlnno, 50jkmo, 50kmo, 50ko, 53gkn, 53, 53j, 53jo, 56, 58hik, 58hiko, 58hiklo, 60ehln, 60eln, 62, 72hijkllmno, 72hijkmno, 72, 77m, 80o, 80ko, 87, 94jklm, 94m, 99efkm, 99efkmn, 103hl, 111ln, 111jln, 118fikl, 121iklm, 121iklmo, 130j, 140, 145jklo, 149no, 152fgjmo, 167hijkn, 171i, 171, 181, 183k, 183, 188egijklmno, 193, 198ln, 198gln, 212ghjmo, 217ln, 217lno, 224, 224n, 236dghino, 243eklmno, 248h, 270

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2.3.5.11 subgroup

Relative: 7, 65, 152, 342, 494, 1171, 11105, 12276, …

2.3.5.7.11.13.19 subgroup

Relative: 5eh, 7, 9, 15, 27e, 41, 53, 72, 130, 152f, 270, 2190, 6079, 8269, 8539, 115175, 148418, 176148, 184417, ….

2.3.5.7.11.13.17.23 subgroup

Relative: 5eg, 8di, 10, 22f, 26i, 29g, 31, 46, 72i, 140, 183, 311, 494, 764, 1578, 2460, 7033, 16808, 102557, ….

sqrt (2).sqrt (3).5.7.11.13 subgroup

Relative: 10e, 10, 34d, 58, 72, 270, 2684, 13112, … (up to 16808).

sqrt (2).sqrt (3).5.7.11.13.17 subgroup

Relative: 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 16808, … (up to 16808).

sqrt (2).sqrt (3).5.7.11.13.17.19.23 subgroup

Relative: 14cf, 34dh, 58hi, 72i, 130, 270, 718, 742i, 814, 954h, 2000, 4038, 12770, 16808, ….

A mathematically defined tier list

The tier of an equal temperament is defined as the number of GPVs before it that is tuned more efficient than it. The sequence of tier-0 equal temperaments is equivalent to the sequence of equal temperaments by efficiency.

Below are the tier lists of equal temperaments in various subgroups by TE relative error, searched up to 16808. GPVs starting with 0 are skipped as they would meddle with the results otherwise. For readability reasons, equal temperaments smaller than 5 are not shown.

3-limit

Tier-0: 5, 12, 41, 53, 306, 665, 15601, ….

Tier-1: 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330, 14936, 16266, ….

Tier-2: 10, 147, 159, 200, 612, 1636, 1995, 14271, ….

Tier-3: 65, 2301, 2660, 13606, ….

Tier-4: 8, 9, 19, 36, 82, 412, 1024, 2966, 3325, 12941, ….

Tier-5: 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990, 12276, ….

5-limit

Tier-0: 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296, ….

Tier-1: 5, 171, 730, 6809, 8592, 16572, ….

Tier-2: 15, 31, 99, 106, 1342, 3684, 11105, 12276, ….

Tier-3: 10, 22, 46, 152, 236, 270, 289, 3125, 5467, 7980, ….

Tier-4: 8, 9, 41, 224, 323, 494, 2954, ….

Tier-5: 5c, 24, 27, 87, 205, 388, 1224, 1901, 9763, 12888, ….

7-limit

Tier-0: 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664, ….

Tier-1: 41, 270, 441, 2954, 6691, 14789, ….

Tier-2: 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566, ….

Tier-3: 7, 27, 140, 1848, 3296, 11835, ….

Tier-4: 14c, 46, 152, 301, 323, 3395, 6520, ….

Tier-5: 5, 58, 118, 494, 513, 783, 1106, 3737, 4973, 5144, 6250, 9816, 15230, ….

11-limit

Tier-0: 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691, ….

Tier-1: 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960, ….

Tier-2: 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770, ….

Tier-3: 5e, 10, 19e, 130, 1106, 1236, 2190, 6349, 7927, ….

Tier-4: 53, 80, 87, 311, 764, 836, 2684, 4843, 6421, ….

Tier-5: 99e, 183, 400, 472, 684, 742, 8539, ….

13-limit

Tier-0: 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618, ….

Tier-1: 5, 7d, 15, 87, 130, 8269, 8539, ….

Tier-2: 10, 34d, 46, 198, 311, 684, 954, 1236, ….

Tier-3: 7df, 29, 103, 581, 1178, 3395, 7033, ….

Tier-4: 5f, 10e, 14cf, 111, 121, 140, 190, 764, 1448, 2460, 11664, 13112, ….

Tier-5: 22, 26, 152f, 183, 373, 422, 742, 1012, 2954, ….

17-limit

Tier-0: 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, ….

Tier-1: 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348, ….

Tier-2: 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269, ….

Tier-3: 14cf, 15g, 17cg, 50, 53, 87, 217, 422, 460, 814, 2000, 2684, 4573, 10428, ….

Tier-4: 7g, 12, 34d, 400, 6079, ….

Tier-5: 10e, 15, 282, 354, 1889, 9033, ….

19-limit

Tier-0: 5egh, 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808, ….

Tier-1: 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311, ….

Tier-2: 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934, ….

Tier-3: 22fh, 72h, 99ef, 190g, 193, 243e, 460, 935, 6961, 11664, ….

Tier-4: 5cf, 5g, 5eg, 10, 15, 34dh, 118f, 282, 954h, 1889, 4349, 6079, ….

Tier-5: 22, 43, 80, 87, 130, 224, 373g, 400, 643, 3178, 3920, 4501, ….

23-limit

Tier-0: 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808, ….

Tier-1: 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889, ….

Tier-2: 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539, ….

Tier-3: 19eghi, 58hi, 99ef, 121i, 624, 814, 4038, 4349, 6079, 6269, ….

Tier-4: 27eg, 68, 718, 742i, 14348, ….

Tier-5: 19egh, 19, 24, 41i, 43, 3395i, 7315, ….

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2.3.5.11 subgroup

Tier-0: 7, 65, 152, 342, 494, 1171, 11105, 12276, ….

Tier-1: 5, 5e, 8, 9, 12e, 12, 15, 22, 31, 53, 72, 87, 118, 270, 677, 1665, 2342, 9934, ….

Tier-2: 19, 34, 183, 407, 559, 5520, 6691, ….

Tier-3: 14, 24, 335, 612, 829, 901, 1513, 2836, 3513, 13347, ….

Tier-4: 41, 46, 159, 224, 2684, 7862, 8763, ….

Tier-5: 10, 19e, 137, 217, 836, 1019, 2072, 3855, ….

sqrt (2).sqrt (3).5.7.11.13 subgroup

Tier-0: 10e, 10, 34d, 58, 72, 270, 2684, 13112, ….

Tier-1: 14cf, 130, 954, 10428, 16808, ….

Tier-2: 24, 38df, 68, 198, 742, 814, 2000, 2954, 14124, ….

Tier-3: 62, 140, 612, 1012, 1730, ….

Tier-4: 14c, 34, 96d, 1084, 3696, 3768, 10158, 10356, 12158, 13040, ….

Tier-5: 10f, 24d, 202, 212, 342f, 13854, ….

sqrt (2).sqrt (3).5.7.11.13.17 subgroup

Tier-0: 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 16808, ….

Tier-1: 14cf, 24, 68, 130, 140, 270, 814, ….

Tier-2: 38df, 212g, 400, 2414, 3638, 12842, ….

Tier-3: 10eg, 14c, 34, 58g, 130g, 328, 414, 684, 11170, 12428, 13112, 13170, ….

Tier-4: 14cfgg, 62, 96d, 198g, 342f, 882, ….

Tier-5: ….