Sequence of equal temperaments by error

Below are the sequences of equal temperaments by TE absolute error (and TE relative error, in bold) in various subgroups, searched up to 196608. These sequences are obtained with the Temperament Evaluator. For a method to iterate through 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, which are noted with an asterisk. Alternatives that pass the test but have a higher error than a previous entry are noted in parentheses.

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

(14c), 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

(19), 22*, 26*, 27eg*, 29g*, 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

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

(53e), 58hi, 62, 68* (68e), 72hi* 72i* (72), 80*, (84), 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

72, 77, 80*, 87*, 94j* 94*, 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

31-limit

99efk, 103h*, 121ik, 130j*, 130, 140k*, 140, 149, 152fgj, 159k* (159), 167hijk*, 181*, 183k*, 183, 190g, 217, 243ek* (243e), 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* (190g), 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* (525j), 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*, (320ij), (354), 373g, 388, 422*, 422l, 525* (525j), 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* (364), 383ij*, 388, 422*, 422l, 472, 487, 525* (525j), 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