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. 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.

These sequences are obtained with the Temperament Evaluator. For a method to iterate thru all GPVs, see Patent val/Properties.

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: 5c, 5, 7d, 8d, 9, 10, 12e, 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, ….

13-limit: 5cf, 5f, 5, 5e, 7df, 7d, 7, 8d, 9, 10, 12, 12f, 14cf, 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, ….

17-limit: 5cf, 5f, 5, 5g, 5eg, 7dfg, 7dg, 7, 8d, 9, 10e, 10, 12, 12f, 14cf, 15, 15g, 16, 17cg, 19eg, 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, ….

19-limit: 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, 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, ….

23-limit: 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, 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, ….

29-limit: Relative: 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808, 148418, ….

31-limit: Relative: 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808, 148418, ….

37-limit: Relative: 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808, 26742, 58973, 65322, 67242, 83096, 148418, ….

41-limit: Relative: 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j, 17461, 29053, 34691, 58973, 65322, 95524, 148418, ….

43-limit: Relative: 80k, 99efkmn, 121iklm, 270i, 311n, 311, 1600, 1920, 4501m, 7361, 14348, 20567, 58973, 109590, 165226, ….

47-limit: Relative: 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348, 23270, 58973, 95524, 109590, ….

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, ….

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: ….