User:FloraC/Mathematically defined ET 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 otherwise meddle with the results. GPVs starting with 1, 2, 3, and 4 may be regarded as trivial, and 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: ….