User:FloraC/Sequence of ETs with lower relative error

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For a method to iterate thru all GPVs, see Patent val/Properties.

The tier of an equal temperament is defined as the number of GPVs before it that is tuned more efficient than it. Efficiency can be determined in various ways, but the most common way is the TE relative error.

GPVs starting with 0 are not accounted as they would meddle with the results. For readability reasons, results are shown in the range of 5 and 16808.

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, 20203
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, 20203
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
29-limit
Tier-0: 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808
31-limit
Tier-0: 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808
37-limit
Tier-0: 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808
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: …