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

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 smaller than 5 are not shown.

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3-limit

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

(continued) 31867, 79335, 111202, 190537…

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…

(continued) 73709, 78005…

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…

(continued) 18355, 84814, 103169…

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…

(continued) 40006, 54624, 121524…

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…

(continued) 73591, 81860, 87939, 96478…

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…

(continued) 20203, 102557…

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…

(continued) 20203, 128215…

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…

(continued) 148418…

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…

(continued) 148418…

31-limit

Tier-0: 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808…

(continued) 148418…

37-limit

Tier-0: 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808…

(continued) 26742, 58973, 65322, 67242, 83096, 148418…

41-limit

Tier-0: 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j…

(continued) 17461, 29053, 34691, 58973, 65322, 95524, 148418…

43-limit

Tier-0: 80k, 99efkmn, 121iklm, 270i, 311n, 311, 1600, 1920, 4501m, 7361, 14348…

(continued) 20567, 58973, 109590, 165226…

47-limit

Tier-0: 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348…

(continued) 23270, 58973, 95524, 109590…

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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: …