User:FloraC/Sequence of equal temperaments by error/Supplement: Difference between revisions
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Below are the sequences of equal | Below are the sequences of [[equal temperament]]s in various [[subgroup]]s by [[TE error|TE absolute error]] and [[TE relative error]], searched up to [[196608edo|196608]]. [[GPV]]s 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 [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator]. For a method to iterate thru all GPVs, see [[Patent val/Properties]]. | These sequences are obtained with the [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator]. For a method to iterate thru all GPVs, see [[Patent val/Properties]]. | ||
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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. | 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 [[16808edo|16808]]. GPVs starting with 0 are skipped as they would meddle with the results otherwise. For readability reasons, | Below are the tier lists of equal temperaments in various subgroups by TE relative error, searched up to [[16808edo|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 | ; 3-limit | ||
Revision as of 18:16, 28 July 2025
Below are the sequences of equal temperaments in various subgroups by TE absolute error and TE relative error, 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.
- 13-limit
- Relative: 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618, …
- 19-limit
- Relative: 5egh, 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808, …
- 23-limit
- Relative: 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808…
- (continued) 148418, ….
- 29-limit
- Relative: 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808…
- (continued) 148418…
- 47-limit
- Relative: 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348…
- 2.3.5.7.11.13.19 subgroup
- Relative: 5eh, 7, 9, 15, 27e, 41, 53, 72, 130, 152f, 270, 2190, 6079, 8269, 8539, …
- 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, …
- (continued) 102557, ….
- sqrt (2).sqrt (3).5.7.11.13.17 subgroup
- Relative: 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 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, …
- (continued) ….
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: see above.
- 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: see above.
- 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: see above.
- 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-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: see above.
- 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: see above.
- 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: see above.
- 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: see above.
- 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: see above.
- 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: see above.
- 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, ….