User:FloraC/Sequence of equal temperaments by error/Supplement: Difference between revisions

Revision as of 18:14, 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, edos 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

Relative: 5, 12, 41, 53, 306, 665, 15601, …

(continued) 31867, 79335, 111202, 190537, ….

5-limit

Relative: 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296…

(continued) 73709, 78005…

7-limit

Relative: 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664…

(continued) 18355, 84814, 103169…

11-limit

Relative: 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691…

(continued) 40006, 54624, 121524…

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, …

(continued) 73591, 81860, 87939, 96478, ….

17-limit

Relative: 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, …

(continued) 20203, 102557, ….

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, …

(continued) 20203, 128215, ….

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…

31-limit

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

(continued) 148418…

37-limit

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

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

41-limit

Relative: 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j…

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

43-limit

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

(continued) 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…

(continued) 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, …

(continued) 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, …

(continued) 102557, ….

sqrt (2).sqrt (3).5.7.11.13 subgroup: Relative: 10e, 10, 34d, 58, 72, 270, 2684, 13112, ….

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, edos 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, ….

sqrt (2).sqrt (3).5.7.11.13.17 subgroup: Tier-0: see above.; 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: ….

@@ Line 1: / Line 1: @@
-For a method to iterate thru all GPVs, see [[Patent val/Properties]].
+Below are the sequences of equal temperaments in various subgroups by [[TE error|TE absolute error]] and [[TE relative error]], searched up to [[196608edo|196608]]. GPVs starting with 0 are skipped as they would meddle with the results otherwise. For readability reasons, edos smaller than 5 are not shown.
-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.
+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]].
-GPVs starting with 0 are not accounted as they would meddle with the results. For readability reasons, results smaller than 5 are not shown.
+; 3-limit
+: Relative: {{EDOs| 5, 12, 41, 53, 306, 665, 15601, …}}
+:: (continued) {{EDOs| 31867, 79335, 111202, 190537, … }}.
-----
+; 5-limit
+: Relative: {{EDOs| 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296 }}…
-; 3-limit:
-: Tier-0: {{EDOs| 5, 12, 41, 53, 306, 665, 15601 }}…
-:: (continued) {{EDOs| 31867, 79335, 111202, 190537 }}…
-: Tier-1: {{EDOs| 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330, 14936, 16266 }}…
-: Tier-2: {{EDOs| 10, 147, 159, 200, 612, 1636, 1995, 14271 }}…
-: Tier-3: {{EDOs| 65, 2301, 2660, 13606 }}…
-: Tier-4: {{EDOs| 8, 9, 19, 36, 82, 412, 1024, 2966, 3325, 12941 }}…
-: Tier-5: {{EDOs| 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990, 12276 }}…
-; 5-limit:
-: Tier-0: {{EDOs| 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296 }}…
 :: (continued) {{EDOs| 73709, 78005 }}…
-: Tier-1: {{EDOs| 5, 171, 730, 6809, 8592, 16572 }}…
-: Tier-2: {{EDOs| 15, 31, 99, 106, 1342, 3684, 11105, 12276 }}…
-: Tier-3: {{EDOs| 10, 22, 46, 152, 236, 270, 289, 3125, 5467, 7980 }}…
-: Tier-4: {{EDOs| 8, 9, 41, 224, 323, 494, 2954 }}…
-: Tier-5: {{EDOs| 5c, 24, 27, 87, 205, 388, 1224, 1901, 9763, 12888 }}…
-; 7-limit:
+; 7-limit
-: Tier-0: {{EDOs| 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664 }}…
+: Relative: {{EDOs| 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664 }}…
 :: (continued) {{EDOs| 18355, 84814, 103169 }}…
-: Tier-1: {{EDOs| 41, 270, 441, 2954, 6691, 14789 }}…
-: Tier-2: {{EDOs| 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566 }}…
-: Tier-3: {{EDOs| 7, 27, 140, 1848, 3296, 11835 }}…
-: Tier-4: {{EDOs| 14c, 46, 152, 301, 323, 3395, 6520 }}…
-: Tier-5: {{EDOs| 5, 58, 118, 494, 513, 783, 1106, 3737, 4973, 5144, 6250, 9816, 15230 }}…
-; 11-limit:
+; 11-limit
-: Tier-0: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691 }}…
+: Relative: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691 }}…
 :: (continued) {{EDOs| 40006, 54624, 121524 }}…
-: Tier-1: {{EDOs| 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960 }}…
-: Tier-2: {{EDOs| 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770 }}…
-: Tier-3: {{EDOs| 5e, 10, 19e, 130, 1106, 1236, 2190, 6349, 7927 }}…
-: Tier-4: {{EDOs| 53, 80, 87, 311, 764, 836, 2684, 4843, 6421 }}…
-: Tier-5: {{EDOs| 99e, 183, 400, 472, 684, 742, 8539 }}…
-; 13-limit:
+; 13-limit
-: Tier-0: {{EDOs| 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618 }}…
+: Relative: {{EDOs| 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618, … }}
-:: (continued) {{EDOs| 73591, 81860, 87939, 96478 }}…
+:: (continued) {{EDOs| 73591, 81860, 87939, 96478, … }}.
-: Tier-1: {{EDOs| 5, 7d, 15, 87, 130, 8269, 8539 }}…
-: Tier-2: {{EDOs| 10, 34d, 46, 198, 311, 684, 954, 1236 }}…
-: Tier-3: {{EDOs| 7df, 29, 103, 581, 1178, 3395, 7033 }}…
-: Tier-4: {{EDOs| 5f, 10e, 14cf, 111, 121, 140, 190, 764, 1448, 2460, 11664, 13112 }}…
-: Tier-5: {{EDOs| 22, 26, 152f, 183, 373, 422, 742, 1012, 2954 }}…
-; 17-limit:
+; 17-limit
-: Tier-0: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808 }}…
+: Relative: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, … }}
-:: (continued) {{EDOs| 20203, 102557 }}…
+:: (continued) {{EDOs| 20203, 102557, … }}.
-: Tier-1: {{EDOs| 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348 }}…
-: Tier-2: {{EDOs| 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269 }}…
-: Tier-3: {{EDOs| 14cf, 15g, 17cg, 50, 53, 87, 217, 422, 460, 814, 2000, 2684, 4573, 10428 }}…
-: Tier-4: {{EDOs| 7g, 12, 34d, 400, 6079 }}…
-: Tier-5: {{EDOs| 10e, 15, 282, 354, 1889, 9033 }}…
-; 19-limit:
+; 19-limit
-: Tier-0: {{EDOs| 5egh, 7, 8d, 8dhh, 9g, 9, 12f, 19egh, 26, 27eg, 46, 72, 217, 270, 581, 742, 1178, 1578, 2000, 2460, 3395, 8269, 8539, 16808 }}…
+: Relative: {{EDOs| 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) {{EDOs| 20203, 128215 }}…
+:: (continued) {{EDOs| 20203, 128215, … }}.
-: Tier-1: {{EDOs| 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311 }}…
-: Tier-2: {{EDOs| 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934 }}…
-: Tier-3: {{EDOs| 22fh, 72h, 99ef, 190g, 193, 243e, 460, 935, 6961, 11664 }}…
-: Tier-4: {{EDOs| 5cf, 5g, 5eg, 10, 15, 34dh, 118f, 282, 954h, 1889, 4349, 6079 }}…
-: Tier-5: {{EDOs| 22, 43, 80, 87, 130, 224, 373g, 400, 643, 3178, 3920, 4501 }}…
-; 23-limit:
+; 23-limit
-: Tier-0: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808 }}…
+: Relative: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808 }}…
-:: (continued) {{EDOs| 148418 }}…
+:: (continued) {{EDOs| 148418, … }}.
-: Tier-1: {{EDOs| 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889 }}…
-: Tier-2: {{EDOs| 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539 }}…
-: Tier-3: {{EDOs| 19eghi, 58hi, 99ef, 121i, 624, 814, 4038, 4349, 6079, 6269 }}…
-: Tier-4: {{EDOs| 27eg, 68, 718, 742i, 14348 }}…
-: Tier-5: {{EDOs| 19egh, 19, 24, 41i, 43, 3395i, 7315 }}…
-; 29-limit:
+; 29-limit
-: Tier-0: {{EDOs| 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808 }}…
+: Relative: {{EDOs| 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808 }}…
 :: (continued) {{EDOs| 148418 }}…
-; 31-limit:
+; 31-limit
-: Tier-0: {{EDOs| 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808 }}…
+: Relative: {{EDOs| 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808 }}…
 :: (continued) {{EDOs| 148418 }}…
-; 37-limit:
+; 37-limit
-: Tier-0: {{EDOs| 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808 }}…
+: Relative: {{EDOs| 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808 }}…
 :: (continued) {{EDOs| 26742, 58973, 65322, 67242, 83096, 148418 }}…
-; 41-limit:
+; 41-limit
-: Tier-0: {{EDOs| 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j }}…
+: Relative: {{EDOs| 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j }}…
 :: (continued) {{EDOs| 17461, 29053, 34691, 58973, 65322, 95524, 148418 }}…
 ; 43-limit
-: Tier-0: {{EDOs| 80k, 99efkmn, 121iklm, 270i, 311n, 311, 1600, 1920, 4501m, 7361, 14348 }}…
+: Relative: {{EDOs| 80k, 99efkmn, 121iklm, 270i, 311n, 311, 1600, 1920, 4501m, 7361, 14348 }}…
 :: (continued) {{EDOs| 20567, 58973, 109590, 165226 }}…
 ; 47-limit
-: Tier-0: {{EDOs| 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348 }}…
+: Relative: {{EDOs| 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348 }}…
 :: (continued) {{EDOs| 23270, 58973, 95524, 109590 }}…
 ----
-; 2.3.5.11 subgroup:
+; 2.3.5.11 subgroup
-: Tier-0: {{EDOs| 7, 65, 152, 342, 494, 1171, 11105, 12276 }}…
+: Relative: {{EDOs| 7, 65, 152, 342, 494, 1171, 11105, 12276, … }}
-: Tier-1: {{EDOs| 5, 5e, 8, 9, 12e, 12, 15, 22, 31, 53, 72, 87, 118, 270, 677, 1665, 2342, 9934 }}…
-: Tier-2: {{EDOs| 19, 34, 183, 407, 559, 5520, 6691 }}…
+; 2.3.5.7.11.13.19 subgroup
-: Tier-3: {{EDOs| 14, 24, 335, 612, 829, 901, 1513, 2836, 3513, 13347 }}…
+: Relative: {{EDOs| 5eh, 7, 9, 15, 27e, 41, 53, 72, 130, 152f, 270, 2190, 6079, 8269, 8539, … }}
-: Tier-4: {{EDOs| 41, 46, 159, 224, 2684, 7862, 8763 }}…
+:: (continued) {{EDOs| 115175, 148418, 176148, 184417, … }}.
-: Tier-5: {{EDOs| 10, 19e, 137, 217, 836, 1019, 2072, 3855 }}…
+; 2.3.5.7.11.13.17.23 subgroup
+: Relative: {{EDOs| 5eg, 8di, 10, 22f, 26i, 29g, 31, 46, 72i, 140, 183, 311, 494, 764, 1578, 2460, 7033, 16808, … }}
+:: (continued) {{EDOs| 102557, … }}.
+; sqrt (2).sqrt (3).5.7.11.13 subgroup
+: Relative: {{EDOs| 10e, 10, 34d, 58, 72, 270, 2684, 13112, … }}.
+; sqrt (2).sqrt (3).5.7.11.13.17 subgroup
+: Relative: {{EDOs| 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: {{EDOs| 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 [[16808edo|16808]]. GPVs starting with 0 are skipped as they would meddle with the results otherwise. For readability reasons, edos smaller than 5 are not shown.
+; 3-limit
+: Tier-0: see above.
+: Tier-1: {{EDOs| 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330, 14936, 16266, … }}.
+: Tier-2: {{EDOs| 10, 147, 159, 200, 612, 1636, 1995, 14271, … }}.
+: Tier-3: {{EDOs| 65, 2301, 2660, 13606, … }}.
+: Tier-4: {{EDOs| 8, 9, 19, 36, 82, 412, 1024, 2966, 3325, 12941, … }}.
+: Tier-5: {{EDOs| 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990, 12276, … }}.
+; 5-limit
+: Tier-0: see above.
+: Tier-1: {{EDOs| 5, 171, 730, 6809, 8592, 16572, … }}.
+: Tier-2: {{EDOs| 15, 31, 99, 106, 1342, 3684, 11105, 12276, … }}.
+: Tier-3: {{EDOs| 10, 22, 46, 152, 236, 270, 289, 3125, 5467, 7980, … }}.
+: Tier-4: {{EDOs| 8, 9, 41, 224, 323, 494, 2954, … }}.
+: Tier-5: {{EDOs| 5c, 24, 27, 87, 205, 388, 1224, 1901, 9763, 12888, … }}.
+; 7-limit
+: Tier-0: see above.
+: Tier-1: {{EDOs| 41, 270, 441, 2954, 6691, 14789, … }}.
+: Tier-2: {{EDOs| 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566, … }}.
+: Tier-3: {{EDOs| 7, 27, 140, 1848, 3296, 11835, … }}.
+: Tier-4: {{EDOs| 14c, 46, 152, 301, 323, 3395, 6520, … }}.
+: Tier-5: {{EDOs| 5, 58, 118, 494, 513, 783, 1106, 3737, 4973, 5144, 6250, 9816, 15230, … }}.
+; 11-limit
+: Tier-1: {{EDOs| 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960, … }}.
+: Tier-2: {{EDOs| 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770, … }}.
+: Tier-3: {{EDOs| 5e, 10, 19e, 130, 1106, 1236, 2190, 6349, 7927, … }}.
+: Tier-4: {{EDOs| 53, 80, 87, 311, 764, 836, 2684, 4843, 6421, … }}.
+: Tier-5: {{EDOs| 99e, 183, 400, 472, 684, 742, 8539, … }}.
+; 13-limit:
+: Tier-0: see above.
+: Tier-1: {{EDOs| 5, 7d, 15, 87, 130, 8269, 8539, … }}.
+: Tier-2: {{EDOs| 10, 34d, 46, 198, 311, 684, 954, 1236, … }}.
+: Tier-3: {{EDOs| 7df, 29, 103, 581, 1178, 3395, 7033, … }}.
+: Tier-4: {{EDOs| 5f, 10e, 14cf, 111, 121, 140, 190, 764, 1448, 2460, 11664, 13112, … }}.
+: Tier-5: {{EDOs| 22, 26, 152f, 183, 373, 422, 742, 1012, 2954, … }}.
+; 17-limit:
+: Tier-0: see above.
+: Tier-1: {{EDOs| 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348, … }}.
+: Tier-2: {{EDOs| 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269, … }}.
+: Tier-3: {{EDOs| 14cf, 15g, 17cg, 50, 53, 87, 217, 422, 460, 814, 2000, 2684, 4573, 10428, … }}.
+: Tier-4: {{EDOs| 7g, 12, 34d, 400, 6079, … }}.
+: Tier-5: {{EDOs| 10e, 15, 282, 354, 1889, 9033, … }}.
+; 19-limit:
+: Tier-0: see above.
+: Tier-1: {{EDOs| 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311, … }}.
+: Tier-2: {{EDOs| 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934, … }}.
+: Tier-3: {{EDOs| 22fh, 72h, 99ef, 190g, 193, 243e, 460, 935, 6961, 11664, … }}.
+: Tier-4: {{EDOs| 5cf, 5g, 5eg, 10, 15, 34dh, 118f, 282, 954h, 1889, 4349, 6079, … }}.
+: Tier-5: {{EDOs| 22, 43, 80, 87, 130, 224, 373g, 400, 643, 3178, 3920, 4501, … }}.
-; 2.3.5.7.11.13.19 subgroup:
+; 23-limit:
-: Tier-0: {{EDOs| 5eh, 7, 9, 15, 27e, 41, 53, 72, 130, 152f, 270, 2190, 6079, 8269, 8539 }}…
+: Tier-0: see above.
-:: (continued) {{EDOs| 115175, 148418, 176148, 184417 }}…
+: Tier-1: {{EDOs| 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889, … }}.
+: Tier-2: {{EDOs| 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539, … }}.
+: Tier-3: {{EDOs| 19eghi, 58hi, 99ef, 121i, 624, 814, 4038, 4349, 6079, 6269, … }}.
+: Tier-4: {{EDOs| 27eg, 68, 718, 742i, 14348, … }}.
+: Tier-5: {{EDOs| 19egh, 19, 24, 41i, 43, 3395i, 7315, … }}.
-; 2.3.5.7.11.13.17.23 subgroup:
+----
-: Tier-0: {{EDOs| 5eg, 8di, 10, 22f, 26i, 29g, 31, 46, 72i, 140, 183, 311, 494, 764, 1578, 2460, 7033, 16808 }}…
-:: (continued) {{EDOs| 102557 }}…
+; 2.3.5.11 subgroup:
+: Tier-0: see above.
+: Tier-1: {{EDOs| 5, 5e, 8, 9, 12e, 12, 15, 22, 31, 53, 72, 87, 118, 270, 677, 1665, 2342, 9934, … }}.
+: Tier-2: {{EDOs| 19, 34, 183, 407, 559, 5520, 6691, … }}.
+: Tier-3: {{EDOs| 14, 24, 335, 612, 829, 901, 1513, 2836, 3513, 13347, … }}.
+: Tier-4: {{EDOs| 41, 46, 159, 224, 2684, 7862, 8763, … }}.
+: Tier-5: {{EDOs| 10, 19e, 137, 217, 836, 1019, 2072, 3855, … }}.
 ; sqrt (2).sqrt (3).5.7.11.13 subgroup:
-: Tier-0: {{EDOs| 10e, 10, 34d, 58, 72, 270, 2684, 13112 }}…
+: Tier-0: see above.
-: Tier-1: {{EDOs| 14cf, 130, 954, 10428, 16808 }}…
+: Tier-1: {{EDOs| 14cf, 130, 954, 10428, 16808, … }}.
-: Tier-2: {{EDOs| 24, 38df, 68, 198, 742, 814, 2000, 2954, 14124 }}…
+: Tier-2: {{EDOs| 24, 38df, 68, 198, 742, 814, 2000, 2954, 14124, … }}.
-: Tier-3: {{EDOs| 62, 140, 612, 1012, 1730 }}…
+: Tier-3: {{EDOs| 62, 140, 612, 1012, 1730, … }}.
-: Tier-4: {{EDOs| 14c, 34, 96d, 1084, 3696, 3768, 10158, 10356, 12158, 13040 }}…
+: Tier-4: {{EDOs| 14c, 34, 96d, 1084, 3696, 3768, 10158, 10356, 12158, 13040, … }}.
-: Tier-5: {{EDOs| 10f, 24d, 202, 212, 342f, 13854 }}…
+: Tier-5: {{EDOs| 10f, 24d, 202, 212, 342f, 13854, … }}.
 ; sqrt (2).sqrt (3).5.7.11.13.17 subgroup:
-: Tier-0: {{EDOs| 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 16808 }}…
+: Tier-0: see above.
-: Tier-1: {{EDOs| 14cf, 24, 68, 130, 140, 270, 814 }}…
+: Tier-1: {{EDOs| 14cf, 24, 68, 130, 140, 270, 814, … }}.
-: Tier-2: {{EDOs| 38df, 212g, 400, 2414, 3638, 12842 }}…
+: Tier-2: {{EDOs| 38df, 212g, 400, 2414, 3638, 12842, … }}.
-: Tier-3: {{EDOs| 10eg, 14c, 34, 58g, 130g, 328, 414, 684, 11170, 12428, 13112, 13170 }}…
+: Tier-3: {{EDOs| 10eg, 14c, 34, 58g, 130g, 328, 414, 684, 11170, 12428, 13112, 13170, … }}.
-: Tier-4: {{EDOs| 14cfgg, 62, 96d, 198g, 342f, 882 }}…
+: Tier-4: {{EDOs| 14cfgg, 62, 96d, 198g, 342f, 882, … }}.
-: Tier-5: {{EDOs|  }}…
+: Tier-5: ….
-; sqrt (2).sqrt (3).5.7.11.13.17.19.23 subgroup:
-: Tier-0: {{EDOs| 14cf, 34dh, 58hi, 72i, 130, 270, 718, 742i, 814, 954h, 2000, 4038, 12770, 16808 }}…
-:: (continued) …

User:FloraC/Sequence of equal temperaments by error/Supplement: Difference between revisions

Revision as of 18:14, 28 July 2025

A mathematically defined tier list

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