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

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Below are the sequences of [[equal temperament]]s in various [[subgroup]]s by [[TE error|TE absolute error]] (and [[TE relative error]], in '''bold'''), 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.  
Below are the sequences of [[equal temperament]]s in various [[subgroup]]s by [[TE error|TE absolute error]] (and [[TE relative error]], in '''bold'''), searched up to [[196608edo|196608]]. 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]].  
[[GPV]]s starting with 0 are ignored 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.
 
(WIP) While looking at these sequences, an important consideration is whether a temperament tunes the relevant intervals [[monotonic]]. If a temperament is meant to be used in the [[harmonic limit|''p''-prime-limit]], it is reasonable to request that the order of size for the [[odd limit|''p''-odd-limit]] should at least be kept. For this reason and for the sake of readability, each main sequence starts with the first temperament that passes the monotonicity test, with the entries before that point relegated to the exotemperaments databox below it. There are usually some more entries in the main sequence that fail the monotonicity test. These are noted with an apostrophe.  


; 3-limit
; 3-limit
Line 10: Line 12:


; 7-limit
; 7-limit
: {{EDOs| '''5''', 7d, 8d, 9, '''10''', '''12''', '''19''', 27, '''31''', 41, 53, '''72''', '''99''', '''171''', 441, 612, 935, 1106, 1277, 1407, 1547, 1578, 1718, 1848, 2019, 2684, 2783, 2954, '''3125''', 6520, 6691, 9816, '''11664''', 14789, '''18355''', 33144, 51499, 51670, 63334, 70025, 81689, '''84814''', '''103169''', … }}.  
: {{EDOs| '''5''', 7d*, 8d, 9, '''10''', '''12''', '''19''', 27, '''31''', 41, 53, '''72''', '''99''', '''171''', 441, 612, 935, 1106, 1277, 1407, 1547, 1578, 1718, 1848, 2019, 2684, 2783, 2954, '''3125''', 6520, 6691, 9816, '''11664''', 14789, '''18355''', 33144, 51499, 51670, 63334, 70025, 81689, '''84814''', '''103169''', … }}.  


; 11-limit
; 11-limit
: {{EDOs| 5c, '''5''', '''7d''', '''8d''', '''9''', 10, 12e, '''12''', 15, 19e, 19, '''22''', 27e, '''31''', 41, 53, 58, '''72''', 118, 130, 152, 224, '''270''', '''342''', 612, 836, 1084, 1106, 1236, 1506, 1578, '''1848''', 2684, 3395, 3737, 4007, 4501, 4843, 5585, '''6079''', '''6691''', 12276, 12770, 14618, 14960, 21309, 25046, 26894, 33315, 39664, '''40006''', 47933, '''54624''', 81860, 94630, 96478, '''121524''', 176148, … }}.  
: {{EDOs| '''12''', 15, 19e*, 19, '''22''', 27e, '''31''', 41, 53, 58, '''72''', 118, 130, 152, 224, '''270''', '''342''', 612, 836, 1084, 1106, 1236, 1506, 1578, '''1848''', 2684, 3395, 3737, 4007, 4501, 4843, 5585, '''6079''', '''6691''', 12276, 12770, 14618, 14960, 21309, 25046, 26894, 33315, 39664, '''40006''', 47933, '''54624''', 81860, 94630, 96478, '''121524''', 176148, … }}.  
: {{Databox|Exotemperaments| 5c, '''5''', '''7d''', '''8d''', '''9''', 10, 12e }}


; 13-limit
; 13-limit
: {{EDOs| 5cf, 5f, 5, '''5e''', 7df, 7d, '''7''', '''8d''', '''9''', 10, 12, '''12f''', 14cf, 15, 17c, '''19e''', '''19''', 22, 26, '''27e''', 29, 31f, '''31''', '''41''', 46, '''53''', '''58''', '''72''', 87, 103, 111, 121, 130, 183, 190, 198, '''224''', '''270''', '''494''', 684, 764, 935, 954, 1178, 1236, 1448, '''1506''', '''2190''', '''2684''', 3395, 4079, 4349, 4843, '''5585''', '''6079''', 8269, 8539, 11664, 13112, 14348, '''14618''', 20203, 22887, 27236, 33315, 40276, 41584, 46355, 63163, 65062, 66558, 69242, '''73591''', '''81860''', '''87939''', '''96478''', 161530, 184417, … }}.  
: {{EDOs| 15, 17c, '''19e'''*, '''19''', 22*, 26, '''27e''', 29, 31f*, '''31''', '''41''', 46, '''53''', '''58''', '''72''', 87, 103, 111, 121, 130, 183, 190, 198, '''224''', '''270''', '''494''', 684, 764, 935, 954, 1178, 1236, 1448, '''1506''', '''2190''', '''2684''', 3395, 4079, 4349, 4843, '''5585''', '''6079''', 8269, 8539, 11664, 13112, 14348, '''14618''', 20203, 22887, 27236, 33315, 40276, 41584, 46355, 63163, 65062, 66558, 69242, '''73591''', '''81860''', '''87939''', '''96478''', 161530, 184417, … }}.  
: {{Databox|Exotemperaments| 5cf, 5f, 5, '''5e''', 7df, 7d, '''7''', '''8d''', '''9''', 10, 12, '''12f''', 14cf }}


; 17-limit
; 17-limit
: {{EDOs| 5cf, 5f, 5, 5g, '''5eg''', 7dfg, 7dg, 7, '''8d''', 9, 10e, '''10''', 12, 12f, 14cf, 15, 15g, 16, 17cg, '''19eg''', '''22''', '''26''', '''27eg''', 29g, '''31''', 38df, 41g, 41, '''46''', 58, '''72''', 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, '''494''', 581, 742, 764, 814, 935, 954, 1178, 1236, 1395, '''1506''', 2000, 2414, 2460, '''3395''', 4573, 5455, 5809, 5855, 6079, '''7033''', 8539, 10428, 11934, 14348, '''16808''', '''20203''', 27236, 41584, 51940, 58973, 60973, 76275, 79176, 82354, 88209, '''102557''', 128215, 129793, 155451, 161530, 169799, 181733, … }}.  
: {{EDOs| '''31''', 38df, 41g*, 41, '''46''', 58, '''72''', 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, '''494''', 581, 742, 764, 814, 935, 954, 1178, 1236, 1395, '''1506''', 2000, 2414, 2460, '''3395''', 4573, 5455, 5809, 5855, 6079, '''7033''', 8539, 10428, 11934, 14348, '''16808''', '''20203''', 27236, 41584, 51940, 58973, 60973, 76275, 79176, 82354, 88209, '''102557''', 128215, 129793, 155451, 161530, 169799, 181733, … }}.  
: {{Databox|Exotemperaments| 5cf, 5f, 5, 5g, '''5eg''', 7dfg, 7dg, 7, '''8d''', 9, 10e, '''10''', 12, 12f, 14cf, 15, 15g, 16, 17cg, '''19eg''', '''22''', '''26''', '''27eg''', 29g }}


; 19-limit
; 19-limit
: {{EDOs| 5cf, 5g, 5eg, '''5egh''', 7deefgh, 7dfgh, '''7''', '''8d''', '''8dhh''', '''9g''', '''9''', 10, 10h, 12e, 12, '''12f''', 14cf, 15, 15g, 17cgh, '''19egh''', 22, 22fh, 24, '''26''', '''27eg''', 29g, 31, 34dh, 38df, 41g, 41, '''46''', 50, 53, 58h, 68, 72h, '''72''', 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, '''217''', 243e, '''270''', 311, 400, 422, 460, 525, '''581''', '''742''', 935, 954h, '''1178''', '''1578''', 1889, '''2000''', '''2460''', 3178, '''3395''', 4349, 4501, 5498, 5527, 5809, 6079, 6961, 7527, '''8269''', '''8539''', 11934, 14348, 15854, 16066, '''16808''', '''20203''', 36269, 37011, 40770, 49039, 50434, 52434, 58973, 60973, 79176, 95984, 99162, 111407, 119365, 119946, '''128215''', 178338, … }}.  
: {{EDOs| 34dh, 38df, 41g*, 41, '''46'''*, 50, 53, 58h, 68, 72h*, '''72''', 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, '''217''', 243e, '''270''', 311, 400, 422, 460, 525, '''581''', '''742''', 935, 954h, '''1178''', '''1578''', 1889, '''2000''', '''2460''', 3178, '''3395''', 4349, 4501, 5498, 5527, 5809, 6079, 6961, 7527, '''8269''', '''8539''', 11934, 14348, 15854, 16066, '''16808''', '''20203''', 36269, 37011, 40770, 49039, 50434, 52434, 58973, 60973, 79176, 95984, 99162, 111407, 119365, 119946, '''128215''', 178338, … }}.  
: {{Databox|Exotemperaments| 5cf, 5g, 5eg, '''5egh''', 7deefgh, 7dfgh, '''7''', '''8d''', '''8dhh''', '''9g''', '''9''', 10, 10h, 12e, 12, '''12f''', 14cf, 15, 15g, 17cgh, '''19egh''', 22, 22fh, 24, '''26''', '''27eg''', 29g, 31 }}


; 23-limit
; 23-limit
: {{EDOs| 5cfi, 5g, 5eg, '''5egh''', 7deefghi, 7dfghi, 7, 8d, 8di, '''8dhhi''', 9gi, 9i, 9, 10e, 10, '''10h''', 12e, 12, 12f, '''12fi''', '''14cf''', 15, '''15g''', 18ehi, 19eghi, 22i, 22fh, 24, '''26i''', 27egi, '''29g''', 31, 34dh, 38dfi, 39dfgi, 41g, 41, 41i, 43, '''46''', 50, 53, 58hi, 62, 68, 72hi, 72i, 80, 87, '''94''', 111, 121i, 130, 140, 152fg, 159, 183, '''190g''', '''193''', '''217''', 243e, '''270''', '''282''', '''311''', 373g, 422, 525, 566g, '''581''', 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, '''1578''', 1889, 2000, '''2460''', 3178, 4038, 4349, 5144, 5395, 5638, 5783, 5958, 6079, 6269, 7315, '''8269''', 8539, 10539, '''11664''', 14348, '''16808''', 28472, 28742, 29053, 32231, 36269, 38189, 44907, 45861, 49039, 58973, 75781, 81176, 95242, 102557, 119365, 140149, '''148418''', 178338, … }}.  
: {{EDOs| 58hi, 62, 68*, 72hi*, 72i*, 80*, 87*, '''94''', 111, 121i, 130, 140, 152fg, 159, 183, '''190g''', '''193''', '''217''', 243e, '''270''', '''282''', '''311''', 373g, 422, 525, 566g, '''581''', 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, '''1578''', 1889, 2000, '''2460''', 3178, 4038, 4349, 5144, 5395, 5638, 5783, 5958, 6079, 6269, 7315, '''8269''', 8539, 10539, '''11664''', 14348, '''16808''', 28472, 28742, 29053, 32231, 36269, 38189, 44907, 45861, 49039, 58973, 75781, 81176, 95242, 102557, 119365, 140149, '''148418''', 178338, … }}.  
: {{Databox|Exotemperaments| 5cfi, 5g, 5eg, '''5egh''', 7deefghi, 7dfghi, 7, 8d, 8di, '''8dhhi''', 9gi, 9i, 9, 10e, 10, '''10h''', 12e, 12, 12f, '''12fi''', '''14cf''', 15, '''15g''', 18ehi, 19eghi, 22i, 22fh, 24, '''26i''', 27egi, '''29g''', 31, 34dh, 38dfi, 39dfgi, 41g, 41, 41i, 43, '''46''', 50, 53 }}
 
----


; 29-limit
; 29-limit

Revision as of 17:16, 5 August 2025

Below are the sequences of equal temperaments in various subgroups by TE absolute error (and TE relative error, in bold), searched up to 196608. These sequences are obtained with the Temperament Evaluator. For a method to iterate thru all GPVs, see Patent val/Properties.

GPVs starting with 0 are ignored 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.

(WIP) While looking at these sequences, an important consideration is whether a temperament tunes the relevant intervals monotonic. If a temperament is meant to be used in the p-prime-limit, it is reasonable to request that the order of size for the p-odd-limit should at least be kept. For this reason and for the sake of readability, each main sequence starts with the first temperament that passes the monotonicity test, with the entries before that point relegated to the exotemperaments databox below it. There are usually some more entries in the main sequence that fail the monotonicity test. These are noted with an apostrophe.

3-limit
5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, 79335, 111202, 190537, … (OEIS: A060528).
5-limit
5, 7, 12, 19, 31, 34, 46, 53, 118, 171, 289, 323, 388, 441, 559, 612, 1171, 1783, 2513, 3684, 4296, 12276, 16572, 20868, 25164, 46032, 48545, 52841, 73709, 78005, 151714, ….
7-limit
5, 7d*, 8d, 9, 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, 935, 1106, 1277, 1407, 1547, 1578, 1718, 1848, 2019, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 51670, 63334, 70025, 81689, 84814, 103169, ….
11-limit
12, 15, 19e*, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 3395, 3737, 4007, 4501, 4843, 5585, 6079, 6691, 12276, 12770, 14618, 14960, 21309, 25046, 26894, 33315, 39664, 40006, 47933, 54624, 81860, 94630, 96478, 121524, 176148, ….
Exotemperaments
5c, 5, 7d, 8d, 9, 10, 12e
13-limit
15, 17c, 19e*, 19, 22*, 26, 27e, 29, 31f*, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, 11664, 13112, 14348, 14618, 20203, 22887, 27236, 33315, 40276, 41584, 46355, 63163, 65062, 66558, 69242, 73591, 81860, 87939, 96478, 161530, 184417, ….
Exotemperaments
5cf, 5f, 5, 5e, 7df, 7d, 7, 8d, 9, 10, 12, 12f, 14cf
17-limit
31, 38df, 41g*, 41, 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954, 1178, 1236, 1395, 1506, 2000, 2414, 2460, 3395, 4573, 5455, 5809, 5855, 6079, 7033, 8539, 10428, 11934, 14348, 16808, 20203, 27236, 41584, 51940, 58973, 60973, 76275, 79176, 82354, 88209, 102557, 128215, 129793, 155451, 161530, 169799, 181733, ….
Exotemperaments
5cf, 5f, 5, 5g, 5eg, 7dfg, 7dg, 7, 8d, 9, 10e, 10, 12, 12f, 14cf, 15, 15g, 16, 17cg, 19eg, 22, 26, 27eg, 29g
19-limit
34dh, 38df, 41g*, 41, 46*, 50, 53, 58h, 68, 72h*, 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h, 1178, 1578, 1889, 2000, 2460, 3178, 3395, 4349, 4501, 5498, 5527, 5809, 6079, 6961, 7527, 8269, 8539, 11934, 14348, 15854, 16066, 16808, 20203, 36269, 37011, 40770, 49039, 50434, 52434, 58973, 60973, 79176, 95984, 99162, 111407, 119365, 119946, 128215, 178338, ….
Exotemperaments
5cf, 5g, 5eg, 5egh, 7deefgh, 7dfgh, 7, 8d, 8dhh, 9g, 9, 10, 10h, 12e, 12, 12f, 14cf, 15, 15g, 17cgh, 19egh, 22, 22fh, 24, 26, 27eg, 29g, 31
23-limit
58hi, 62, 68*, 72hi*, 72i*, 80*, 87*, 94, 111, 121i, 130, 140, 152fg, 159, 183, 190g, 193, 217, 243e, 270, 282, 311, 373g, 422, 525, 566g, 581, 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, 1578, 1889, 2000, 2460, 3178, 4038, 4349, 5144, 5395, 5638, 5783, 5958, 6079, 6269, 7315, 8269, 8539, 10539, 11664, 14348, 16808, 28472, 28742, 29053, 32231, 36269, 38189, 44907, 45861, 49039, 58973, 75781, 81176, 95242, 102557, 119365, 140149, 148418, 178338, ….
Exotemperaments
5cfi, 5g, 5eg, 5egh, 7deefghi, 7dfghi, 7, 8d, 8di, 8dhhi, 9gi, 9i, 9, 10e, 10, 10h, 12e, 12, 12f, 12fi, 14cf, 15, 15g, 18ehi, 19eghi, 22i, 22fh, 24, 26i, 27egi, 29g, 31, 34dh, 38dfi, 39dfgi, 41g, 41, 41i, 43, 46, 50, 53
29-limit
5cfi, 5egj, 5eghj, 7deefghijj, 7dfghijj, 7, 8di, 8dhhijj, 9gij, 9, 10ej, 10, 10h, 12de, 12e, 12fij, 14cfjj, 15, 15g, 17cgh, 18ehij, 19eghi, 22i, 22h, 22fh, 26i, 29gj, 29g, 31j, 31, 34dh, 34dhj, 38dfij, 39dfgij, 41g, 41, 43, 46, 50, 53, 58hi, 62, 68j, 72hij, 72ij, 72, 77, 80, 87, 94j, 94, 99ef, 103h, 113, 118, 121i, 130, 140h, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578, 1920, 2000, 2460, 2742, 3041, 3178, 3395i, 4038j, 4190g, 4349, 4501, 5395, 5783, 6079, 6893, 7315, 7361, 7527, 7847j, 8269, 8539, 9204, 9447, 9934, 10729, 11664j, 11934, 12245, 13112, 14066, 15230, 16808, 28742, 29053, 36269, 45861, 48945, 52704, 53077, 56513, 58392, 58973, 60973, 65322, 67242, 69512, 78716, 81176j, 83096, 87445, 92782, 102557, 119365, 148418, ….
31-limit
5cfijjk, 5cfik, 5g, 5gj, 5egj, 5eghj, 7deefghijjk, 7dfghijjk, 7, 8d, 8di, 9gijk, 10ejk, 10h, 12de, 12e, 12fijk, 14cfjj, 15, 15k, 15gk, 17cghk, 18ehijk, 19eghikk, 19eghi, 19egh, 19, 22i, 22h, 22fh, 24, 26ik, 27egij, 29gjk, 31, 34dhk, 34dhjk, 38dfij, 39dfgijk, 41g, 41, 43, 46, 58hi, 58hik, 62, 68, 68j, 72hijk, 77, 80k, 87, 94, 99efk, 103h, 121ik, 130j, 130, 140k, 140, 149, 152fgj, 159k, 167hijk, 181, 183k, 183, 190g, 217, 243ek, 270, 311, 388, 422, 525, 566gj, 571, 624jk, 639hj, 643ijk, 653, 692ik, 718, 742i, 863efgjk, 882, 908, 954hj, 1205g, 1289, 1308, 1395, 1578, 1920, 2311, 2460j, 2460, 2684, 2742k, 2901, 3041, 3323, 3395ik, 3759, 4038j, 4349, 4501, 5395, 5455, 5809, 6079, 6219, 6893, 7315, 7361, 7527, 8539, 9447, 9934, 11353, 11664j, 12245, 13823, 14066k, 14348, 15112, 16808, 29053, 31920, 32231, 34691, 36269, 40078, 47309, 48514, 48945, 50434, 51499, 52704, 58973, 64368, 65322, 67242, 78716, 81176j, 83096, 92782, 109590, 119365, 131610, 148418, ….
37-limit
5cfijjkll, 5cfikll, 5cfik, 5, 5g, 5egj, 5eghj, 5eghjl, 6fhijl, 7deefghijjkll, 7dfghijjkll, 7dfghijjk, 7l, 8d, 8di, 9gijkl, 10ejk, 10h, 12del, 12el, 12fijk, 14cfjjl, 15, 15gk, 15gkl, 16j, 17cgk, 17cghk, 19eghikkl, 19eghil, 19egh, 19gh, 19g, 19, 22il, 22i, 22, 22h, 22fh, 24, 26ik, 27egij, 29gjk, 29gk, 31fghjk, 31l, 34dhk, 34dhjkl, 38dfijkkl, 38dfijl, 39dfgijkl, 43, 46l, 46, 50jk, 50k, 53, 53j, 58hik, 58hikl, 60el, 62, 65d, 68, 68j, 68jl, 72hijkll, 72hijk, 72ijk, 72, 77, 80k, 94, 99ek, 99efk, 103hl, 121ikl, 130j, 140k, 149, 152fgj, 159k, 181, 183k, 183, 190gl, 198gl, 212ghj, 212gh, 217, 217l, 224, 239, 243ekl, 248h, 270l, 270, 270i, 296, 311k, 311, 388, 422l, 487, 525, 566gjl, 571, 597ik, 624jk, 643ijkl, 653, 692ik, 718, 742il, 814k, 814, 863efgjk, 882, 908, 954hjl, 1106, 1164fi, 1205gl, 1224, 1289, 1308, 1395, 1578l, 1600, 1696, 1821efk, 1920, 2460j, 2460, 2684, 2742kl, 2901, 3041, 3395ik, 3759, 3949i, 3992, 4190g, 4349, 4501, 6219, 6349, 6893, 7033, 7361, 7847jkl, 8269, 8893, 9281, 9934, 10652, 10729k, 11025, 11353, 12245, 13823, 14066k, 14348, 15854j, 16808, 20567, 23270, 25347, 26742, 31920, 32231, 34691, 40078, 47309, 48945, 52704, 53077, 57578, 58973, 65322, 67242, 81176j, 83096, 95524, 109590, 131610, 148418, ….
41-limit
5cfijjkllm, 5cfikllm, 5cfikm, 5f, 5, 5g, 5egj, 5eghj, 5eghjl, 6fhijlm, 7dfghijjkllmmm, 7dfghijjkllm, 7dfghijjkm, 7l, 8d, 8di, 9gijklmm, 9gijkl, 10h, 12del, 12el, 12fijkm, 14cfjjlmm, 15, 15km, 15gkm, 15gklm, 16jm, 17cghk, 17cghkm, 18ehijkm, 19eghikklm, 19eghilm, 19, 22il, 22i, 22, 22h, 22fh, 24, 26ik, 27egij, 29gjk, 29gk, 31, 31l, 34dhjklm, 38dfijkklm, 38dfijlm, 39dfgijklmm, 41, 41i, 43, 43m, 46l, 46, 49fghjl, 50km, 50k, 53gk, 53k, 53, 53j, 58hik, 58hikl, 60ehl, 60el, 62, 72hijkllm, 72hijkm, 72, 77m, 80, 80k, 99ekm, 99efkm, 118fikl, 121iklm, 130j, 137, 140hk, 140k, 149, 159k, 166gl, 171, 181, 183k, 183, 188egijklm, 193, 193l, 198l, 198gl, 217, 217l, 224, 248h, 270, 270i, 311k, 311, 388, 422, 422l, 525, 540, 566gjl, 597ik, 624jk, 639hjm, 653, 692ikm, 718, 764, 764j, 814k, 814, 863efgjk, 908m, 954hjl, 1012, 1147i, 1164fim, 1178, 1205gl, 1289, 1385k, 1395, 1578l, 1696, 1817hi, 1920, 2159, 2270il, 2460jm, 2460m, 2460, 2554, 2581ij, 2703, 2742klm, 3041, 3084, 3337, 3395ik, 3759, 3992, 4349, 4501m, 5144, 5638, 5809, 6151, 6349, 7204, 7216, 7361, 7847jklm, 8893, 9717, 9934, 10729k, 11353, 11534, 12245, 12428, 13584, 14066k, 14348, 15854j, 16808, 17461, 19933, 20203, 20567, 23270, 25347, 27730, 28414hjk, 28742, 29053, 31920, 34691, 40078, 42052, 44268, 47309, 49256, 51176, 53077, 53612, 56354m, 57578, 58392, 58973, 65322, 81176j, 95524, 109590, 133431, 148418, 165226 ….
43-limit
5cfijjkllmnn, 5cfikllmnn, 5cfikllm, 5cfikm, 5f, 5, 5g, 5eghj, 5eghjn, 5eghjln, 6fhijlm, 7dfghijjkllmmmn, 7dfghijjkllmn, 7dfghijjkmn, 7l, 8dn, 8din, 9gijkln, 10egjkm, 10ejkm, 10h, 12del, 12el, 12fijkmn, 14cfjjkklmmn, 14cfjjlmmn, 15mn, 15kmn, 15gkmn, 15gklmn, 17cghkn, 17cghkmn, 18ehijkm, 19eghikklmnn, 19g, 19, 22il, 22i, 22, 22h, 22hn, 22fhn, 24, 26iknn, 26ik, 27egj, 27egij, 29gjk, 29gk, 31l, 34dhkn, 34dhjkmn, 34dhjklmn, 38dfijkklmnn, 38dfijlmnn, 38dfijlm, 39dfgijkl, 39dfgijklmm, 41in, 43, 43m, 43mn, 46ln, 49fghjlnn, 50jkm, 50km, 50k, 53, 53j, 58hi, 58hik, 58hikl, 60ehln, 60eln, 62, 72hijkllmn, 72hijkmn, 77m, 80, 80k, 84, 87, 94, 99efkm, 99efkmn, 111jln, 113l, 113, 118fikl, 121iklm, 140kn, 140k, 140, 145jkl, 149n, 152fgjm, 171, 181, 183k, 183, 193, 198ln, 198gln, 217n, 217ln, 236dghin, 239, 243eklmn, 248h, 270, 270i, 296, 311n, 311, 373g, 388, 422, 422l, 525, 540, 566gjl, 597ik, 624jk, 639hjmn, 653, 711, 718, 764hj, 814k, 814, 863efgjk, 882, 908m, 954hjln, 1065, 1164fimn, 1178j, 1178, 1205gl, 1323, 1323k, 1357efhijlmn, 1385k, 1395, 1578kln, 1578ln, 1578l, 1600, 1696, 1817hi, 1920, 2159, 2270il, 2554n, 2581ijn, 2684m, 2684, 2703, 2742klm, 3041, 3084, 3337, 3395ik, 3759, 3949i, 3976j, 4132k, 4349, 4380, 4501m, 5395, 5638, 5809, 5927fk, 6349, 6597, 7033, 7204, 7216, 7361, 8893, 9717, 10729k, 11664jm, 12245, 12428, 12999, 13112, 14066kn, 14348, 16066i, 17461, 19933, 20203, 20567, 23270, 26742m, 26742, 28742, 29053, 32231, 34691, 36269, 37592, 37946, 40649, 44268, 47309, 52704m, 53612, 54997, 55407, 56354m, 58973, 73815, 81176jn, 83096, 95524, 109590, 135800g, 148418, 157865, 159070, 165226, ….
47-limit
Relative: 5cfijjkllmnno, 5cfikllmnno, 5cfikllmo, 5cfikmo, 5f, 5, 5g, 5eghj, 5eghjn, 5eghjln, 6fhijlmo, 7dfghijjkllmmmno, 7dfghijjkllmno, 7dfghijjkmno, 7l, 8dno, 8dino, 9gijklno, 9k, 9, 10egjkmo, 10ejkmo, 10h, 12delo, 12elo, 12fijkm, 12fijkmn, 14cfjjkklmmno, 14cfjjlmmno, 15gkmno, 15gklmno, 17cghkno, 17cghkmno, 19eghikklmnno, 19eghilmnno, 19eghmo, 19go, 19o, 22ilm, 22il, 22i, 22fhn, 22fhno, 24, 26iknn, 26ik, 29gjk, 29gk, 31l, 34dhkn, 34dhjklmn, 38dfijkklmnnoo, 38dfijlmnnoo, 39dfgijkl, 39dfgijklmm, 41in, 43, 43m, 43mn, 46lno, 49fghjlnno, 50jkmo, 50kmo, 50ko, 53gkn, 53, 53j, 53jo, 56, 58hik, 58hiko, 58hiklo, 60ehln, 60eln, 62, 72hijkllmno, 72hijkmno, 72, 77m, 80o, 80ko, 87, 94jklm, 94m, 99efkm, 99efkmn, 103hl, 111ln, 111jln, 118fikl, 121iklm, 121iklmo, 130j, 140, 145jklo, 149no, 152fgjmo, 167hijkn, 171i, 171, 181, 183k, 183, 188egijklmno, 193, 198ln, 198gln, 212ghjmo, 217ln, 217lno, 224, 224n, 236dghino, 243eklmno, 248h, 270, 270i, 296, 311n, 311, 311o, 320ij, 354, 364i, 383ij, 388, 422, 422l, 472, 487, 525, 566gjl, 597ik, 639hjmno, 653, 711, 718, 764hj, 814k, 814, 863efgjk, 908mo, 954hjln, 1065, 1121, 1164fimn, 1178, 1289n, 1323k, 1357efhijlmn, 1385k, 1578kln, 1578ln, 1578l, 1646, 1696o, 1817h, 1817hi, 1920, 2072, 2132ehln, 2159, 2270il, 2554n, 2581ijno, , 2619l, 2684o, 2703, 2742klm, 3084o, 3224, 3337o, 3395ik, 3889, 3949i, 4038j, 4132ko, 4349, 4380, 4973, 5091, 4387ho, 5395, 5455l, 5809, 5927fk, 6151, 6219, 6349, 7204, 7216, 7315m, 7361, 7361o, 8269, 8539l, 8539, 8893, 9717, 10257, 10729k, 11664jm, 12245, 13823, 14124, 14348, 16066i, 17461o, 17883, 19381jm, 19849, 20567, 23270, 28742, 30631, 31243, 32137, 32231, 36269, 37592, 40649, 44268o, 45861, 47309, 52704m, 53077, 55407, 58973, 73815, 81176jno, 83096, 89023, 95524, 109590, 136173, 148418, 157865, 166431, 182687, 191580, ….
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, 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, 102557, ….
sqrt (2).sqrt (3).5.7.11.13 subgroup
Relative: 10e, 10, 34d, 58, 72, 270, 2684, 13112, … (up to 16808).
sqrt (2).sqrt (3).5.7.11.13.17 subgroup
Relative: 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 16808, … (up to 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, ….

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: 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, ….
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: ….
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