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.
{{Breadcrumb|Sequence of equal temperaments by 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]].  
; 2.3.5.11 subgroup
 
: {{EDOs| '''7''', 12, 15, 19, 22, 31, 34, 41, 46, 53, '''65''', 87, 118, '''152''', 224, 270, 335, '''342''', '''494''', 677, 1019, '''1171''', 3855, 4349, 5026, 5520, 6691, 7862, 8763, 9934, '''11105''', '''12276''', 35657, 41177, 42348, 53453, '''54624''', 66900, 121524, 149931, 161036, 173312, 192279, … }}
; 3-limit
: {{EDOs| '''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
: {{Databox|Exotemperaments*| 5c, 5, 5e }}
: {{EDOs| 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, … }}.
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 9, 11}  


; 7-limit
; 2.3.7.11 subgroup
: {{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''', 9, 10, 12, 14, '''17''', 31, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, 477, 670, 701, 836, 877, 1012, 1506, 1713, '''1848''', 4573, '''4708''', 6556, 11264, 17478, 19326, 22186, 24034, '''28742''', '''35298''', 59332, '''64040''', 94630, 123372, 158670, … }}


; 11-limit
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 7, 9, 11}  
: {{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, … }}.


; 13-limit
; 2.3.5.13 subgroup
: {{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| '''5''', '''7''', 10, 12, 15, '''19''', '''34''', '''53''', 130, 140, 164, 183, 217, 270, 354, 388, 407, '''441''', 494, 848, 901, 1289, '''1342''', 2901, 3395, 3802, 4243, 4737, 5638, '''6079''', 11717, 15960, 17302, 22039, '''23381''', 35098, 45420, '''57137''', 80518, 137655, '''143734''', 189154, … }}


; 17-limit
: {{Databox|Exotemperaments*| 5f }}
: {{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, … }}.
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 9, 13}  


; 19-limit
; 2.3.7.13 subgroup
: {{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| '''5''', 7, 9, '''10''', '''17''', 26, 27, '''36''', 77, '''94''', '''130''', 207, 234, 270, 347, 364, 441, '''477''', '''571''', 1048, 1412, '''1983''', 5144, 6079, 6556, 7127, 8539, '''9110''', '''20203''', 49516, 58055, 67165, 69148, 78258, 87368, 98461, 107571, 127774, 179750, 190843, … }}


; 23-limit
: {{Databox|Exotemperaments*| '''5f''' }}
: {{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, … }}.
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 7, 9, 13}  


----
----


; 29-limit
; 2.3.5.7.11.13.19 subgroup
: Relative: {{EDOs| 5eghj, 10h, 12fij, 14cfjj, 15, 15g, 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808, 148418, … }}.  
: {{EDOs| '''27e''', 31, 34dh*, 38df*, 41f*, '''41''', 50*, '''53''', 58h, '''72''', 87, 94, 103h, 111, 121, '''130''', '''152f''', 190, 217, 224, '''270''', 552, 581, 684h, 742, 935, 954h, 1012, 1178, 1308, 1448, 1578, 1730, 2000, '''2190''', 3125, 3395, 3768, 4079, 4190, 4349, 4771, 5585, '''6079''', '''8269''', '''8539''', 14618, 16808, 20203, 28472, 35999, 44538, 46355, 47663, 47933, 52434, 56202, 58973, 60973, 69242, 85255, 89445, 94972, '''115175''', 128215, '''148418''', '''176148''', '''184417''', … }}.  


; 31-limit
: {{Databox|Exotemperaments*| 5cf, 5f, 5, '''5eh''', 7dfh, 7dh, '''7''', 8d, 8dhh, '''9''', 12e, 12, 12f, 14cf, '''15''', 19eh, 22, 22fh, 24, 26 }}
: Relative: {{EDOs| 5eghj, 7, 12fijk, 14cfjj, 15gk, 22fh, 26ik, 31, 39dfgijk, 41, 43, 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808, 148418, … }}.
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21}  


; 37-limit
; 2.3.5.7.11.13.19.29 subgroup
: Relative: {{EDOs| 5eghj, 5eghjl, 7dfghijjk, 8di, 10h, 12fijk, 19, 22fh, 26ik, 39dfgijkl, 43, 46, 58hikl, 77, 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808, 26742, 58973, 65322, 67242, 83096, 148418, … }}.  
: {{EDOs| '''41''', 53*, '''58h''', 72, 77, 80, 87* (94), 99ef, '''103h'''* (113, 118f, 118), 121, 130, 140, '''152fj''', 159, 183, 190, 198, 217, 224, 243e, 270j*, '''270''', 422, 472, 494, 494h, 552, 590, 612, 684h, 692, 742, 795, 908, 935, 954hj, 1106, 1205, 1308, 1448, 1578, 1920, '''2000''', '''2190''', 3395, 4190, 4501, 5585, 6079, 6539, 7527, 8539, 10729, 11934, 12618, 13040, 14124, '''14618''', '''16808''', '''20203''', 28742, 33243, 36269, 40770, 46355, 47663, 54894, 56202, 60703, 60973, 67512, 80906, 87445, 94972, 115175, '''128215''', '''148418''', … }}.  


; 41-limit
: {{Databox|Exotemperaments*| 5cf, 5f, 5, '''5ehj''', 7deefhjj, 7dfhjj, 7dh, '''7''', 8d, 8dhhjj, '''9j''', 10h, 12e, 12, 12f, 12fj, 14cfjj, '''15''', 19eh, 22, 22fh, 24, 26, 27ej, 31j, 31, 34dhj, 38dfj, 41f }}
: Relative: {{EDOs| 5eghj, 5eghjl, 7dfghijjkm, 10h, 12fijkm, 19, 22fh, 26ik, 58hikl, 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j, 17461, 29053, 34691, 58973, 65322, 95524, 148418, … }}.
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} (omission of 25 is intentional)


; 43-limit
; 2.3.5.7.11.13.17.23 subgroup
: Relative: {{EDOs| 5eghjn, 5eghjln, 7dfghijjkmn, 12el, 12fijkmn, 14cfjjlmmn, 15gkmn, 15gklmn, 19, 22fhn, 26ik, 27egij, 58hik, 58hikl, 80k, 99efkmn, 121iklm, 270i, 311n, 311, 1600, 1920, 4501m, 7361, 14348, 20567, 58973, 109590, 165226, … }}.  
: {{EDOs| '''31''', 34d, 39dfgi*, 41g* 41* 41i*, 43*, '''46''', 58i, 68, '''72i'''* (72), 94, 103, 111, 121i, 130, '''140''', 159, '''183''', 217, 224, 243e, 270, '''311''', 373g, 422, '''494''', 581, 684, 742i, '''764''', 814, 954, 1075, 1236, 1506i, '''1578''', 1889, 2414, '''2460''', 3696, 3809, 3855, 3992, 4038, 4349, 4573, 5455, 6079, '''7033''', 11664, 13112, 14348, '''16808''', 29053, 29236, 31156, 32231, 35132, 35505, 36269, 38189, 41584, 45861, 49039, 51940, 58973, 68748, 75781, 85749, 88209, '''102557''', 161530, 178338, … }}.  


; 47-limit
: {{Databox|Exotemperaments*| 5cfi, 5, 5g, '''5eg''', 7dfgi, 7dgi, 7, 8d, '''8di''', 10e, '''10''', 12e, 12, 12f, 12fi, 14cf, 15, 15g, 16, 17cg, 19egi, 19, 22i, 22, '''22f''', '''26i''', 27egi, '''29g''' }}
: Relative: {{EDOs| 5eghjln, 7dfghijjkmno, 12fijkmn, 14cfjjlmmno, 15gklmno, 22fhn, 22fhno, 26ik, 58hiklo, 60ehln, 60eln, 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348, 23270, 58973, 95524, 109590, … }}.
: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 17, 23}  


----
----
; 2.3.5.11 subgroup
: Relative: {{EDOs| 7, 65, 152, 342, 494, 1171, 11105, 12276, … }}
; 2.3.5.7.11.13.19 subgroup
: Relative: {{EDOs| 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: {{EDOs| 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
; sqrt (2).sqrt (3).5.7.11.13 subgroup
: Relative: {{EDOs| 10e, 10, 34d, 58, 72, 270, 2684, 13112, … }} (up to 16808).  
: Relative: {{EDOs| 10e, 10, 34d, 58, 72, 270, 2684, 13112, … }}.  


; sqrt (2).sqrt (3).5.7.11.13.17 subgroup
; sqrt (2).sqrt (3).5.7.11.13.17 subgroup
: Relative: {{EDOs| 10e, 10, 34d, 58, 72, 742, 954, 2000, 2684, 10356, 10428, 16808, … }} (up to 16808).  
: 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
; 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, … }}.
: Relative: {{EDOs| 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 [[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
: Tier-0: {{EDOs| 5, 12, 41, 53, 306, 665, 15601, … }}.
: 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, … }}.
: 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: {{EDOs| 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664, … }}.
: 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-0: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691, … }}.
: 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: {{EDOs| 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618, … }}.
: 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: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, … }}.
: 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: {{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, … }}.
: 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:
: Tier-0: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808, … }}.
: 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, … }}.
 
----
 
; 2.3.5.11 subgroup:
: Tier-0: {{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, … }}.
: 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-1: {{EDOs| 14cf, 130, 954, 10428, 16808, … }}.
: Tier-2: {{EDOs| 24, 38df, 68, 198, 742, 814, 2000, 2954, 14124, … }}.
: Tier-3: {{EDOs| 62, 140, 612, 1012, 1730, … }}.
: Tier-4: {{EDOs| 14c, 34, 96d, 1084, 3696, 3768, 10158, 10356, 12158, 13040, … }}.
: 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-1: {{EDOs| 14cf, 24, 68, 130, 140, 270, 814, … }}.
: 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-4: {{EDOs| 14cfgg, 62, 96d, 198g, 342f, 882, … }}.
: Tier-5: ….

Latest revision as of 11:25, 10 June 2026

< Sequence of equal temperaments by error
2.3.5.11 subgroup
7, 12, 15, 19, 22, 31, 34, 41, 46, 53, 65, 87, 118, 152, 224, 270, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 8763, 9934, 11105, 12276, 35657, 41177, 42348, 53453, 54624, 66900, 121524, 149931, 161036, 173312, 192279, …
Exotemperaments*
5c, 5, 5e
* Monotonicity considered for {1, 3, 5, 9, 11}
2.3.7.11 subgroup
5, 9, 10, 12, 14, 17, 31, 41, 58, 63, 72, 94, 118, 130, 135, 342, 477, 670, 701, 836, 877, 1012, 1506, 1713, 1848, 4573, 4708, 6556, 11264, 17478, 19326, 22186, 24034, 28742, 35298, 59332, 64040, 94630, 123372, 158670, …
* Monotonicity considered for {1, 3, 7, 9, 11}
2.3.5.13 subgroup
5, 7, 10, 12, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270, 354, 388, 407, 441, 494, 848, 901, 1289, 1342, 2901, 3395, 3802, 4243, 4737, 5638, 6079, 11717, 15960, 17302, 22039, 23381, 35098, 45420, 57137, 80518, 137655, 143734, 189154, …
Exotemperaments*
5f
* Monotonicity considered for {1, 3, 5, 9, 13}
2.3.7.13 subgroup
5, 7, 9, 10, 17, 26, 27, 36, 77, 94, 130, 207, 234, 270, 347, 364, 441, 477, 571, 1048, 1412, 1983, 5144, 6079, 6556, 7127, 8539, 9110, 20203, 49516, 58055, 67165, 69148, 78258, 87368, 98461, 107571, 127774, 179750, 190843, …
Exotemperaments*
5f
* Monotonicity considered for {1, 3, 7, 9, 13}
2.3.5.7.11.13.19 subgroup
27e, 31, 34dh*, 38df*, 41f*, 41, 50*, 53, 58h, 72, 87, 94, 103h, 111, 121, 130, 152f, 190, 217, 224, 270, 552, 581, 684h, 742, 935, 954h, 1012, 1178, 1308, 1448, 1578, 1730, 2000, 2190, 3125, 3395, 3768, 4079, 4190, 4349, 4771, 5585, 6079, 8269, 8539, 14618, 16808, 20203, 28472, 35999, 44538, 46355, 47663, 47933, 52434, 56202, 58973, 60973, 69242, 85255, 89445, 94972, 115175, 128215, 148418, 176148, 184417, ….
Exotemperaments*
5cf, 5f, 5, 5eh, 7dfh, 7dh, 7, 8d, 8dhh, 9, 12e, 12, 12f, 14cf, 15, 19eh, 22, 22fh, 24, 26
* Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21}
2.3.5.7.11.13.19.29 subgroup
41, 53*, 58h, 72, 77, 80, 87* (94), 99ef, 103h* (113, 118f, 118), 121, 130, 140, 152fj, 159, 183, 190, 198, 217, 224, 243e, 270j*, 270, 422, 472, 494, 494h, 552, 590, 612, 684h, 692, 742, 795, 908, 935, 954hj, 1106, 1205, 1308, 1448, 1578, 1920, 2000, 2190, 3395, 4190, 4501, 5585, 6079, 6539, 7527, 8539, 10729, 11934, 12618, 13040, 14124, 14618, 16808, 20203, 28742, 33243, 36269, 40770, 46355, 47663, 54894, 56202, 60703, 60973, 67512, 80906, 87445, 94972, 115175, 128215, 148418, ….
Exotemperaments*
5cf, 5f, 5, 5ehj, 7deefhjj, 7dfhjj, 7dh, 7, 8d, 8dhhjj, 9j, 10h, 12e, 12, 12f, 12fj, 14cfjj, 15, 19eh, 22, 22fh, 24, 26, 27ej, 31j, 31, 34dhj, 38dfj, 41f
* Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} (omission of 25 is intentional)
2.3.5.7.11.13.17.23 subgroup
31, 34d, 39dfgi*, 41g* 41* 41i*, 43*, 46, 58i, 68, 72i* (72), 94, 103, 111, 121i, 130, 140, 159, 183, 217, 224, 243e, 270, 311, 373g, 422, 494, 581, 684, 742i, 764, 814, 954, 1075, 1236, 1506i, 1578, 1889, 2414, 2460, 3696, 3809, 3855, 3992, 4038, 4349, 4573, 5455, 6079, 7033, 11664, 13112, 14348, 16808, 29053, 29236, 31156, 32231, 35132, 35505, 36269, 38189, 41584, 45861, 49039, 51940, 58973, 68748, 75781, 85749, 88209, 102557, 161530, 178338, ….
Exotemperaments*
5cfi, 5, 5g, 5eg, 7dfgi, 7dgi, 7, 8d, 8di, 10e, 10, 12e, 12, 12f, 12fi, 14cf, 15, 15g, 16, 17cg, 19egi, 19, 22i, 22, 22f, 26i, 27egi, 29g
* Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 17, 23}
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, ….
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