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| 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.
| | {{Breadcrumb|Sequence of equal temperaments by error}} |
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| 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, … }} |
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| ; 3-limit
| | : {{Databox|Exotemperaments*| 5c, 5, 5e }} |
| : Relative: {{EDOs| 5, 12, 41, 53, 306, 665, 15601, …}}
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 9, 11} |
| :: (continued) {{EDOs| 31867, 79335, 111202, 190537, … }}. | |
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|
| ; 5-limit | | ; 2.3.7.11 subgroup |
| : Relative: {{EDOs| 7, 12, 19, 34, 53, 118, 441, 559, 612, 1171, 1783, 2513, 4296 }}…
| | : {{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, … }} |
| :: (continued) {{EDOs| 73709, 78005 }}…
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|
| ; 7-limit
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 7, 9, 11} |
| : Relative: {{EDOs| 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664 }}…
| |
| :: (continued) {{EDOs| 18355, 84814, 103169 }}…
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|
| ; 11-limit | | ; 2.3.5.13 subgroup |
| : Relative: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691 }}…
| | : {{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, … }} |
| :: (continued) {{EDOs| 40006, 54624, 121524 }}…
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| ; 13-limit
| | : {{Databox|Exotemperaments*| 5f }} |
| : Relative: {{EDOs| 5e, 7, 8d, 9, 12f, 19e, 19, 27e, 31, 41, 53, 58, 72, 224, 270, 494, 1506, 2190, 2684, 5585, 6079, 14618, … }}
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 9, 13} |
| :: (continued) {{EDOs| 73591, 81860, 87939, 96478, … }}. | |
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|
| ; 17-limit | | ; 2.3.7.13 subgroup |
| : Relative: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, … }}
| | : {{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, … }} |
| :: (continued) {{EDOs| 20203, 102557, … }}.
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|
| ; 19-limit
| | : {{Databox|Exotemperaments*| '''5f''' }} |
| : 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, … }}
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 7, 9, 13} |
| :: (continued) {{EDOs| 20203, 128215, … }}. | |
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| ; 23-limit
| | ---- |
| : Relative: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 16808 }}…
| |
| :: (continued) {{EDOs| 148418, … }}.
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| ; 29-limit | | ; 2.3.5.7.11.13.19 subgroup |
| : Relative: {{EDOs| 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808 }}…
| | : {{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''', … }}. |
| :: (continued) {{EDOs| 148418 }}…
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| ; 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| 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 4501, 16808 }}…
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21} |
| :: (continued) {{EDOs| 148418 }}… | |
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| ; 37-limit | | ; 2.3.5.7.11.13.19.29 subgroup |
| : Relative: {{EDOs| 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808 }}…
| | : {{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''', … }}. |
| :: (continued) {{EDOs| 26742, 58973, 65322, 67242, 83096, 148418 }}…
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| ; 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| 80k, 270, 270i, 311, 1920, 4501m, 7361, 14348, 15854j }}…
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} (omission of 25 is intentional) |
| :: (continued) {{EDOs| 17461, 29053, 34691, 58973, 65322, 95524, 148418 }}… | |
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| ; 43-limit | | ; 2.3.5.7.11.13.17.23 subgroup |
| : Relative: {{EDOs| 80k, 99efkmn, 121iklm, 270i, 311n, 311, 1600, 1920, 4501m, 7361, 14348 }}…
| | : {{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, … }}. |
| :: (continued) {{EDOs| 20567, 58973, 109590, 165226 }}…
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| ; 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| 62, 99efkmn, 121iklmo, 198gln, 248h, 270, 270i, 311, 311o, 422l, 1178, 1385k, 1817hi, 1920, 2742klm, 4380, 7361o, 8893, 14348 }}…
| | : <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 17, 23} |
| :: (continued) {{EDOs| 23270, 58973, 95524, 109590 }}… | |
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|
| ---- | | ---- |
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| ; 2.3.5.11 subgroup
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| : Relative: {{EDOs| 7, 65, 152, 342, 494, 1171, 11105, 12276, … }}
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|
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| ; 2.3.5.7.11.13.19 subgroup
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| : Relative: {{EDOs| 5eh, 7, 9, 15, 27e, 41, 53, 72, 130, 152f, 270, 2190, 6079, 8269, 8539, … }}
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| :: (continued) {{EDOs| 115175, 148418, 176148, 184417, … }}.
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|
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| ; 2.3.5.7.11.13.17.23 subgroup
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| : Relative: {{EDOs| 5eg, 8di, 10, 22f, 26i, 29g, 31, 46, 72i, 140, 183, 311, 494, 764, 1578, 2460, 7033, 16808, … }}
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| :: (continued) {{EDOs| 102557, … }}.
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| ; sqrt (2).sqrt (3).5.7.11.13 subgroup | | ; sqrt (2).sqrt (3).5.7.11.13 subgroup |
| Line 79: |
Line 53: |
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| ; 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, … }}. |
| :: (continued) ….
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| == A mathematically defined tier list ==
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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.
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| 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.
| |
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| ; 3-limit
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 7, 17, 24, 29, 94, 106, 253, 359, 971, 1330, 14936, 16266, … }}.
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| : Tier-2: {{EDOs| 10, 147, 159, 200, 612, 1636, 1995, 14271, … }}.
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| : Tier-3: {{EDOs| 65, 2301, 2660, 13606, … }}.
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| : Tier-4: {{EDOs| 8, 9, 19, 36, 82, 412, 1024, 2966, 3325, 12941, … }}.
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| : Tier-5: {{EDOs| 6b, 6, 14, 22, 212, 559, 718, 1277, 3631, 3990, 12276, … }}.
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| ; 5-limit
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 5, 171, 730, 6809, 8592, 16572, … }}.
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| : Tier-2: {{EDOs| 15, 31, 99, 106, 1342, 3684, 11105, 12276, … }}.
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| : Tier-3: {{EDOs| 10, 22, 46, 152, 236, 270, 289, 3125, 5467, 7980, … }}.
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| : Tier-4: {{EDOs| 8, 9, 41, 224, 323, 494, 2954, … }}.
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| : Tier-5: {{EDOs| 5c, 24, 27, 87, 205, 388, 1224, 1901, 9763, 12888, … }}.
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| ; 7-limit
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 41, 270, 441, 2954, 6691, 14789, … }}.
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| : Tier-2: {{EDOs| 7d, 8d, 9, 15, 22, 53, 130, 342, 612, 2019, 3566, … }}.
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| : Tier-3: {{EDOs| 7, 27, 140, 1848, 3296, 11835, … }}.
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| : Tier-4: {{EDOs| 14c, 46, 152, 301, 323, 3395, 6520, … }}.
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| : Tier-5: {{EDOs| 5, 58, 118, 494, 513, 783, 1106, 3737, 4973, 5144, 6250, 9816, 15230, … }}.
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| ; 11-limit
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| : Tier-1: {{EDOs| 7, 14c, 15, 27e, 41, 152, 1578, 8269, 14618, 14960, … }}.
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| : Tier-2: {{EDOs| 19, 46, 58, 118, 224, 494, 612, 5585, 12276, 12770, … }}.
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| : Tier-3: {{EDOs| 5e, 10, 19e, 130, 1106, 1236, 2190, 6349, 7927, … }}.
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| : Tier-4: {{EDOs| 53, 80, 87, 311, 764, 836, 2684, 4843, 6421, … }}.
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| : Tier-5: {{EDOs| 99e, 183, 400, 472, 684, 742, 8539, … }}.
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| ; 13-limit:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 5, 7d, 15, 87, 130, 8269, 8539, … }}.
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| : Tier-2: {{EDOs| 10, 34d, 46, 198, 311, 684, 954, 1236, … }}.
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| : Tier-3: {{EDOs| 7df, 29, 103, 581, 1178, 3395, 7033, … }}.
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| : Tier-4: {{EDOs| 5f, 10e, 14cf, 111, 121, 140, 190, 764, 1448, 2460, 11664, 13112, … }}.
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| : Tier-5: {{EDOs| 22, 26, 152f, 183, 373, 422, 742, 1012, 2954, … }}.
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| ; 17-limit:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 5g, 7dg, 7, 9, 12f, 41, 58, 111, 140, 183, 270, 581, 742, 764, 954, 1178, 1236, 1578, 2460, 8539, 14348, … }}.
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| : Tier-2: {{EDOs| 7dfg, 9g, 22f, 29g, 94, 103, 121, 224, 311, 1817, 5855, 8269, … }}.
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| : Tier-3: {{EDOs| 14cf, 15g, 17cg, 50, 53, 87, 217, 422, 460, 814, 2000, 2684, 4573, 10428, … }}.
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| : Tier-4: {{EDOs| 7g, 12, 34d, 400, 6079, … }}.
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| : Tier-5: {{EDOs| 10e, 15, 282, 354, 1889, 9033, … }}.
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| ; 19-limit:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 7dfgh, 10h, 14cf, 15g, 24, 29g, 31, 41, 53, 58h, 94, 111, 311, … }}.
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| : Tier-2: {{EDOs| 7dgh, 12, 50, 103h, 121, 140, 183, 354, 422, 11934, … }}.
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| : Tier-3: {{EDOs| 22fh, 72h, 99ef, 190g, 193, 243e, 460, 935, 6961, 11664, … }}.
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| : Tier-4: {{EDOs| 5cf, 5g, 5eg, 10, 15, 34dh, 118f, 282, 954h, 1889, 4349, 6079, … }}.
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| : Tier-5: {{EDOs| 22, 43, 80, 87, 130, 224, 373g, 400, 643, 3178, 3920, 4501, … }}.
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| ; 23-limit:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 15, 27egi, 31, 50, 72i, 130, 183, 243e, 373g, 422, 954h, 1889, … }}.
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| : Tier-2: {{EDOs| 22fh, 34dh, 53, 72, 80, 111, 140, 152fg, 159, 525, 935, 1920, 2000, 3178, 8539, … }}.
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| : Tier-3: {{EDOs| 19eghi, 58hi, 99ef, 121i, 624, 814, 4038, 4349, 6079, 6269, … }}.
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| : Tier-4: {{EDOs| 27eg, 68, 718, 742i, 14348, … }}.
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| : Tier-5: {{EDOs| 19egh, 19, 24, 41i, 43, 3395i, 7315, … }}.
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| | |
| ----
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| | |
| ; 2.3.5.11 subgroup:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 5, 5e, 8, 9, 12e, 12, 15, 22, 31, 53, 72, 87, 118, 270, 677, 1665, 2342, 9934, … }}.
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| : Tier-2: {{EDOs| 19, 34, 183, 407, 559, 5520, 6691, … }}.
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| : Tier-3: {{EDOs| 14, 24, 335, 612, 829, 901, 1513, 2836, 3513, 13347, … }}.
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| : Tier-4: {{EDOs| 41, 46, 159, 224, 2684, 7862, 8763, … }}.
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| : Tier-5: {{EDOs| 10, 19e, 137, 217, 836, 1019, 2072, 3855, … }}.
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| | |
| ; sqrt (2).sqrt (3).5.7.11.13 subgroup:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 14cf, 130, 954, 10428, 16808, … }}.
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| : Tier-2: {{EDOs| 24, 38df, 68, 198, 742, 814, 2000, 2954, 14124, … }}.
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| : Tier-3: {{EDOs| 62, 140, 612, 1012, 1730, … }}.
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| : Tier-4: {{EDOs| 14c, 34, 96d, 1084, 3696, 3768, 10158, 10356, 12158, 13040, … }}.
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| : Tier-5: {{EDOs| 10f, 24d, 202, 212, 342f, 13854, … }}.
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| | |
| ; sqrt (2).sqrt (3).5.7.11.13.17 subgroup:
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| : Tier-0: see above.
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| : Tier-1: {{EDOs| 14cf, 24, 68, 130, 140, 270, 814, … }}.
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| : Tier-2: {{EDOs| 38df, 212g, 400, 2414, 3638, 12842, … }}.
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| : Tier-3: {{EDOs| 10eg, 14c, 34, 58g, 130g, 328, 414, 684, 11170, 12428, 13112, 13170, … }}.
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| : Tier-4: {{EDOs| 14cfgg, 62, 96d, 198g, 342f, 882, … }}.
| |
| : Tier-5: ….
| |