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

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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, ….

@@ Line 1: / Line 1: @@
-For a method to iterate thru all GPVs, see [[Patent val/Properties]].
+{{Breadcrumb|Sequence of equal temperaments by error}}
-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.
+; 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, … }}
-GPVs starting with 0 are not accounted as they would meddle with the results. For readability reasons, results are shown in the range of 5 and 16808.
+: {{Databox|Exotemperaments*| 5c, 5, 5e }}
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 9, 11}
-; 3-limit:
+; 2.3.7.11 subgroup
-: Tier-0: {{EDOs| 5, 12, 41, 53, 306, 665, 15601 }}…
+: {{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, … }}
-: 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:
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 7, 9, 11}
-: 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:
+; 2.3.5.13 subgroup
-: Tier-0: {{EDOs| 5, 10, 12, 19, 31, 72, 99, 171, 3125, 11664 }}…
+: {{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, … }}
-: 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:
+: {{Databox|Exotemperaments*| 5f }}
-: Tier-0: {{EDOs| 5, 7, 8d, 9, 12, 22, 31, 72, 270, 342, 1848, 6079, 6691 }}…
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 9, 13}
-: 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:
+; 2.3.7.13 subgroup
-: 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 }}…
+: {{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, … }}
-: 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:
+: {{Databox|Exotemperaments*| '''5f''' }}
-: Tier-0: {{EDOs| 5eg, 8d, 10, 19eg, 22, 26, 27eg, 31, 46, 72, 494, 1506, 3395, 7033, 16808, 20203 }}…
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 7, 9, 13}
-: 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, 20203 }}…
-: 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:
+; 2.3.5.7.11.13.19 subgroup
-: Tier-0: {{EDOs| 14cf, 15g, 26i, 29g, 46, 94, 190g, 193, 217, 270, 282, 311, 581, 1578, 2460, 8269, 11664, 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''', … }}.
-: 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:
+: {{Databox|Exotemperaments*| 5cf, 5f, 5, '''5eh''', 7dfh, 7dh, '''7''', 8d, 8dhh, '''9''', 12e, 12, 12f, 14cf, '''15''', 19eh, 22, 22fh, 24, 26 }}
-: Tier-0: {{EDOs| 26i, 46, 58hi, 80, 140, 159, 183, 243e, 311, 954hj, 1578, 16808 }}…
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21}
-; 31-limit:
+; 2.3.5.7.11.13.19.29 subgroup
-: Tier-0: {{EDOs| 46, 99efk, 103h, 183, 190g, 217, 311, 954hj, 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''', … }}.
-; 37-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 }}
-: Tier-0: {{EDOs| 80k, 99efk, 103hl, 270, 270i, 311, 1395, 1920, 4501, 16808 }}…
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} (omission of 25 is intentional)
-; 2.3.5.11 subgroup:
+; 2.3.5.7.11.13.17.23 subgroup
-: Tier-0: {{EDOs| 7, 65, 152, 342, 494, 1171, 11105, 12276 }}…
+: {{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, … }}.
-: 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:
+: {{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''' }}
-: Tier-0: {{EDOs| 10e, 10, 34d, 58, 72, 270, 2684, 13112 }}…
+: <nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 17, 23}
-: 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 }}…
+; sqrt (2).sqrt (3).5.7.11.13 subgroup
-: Tier-2: {{EDOs| 38df, 212g, 400, 2414, 3638, 12842 }}…
+: Relative: {{EDOs| 10e, 10, 34d, 58, 72, 270, 2684, 13112, … }}.
-: Tier-3: {{EDOs| 10eg, 14c, 34, 58g, 130g, 328, 414, 684, 11170, 12428, 13112, 13170 }}…
-: Tier-4: {{EDOs| 14cfgg, 62, 96d, 198g, 342f, 882 }}…
+; sqrt (2).sqrt (3).5.7.11.13.17 subgroup
-: Tier-5: {{EDOs|  }}…
+: 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, … }}.

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

Latest revision as of 11:25, 10 June 2026

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