View source for User:FloraC/Mathematically defined ET tier list

The '''tier''' of an [[equal temperament]] is defined as the number of [[GPV]]s 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 otherwise meddle with the results. GPVs starting with 1, 2, 3, and 4 may be regarded as trivial, and 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: ….