Very high accuracy temperaments
Below are listed some very high accuracy temperaments (TE error < 0.005 cents/octave).
Rank-2 temperaments
Kwazy
Kwazy (118 & 612) tempers out the kwazy comma, [-53 10 16⟩. 7-limit extensions of the kwazy temperament include bisupermajor and gwazy.
Subgroup: 2.3.5
Comma list: [-53 10 16⟩ = 9010162353515625 / 9007199254740992
Mapping: [⟨2 1 6], ⟨0 8 -5]]
- mapping generators: ~94921875/67108864, ~1125/1024
Optimal tuning (CTE): ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433
Optimal ET sequence: 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934
Badness: 0.014091
Astro
Astro temperament tempers out the astro comma, [91 -12 -31⟩. 7-limit extensions of the astro temperament include kastro.
Subgroup: 2.3.5
Comma list: [91 -12 -31⟩
Mapping: [⟨1 5 1], ⟨0 -31 12]]
- mapping generators: ~2, ~[-38 5 13⟩
Optimal tuning (CTE): ~2 = 1\1, ~[-38 5 13⟩ = 132.1949
Optimal ET sequence: 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c
Badness: 0.062778
Gaster
Subgroup: 2.3.5
Comma list: [-70 72 -19⟩
Mapping: [⟨1 11 38], ⟨0 -19 -72]]
- mapping generators: ~2, ~[19 -19 5⟩
Optimal tuning (CTE): ~2 = 1\1, ~[19 -19 5⟩ = 594.6345
Optimal ET sequence: 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630
Badness: 0.088530
Whoosh
Whoosh (152 & 289) tempers out the whoosh comma, [37 25 -33⟩. 7-limit extensions of the whoosh temperament include whoops.
Subgroup: 2.3.5
Comma list: [37 25 -33⟩
Mapping: [⟨1 17 14], ⟨0 -33 -25]]
- mapping generators: ~2, ~864/625
Optimal tuning (CTE): ~2 = 1\1, ~864/625 = 560.546826
Optimal ET sequence: 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c
Badness: 0.025653
Monzismic
Monzismic (53 & 612) tempers out the monzisma, [54 -37 2⟩. For its 7-limit extension, see Ragismic microtemperaments #Monzismic.
Subgroup: 2.3.5
Comma list: [54 -37 2⟩ = 450359962737049600 / 450283905890997363
Mapping: [⟨1 0 -27], ⟨0 2 37]]
- mapping generators: ~2, ~[27 -18 1⟩
Optimal tuning (CTE): ~2 = 1\1, ~[27 -18 1⟩ = 950.9814
Optimal ET sequence: 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520
Badness: 0.010391
Egads
Subgroup: 2.3.5
Comma list: [-36 -52 51⟩
Mapping: [⟨1 15 16], ⟨0 -51 -52]]
- mapping generators: ~2, ~6/5
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 315.6479
Optimal ET sequence: 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127
Badness: 0.041819
Hemiegads
Subgroup: 2.3.5.7
Comma list: 78125000/78121827, [-48 0 11 8⟩
Mapping: [⟨1 15 16 -16], ⟨0 -102 -104 143]]
Optimal tuning (CTE): ~2 = 1\1, ~36756909/33554432 = 157.8240
Optimal ET sequence: 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c
Badness: 0.0124
Fortune
Fortune tempers out the fortune comma, [-107 47 14⟩. It can be described as 612 & 2513 temperament, which tempers out 78125000/78121827 (euzenius comma) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.
Subgroup: 2.3.5
Comma list: [-107 47 14⟩
Mapping: [⟨1 13 -36], ⟨0 -14 47]]
- mapping generators: ~2, 7381125/4194304
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5679
Optimal ET sequence: 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc
Badness: 0.027876
7-limit
Subgroup: 2.3.5.7
Comma list: 78125000/78121827, [-52 17 12 -1⟩
Mapping: [⟨1 13 -36 -263], ⟨0 -14 47 326]]
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5680
Optimal ET sequence: 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323
Badness: 0.023704
11-limit
Subgroup: 2.3.5.7.11
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752
Mapping: [⟨1 13 -36 -263 -573], ⟨0 -14 47 326 707]]
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5681
Optimal ET sequence: 612, 2513, 3125, 3737, 8086, 19909d
Badness: 0.036641
Senior
Senior (171 & 1171) tempers out the senior comma, [-17 62 -35⟩. 7-limit extensions of the senior temperament include seniority.
Subgroup: 2.3.5
Comma list: [-17 62 -35⟩
Mapping: [⟨1 11 19], ⟨0 -35 -62]]
- mapping generators: ~2, ~[-6 23 -13⟩
Optimal tuning (CTE): ~2 = 1\1, ~[-6 23 -13⟩ = 322.8013
Optimal ET sequence: 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680
Badness: 0.021220
Gross
Gross (118 & 1783) tempers out the gross comma, [144 -22 -47⟩, by which a stack of 22 3rd harmonics and 47 5th harmonics falls short of 144 (gross) octaves.
Subgroup: 2.3.5
Comma list: [144 -22 -47⟩
Mapping: [⟨1 45 -18], ⟨0 -47 22]]
- mapping generators: ~2, ~[-45 7 15⟩
Optimal tuning (CTE): ~2 = 1\1, ~[46 -7 -15⟩ = 91.5310
Optimal ET sequence: 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c
Badness: 0.046315
Quectismic
Subgroup: 2.3.5
Comma list: [554 -351 1⟩
Mapping: [⟨1 0 -554], ⟨0 1 351]]
- mapping generators: ~2, ~3
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9553
Optimal ET sequence: 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551
Badness: 0.372
7-limit
Subgroup: 2.3.5.7
Comma list: [33 -26 12 -7⟩, [-32 13 17 -10⟩
Mapping: [⟨1 0 -554 -945], ⟨0 1 351 598]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548
Optimal ET sequence: 306d, 359cd, 665
Badness: 2.17
Nanoquectismic
Subgroup: 2.3.5.7
Comma list: 43046721/43025920, [82 -19 -3 -16⟩
Mapping: [⟨1 0 -554 109], ⟨0 1 351 -67]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955
Optimal ET sequence: 306, 665, 1636, 2301, 2966
Badness: 0.604
Raider
Subgroup: 2.3.5
Comma list: [71 -99 37⟩
Mapping: [⟨1 28 73], ⟨0 -37 -99]]
- mapping generators: ~2, ~6561/4000
Optimal tuning (CTE): ~2 = 1\1, ~8000/6561 = 343.2961
Optimal ET sequence: 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860
Badness: 0.012867
Pirate
Subgroup: 2.3.5
Comma list: [-90 -15 49⟩
Mapping: [⟨1 43 15], ⟨0 -49 -15]]
- mapping generators: ~2, ~[-23 -4 13⟩
Optimal tuning (CTE): ~2 = 1\1, ~[24 4 -13⟩ = 185.7542
Optimal ET sequence: 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 …
Badness: 0.005656
Atomic
Atomic extensions discussed elsewhere include minutes and hafnium.
Subgroup: 2.3.5
Comma list: [161 -84 -12⟩
Mapping: [⟨12 0 161], ⟨0 1 -7]]
- mapping generators: ~[-67 35 5⟩, ~3
Optimal tuning (CTE): ~[-67 35 5⟩ = 1\12, ~3/2 = 701.9552 (or ~32805/32768 = 1.9552)
Optimal ET sequence: 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032
Badness: 0.003803
7-limit
Subgroup: 2.3.5.7
Comma list: 250047/250000, [-55 30 2 1⟩
Mapping: [⟨12 0 161 338], ⟨0 1 -7 -16]]
Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
Optimal ET sequence: 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd
Badness: 0.045818
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 151263/151250, 184549376/184528125
Mapping: [⟨12 0 161 338 517], ⟨0 1 -7 -16 -25]]
Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
Optimal ET sequence: 12, …, 600e, 612, 1236, 1848
Badness: 0.016029
Niobium
Niobium can be viewed as the 41st-octave counterpart of atomic.
Subgroup: 2.3.5
Comma list: [-875 492 41⟩
Mapping: [⟨41 0 875], ⟨0 1 -12]]
- mapping generators: ~[-64 36 3⟩, ~3
Optimal tuning (CTE): ~[-64 36 3⟩ = 1\41, ~3/2 = 701.9536
Optimal ET sequence: 41, …, 2378, 2419, 2460, 4961, 7421, 17302, 24723, 66748, 91471, 116194
Badness: 5.57
Revopent
Described as the 1848 & 3125 temperament, revopent is named because the year 1848 is known for revolutions while 3125 is 5 to the 5.
Subgroup: 2.3.5.7
Comma list: 645700815/645657712, [51 -13 -1 -10⟩
Mapping: [⟨1 12 -155 5], ⟨0 -19 287 -4]]
Optimal tuning (CTE): ~24576/16807 = 657.792
Optimal ET sequence: 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d
Badness: 0.263
Francium
Francium tempers out the euzenius comma and the 87-126/125 comma in the 7-limit, hereby dividing the octave into 87 parts.
Subgroup: 2.3.5.7
Comma list: 78125000/78121827, [152 -112 -41 43⟩
Mapping: [⟨87 1 -106 -406], ⟨0 4 9 19]]
- mapping generators: ~126/125, ~[37 -25 -12 11⟩
Optimal tuning (CTE): ~126/125 = 1\87, ~[37 -25 -12 11⟩ = 472.043
Optimal ET sequence: 783, …, 8178, 8961, 9744, 10527, 20271, 30798
Badness: 0.762
11-limit
Subgroup: 2.3.5.7.11
Comma list: 78125000/78121827, 199297406/199290375, [74 -17 -5 -4 -7⟩
Mapping: [⟨87 1 -106 -406 1225], ⟨0 4 9 19 -27]]
- mapping generators: ~126/125, ~[35 1 -6 -3 -4⟩
Optimal tuning (CTE): ~126/125 = 1\87, ~[35 1 -6 -3 -4⟩ = 472.043
Optimal ET sequence: 783, 8178, 8961, 9744, 28449de, 38193bcde, 47937bcdee
Badness: 0.352
Rank-3 temperaments
Sasaquinbizo-atriyo (171 & 1106 & 3566)
Subgroup: 2.3.5.7
Comma list: 282475249000/282429536481
Mapping: [⟨1 0 9 -3], ⟨0 1 8 0], ⟨0 0 10 -3]]
Optimal ET sequence: 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737
Badness: 0.000281
Saquadtrizo-asepgu (171 & 2019 & 3566)
Subgroup: 2.3.5.7
Comma list: 13841287201/13839609375
Mapping: [⟨1 0 0 0], ⟨0 1 7 5], ⟨0 0 12 7]]
Optimal ET sequence: 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737
Badness: 0.000135
Laquinzo-aquadquadgu (171 & 441 & 4973)
Subgroup: 2.3.5.7
Comma list: [-7 19 -16 5⟩ = 19534128475869/19531250000000
Mapping: [⟨1 0 3 11], ⟨0 1 4 9], ⟨0 0 5 16]]
Optimal ET sequence: 171, 270, 441, 612, 783, 4802, 4973, 5144
Badness: 0.000297
Laleruyo (171 & 1547 & 3125)
Subgroup: 2.3.5.7
Comma list: 3955078125/3954653486
Mapping: [⟨1 3 0 1], ⟨0 11 0 4], ⟨0 0 1 1]]
Optimal ET sequence: 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296
Badness: 0.000114
Starscape or Latritriru-ayo
Subgroup: 2.3.5.7
Comma list: 645700815/645657712
Mapping: [⟨1 0 4 0], ⟨0 1 1 2], ⟨0 0 9 1]]
Optimal ET sequence: 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634, …
Badness: 0.000545
Sepbizo-asegu (171 & 3566 & 4973)
Subgroup: 2.3.5.7
Comma list: 6104007655641/6103515625000
Mapping: [⟨1 0 13 16], ⟨0 1 10 12], ⟨0 0 14 17]]
Optimal ET sequence: 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710
Badness: 0.000223
Euzenius or Sabiruquinyo
Subgroup: 2.3.5.7
Comma list: 78125000/78121827
Mapping: [⟨1 0 0 -5], ⟨0 2 0 -13], ⟨0 0 1 5]]
Optimal ET sequence: 171, 441, 612, 1730, 1901, 2072, 2171, 2342, 2513, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 16336, 16507, 23027, 26152, 32843, 42659, 58995, 68811, 101654, …
Badness: 0.000205
Quinla-zoyoyo (612 & 1848 & 3125)
Subgroup: 2.3.5.7
Comma list: [-55 30 2 1⟩
Mapping: [⟨1 0 0 55], ⟨0 1 0 -30], ⟨0 0 1 -2]]
Optimal ET sequence: 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795
Badness: 0.000366
Satwethezo-atritribigu (171 & 1547 & 4973)
Subgroup: 2.3.5.7
Comma list: [1 -15 -18 23⟩
Mapping: [⟨1 0 9 7], ⟨0 1 3 3], ⟨0 0 23 18]]
Optimal ET sequence: 171, 1547, 4802, 4973, 5144, 6520, 6691, 6862, 11493, 11664, 11835, 18184, 18355, 18526, 25046, 30019, 30190, 48545
Badness: 0.000114
Trisa-quinbiru-agu (441 & 1848 & 3125)
Subgroup: 2.3.5.7
Comma list: [51 -13 -1 -10⟩
Mapping: [⟨1 0 1 5], ⟨0 1 7 -2], ⟨0 0 10 -1]]
Optimal ET sequence: 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499
Badness: 0.000263
Quinbisa-quinbizo-alegu (21480 & 33144 & 103169)
Subgroup: 2.3.5.7
Comma list: [110 -71 -11 10⟩
Mapping: [⟨1 0 0 -11], ⟨0 1 -1 6], ⟨0 0 10 11]]
Optimal ET sequence: 11664, 21480, 33144, 36881, 48545, 54624, 70025, 81689, 91505, 103169