Very high accuracy temperaments

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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 tunings:

  • CTE: ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433
  • POTE: ~94921875/67108864 = 1\2, ~1125/1024 = 162.742626

Optimal ET sequence22, 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 tunings:

  • CTE: ~2 = 1\1, ~[-38 5 13 = 132.1949
  • POTE: ~2 = 1\1, ~[-38 5 13 = 132.194668

Optimal ET sequence9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c

Badness: 0.062778

Gaster

See also: Mirkwai clan #Gaster

Subgroup: 2.3.5

Comma list: [-70 72 -19

Mapping[1 11 38], 0 -19 -72]]

mapping generators: ~2, ~[19 -19 5

Optimal tunings:

  • CTE: ~2 = 1\1, ~[19 -19 5 = 594.6345
  • POTE: ~2 = 1\1, ~[19 -19 5 = 594.634565

Optimal ET sequence111, 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 tunings:

  • CTE: ~2 = 1\1, ~864/625 = 560.5470
  • POTE: ~2 = 1\1, ~864/625 = 560.546826

Optimal ET sequence15, 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 tunings:

  • CTE: ~2 = 1\1, ~[27 -18 1 = 950.9814
  • POTE: ~2 = 1\1, ~[27 -18 1 = 950.981594

Optimal ET sequence53, 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 tunings:

  • CTE: ~2 = 1\1, ~6/5 = 315.6479
  • POTE: ~2 = 1\1, ~6/5 = 315.647919

Optimal ET sequence19, 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 tunings:

  • CTE: ~2 = 1\1, ~36756909/33554432 = 157.8240

Optimal ET sequence38, 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 tunings:

  • CTE: ~2 = 1\1, ~8388608/7381125 = 221.5679
  • POTE: ~2 = 1\1, ~8388608/7381125 = 221.567899

Optimal ET sequence65, 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 tunings:

  • CTE: ~2 = 1\1, ~8388608/7381125 = 221.5680
  • POTE: ~2 = 1\1, ~8388608/7381125 = 221.568031

Optimal ET sequence612, 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 tunings:

  • CTE: ~2 = 1\1, ~8388608/7381125 = 221.5681
  • POTE: ~2 = 1\1, ~8388608/7381125 = 221.568139

Optimal ET sequence612, 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 tunings:

  • CTE: ~2 = 1\1, ~[-6 23 -13 = 322.8013
  • POTE: ~2 = 1\1, ~[-6 23 -13 = 322.801358

Optimal ET sequence171, 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 tunings:

  • CTE: ~2 = 1\1, ~[46 -7 -15 = 91.5310
  • POTE: ~2 = 1\1, ~[46 -7 -15 = 91.531001

Optimal ET sequence13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c

Badness: 0.046315

Music

Quectismic

See also: Quectisma

Subgroup: 2.3.5

Comma list: [554 -351 1

Mapping[1 0 -554], 0 1 351]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.9553

Optimal ET sequence306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551

Badness: 0.372

Nanoquectismic

Subgroup: 2.3.5.7

Comma list: 43046721/43025920, [82 -19 -3 -16

Mapping: [1 0 -554 109], 0 1 351 -67]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.955

Optimal ET sequence306, 665, 1636, 2301, 2966

Badness: 0.604

Conanoquectismic

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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.9548

Optimal ET sequence306d, 359cd, 665

Badness: 2.17

Raider

See also: Syntonic-chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37

Mapping[1 28 73], 0 -37 -99]]

mapping generators: ~2, ~6561/4000

Optimal tunings:

  • CTE: ~2 = 1\1, ~8000/6561 = 343.2961
  • POTE: ~2 = 1\1, ~8000/6561 = 343.296098

Optimal ET sequence388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860

Badness: 0.012867

Pirate

See also: Hemfiness temperaments #Piracy

Subgroup: 2.3.5

Comma list: [-90 -15 49

Mapping[1 43 15], 0 -49 -15]]

Mapping generators: ~2, ~[-23 -4 13

Optimal tunings:

  • CTE: ~2 = 1\1, ~[24 4 -13 = 185.754186
  • POTE: ~2 = 1\1, ~[24 4 -13 = 185.754179

Optimal ET sequence84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832

Badness: 0.005656

Atomic

See also: Kirnberger's atom

Atomic extensions discussed elsewhere include minutes and hafnium.

Atomic does not temper out the schisma, so 3/2 is one schisma sharp of its 12edo value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the Pythagorean comma is twelve schismas, and hence 81/80, the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.

Subgroup: 2.3.5

Comma list: [161 -84 -12

Mapping[12 0 161], 0 1 -7]]

mapping generators: ~[-67 35 5, ~3

Optimal tunings:

  • CTE: ~[-67 35 5 = 1\12, ~3/2 = 701.9552 (~32805/32768 = 1.9552)
  • POTE: ~[-67 35 5 = 1\12, ~3/2 = 701.955162 (~32805/32768 = 1.955162)

Optimal ET sequence12, …, 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 tunings:

  • CTE: ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
  • POTE: ~30375/28672 = 1\12, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)

Optimal ET sequence12, …, 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 tunings:

  • CTE: ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
  • POTE: ~30375/28672 = 1\12, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)

Optimal ET sequence12, …, 600e, 612, 1236, 1848

Badness: 0.016029

Revopentic (2.3.7)

See also: Revopentisma

Subgroup: 2.3.7

Comma list: 11399736556781568/11398895185373143

Sval mapping[1 12 5], 0 -19 -4]]

mapping generators: ~2 = 1\1, 24576/16807

Optimal tunings:

  • CTE: ~2 = 1\1, ~24576/16807 = 657.792

Supporting ETs: 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794

Badness (Sintel's TE logflat): 0.070

Revopent

Revopent can be described as the 1848 & 3125 temperament.

Subgroup: 2.3.5.7

Comma list: 645700815/645657712, [51 -13 -1 -10

Mapping: [1 12 -155 5], 0 -19 287 -4]]

Optimal tunings:

  • CTE: ~24576/16807 = 657.792

Optimal ET sequence135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d

Badness: 0.263

Nanic

See also: Nanisma

Subgroup: 2.3.7

Comma list: [109 -67 -1

Mapping[1 0 109], 0 1 -67]]

sval mapping generators: ~2, ~3

Optimal tunings:

  • POTE: ~2 = 1\1, ~3/2 = 701.9578

Optimal ET sequence53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733

Badness: 0.0138

Icarus

Named after a very distant star. Curiously, the names of two even more distant stars coincide with named temperaments, those being godzilla and mothra.

Subgroup: 3.5.7

Comma list: [-32 -64 71

Mapping[1 35 32], 0 -71 -64]]

Optimal tunings:

  • CTE: ~3 = 1\1<3/1>, ~[5 9 -10 = 898.340

Optimal ET sequence: b163cd, b199c, b235, b271, b4643, b4914, …, b8708, b8979, b18229, b27208, b99853, b127061, b154269, b181477, b208685, b390162

Badness: 0.493 × 10-3

Rank-3 temperaments

Sasaquinbizo-atriyo (171 & 1106 & 3566)

Subgroup: 2.3.5.7

Comma list: [3 -24 3 10 = 282475249000/282429536481

Mapping[1 0 9 -3], 0 1 8 0], 0 0 10 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9624, ~6561/3430 = 1122.9387

Optimal ET sequence171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069

Badness: 0.281 × 10-3

Saquadtrizo-asepgu (171 & 2019 & 3566)

Subgroup: 2.3.5.7

Comma list: [0 -11 -7 12 = 13841287201/13839609375

Mapping[1 0 0 0], 0 1 7 5], 0 0 -12 -7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9584, ~16807/10125 = 877.2825

Optimal ET sequence171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d

Badness: 0.135 × 10-3

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.9501, ~250/189 = 484.2956

Optimal ET sequence171, 441, 612, 2696, 2795, 2867, 2966, 3137, 3308, 3407, 3479, 3578, 3749, 3920, 4091, 4190, 4361, 4532, 4703, 4802, 4973, 5144, 10117, 10558, 15531, 15702, 52079, 62637, 67781, 78339

Badness: 0.297 × 10-3

Laleruyo (171 & 1547 & 3125)

Subgroup: 2.3.5.7

Comma list: [-1 4 11 -11 = 3955078125/3954653486

Mapping[1 3 0 1], 0 -11 0 -4], 0 0 1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~375/343 = 154.3678, ~5/4 = 386.3070

Optimal ET sequence171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987

Badness: 0.114 × 10-3

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.9514, ~9/7 = 435.0709

Optimal ET sequence171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634

Badness: 0.545 × 10-3

Sepbizo-asegu (171 & 3566 & 4973)

Subgroup: 2.3.5.7

Comma list: [-3 2 -17 14 = 6104007655641/6103515625000

Mapping[1 0 13 16], 0 1 10 12], 0 0 -14 -17]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548, ~31250/16807 = 1073.8021

Optimal ET sequence171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653

Badness: 0.223 × 10-3

Euzenius or Sabiruquinyo

See also: Euzenius comma

Subgroup: 2.3.5.7

Comma list: 78125000/78121827

Mapping[1 1 0 -5], 0 2 0 -13], 0 0 1 5]]

Optimal tuning (CTE): ~2 = 1\1, ~6250/5103 = 350.9787, ~5/4 = 386.3099

Optimal ET sequence171, 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: 0.205 × 10-3

Nommismic or Quinla-zoyoyo

See also: Nommisma

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.9516, ~5/4 = 386.3132

Optimal ET sequence612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, …

Badness: 0.366 × 10-3

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.9551, ~56953125/40353607 = 596.5022

Optimal ET sequence171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, …

Badness: 0.114 × 10-3

Technologismic or 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.9553, ~63/32 = 1172.7374

Optimal ET sequence441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, …

Badness: 0.263 × 10-3

Quinbisa-quinbizo-alegu (1848 & 11664 & 21480)

Subgroup: 2.3.5.7

Comma list: [110 -71 -11 10

Mapping[1 0 0 -11], 0 1 9 17], 0 0 -10 -11]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955, ~32805/28672 = 233.128

Optimal ET sequence1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, …

Badness: 0.122 × 10-3