Very high accuracy temperaments

Below are listed some very high accuracy temperaments (TE error < 0.005 cents/octave).

Rank-2 temperaments

Kwazy

Kwazy temperament (118&612) tempers out the kwazy comma, [-53 10 16⟩ (quadla-quadquadyo). 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]]

POTE generator: ~1125/1024 = 162.742626

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⟩ (tribisa-thiwegu). 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]]

POTE generator: ~296630859375/274877906944 = 132.194668

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

Main article: Gaster temperament

See also: Mirkwai clan § Gaster

Subgroup: 2.3.5

Comma: [-70 72 -19⟩

Mapping: [⟨1 11 38], ⟨0 -19 -72]]

POTE generator: ~1638400000/1162261467 = 594.634565

Optimal ET sequence: 111, 224, 335, 559, 1342, 1901, 3243

Badness: 0.088530

Whoosh

Whoosh temperament (152&289) tempers out the whoosh comma, [37 25 -33⟩ (saletrigu). 7-limit extensions of the whoosh temperament include whoops.

Subgroup: 2.3.5

Comma list: [37 25 -33⟩

Mapping: [⟨1 -16 -11], ⟨0 33 25]]

POTE generator: ~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]]

POTE generator: ~387420489/335544320 = 249.018406

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]]

Optimal tuning (CTE): ~6/5 = 315.647919

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, 281484423828125/281474976710656

Mapping: [⟨1 15 16 -16], ⟨0 -102 -104 143]]

Optimal tuning (CTE): ~36756909/33554432 = 157.8240

Vals: 38, 403, 441, 479, 844, 2684, 3125, 3566, 4007, 6691, ...

Fortune

Fortune temperament tempers out the fortune comma, [-107 47 14⟩ (tritrila-sepbiyo). 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 -1 11], ⟨0 14 -47]]

POTE generator: ~8388608/7381125 = 221.567899

Optimal ET sequence: 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151

Badness: 0.027876

7-limit

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [-52 17 12 -1⟩

Mapping: [⟨1 -1 11 63], ⟨0 14 -47 -326]]

POTE generator: ~8388608/7381125 = 221.568031

Optimal GPV sequence: Template:Val list

Badness: 0.023704

11-limit

Subgroup: 2.3.5.7.11

Comma list: 151263/151250, 21437500/21434787, 369140625/369098752

Mapping: [⟨1 -1 11 63 134], ⟨0 14 -47 -326 -707]]

POTE generator: ~8388608/7381125 = 221.568139

Optimal GPV sequence: Template:Val list

Badness: 0.036641

Senior

Senior temperament (171&1171) tempers out the senior comma, [-17 62 -35⟩ (quadla-sepquingu). 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]]

POTE generator: ~94143178827/78125000000 = 322.801358

Optimal ET sequence: 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680

Badness: 0.021220

Gross

Gross temperament (118&1783) tempers out the gross comma, [144 -22 -47⟩, by which a stack of 22 tritaves (third harmonic) and 47 pentaves (fifth harmonic) falls short of 144 (gross) octaves.

Subgroup: 2.3.5

Comma list: [144 -22 -47⟩

Mapping: [⟨1 -2 4], ⟨0 47 -22]]

POTE generator: ~1953125000000000/1853020188851841 = 91.531001

Optimal ET sequence: 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467

Badness: 0.046315

Raider

See also: Syntonic-chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37⟩

Mapping: [⟨1 -9 -26], ⟨0 37 99]]

POTE generator: ~8000/6561 = 343.296098

Optimal ET sequence: 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355

Badness: 0.012867

Pirate

Subgroup: 2.3.5

Comma list: [-90 -15 49⟩

Mapping: [⟨1 -6 0], ⟨0 49 15]]

POTE generator: ~[24 4 -13⟩ = 185.754179

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.

See also: Kirnberger's atom

Subgroup: 2.3.5

Comma list: [161 -84 -12⟩

Mapping: [⟨12 0 161], ⟨0 1 -7]]

POTE generator: ~32805/32768 = 1.955162

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]]

POTE generator: ~32805/32768 = 1.949608

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~32805/32768 = 1.947988

Optimal GPV sequence: Template:Val list

Badness: 0.016029

Niobium

Can be viewed as the 41st-octave counterpart of atomic.

Subgroup: 2.3.5

Comma list: [-875 492 41⟩

Mapping: [⟨41 41 383], ⟨0 -1 12]]

Mapping generators: ~18761829412124890125/18446744073709551616, ~3/2

Optimal tuning (CTE): ~3/2 = 701.9536

Vals: 41, 2460, 4961, 7421, 9881, ...

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]]

Vals: Template:Val list

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]]

Vals: Template:Val list

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]]

Vals: Template:Val list

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]]

Vals: Template:Val list

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]]

Template:Val list …

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]]

Vals: Template:Val list

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]]

Template:Val list …

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]]

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]]

Vals: Template:Val list

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]]

Vals: Template:Val list

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]]

Vals: Template:Val list