Very high accuracy temperaments: Difference between revisions

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Line 7: Line 7:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992
[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992


Mapping: [{{val| 2 1 6 }}, {{val| 0 8 -5 }}]
[[Mapping]]: [{{val| 2 1 6 }}, {{val| 0 8 -5 }}]


POTE generator: ~1125/1024 = 162.742626
[[POTE generator]]: ~1125/1024 = 162.742626


Vals: {{Val list| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}
{{Optimal GPV sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}


Badness: 0.014091
[[Badness]]: 0.014091


== Astro ==
== Astro ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 91 -12 -31 }}
[[Comma list]]: {{monzo| 91 -12 -31 }}


Mapping: [{{val| 1 5 1 }}, {{val| 0 -31 12 }}]
[[Mapping]]: [{{val| 1 5 1 }}, {{val| 0 -31 12 }}]


POTE generator: ~296630859375/274877906944 = 132.194668
[[POTE generator]]: ~296630859375/274877906944 = 132.194668


Vals: {{Val list| 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}
{{Optimal GPV sequence|legend=1| 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}


Badness: 0.062778
[[Badness]]: 0.062778
 
== Gaster ==
{{main| Gaster temperament }}
{{see also| Mirkwai clan #Gaster }}
 
Subgroup: 2.3.5
 
[[Comma]]: {{monzo|-70 72 -19}}
 
[[Mapping]]: [{{val|1 11 38}}, {{val|0 -19 -72}}]
 
[[POTE generator]]: ~1638400000/1162261467 = 594.634565
 
{{Optimal GPV sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243 }}
 
[[Badness]]: 0.088530


== Whoosh ==
== Whoosh ==
Line 35: Line 51:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 37 25 -33 }}
[[Comma list]]: {{monzo| 37 25 -33 }}


Mapping: [{{val| 1 -16 -11 }}, {{val| 0 33 25 }}]
[[Mapping]]: [{{val| 1 -16 -11 }}, {{val| 0 33 25 }}]


POTE generator: ~864/625 = 560.546826
[[POTE generator]]: ~864/625 = 560.546826


Vals: {{Val list| 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
{{Optimal GPV sequence|legend=1| 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}


Badness: 0.025653
[[Badness]]: 0.025653


== Monzismic ==
== Monzismic ==
Line 50: Line 66:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 54 -37 2 }} = 450359962737049600 / 450283905890997363
[[Comma list]]: {{monzo| 54 -37 2 }} = 450359962737049600 / 450283905890997363


Mapping: [{{val| 1 0 -27 }}, {{val| 0 2 37 }}]
[[Mapping]]: [{{val| 1 0 -27 }}, {{val| 0 2 37 }}]


POTE generator: ~387420489/335544320 = 249.018406
[[POTE generator]]: ~387420489/335544320 = 249.018406


Vals: {{Val list| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
{{Optimal GPV sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}


Badness: 0.010391
[[Badness]]: 0.010391


== Egads ==
== Egads ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| -36 -52 51 }}
[[Comma list]]: {{monzo| -36 -52 51 }}


Mapping: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]
[[Mapping]]: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]


POTE generator: ~6/5 = 315.647919
[[POTE generator]]: ~6/5 = 315.647919


Vals: {{Val list| 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}
{{Optimal GPV sequence|legend=1| 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}


Badness: 0.041819
[[Badness]]: 0.041819


== Fortune ==
== Fortune ==
Line 78: Line 94:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| -107 47 14 }}
[[Comma list]]: {{monzo| -107 47 14 }}


Mapping: [{{val| 1 -1 11 }}, {{val| 0 14 -47 }}]
[[Mapping]]: [{{val| 1 -1 11 }}, {{val| 0 14 -47 }}]


POTE generator: ~8388608/7381125 = 221.567899
[[POTE generator]]: ~8388608/7381125 = 221.567899


Vals: {{Val list| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151 }}
{{Optimal GPV sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151 }}


Badness: 0.027876
[[Badness]]: 0.027876


=== 7-limit ===
=== 7-limit ===
Line 97: Line 113:
POTE generator: ~8388608/7381125 = 221.568031
POTE generator: ~8388608/7381125 = 221.568031


Vals: {{Val list| 612, 1901, 2513, 3125, 6862, 9987 }}
Optimal GPV sequence: {{Val list| 612, 1901, 2513, 3125, 6862, 9987 }}


Badness: 0.023704
Badness: 0.023704
Line 110: Line 126:
POTE generator: ~8388608/7381125 = 221.568139
POTE generator: ~8388608/7381125 = 221.568139


Vals: {{Val list| 612, 2513, 3125, 3737, 8086 }}
Optimal GPV sequence: {{Val list| 612, 2513, 3125, 3737, 8086 }}


Badness: 0.036641
Badness: 0.036641
Line 119: Line 135:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| -17 62 -35 }}
[[Comma list]]: {{monzo| -17 62 -35 }}


Mapping: [{{val| 1 11 19 }}, {{val| 0 -35 -62 }}]
[[Mapping]]: [{{val| 1 11 19 }}, {{val| 0 -35 -62 }}]


POTE generator: ~94143178827/78125000000 = 322.801358
[[POTE generator]]: ~94143178827/78125000000 = 322.801358


Vals: {{Val list| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
{{Optimal GPV sequence|legend=1| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}


Badness: 0.021220
[[Badness]]: 0.021220


== Gross ==
== Gross ==
Line 134: Line 150:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 144 -22 -47 }}
[[Comma list]]: {{monzo| 144 -22 -47 }}


Mapping: [{{val| 1 -2 4 }}, {{val| 0 47 -22 }}]
[[Mapping]]: [{{val| 1 -2 4 }}, {{val| 0 47 -22 }}]


POTE generator: ~1953125000000000/1853020188851841 = 91.531001
[[POTE generator]]: ~1953125000000000/1853020188851841 = 91.531001


Vals: {{Val list| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467 }}
{{Optimal GPV sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467 }}


Badness: 0.046315
[[Badness]]: 0.046315


== Raider ==
== Raider ==
Line 149: Line 165:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 71 -99 37 }}
[[Comma list]]: {{monzo| 71 -99 37 }}


Mapping: [{{val| 1 -9 -26 }}, {{val| 0 37 99 }}]
[[Mapping]]: [{{val| 1 -9 -26 }}, {{val| 0 37 99 }}]


POTE generator: ~8000/6561 = 343.2961
[[POTE generator]]: ~8000/6561 = 343.296098


Vals: {{Val list| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355 }}
{{Optimal GPV sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355 }}


Badness: 0.0129
[[Badness]]: 0.012867


== Pirate ==
== Pirate ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| -90 -15 49 }}
[[Comma list]]: {{monzo| -90 -15 49 }}


Mapping: [{{val| 1 -6 0 }}, {{val| 0 49 15 }}]
[[Mapping]]: [{{val| 1 -6 0 }}, {{val| 0 49 15 }}]


POTE generator: ~1358954496/1220703125 = 185.7542
[[POTE generator]]: ~1358954496/1220703125 = 185.754179


Vals: {{Val list| 730, 1783, 2513, 4296, 6079, 6809, 10375, 11105 }}
{{Optimal GPV sequence|legend=1| 730, 1783, 2513, 4296, 6079, 6809, 10375, 11105 }}


Badness: 0.0057
[[Badness]]: 0.005656


== Atomic ==
== Atomic ==
Line 177: Line 193:
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 161 -84 -12 }}
[[Comma list]]: {{monzo| 161 -84 -12 }}


Mapping: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]
[[Mapping]]: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]


POTE generator: ~32805/32768 = 1.95516
[[POTE generator]]: ~32805/32768 = 1.955162


Vals: {{Val list| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 3684, 4296, 4308, 4908, 7980, 12276, 16572, 20868, 25164, 29460, 33756, 46032 }}
{{Optimal GPV sequence|legend=1| 12, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}


Badness: 0.0038
[[Badness]]: 0.003803


=== 7-limit ===
=== 7-limit ===
Line 194: Line 210:
Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]
Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]


POTE generator: ~32805/32768 = 1.94961
POTE generator: ~32805/32768 = 1.949608


Vals: {{Val list| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 4308 }}
Optimal GPV sequence: {{Val list| 12, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}


Badness: 0.0458
Badness: 0.045818


=== 11-limit ===
=== 11-limit ===
Line 207: Line 223:
Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]
Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]


POTE generator: ~32805/32768 = 1.94799
POTE generator: ~32805/32768 = 1.947988


Vals: {{Val list| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 4308 }}
Optimal GPV sequence: {{Val list| 12, 600e, 612, 1236, 1848 }}


Badness: 0.0160
Badness: 0.016029


= Rank-3 temperaments =
= Rank-3 temperaments =

Revision as of 02:06, 16 January 2022

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

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 temperament (53&559) tempers out the monzisma, [54 -37 2⟩. 7-limit extensions of the monzismic temperament include monzism.

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]]

POTE generator: ~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

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: ~1358954496/1220703125 = 185.754179

Optimal ET sequence: 730, 1783, 2513, 4296, 6079, 6809, 10375, 11105

Badness: 0.005656

Atomic

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

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]]

Vals: 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]]

Vals: 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]]

Vals: Template:Val list

Badness: 0.000366

Sa-twentythezo-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

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