Mirkwai clan

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Temperaments of the mirkwai clan temper out the mirkwai comma, [0 3 4 -5 = 16875/16807, a no-twos comma. Members of the clan include grendel, kwai, pluto and mirkat considered below, as well as these considered elsewhere:

Canopus

Main article: Canopus

Subgroup: 3.5.7

Comma list: 16875/16807

Sval mapping: [1 3 3], 0 -5 -4]]

Sval mapping generators: ~3, ~7/5

POTE generator: ~7/5 = 583.9584

Vals: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697

Grendel

Subgroup: 2.3.5.7

Comma list: 6144/6125, 16875/16807

Mapping: [1 9 2 7], 0 -23 1 -13]]

Wedgie⟨⟨23 -1 13 -55 -44 33]]

POTE generator: ~5/4 = 386.863

Vals31, 90, 121, 152, 335d

Badness: 0.051834

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5632/5625

Mapping: [1 9 2 7 17], 0 -23 1 -13 -42]]

POTE generator: ~5/4 = 386.856

Vals: 31, 90e, 121, 152, 335d, 487d

Badness: 0.019845

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 1375/1372

Map: [1 9 2 7 17 -5], 0 -23 1 -13 -42 27]]

POTE generator: ~5/4 = 386.826

Vals: 31, 121, 152f, 425deff

Badness: 0.024839

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274

Mapping: [1 9 2 7 17 -5 -3], 0 -23 1 -13 -42 27 22]]

POTE generator: ~5/4 = 386.812

Vals: 31, 121, 273defgg

Badness: 0.021400

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714

POTE generator: ~5/4 = 386.819

Mapping: [1 9 2 7 17 -5 -3 -8], 0 -23 1 -13 -42 27 22 38]]

Vals: 31, 121, 152fg, 273defgg

Badness: 0.018413

Kwai

Subgroup: 2.3.5.7

Comma list: 5120/5103, 16875/16807

Mapping: [1 0 -50 -40], 0 1 33 27]]

Wedgie⟨⟨1 33 27 50 40 -30]]

POTE generator: ~3/2 = 702.616

Vals41, 111, 152, 345, 497d

Badness: 0.054476

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 16384/16335

Mapping: [1 0 -50 -40 32], 0 1 33 27 -18]]

POTE generator: ~3/2 = 702.623

Vals: 29cd, 41, 111, 152

Badness: 0.026219

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1375/1372

Mapping: [1 0 -50 -40 32 27], 0 1 33 27 -18 -21]]

POTE generator: ~3/2 = 702.644

Vals: 29cd, 41, 111, 152f

Badness: 0.024555

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088

Mapping: [1 0 -50 -40 32 27 58], 0 1 33 27 -18 -21 -34]]

POTE generator: ~3/2 = 702.660

Vals: 29cdg, 41, 111, 152fg, 263dfg

Badness: 0.021950

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845

Mapping: [1 0 -50 -40 32 27 58 -56], 0 1 33 27 -18 -21 -34 38]]

POTE generator: ~3/2 = 702.657

Vals: 29cdgh, 41, 111, 152fg, 263dfgh

Badness: 0.016957

Pluto

Subgroup: 2.3.5.7

Comma list: 4000/3969, 10976/10935

Mapping: [1 5 15 15], 0 -7 -26 -25]]

Wedgie⟨⟨7 26 25 25 20 -15]]

POTE generator: ~7/5 = 585.147

Vals39d, 41, 80, 121, 404bd

Badness: 0.057514

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 1375/1372

Mapping: [1 5 15 15 2], 0 -7 -26 -25 3]]

POTE generator: ~7/5 = 585.114

Vals: 39d, 41, 80, 121

Badness: 0.029844

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 540/539

Mapping: [1 5 15 15 2 -8], 0 -7 -26 -25 3 24]]

POTE generator: ~7/5 = 585.123

Vals: 39d, 41, 80, 121

Badness: 0.025717

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 540/539

Mapping: [1 5 15 15 2 -8 -12], 0 -7 -26 -25 3 24 33]]

POTE generator: ~7/5 = 585.116

Vals: 39d, 41, 80, 121

Badness: 0.021463

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594

Mapping: [1 5 15 15 2 -8 -12 14], 0 -7 -26 -25 3 24 33 -20]]

POTE generator: ~7/5 = 585.109

Vals: 39d, 41, 80, 121

Badness: 0.017650

Orcus

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 275/273, 896/891

Mapping: [1 5 15 15 2 12], 0 -7 -26 -25 3 -17]]

POTE generator: ~7/5 = 585.111

Vals: 41, 80f, 121ff

Badness: 0.033441

Plutino

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 10976/10935

Mapping: [1 5 15 15 22], 0 -7 -26 -25 -38]]

POTE generator: ~7/5 = 585.283

Vals: 39dee, 41

Badness: 0.057966

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 245/242, 729/728

Mapping: [1 5 15 15 22 12], 0 -7 -26 -25 -38 -17]]

POTE generator: ~7/5 = 585.232

Vals: 39deef, 41

Badness: 0.040182

Eris

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65625/65536

Mapping: [1 10 0 6], 0 -29 8 -11]]

Wedgie⟨⟨29 -8 11 -80 -64 48]]

POTE generator: ~60/49 = 348.216

Vals31, 131, 162, 193, 224, 1823cd, 2271cd

Badness: 0.074719

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65625/65536

Mapping: [1 10 0 6 20], 0 -29 8 -11 -57]]

POTE generator: ~11/9 = 348.219

Vals: 31, 193, 224, 703, 927d, 1151cd

Badness: 0.027621

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372, 4096/4095

Mapping: [1 10 0 6 20 -14], 0 -29 8 -11 -57 61]]

POTE generator: ~11/9 = 348.213

Vals: 31, 193, 224

Badness: 0.025137

Mirkat

Subgroup: 2.3.5.7

Comma list: 16875/16807, 19683/19600

Mapping: [3 2 1 2], 0 6 13 14]]

Wedgie⟨⟨18 39 42 20 16 -12]]

POTE generator: ~10/9 = 183.539

Vals39d, 72, 111, 183, 255

Badness: 0.059376

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 8019/8000

Mapping: [3 2 1 2 9], 0 6 13 14 3]]

POTE generator: ~10/9 = 183.528

Vals: 39d, 72, 111, 183, 255

Badness: 0.022126

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 1375/1372

Mapping: [3 2 1 2 9 1], 0 6 13 14 3 22]]

POTE generator: ~10/9 = 183.577

Vals: 39df, 72, 111, 183, 255f

Badness: 0.018632

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 540/539, 561/560, 715/714

Mapping: [3 2 1 2 9 1 4], 0 6 13 14 3 22 18]]

POTE generator: ~10/9 = 183.579

Vals: 39dfg, 72, 111, 183, 255f

Badness: 0.011775

Subsemifourth

Subgroup: 2.3.5.7

Comma list: 16875/16807, 26873856/26796875

Mapping: [1 -8 -4 -8], 0 47 31 53]]

Wedgie⟨⟨47 31 53 -60 -48 36]]

POTE generator: ~144/125 = 244.719

Vals49, 103, 152, 255, 407

Badness: 0.135173

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 234375/234256

Mapping: [1 -8 -4 -8 -10], 0 47 31 53 66]]

POTE generator: ~121/105 = 244.719

Vals: 49, 103, 152, 255, 407, 966d

Badness: 0.034276

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845, 1375/1372, 1575/1573

Mapping: [1 -8 -4 -8 -10 -12], 0 47 31 53 66 77]]

POTE generator: ~15/13 = 244.714

Vals: 49f, 103, 152f, 255, 407f, 662df

Badness: 0.028387

Hemiseptisix

The name hemiseptisix means a half of septimal major sixth (12/7). The hemiseptisix temperament (103&121) tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma, lazozotritriyo) in the 7-limit. 224EDO provides an excellent tuning for 7, 11, and 13-limit hemiseptisix.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 95703125/95551488

Mapping: [1 -19 -7 -17], 0 53 24 51]]

Wedgie⟨⟨53 24 51 -85 -68 51]]

POTE generator: ~98/75 = 466.071

Vals18, 103, 121, 224

Badness: 0.162826

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 2734375/2725888

Mapping: [1 -19 -7 -17 -28], 0 53 24 51 81]]

POTE generator: ~55/42 = 466.070

Vals: 103, 121, 224

Badness: 0.043381

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372, 2200/2197

Mapping: [1 -19 -7 -17 -28 -13], 0 53 24 51 81 43]]

POTE generator: ~55/42 = 466.071

Vals: 103, 121, 224

Badness: 0.021127

Septendesemi

The name septendesemi means a septendecimal semitone (17/16). The septendesemi temperament (80&103) tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma, zotritrigu) in the 7-limit. 183EDO provides an excellent tuning for 7, 11, 13, and 17-limit septendesemi.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1959552/1953125

Mapping: [1 -2 -1 -2], 0 41 38 55]]

Wedgie⟨⟨41 38 55 -35 -28 21]]

POTE generator: ~343/324 = 104.916

Vals80, 103, 183

Badness: 0.146795

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 43923/43750

Mapping: [1 -2 -1 -2 -1], 0 41 38 55 51]]

POTE generator: ~35/33 = 104.916

Vals: 80, 103, 183

Badness: 0.041554

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372, 4225/4224

Mapping: [1 -2 -1 -2 -1 3], 0 41 38 55 51 8]]

POTE generator: ~35/33 = 104.908

Vals: 80, 103, 183, 469f, 652def

Badness: 0.027908

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224

Mapping: [1 -2 -1 -2 -1 3 4], 0 41 38 55 51 8 1]]

POTE generator: ~17/16 = 104.909

Vals: 80, 103, 183, 469f, 652def

Badness: 0.020128

Subsedia

The generator for subsedia (10&111) is 0.5 cents flat of 15/14-wide semitone and tempers out the mirkwai comma and 65536/64827 (buzzardisma, saquadru comma). In this temperament, three generators makes ~16/13, five of them equals ~24/17, twelve of them equals ~16/7, sixteen of them equals ~3/1, and 45 of them equals ~22/1.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 65536/64827

Mapping: [1 0 5 4], 0 16 -27 -12]]

Wedgie⟨⟨16 -27 -12 -80 -64 48]]

POTE generator: ~15/14 = 118.965

Vals10, 101, 111, 121, 232d

Badness: 0.157658

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65536/64827

Mapping: [1 0 5 4 -1], 0 16 -27 -12 45]]

POTE generator: ~15/14 = 118.968

Vals: 10, 101, 111, 121, 232d

Badness: 0.066838

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 676/675, 1375/1372

Mapping: [1 0 5 4 -1 4], 0 16 -27 -12 45 -3]]

POTE generator: ~15/14 = 118.968

Vals: 10, 101, 111, 121, 232d

Badness: 0.031635

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 442/441, 540/539, 715/714

Mapping: [1 0 5 4 -1 4 3], 0 16 -27 -12 45 -3 11]]

POTE generator: ~15/14 = 118.968

Vals: 10, 101, 111, 121, 232dg

Badness: 0.019707

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714

Mapping: [1 0 5 4 -1 4 3 10], 0 16 -27 -12 45 -3 11 -58]]

POTE generator: ~15/14 = 118.964

Vals: 10, 101h, 111, 121, 232dg

Badness: 0.017935