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.

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

Optimal tuning (POTE): ~3 = 1\1edt, ~7/5 = 583.9584

Optimal ET sequence: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697

Overview to extensions

The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid.

The others are weak extensions. Mirkat tempers out 19683/19600, splitting the generator in two with a semitwelfth period. Sqrtphi tempers out 15625/15552, splitting the period in six. Miracle tempers out 225/224. Pluto tempers out 4000/3969. These split the generator in five. Quanharuk tempers out 32805/32768, splitting the generator in three with a 1/5-twelfth period. Semisept tempers out 1728/1715 and 3136/3125, splitting the generator in six. Kwai tempers out 5120/5103, splitting the generator in ten. Grendel tempers out 6144/6125, splitting the generator in eleven. Finally, eris tempers out 65625/65536, splitting the generator in sixteen.

Members of the clan considered below are grendel, kwai, pluto, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer. Discussed elsewhere are:

For no-twos extensions, see No-twos subgroup temperaments#Canopus.

Grendel

For the 5-limit version of this temperament, see Syntonic-31 equivalence continuum #Counterwürschmidt.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 16875/16807

Mapping: [⟨1 9 2 7], ⟨0 -23 1 -13]]

mapping generators: ~2, ~5/4

Wedgie: ⟨⟨ 23 -1 13 -55 -44 33 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 386.863

Optimal ET sequence: 31, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 386.856

Optimal ET sequence: 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

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 386.826

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 386.812

Optimal ET sequence: 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

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 386.819

Optimal ET sequence: 31, 121, 152fg, 273defgg

Badness: 0.018413

Kwai

For the 5-limit version of this temperament, see High badness temperaments #Kwai.

Named by Gene Ward Smith in 2004 for its "bridgeability"[1], kwai is generated by a fifth, and can be described as 41 & 70.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 16875/16807

Mapping: [⟨1 0 -50 -40], ⟨0 1 33 27]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 33 27 50 40 -30 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.616

Optimal ET sequence: 41, 111, 152, 345, 497d

Badness: 0.054476

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5120/5103

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.623

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.644

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.660

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.657

Optimal ET sequence: 29cdgh, 41, 111, 152fg, 263dfgh

Badness: 0.016957

Hemikwai

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 676/675, 1375/1372, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51], ⟨0 2 66 54 -36 69]]

mapping generators: ~2, ~26/15

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.314

Optimal ET sequence: 82, 111, 193, 304d

Badness: 0.044108

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51 -30], ⟨0 2 66 54 -36 69 43]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.314

Optimal ET sequence: 82, 111, 193, 304d

Badness: 0.025806

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444

Mapping: [⟨1 0 -50 -40 32 -51 -30 -56], ⟨0 2 66 54 -36 69 43 76]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.313

Optimal ET sequence: 82, 111, 193, 304dh

Badness: 0.019146

Pluto

Not to be confused with plutus.

Pluto, named by Gene Ward Smith in 2010[2], can be described as the 41 & 80 temperament. It is generated by a sharpened 7/5, and 59\121 is about perfect as a tuning.

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.147

Optimal ET sequence: 39d, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.114

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.123

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.116

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.109

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.111

Optimal ET sequence: 39df, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.283

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.232

Optimal ET sequence: 39deef, 41

Badness: 0.040182

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.539

Optimal ET sequence: 39d, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.528

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.577

Optimal ET sequence: 39df, 72, 111, 183

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]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.579

Optimal ET sequence: 39dfg, 72, 111, 183

Badness: 0.011775

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~60/49 = 348.216

Optimal ET sequence: 31, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.219

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.213

Optimal ET sequence: 31, 193, 224

Badness: 0.025137

Subsemifourth

Subgroup: 2.3.5.7

Comma list: 16875/16807, 26873856/26796875

Mapping: [⟨1 39 27 45], ⟨0 -47 -31 -53]]

mapping generators: ~2, ~125/72

Wedgie: ⟨⟨ 47 31 53 -60 -48 36 ]]

Optimal tuning (POTE): ~2 = 1\1, ~144/125 = 244.719

Optimal ET sequence: 49, 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 39 27 45 56], ⟨0 -47 -31 -53 -66]]

Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 244.719

Optimal ET sequence: 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 39 27 45 56 65], ⟨0 -47 -31 -53 -66 -77]]

Optimal tuning (POTE): ~2 = 1\1, ~15/13 = 244.714

Optimal ET sequence: 49f, 103, 152f, 255, 407f, 662df

Badness: 0.028387

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 39 37 53], ⟨0 -41 -38 -55]]

mapping generators: ~2, ~648/343

Wedgie: ⟨⟨ 41 38 55 -35 -28 21 ]]

Optimal tuning (POTE): ~2 = 1\1, ~343/324 = 104.916

Optimal ET sequence: 80, 103, 183

Badness: 0.146795

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 39 37 53 50], ⟨0 -41 -38 -55 -51]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 104.916

Optimal ET sequence: 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 39 37 53 50 11], ⟨0 -41 -38 -55 -51 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 104.908

Optimal ET sequence: 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 39 37 53 50 11 5], ⟨0 -41 -38 -55 -51 -8 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~17/16 = 104.909

Optimal ET sequence: 80, 103, 183, 469f, 652def

Badness: 0.020128

Gaster

For the 5-limit version of this temperament, see Very high accuracy temperaments #Gaster.
Main article: Gaster temperament

The gaster temperament (111 & 113) tempers out [-70 72 -19⟩ (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and skeetsma (14348907/14336000) in the 7-limit. The word "gaster" means abdomen or stomach, but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals 111, 113 and 224 support it even in the 41-limit.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 14348907/14336000

Mapping: [⟨1 11 38 37], ⟨0 -19 -72 -69]]

Wedgie: ⟨⟨ 19 72 69 70 56 -42 ]]

Optimal tuning (POTE): ~2 = 1\1, ~800/567 = 594.641

Optimal ET sequence: 111, 224

Badness: 0.154521

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 14348907/14336000

Mapping: [⟨1 11 38 37 -1], ⟨0 -19 -72 -69 9]]

Optimal tuning (POTE): ~2 = 1\1, ~512/363 = 594.639

Optimal ET sequence: 111, 224, 783d, 1007d, 1231dd

Badness: 0.054060

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 2200/2197

Mapping: [⟨1 11 38 37 -1 26], ⟨0 -19 -72 -69 9 -45]]

Optimal tuning (POTE): ~2 = 1\1, ~55/39 = 594.639

Optimal ET sequence: 111, 224, 783df, 1007df, 1231ddf

Badness: 0.024882

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197

Mapping: [⟨1 11 38 37 -1 26 14], ⟨0 -19 -72 -69 9 -45 -20]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.636

Optimal ET sequence: 111, 224, 559dgg

Badness: 0.021436

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444

Mapping: [⟨1 11 38 37 -1 26 14 32], ⟨0 -19 -72 -69 9 -45 -20 -56]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.636

Optimal ET sequence: 111, 224

Badness: 0.018370

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7], ⟨0 -19 -72 -69 9 -45 -20 -56 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.641

Optimal ET sequence: 111, 224

Badness: 0.017619

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.646

Optimal ET sequence: 111, 113, 224

Badness: 0.016815

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.644

Optimal ET sequence: 111, 113, 224

Badness: 0.014790

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.644

Optimal ET sequence: 111, 113, 224

Badness: 0.014377

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666

Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27 45], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80]]

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.643

Optimal ET sequence: 111, 113, 224

Badness: 0.012858

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.965

Optimal ET sequence: 10, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.968

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.968

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.968

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.964

Optimal ET sequence: 10, 101h, 111, 121, 232dg

Badness: 0.017935

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 34 17 34], ⟨0 -53 -24 -51]]

mapping generators: ~2, ~75/49

Wedgie: ⟨⟨ 53 24 51 -85 -68 51 ]]

Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 466.071

Optimal ET sequence: 103, 121, 224

Badness: 0.162826

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 34 17 34 53], ⟨0 -53 -24 -51 -81]]

Optimal tuning (POTE): ~2 = 1\1, ~55/42 = 466.070

Optimal ET sequence: 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 34 17 34 53 30], ⟨0 -53 -24 -51 -81 -43]]

Optimal tuning (POTE): ~2 = 1\1, ~55/42 = 466.071

Optimal ET sequence: 103, 121, 224

Badness: 0.021127

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197

Mapping: [⟨1 34 17 34 53 30 31], ⟨0 -53 -24 -51 -81 -43 -44]]

Optimal tuning (POTE): ~2 = 1\1, ~17/13 = 466.074

Optimal ET sequence: 103, 121, 224

Badness: 0.018611

Browser

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 16875/16807, 78732/78125

Mapping: [⟨1 6 8 10], ⟨0 -35 -45 -57]]

Wedgie: ⟨⟨ 35 45 57 -10 -8 6 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/45 = 151.399

Optimal ET sequence: 103, 111, 214

Badness: 0.180645

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 78732/78125

Mapping: [⟨1 6 8 10 8], ⟨0 -35 -45 -57 -36]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.405

Optimal ET sequence: 103, 214

Badness: 0.057634

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 1375/1372

Mapping: [⟨1 6 8 10 8 9], ⟨0 -35 -45 -57 -36 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.403

Optimal ET sequence: 103, 111, 214

Badness: 0.028822

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 847/845

Mapping: [⟨1 6 8 10 8 9 8], ⟨0 -35 -45 -57 -36 -42 -31]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.397

Optimal ET sequence: 103, 111, 214

Badness: 0.020384

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714

Mapping: [⟨1 6 8 10 8 9 8 18], ⟨0 -35 -45 -57 -36 -42 -31 -109]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 151.396

Optimal ET sequence: 103h, 111, 214

Badness: 0.017570

Grazer

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1071875/1062882

Mapping: [⟨1 34 47 58], ⟨0 -37 -51 -63]]

mapping generators: ~2, ~90/49

Wedgie: ⟨⟨ 37 51 63 -5 -4 3 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/45 = 148.719

Optimal ET sequence: 113, 121, 234

Badness: 0.217166

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 218750/216513

Mapping: [⟨1 34 47 58 35], ⟨0 -37 -51 -63 -36]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.729

Optimal ET sequence: 113, 121, 234, 355e, 589cee

Badness: 0.076062

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539, 2200/2197

Mapping: [⟨1 34 47 58 35 44], ⟨0 -37 -51 -63 -36 -46]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.729

Optimal ET sequence: 113, 121, 234, 355e, 589cee

Badness: 0.036248

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989

Mapping: [⟨1 34 47 58 35 44 33], ⟨0 -37 -51 -63 -36 -46 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.735

Optimal ET sequence: 113, 121, 234g, 355eg

Badness: 0.025410

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663

Mapping: [⟨1 34 47 58 35 44 33 6], ⟨0 -37 -51 -63 -36 -46 -33 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.727

Optimal ET sequence: 113, 121, 234g, 355eg, 589ceegg

Badness: 0.022574

Notes

  1. ↑ Yahoo! Tuning Group | Kwai
  2. ↑ Yahoo! Tuning Group | 19-limit Pluto temperament
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