# Mirkwai clan

Temperaments of the mirkwai clan temper out the mirkwai comma, [0 3 4 -5 = 16875/16807, a no-twos comma.

## 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

### 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:

## 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

### 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

### 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

### 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

### 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

## 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

### 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

#### 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

##### 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

##### 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

#### 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

##### 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

##### 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

## 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

### 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

#### 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

##### 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

##### 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

#### 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

### 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

#### 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

## 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

### 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

### 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

### 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

## 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

### 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

### 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

## 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

### 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

### 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

## 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

### 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

### 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

### 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

## Gaster

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

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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

## 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

### 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

### 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

### 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

### 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

## 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

### 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

### 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

### 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

## Browser

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

### 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

### 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

### 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

### 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

## 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

### 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

### 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

### 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

### 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