# Porwell temperaments

(Redirected from Hendecatonic)

This family of temperaments tempers out the porwell comma, [11 1 -3 -2 = 6144/6125, and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.

Discussed elsewhere are:

## Hendecatonic

The hendecatonic temperament has a period of 1/11 octave, which represents 16/15 and four times of which represent 9/7.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 10976/10935

Mapping[11 0 43 -4], 0 1 -1 2]]

mapping generators: ~16/15, ~3

Wedgie⟨⟨11 -11 22 -43 4 82]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.054

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 10976/10935

Mapping: [11 0 43 -4 38], 0 1 -1 2 0]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 4459/4455

Mapping: [11 0 43 -4 38 93], 0 1 -1 2 0 -3]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023

Mapping: [11 0 43 -4 38 93 45], 0 1 -1 2 0 -3 0]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301

### Cohendecatonic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 4375/4356

Mapping: [11 0 43 -4 73], 0 1 -1 2 -2]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 625/624

Mapping: [11 0 43 -4 73 128], 0 1 -1 2 -2 -5]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 375/374, 540/539

Mapping: [11 0 43 -4 73 128 45], 0 1 -1 2 -2 -5 0]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877

### Icosidillic

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 6144/6125, 9801/9800

Mapping: [22 0 86 -8 111], 0 1 -1 2 -1]]

mapping generators: ~33/32, ~3

Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914

## Twothirdtonic

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping[1 3 2 4], 0 -13 3 -11]]

mapping generators: ~2, ~15/14

Wedgie⟨⟨13 -3 11 -35 -19 34]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.401

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 686/675

Mapping: [1 3 2 4 4], 0 -13 3 -11 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

Mapping: [1 3 2 4 4 5], 0 -13 3 -11 -5 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409

## Semaja

Subgroup: 2.3.5.7

Comma list: 3125/3087, 6144/6125

Mapping[1 -2 1 3], 0 19 7 -1]]

mapping generators: ~2, ~8/7

Wedgie⟨⟨19 7 -1 -33 -55 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4834

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 3125/3087

Mapping: [1 -2 1 3 1], 0 19 7 -1 13]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 176/175, 275/273

Mapping: [1 -2 1 3 1 2], 0 19 7 -1 13 9]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794

## Nessafof

Cryptically named by Petr Pařízek in 2011[1], nessafof adds the landscape comma and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked 5 times, makes 5/1[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 250047/250000

Mapping[3 2 5 10], 0 7 5 -4]]

mapping generators: ~63/50, ~35/32

Wedgie⟨⟨21 15 -12 -25 -78 -70]]

Optimal tuning (POTE): ~63/50 = 1\3, ~35/32 = 157.480

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 250047/250000

Mapping: [3 2 5 10 8], 0 7 5 -4 6]]

Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520

### Nessa

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 4375/4356

Mapping: [3 2 5 10 10], 0 7 5 -4 1]]

Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 364/363, 441/440, 625/624

Mapping: [3 2 5 10 10 6], 0 7 5 -4 1 13]]

Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429

## Aufo

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

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping[1 6 -7 19], 0 -9 19 -33]]

mapping generators: ~2, ~45/32

Wedgie⟨⟨9 -19 33 -51 27 130]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.782

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 177147/175616

Mapping: [1 6 -7 19 1], 0 -9 19 -33 5]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 58806/57967

Mapping: [1 6 -7 19 1 -12], 0 -9 19 -33 5 32]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788

### Aufic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 72171/71680

Mapping: [1 6 -7 19 -25], 0 -9 19 -33 58]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 4096/4095

Mapping: [1 6 -7 19 -25 -12], 0 -9 19 -33 58 32]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796

## Whoops

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

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping[1 17 14 -7], 0 -33 -25 21]]

mapping generators: ~2, ~441/320

Wedgie⟨⟨33 25 -21 -37 -126 -119]]

Optimal tuning (POTE): ~2 = 1\1, ~441/320 = 560.519

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 6144/6125

Mapping: [1 17 14 -7 10], 0 -33 -25 21 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519

## Polypyth

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

Polypyth (46 & 121) tempers out the same 5-limit comma as the leapday temperament (29 & 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 179200/177147

Mapping[1 0 -31 52], 0 1 21 -31]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.174

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 6144/6125

Mapping: [1 0 -31 52 59], 0 1 21 -31 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1716/1715

Mapping: [1 0 -31 52 59 64], 0 1 21 -31 -35 -38]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715

Mapping: [1 0 -31 52 59 64 39], 0 1 21 -31 -35 -38 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168

## Icositritonic

The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents 6/5 and nine period represents 21/16.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 9920232/9765625

Mapping[23 0 17 101], 0 1 1 -1]]

mapping generators: ~1323/1280, ~3

Wedgie⟨⟨23 23 -23 -17 -101 -118]]

Optimal tuning (POTE): ~1323/1280 = 1\23, ~64/63 = 29.3586

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 35937/35840

Mapping: [23 0 17 101 116], 0 1 1 -1 -1]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 847/845, 3584/3575

Mapping: [23 0 17 101 116 158], 0 1 1 -1 -1 -2]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088

Mapping: [23 0 17 101 116 158 94], 0 1 1 -1 -1 -2 0]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845

Mapping: [23 0 17 101 116 158 94 207], 0 1 1 -1 -1 -2 0 -3]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760

### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845

Mapping: [23 0 17 101 116 158 94 207 104], 0 1 1 -1 -1 -2 0 -3 0]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471

## Countermiracle

The countermiracle temperament (31 & 145) tempers out the trimyna, 50421/50000 and the quince comma, 823543/819200.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping[1 4 3 3], 0 -25 -7 -2]]

mapping generators: ~2, ~343/320

Wedgie⟨⟨25 7 2 -47 -67 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9169

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 6144/6125

Mapping: [1 4 3 3 8], 0 -25 -7 -2 -47]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158

#### Countermiraculous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000, 6144/6125

Mapping: [1 4 3 3 8 1], 0 -25 -7 -2 -47 28]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224

Mapping: [1 4 3 3 8 1 1], 0 -25 -7 -2 -47 28 32]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756

#### Counterbenediction

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 3146/3125, 3584/3575

Mapping: [1 4 3 3 8 -2], 0 -25 -7 -2 -47 59]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125

Mapping: [1 4 3 3 8 -2 -2], 0 -25 -7 -2 -47 59 63]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391

#### Countermanna

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3388/3375, 6144/6125

Mapping: [1 4 3 3 8 15 0 -25 -7 -2 -47 -117]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375

Mapping: [1 4 3 3 8 15 15], 0 -25 -7 -2 -47 -117 -113]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832

### Counterrevelation

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 50421/50000

Mapping: [1 4 3 3 5], 0 -25 -7 -2 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 13750/13689

Mapping: [1 4 3 3 5 1], 0 -25 -7 -2 -16 28]]

Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625

Mapping: [1 4 3 3 5 1 1], 0 -25 -7 -2 -16 28 32]]

Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527

## Absurdity

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

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175000

Mapping[7 0 -17 64], 0 1 3 -4]]

mapping generators: ~972/875, ~3

Optimal tuning (POTE): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 72171/71680

Mapping[7 0 -17 64 124], 0 1 3 -4 -9]]

Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 700.6354 (or ~10/9 = 186.3497)

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 1188/1183, 3584/3575

Mapping[7 0 -17 64 124 37], 0 1 3 -4 -9 -1]]

Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625

Mapping[7 0 -17 64 124 37 -49], 0 1 3 -4 -9 -1 7]]

Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

Mapping[7 0 -17 64 124 37 -49 63], 0 1 3 -4 -9 -1 7 -3]]

Optimal tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)

## Dodifo

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

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping[1 12 5 4], 0 -35 -9 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.070

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2560/2541, 4375/4356

Mapping: [1 12 5 4 -1], 0 -35 -9 -4 15]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048