# Gamelismic clan

(Redirected from Oracle)

The 2.3.7 subgroup comma for the gamelismic clan is the gamelisma, 1029/1024, with monzo [-10 1 0 3. For any member of the clan, for the rank-3 gamelismic temperament itself, and for the rank-2 2.3.7 temperament slendric, this means three ~8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)3 × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. 36edo is a good tuning for slendric, though if the full 7-limit is desired, 72edo, 77edo or 118edo might be preferred.

To the gamelisma itself we need to add the comma which appears next on the modified normal comma list for the full 7-limit. The second comma on the list for mothra is 81/80, for rodan 245/243, for guiron 32805/32768, for gorgo 36/35, and for gidorah 256/245. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.

Miracle adds 33075/32768 and uses the secor, half an ~8/7, as generator. Lemba adds 525/512 to the list, and has a half-octave period. Valentine adds 6144/6125 with a generator of ~21/20 and superkleismic adds 875/864 with a generator of ~6/5. Unidec adds 4375/4374, and has a generator of ~10/9 with a half-octave period. Hemithirds adds 65625/65536 with a generator half of a classical major third. Finally, tritikleismic adds 15625/15552 and has a generator of 6/5 with a 1/3-octave period.

Full 7-limit temperaments discussed elsewhere are:

The rest are considered below.

No-five subgroup extensions of slendric include radon, a 2.3.7.11 extension that may be viewed as no-five rodan, and baladic, a 2.3.7.13.17 extension, considered below. Dicussed elsewhere is gigapyth in the 2.3.7.85 subgroup.

## Slendric

Subgroup: 2.3.7

Comma list: 1029/1024

Sval mapping[1 1 3], 0 3 -1]]

sval mapping generators: ~2, ~8/7

Gencom mapping[1 1 0 3], 0 3 0 -1]]

gencom: [2 8/7; 1029/1024]
• CTE: ~2 = 1\1, ~8/7 = 233.889
• POTE: ~2 = 1\1, ~8/7 = 233.688

Subgroup: 2.3.7.11

Comma list: 896/891, 1029/1024

Sval mapping[1 1 3 6], 0 3 -1 -13]]

Gencom mapping[1 1 0 3 6], 0 3 0 -1 -13]]

gencom: [2 8/7; 896/891 1029/1024]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.381

Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning.

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Sval mapping[2 2 6 7 7], 0 3 -1 1 3]]

sval mapping generators: ~17/12, ~8/7

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6155

## Rodan

Rodan tempers out 245/243 and can be described as the 41 & 46 temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1024

Mapping[1 1 -1 3], 0 3 17 -1]]

Wedgie⟨⟨3 17 -1 20 -10 -50]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.417

[[1 0 0 0, [5/3 0 1/6 -1/6, [25/9 0 17/18 -17/18, [25/9 0 -1/18 1/18]
Eigenmonzo (unchanged-interval) basis: 2.7/5

Algebraic generator: larger root of 20x2 - 36x + 15, or (9 + √6)/10.

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 441/440

Mapping: [1 1 -1 3 6], 0 3 17 -1 -13]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.459

Minimax tuning:

• 11-odd-limit: ~8/7 = [4/19 2/19 0 0 -1/19
[[1 0 0 0 0, [31/19 6/19 0 0 -3/19, [49/19 34/19 0 0 -17/19, [53/19 -2/19 0 0 1/19, [62/19 -26/19 0 0 13/19]
Eigenmonzo (unchanged-interval) basis: 2.11/9

Algebraic generator: positive root of x2 + 16x - 31, or √95 - 8.

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351, 364/363

Mapping: [1 1 -1 3 6 8], 0 3 17 -1 -13 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.482

Minimax tuning:

• 13- and 15-odd-limit: ~8/7 = [3/14 1/14 0 0 0 -1/28
Eigenmonzos (unchanged-intervals): 2, 13/9

Algebraic generator: Gatetone, positive root of 4x6 - 7x - 1. Recurrence converges slowly.

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 196/195, 245/243, 256/255, 273/272

Mapping: [1 1 -1 3 6 8 8], 0 3 17 -1 -13 -22 -20]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.524

Minimax tuning:

• 17-odd-limit: ~8/7 = [3/13 1/13 0 0 0 0 -1/26
Eigenmonzos (unchanged-intervals): 2, 18/17

#### Aerodactyl

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 245/243, 385/384, 441/440

Mapping: [1 1 -1 3 6 -1], 0 3 17 -1 -13 24]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.639

### Aerodino

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1029/1024

Mapping: [1 1 -1 3 -3], 0 3 17 -1 33]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.728

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243, 847/845

Mapping: [1 1 -1 3 -3 -1], 0 3 17 -1 33 24]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.782

### Varan

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1029/1024

Mapping: [1 1 -1 3 -2], 0 3 17 -1 28]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.145

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 245/243, 352/351

Mapping: [1 1 -1 3 -2 0], 0 3 17 -1 28 19]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.089

## Guiron

Guiron tempers out the schisma, and finds the prime 5 at the diminished fourth as does any temperament in the schismatic family. It can be described as 36 & 41. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 10976/10935

Mapping[1 1 7 3], 0 3 -24 -1]]

mapping generators: ~2, ~8/7

Wedgie⟨⟨3 -24 -1 -45 -10 65]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.930

[[1 0 0 0, [15/8 0 -1/8 0, [0 0 1 0, [65/24 0 1/24 0]
Eigenmonzo (unchanged-interval) basis: 2.5

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 10976/10935

Mapping: [1 1 7 3 -2], 0 3 -24 -1 28]]

mapping generators: ~2, ~8/7

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.931

Minimax tuning:

• 11-odd-limit: ~8/7 = [7/24 0 -1/24
[[1 0 0 0 0, [15/8 0 -1/8 0 0, [0 0 1 0 0, [65/24 0 1/24 0 0, [37/6 0 -7/6 0 0]
Eigenmonzo (unchanged-interval) basis: 2.5

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 729/728

Mapping: [1 1 7 3 -2 0], 0 3 -24 -1 28 19]]

mapping generators: ~2, ~8/7

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.890

## Mothra

Mothra tempers out 81/80 and finds the prime 5 at a stack of four fifths as does any temperament in the meantone family. It also tempers out 1728/1715, the orwellisma. It can be described as 26 & 31. Using 31edo with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential[1], so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo in the article on strictly proper 7-tone 31edo scales.

Note that mothra is also called cynder in the 7-limit, which can be a little confusing sometimes.

Its S-expression-based comma list is {S6/S7, S7/S8(, S6/S8 = S9)}, taking advantage of the fact that 81/80 is a semiparticular.

Subgroup: 2.3.5.7

Comma list: 81/80, 1029/1024

Mapping[1 1 0 3], 0 3 12 -1]]

mapping generators: ~2, ~8/7

Wedgie⟨⟨3 12 -1 12 -10 -36]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.193

Algebraic generator: Rabrindanath, largest real root of x8 - 3x2 + 1, or 232.0774 cents.

[[1 0 0 0, [1 0 1/4 0, [0 0 1 0, [3 0 -1/12 0]
Eigenmonzo (unchanged-interval) basis: 2.5

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 385/384

Mapping: [1 1 0 3 5], 0 3 12 -1 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.031

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 144/143

Mapping: [1 1 0 3 5 1], 0 3 12 -1 -8 14]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.811

Music

### Cynder

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1029/1024

Mapping: [1 1 0 3 0], 0 3 12 -1 18]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.317

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 81/80, 640/637

Mapping: [1 1 0 3 0 1], 0 3 12 -1 18 14]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.293

### Mosura

The S-expression-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {S6/S7, S7/S8, (S6/S8 = S9,) S8/S10}.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 540/539

Mapping: [1 1 0 3 -1], 0 3 12 -1 23]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.419

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: [1 1 0 3 -1 7], 0 3 12 -1 23 -17]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.640

## Gorgo

In the 5-limit, gorgo tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and a 3/2. Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in 16edo and 21edo. The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list.

### 5-limit (laconic)

Subgroup: 2.3.5

Comma list: 2187/2000

Mapping[1 1 1], 0 3 7]]

Wedgie⟨⟨3 7 4]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 227.426

### 7-limit

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1024

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

Wedgie⟨⟨3 7 -1 4 -10 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 228.334

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 1029/1024

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.373

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 507/500

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.230

### Spartan

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1024

Mapping: [1 1 1 3 5], 0 3 7 -1 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 229.535

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 56/55, 507/500

Mapping: [1 1 1 3 5 2], 0 3 7 -1 -8 9]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 229.059

Music

## Gidorah

### 5-limit (university)

Subgroup: 2.3.5

Comma list: 144/125

Mapping[1 1 2], 0 3 2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 235.4416

Optimal ET sequence5, 31cccc, 36…, 41…, 46…, 51

### 7-limit

Subgroup: 2.3.5.7

Comma list: 21/20, 144/125

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

Wedgie⟨⟨3 2 -1 -4 -10 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 230.762

## Oncle

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2430/2401

Mapping[1 1 6 3], 0 3 -19 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.498

## Archaeotherium

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

Subgroup: 2.3.5.7

Comma list: 405/392, 1029/1024

Mapping[1 1 5 3], 0 3 -14 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 230.258

## Clyndro

Subgroup: 2.3.5.7

Comma list: 135/128, 360/343

Mapping[1 1 4 3], 0 3 -9 -1]]

Wedgie⟨⟨3 -9 -1 -21 -10 23]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.469

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 352/343

Mapping: [1 1 4 3 4], 0 3 -9 -1 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.428

## Miracle

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1024

Mapping[1 1 3 3], 0 6 -7 -2]]

mapping generator: ~2, ~15/14

Wedgie⟨⟨6 -7 -2 -25 -20 15]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.675

[[1 0 0 0, [25/13 6/13 -6/13 0, [25/13 -7/13 7/13 0, [35/13 -2/13 2/13 0]
Eigenmonzo (unchanged-interval) basis: 2.5/3
[[1 0 0 0, [25/19 12/19 -6/19 0, [50/19 -14/19 7/19 0, [55/19 -4/19 2/19 0]
Eigenmonzo (unchanged-interval) basis: 2.9/5
• 7-odd-limit diamond monotone: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
• 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
• 7- and 9-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]
• 7-odd-limit diamond monotone and tradeoff: ~15/14 = [115.587, 116.993]
• 9-odd-limit diamond monotone and tradeoff: ~15/14 = [116.129, 116.993]

Algebraic generator: Secor59, positive root of 15x6 - 8x4 - 12

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 385/384

Mapping: [1 1 3 3 2], 0 6 -7 -2 15]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.633

Minimax tuning:

• 11-odd-limit: ~15/14 = [1/19 2/19 -1/19
[[1 0 0 0 0, [25/19 12/19 -6/19 0 0, [50/19 -14/19 7/19 0 0, [55/19 -4/19 2/19 0 0, [53/19 30/19 -15/19 0 0]
Eigenmonzo (unchanged-interval) basis: 2.9/5

Tuning ranges:

• 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
• 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]
• 11-odd-limit diamond monotone and tradeoff: ~15/14 = [116.129, 116.993]

Algebraic generator: Secor59

#### Miraculous

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 243/242

Mapping: [1 1 3 3 2 4], 0 6 -7 -2 15 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.747

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 120/119, 144/143, 154/153, 170/169

Mapping: [1 1 3 3 2 4 4], 0 6 -7 -2 15 -3 1]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.769

#### Benediction

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350, 385/384

Mapping: [1 1 3 3 2 7], 0 6 -7 -2 15 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.574

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 351/350, 375/374

Mapping: [1 1 3 3 2 7 7], 0 6 -7 -2 15 -34 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.585

#### Manna

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 385/384

Mapping: [1 1 3 3 2 0], 0 6 -7 -2 15 38]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.739

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 325/324, 385/384

Mapping: [1 1 3 3 2 0 0], 0 6 -7 -2 15 38 42]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.727

#### Semimiracle

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 385/384

Mapping: [2 2 6 6 4 7], 0 6 -7 -2 15 2]]

Optimal tuning (POTE): ~99/70 = 1\2, ~15/14 = 116.624

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 273/272

Mapping: [2 2 6 6 4 7 7], 0 6 -7 -2 15 2 6]]

Optimal tuning (POTE): ~2 = 17\12, ~15/14 = 116.628

#### Hemisecordite

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 847/845

Mapping: [1 1 3 3 2 2], 0 12 -14 -4 30 35]]

Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.288

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384, 847/845

Mapping: [1 1 3 3 2 2 2], 0 12 -14 -4 30 35 43]]

Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.261

##### Semihemisecordite

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 289/288, 385/384, 847/845

Mapping: [2 2 6 6 4 4 7], 0 12 -14 -4 30 35 12]]

Optimal tuning (POTE): ~17/12 = 1\2, ~27/26 = 58.288

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384

Mapping: [2 2 6 6 4 4 7 8], 0 12 -14 -4 30 35 12 5]]

Optimal tuning (POTE): ~17/12 = 1\2, ~27/26 = 58.283

###### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384

Mapping: [2 2 6 6 4 4 7 8 7], 0 12 -14 -4 30 35 12 5 21]]

Optimal tuning (POTE): ~17/12 = 1\2, ~27/26 = 58.283

#### Phicordial

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 2200/2197

Mapping: [1 7 -4 1 17 4], 0 -18 21 6 -45 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 361.121

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197

Mapping: [1 7 -4 1 17 4 8], 0 -18 21 6 -45 -1 -13]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 361.123

### Revelation

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1029/1024

Mapping: [1 1 3 3 5], 0 6 -7 -2 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.277

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 512/507

Mapping: [1 1 3 3 5 4], 0 6 -7 -2 -16 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.268

### Hemimiracle

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 1029/1024

Mapping: [1 1 3 3 4], 0 12 -14 -4 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 58.408

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/242, 512/507

Mapping: [1 1 3 3 4 4], 0 12 -14 -4 -11 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 58.430

### Oracle

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 1029/1024

Mapping: [1 7 -4 1 3], 0 -12 14 4 1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 541.668

## Hemiseven

Subgroup: 2.3.5.7

Comma list: 1029/1024, 19683/19600

Mapping[1 4 14 2], 0 -6 -29 2]]

Wedgie⟨⟨6 29 -2 32 -20 -86]]

Optimal tuning (POTE): ~2 = 1\1, ~320/243 = 483.267

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 19683/19600

Mapping: [1 4 14 2 -5], 0 -6 -29 2 21]]

Optimal tuning (POTE): ~2 = 1\1, ~320/243 = 483.276

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440, 676/675

Mapping: [1 4 14 2 -5 19], 0 -6 -29 2 21 -38]]

Optimal tuning (POTE): ~2 = 1\1, ~120/91 = 483.256

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 676/675

Mapping: [1 4 14 2 -5 19 21], 0 -6 -29 2 21 -38 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~45/34 = 483.261

## Unidec

### 5-limit (unidecmic)

Subgroup: 2.3.5

Comma list: 31381059609/31250000000

Mapping[2 5 8], 0 -6 -11]]

mapping generators: ~177147/125000, ~10/9

Optimal tuning (POTE): ~177147/125000 = 1\2, ~10/9 = 183.047

### 7-limit

Subgroup: 2.3.5.7

Comma list: 1029/1024, 4375/4374

Mapping[2 5 8 5], 0 -6 -11 2]]

Wedgie⟨⟨12 22 -4 7 -40 -71]]

Optimal tuning (POTE): ~1225/864 = 1\2, ~10/9 = 183.161

[[1 0 0 0, [47/26 0 6/13 -6/13, [71/26 0 11/13 -11/13, [71/26 0 -2/13 2/13]
Eigenmonzo (unchanged-interval) basis: 2.7/5
[[1 0 0 0, [10/7 6/7 0 -3/7, [57/28 11/7 0 -11/14, [20/7 -2/7 0 1/7]
Eigenmonzo (unchanged-interval) basis: 2.9/7

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4375/4374

Mapping: [2 5 8 5 6], 0 -6 -11 2 3]]

Minimax tuning:

[[1 0 0 0 0, [10/7 6/7 0 -3/7 0, [57/28 11/7 0 -11/14 0, [20/7 -2/7 0 1/7 0, [99/28 -3/7 0 3/14 0]
Eigenmonzo (unchanged-interval) basis: 2.9/7

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 729/728

Mapping: [2 5 8 5 6 19], 0 -6 -11 2 3 -38]]

Optimal tuning (POTE): ~99/70 = 1\2, ~10/9 = 183.187

#### Hendec

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 385/384

Mapping: [2 5 8 5 6 8], 0 -6 -11 2 3 -2]]

Optimal tuning (POTE): ~91/64 = 1\2, ~10/9 = 183.198

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 273/272, 325/324, 364/363

Mapping: [2 5 8 5 6 8 10], 0 -6 -11 2 3 -2 -6]]

Optimal tuning (POTE): ~17/12 = 1\2, ~10/9 = 183.196

## Superkleismic

The S-expression-based comma list of superkleismic is {S5/S6, S7/S8, S10, S12(, S21)}, from which (through careful observation of the equivalences therein) one can derive that a sharpened ~6/5 is the generator as well as the mapping of the full 13-limit.

Subgroup: 2.3.5.7

Comma list: 875/864, 1029/1024

Mapping[1 4 5 2], 0 -9 -10 3]]

Wedgie⟨⟨9 10 -3 -5 -30 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 321.930

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 385/384

Mapping: [1 4 5 2 4], 0 -9 -10 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 321.847

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 245/243

Mapping: [1 4 5 2 4 8], 0 -9 -10 3 -2 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 321.994

## Lagaca

Subgroup: 2.3.5.7

Comma list: 1029/1024, 11529602/11390625

Mapping[2 5 2 5], 0 -9 13 3]]

Wedgie⟨⟨18 -26 -6 -83 -60 59]]

Optimal tuning (POTE): ~3375/2401 = 1\2, ~15/14 = 122.027

## Necromanteion

Subgroup: 2.3.5.7

Comma list: 1029/1024, 5103/5000

Mapping[1 7 10 1], 0 -12 -17 4]]

Wedgie⟨⟨12 17 -4 -1 -40 -57]]

Optimal tuning (POTE): ~2 = 1\1, ~48/35 = 541.779

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242, 1029/1024

Mapping: [1 7 10 1 17], 0 -12 -17 4 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.729

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242, 343/338

Mapping: [1 7 10 1 17 1], 0 -12 -17 4 -30 6]]

Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.606

## Restles

Subgroup: 2.3.5.7

Comma list: 1029/1024, 153664/151875

Mapping[1 -2 8 4], 0 12 -19 -4]]

Wedgie⟨⟨12 -19 -4 -58 -40 44]]

Optimal tuning (POTE): ~2 = 1\1, ~315/256 = 358.5485

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 153664/151875

Mapping: [1 -2 8 4 -7], 0 12 -19 -4 35]]

Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 358.5713

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 676/675

Mapping: [1 -2 8 4 -7 4], 0 12 -19 -4 35 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 358.5739

## Quartemka

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1250000/1240029

Mapping[1 4 6 2], 0 -21 -32 7]]

Wedgie⟨⟨21 32 -7 2 -70 -106]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 138.006

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 800000/793881

Mapping: [1 4 6 2 3], 0 -21 -32 7 4]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 137.990

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 2200/2197

Mapping: [1 4 6 2 3 6], 0 -21 -32 7 4 -20]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 137.990

## Tritriple

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1959552/1953125

Mapping[1 -11 -7 7], 0 27 20 -9]]

Wedgie⟨⟨27 20 -9 -31 -90 -77]]

Optimal tuning (POTE): ~2 = 1\1, ~864/625 = 559.295

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 43923/43750

Mapping: [1 -11 -7 7 -4], 0 27 20 -9 16]]

Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 559.293

## Widefourth

Subgroup: 2.3.5.7

Comma list: 1029/1024, 48828125/48771072

Mapping[1 16 8 -2], 0 -33 -13 11]]

Wedgie⟨⟨33 13 -11 -56 -110 -62]]

Optimal tuning (POTE): ~2 = 1\1, ~3125/2304 = 524.210

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 234375/234256

Mapping: [1 16 8 -2 17], 0 -33 -13 11 -31]]

Optimal tuning (POTE): ~2 = 1\1, ~847/625 = 524.210