# Kleismic family

(Redirected from Catakleismic temperament)

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is [-6 -5 6, and flipping that yields ⟨⟨6 5 -6]] for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)5 = 5/2 × 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called hanson, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.

## Hanson

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [1 0 1], 0 6 5]]

POTE generator: ~6/5 = 317.007

• 5-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
• 5-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.263]
• 5-odd-limit diamond monotone and tradeoff: ~6/5 = [315.641, 317.263]

Optimal GPV sequence15, 19, 34, 53, 458, 511c, …, 882c

Music
Analysis and diagrams

### Cata

Main article: Catakleismic

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Sval mapping: [1 0 1 0], 0 6 5 14]]

POTE generator: ~6/5 = 317.0756

Optimal GPV sequence: 15, 19, 34, 53, 140, 193, 246

## Keemun

Main article: Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [1 0 1 2], 0 6 5 3]]

Wedgie⟨⟨6 5 3 -6 -12 -7]]

POTE generator: ~6/5 = 316.473

• 7-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
• 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [308.744, 322.942]
• 7-odd-limit diamond monotone and tradeoff: ~6/5 = [308.744, 322.942]
• 9-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 100/99

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

POTE generator: ~6/5 = 317.576

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]

Optimal GPV sequence: 4, 15, 19, 34

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 65/64, 100/99

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

POTE generator: ~6/5 = 316.611

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~6/5 = 315.789

Optimal GPV sequence: 4, 15f, 19, 53def, 72def

#### Kema

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 100/99

Mapping: [1 0 1 2 4 0], 0 6 5 3 -2 14]]

POTE generator: ~6/5 = 317.423

Tuning ranges:

• 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
• 13-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]
• 15-odd-limit diamond monotone and tradeoff: ~6/5 = 315.789

Optimal GPV sequence: 15, 19, 34, 87ddee

#### Kumbaya

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 56/55, 66/65

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

POTE generator: ~6/5 = 318.595

Optimal GPV sequence: 4, 15, 19f, 34ff

### Qeema

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 126/125

Mapping: [1 0 1 2 -1], 0 6 5 3 17]]

POTE generator: ~6/5 = 314.730

Optimal GPV sequence: 4e, 19, 42bcd, 61bcdd

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 126/125

Mapping: [1 0 1 2 -1 0], 0 6 5 3 17 14]]

POTE generator: ~6/5 = 315.044

Optimal GPV sequence: 4ef, 19

### Darjeeling

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 77/75

Mapping: [1 0 1 2 0], 0 6 5 3 13]]

POTE generator: ~6/5 = 317.656

Optimal GPV sequence: 15, 19e, 34e

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 66/65, 77/75

Mapping: [1 0 1 2 0 0], 0 6 5 3 13 14]]

POTE generator: ~6/5 = 317.298

Optimal GPV sequence: 15, 19e, 34e, 53dee

## Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [1 0 1 6], 0 6 5 -12]]

Wedgie⟨⟨6 5 -12 -6 -36 -42]]

POTE generator: ~6/5 = 318.267

• 7- and 9-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [315.641, 319.265]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [317.647, 319.265]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1331/1323

Mapping: [1 0 1 6 4], 0 6 5 -12 -2]]

POTE generator: ~6/5 = 318.282

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
• 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [317.647, 320.000]

Optimal GPV sequence: 15, 34d, 49

## Catakleismic

Main article: Catakleismic

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping: [1 0 1 -3], 0 6 5 22]]

Wedgie⟨⟨6 5 22 -6 18 37]]

POTE generator: ~6/5 = 316.732

• 7- and 9-odd-limit diamond monotone: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.263]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 317.263]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4375/4374

Mapping: [1 0 1 -3 9], 0 6 5 22 -21]]

POTE generator: ~6/5 = 316.719

Tuning ranges:

• 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
• 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 316.981]

Optimal GPV sequence: 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 385/384

Mapping: [1 0 1 -3 9 0], 0 6 5 22 -21 14]]

POTE generator: ~6/5 = 316.738

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 316.981]

Optimal GPV sequence: 19, 34de, 53, 72, 125f, 197ef, 269cef

### Cataclysmic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2200/2187

Mapping: [1 0 1 -3 -5], 0 6 5 22 32]]

POTE generator: ~6/5 = 317.042

Optimal GPV sequence: 19e, 34d, 53

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 275/273

Mapping: [1 0 1 -3 -5 0], 0 6 5 22 32 14]]

POTE generator: ~6/5 = 317.036

Optimal GPV sequence: 19e, 34d, 53

### Catalytic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4374

Mapping: [1 0 1 -3 -10], 0 6 5 22 51]]

POTE generator: ~6/5 = 316.653

Optimal GPV sequence: 19e, 53e, 72

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 1716/1715

Mapping: [1 0 1 -3 -10 0], 0 6 5 22 51 14]]

POTE generator: ~6/5 = 316.639

Optimal GPV sequence: 19e, 53e, 72

### Cataleptic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 864/847

Mapping: [1 0 1 -3 4], 0 6 5 22 -2]]

POTE generator: ~6/5 = 317.083

Optimal GPV sequence: 19, 34d, 53e

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 676/675

Mapping: [1 0 1 -3 4 0], 0 6 5 22 -2 14]]

POTE generator: ~6/5 = 317.118

Optimal GPV sequence: 19, 34d, 53e, 87dee

### Bikleismic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4375/4356

Mapping: [2 0 2 -6 -1], 0 6 5 22 15]]

POTE generator: ~6/5 = 316.721

Optimal GPV sequence: 34d, 72, 322c, …, 610bcc

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 325/324

Mapping: [2 0 2 -6 -1 0], 0 6 5 22 15 14]]

POTE generator: ~6/5 = 316.726

Optimal GPV sequence: 34d, 72

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [2 0 2 -6 -1 0 5], 0 6 5 22 15 14 6]]

POTE generator: ~6/5 = 316.726

Optimal GPV sequence: 34d, 38df, 72

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [2 0 2 -6 -1 0 5 -1], 0 6 5 22 15 14 6 18]]

POTE generator: ~6/5 = 316.726

Optimal GPV sequence: 34dh, 38df, 72

## Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [1 0 1 11], 0 6 5 -31]]

Wedgie⟨⟨6 5 -31 -6 -66 -86]]

POTE generator: ~6/5 = 317.121

• 7- and 9-odd-limit diamond monotone: ~6/5 = [316.667, 317.647] (19\72 to 9\34)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.263]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [316.667, 317.263]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3388/3375

Mapping: [1 0 1 11 -5], 0 6 5 -31 32]]

POTE generator: ~6/5 = 317.162

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
• 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]
• 11-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.370]

Optimal GPV sequence: 34, 53, 87, 140, 227, 367e, 507e

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 625/624

Mapping: [1 0 1 11 -5 0], 0 6 5 -31 32 14]]

POTE generator: ~6/5 = 317.162

Tuning ranges:

• 13-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
• 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
• 13-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.647]
• 15-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.241]

Optimal GPV sequence: 34, 53, 87, 140, 367e, 507e

## Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [1 0 1 -12], 0 6 5 56]]

Wedgie⟨⟨6 5 56 -6 72 116]]

POTE generator: ~6/5 = 317.314

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 14700/14641

Mapping: [1 0 1 -12 -5], 0 6 5 56 32]]

POTE generator: ~6/5 = 317.311

Optimal GPV sequence: 34d, 53d, 87, 121, 208

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 625/624

Mapping: [1 0 1 -12 -5 0], 0 6 5 56 32 14]]

POTE generator: ~6/5 = 317.311

Optimal GPV sequence: 34d, 53d, 87, 121, 208

## Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [1 0 1 4], 0 12 10 -9]]

Wedgie⟨⟨12 10 -9 -12 -48 -49]]

POTE generator: ~35/32 = 158.649

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 4000/3969

Mapping: [1 0 1 4 2], 0 12 10 -9 11]]

POTE generator: ~11/10 = 158.677

Optimal GPV sequence: 15, 38, 53, 68, 121e

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 275/273, 325/324

POTE generator: ~11/10 = 158.655

Mapping: [1 0 1 4 2 0], 0 12 10 -9 11 28]]

Optimal GPV sequence: 15, 38f, 53, 121e

## Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping: [1 6 6 12], 0 -12 -10 -25]]

Mapping generators: ~2, ~9/7

Wedgie⟨⟨12 10 25 -12 6 30]]

POTE generator: ~9/7 = 441.335

[[1 0 0 0, [6/25 0 0 12/25, [6/5 0 0 2/5, [0 0 0 1]
Eigenmonzos (unchanged intervals): 2, 7

Algebraic generator: real root of 5x3 - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3136/3125

Mapping: [1 6 6 12 -5], 0 -12 -10 -25 23]]

POTE generator: ~9/7 = 441.355

Optimal GPV sequence: 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 625/624

Mapping: [1 6 6 12 -5 14], 0 -12 -10 -25 23 -28]]

POTE generator: ~9/7 = 441.363

Optimal GPV sequence: 19, 49ef, 68, 87, 503bdf, 590bdf

## Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

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

Mapping generators: ~63/50, ~6/5

Wedgie⟨⟨18 15 -6 -18 -60 -56]]

POTE generator: ~6/5 = 316.872

[[1 0 0 0, [2 0 6/7 -6/7, [8/3 0 5/7 -5/7, [8/3 0 -2/7 2/7]
Eigenmonzos (unchanged intervals): 2, 7/5
[[1 0 0 0, [10/7 6/7 0 -3/7, [46/21 5/7 0 -5/14, [20/7 -2/7 0 1/7]
Eigenmonzos (unchanged intervals): 2, 9/7

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

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

Mapping generators: ~44/35, ~6/5

POTE generator: ~6/5 = 316.881

Minimax tuning:

• 11-odd-limit: ~6/5 = [5/21 1/7 0 -1/14
[[1 0 0 0 0, [10/7 6/7 0 -3/7 0, [46/21 5/7 0 -5/14 0, [20/7 -2/7 0 1/7 0, [71/21 3/7 0 -3/14 0]
Eigenmonzos (unchanged intervals): 2, 9/7

Optimal GPV sequence: 15, 42bc, 57, 72, 159, 231

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

Mapping: [3 0 3 10 8 0], 0 6 5 -2 3 14]]

Mapping generators: ~44/35, ~6/5

POTE generator: ~6/5 = 316.9585

Optimal GPV sequence: 72, 87, 159

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [3 0 3 10 8 0 -2], 0 6 5 -2 3 14 18]]

Mapping generators: ~34/27, ~6/5

POTE generator: ~6/5 = 316.9082

Optimal GPV sequence: 72, 159, 231f

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping: [4 0 4 7], 0 6 5 4]]

Wedgie⟨⟨24 20 16 -24 -42 -19]]

POTE generator: ~6/5 = 316.9999

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 6250/6237

Mapping: [4 0 4 7 17], 0 6 5 4 -3]]

POTE generator: ~6/5 = 316.9247

Optimal GPV sequence: 68, 72, 140, 212, 284, 496ce, 780ccdee

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 1375/1372

Mapping: [4 0 4 7 17 0], 0 6 5 4 -3 14]]

POTE generator: ~6/5 = 316.9887

Optimal GPV sequence: 68, 72, 140, 212

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 625/624

Mapping: [4 0 4 7 17 0 10], 0 6 5 4 -3 14 6]]

POTE generator: ~6/5 = 316.9846

Optimal GPV sequence: 68, 72, 140, 212g

## Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [1 6 6 6], 0 -18 -15 -13]]

Wedgie⟨⟨18 15 13 -18 -30 -12]]

POTE generator: ~25/21 = 294.303

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 3125/3087

Mapping: [1 6 6 6 14], 0 -18 -15 -13 -43]]

POTE generator: ~25/21 = 294.181

Optimal GPV sequence: 49, 53, 102d, 155d

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 275/273, 325/324, 540/539

Mapping: [1 6 6 6 14 14], 0 -18 -15 -13 -43 -42]]

POTE generator: ~13/11 = 294.187

Optimal GPV sequence: 49f, 53, 102df, 155d

## Marfifths

The marfifths temperament (19&140) tempers out the hemimage comma, 10976/10935. It splits the interval of major tenth (~10/3) into three marvelous fifth (112/75) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 15625/15552

Mapping: [1 -6 -4 -17], 0 18 15 47]]

Wedgie⟨⟨18 15 47 -18 24 67]]

POTE generator: ~75/56 = 505.705

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 10976/10935

Mapping: [1 -6 -4 -17 22], 0 18 15 47 -44]]

POTE generator: ~75/56 = 505.684

Optimal GPV sequence: 19, 140, 159, 299

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 10976/10935

Mapping: [1 -6 -4 -17 22 -14], 0 18 15 47 -44 42]]

POTE generator: ~75/56 = 505.686

Optimal GPV sequence: 19, 140, 159, 299

### Diatessic

The diatessic temperament (121&140) is closely related to the diatess tuning (generator: 505.727281 cents).

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2200/2187, 5632/5625

Mapping: [1 -6 -4 -17 -37], 0 18 15 47 96]]

POTE generator: ~75/56 = 505.740

Optimal GPV sequence: 19e, 121, 140, 261, 401, 662bd

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 625/624, 1375/1372

Mapping: [1 -6 -4 -17 -37 -14], 0 18 15 47 96 42]]

POTE generator: ~75/56 = 505.740

Optimal GPV sequence: 19e, 121, 140, 261, 401, 662bd

### Marf

The marf temperament (19&121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 15625/15552

Mapping: [1 -6 -4 -17 14], 0 18 15 47 -25]]

POTE generator: ~75/56 = 505.769

Optimal GPV sequence: 19, 102d, 121

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539, 625/624, 896/891

Mapping: [1 -6 -4 -17 14 -14], 0 18 15 47 -25 42]]

POTE generator: ~75/56 = 505.771

Optimal GPV sequence: 19, 121

## Marthirds

The marthirds temperament (19&193) tempers out the sesquiquartisma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third (56/45) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 2460375/2458624

Mapping: [1 -6 -4 -19], 0 24 20 69]]

Wedgie⟨⟨24 20 69 -24 42 104]]

POTE generator: ~56/45 = 379.252

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 15625/15552, 19712/19683

Mapping: [1 -6 -4 -19 -43], 0 24 20 69 147]]

POTE generator: ~56/45 = 379.257

Optimal GPV sequence: 19e, 193, 212, 405, 617c, 1022cce

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 19712/19683

Mapping: [1 -6 -4 -19 -43 -14], 0 24 20 69 147 56]]

POTE generator: ~56/45 = 379.256

Optimal GPV sequence: 19e, 193, 212, 405f, 617cff

## Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping: [9 0 9 11], 0 6 5 6]]

Wedgie⟨⟨54 45 54 -54 -66 -1]]

POTE generator: ~6/5 = 317.005

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4000/3993, 15625/15552

Mapping: [9 0 9 11 24], 0 6 5 6 3]]

POTE generator: ~6/5 = 317.010

Optimal GPV sequence: 72, 261, 333, 405, 882c

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 4000/3993

Mapping: [9 0 9 11 24 0], 0 6 5 6 3 14]]

POTE generator: ~6/5 = 317.086

Optimal GPV sequence: 72, 261, 333, 738cf, 1071bcff

## Sqrtphi

Main article: Sqrtphi

The just value of sqrt (φ) is 416.545 cents.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 16875/16807

Mapping: [1 12 11 16], 0 -30 -25 -38]]

POTE generator: ~125/98 = 416.603

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4375/4356

Mapping: [1 12 11 16 17], 0 -30 -25 -38 -39]]

POTE generator: ~14/11 = 416.604

Optimal GPV sequence: 49, 72, 193, 265

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624, 1375/1372

Mapping: [1 12 11 16 17 28], 0 -30 -25 -38 -39 -70]]

POTE generator: ~14/11 = 416.585

Optimal GPV sequence: 72, 121, 193

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

Mapping: [1 12 11 16 17 28 27], 0 -30 -25 -38 -39 -70 -66]]

POTE generator: ~14/11 = 416.585

Optimal GPV sequence: 72, 121, 193

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

Mapping: [1 12 11 16 17 28 27 -2], 0 -30 -25 -38 -39 -70 -66 18]]

POTE generator: ~14/11 = 416.580

Optimal GPV sequence: 72, 121, 193