# Compton family

The **Compton family** tempers out the Pythagorean comma, 531441/524288 = [-19 12⟩, and hence the fifths form a closed 12-note circle of fifths, identical to 12EDO. While the tuning of the fifth will be that of 12EDO, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

## Compton

Subgroup: 2.3.5

Comma: 531441/524288

Mapping: [⟨12 19 0], ⟨0 0 1]

POTE generator: ~5/4 = 384.884 or ~81/80 = 15.116

Optimal GPV sequence: 12, 72, 84, 156, 240, 396b

Badness: 0.094494

## Septimal compton

In terms of the normal list, compton adds 413343/409600 = [-14 10 -2 1⟩ to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament (also called as *waage*); in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and 72EDO, 84EDO or 240EDO make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.

In either the 5 or 7-limit, 240EDO is an excellent tuning, with 81/80 coming in at 15 cents exactly. In the 12EDO, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this 72EDO can be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 250047/250000

Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]

POTE generator: ~5/4 = 383.7752

Optimal GPV sequence: 12, 60, 72, 228, 300c, 372bc, 444bc

Badness: 0.035686

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4356

Mapping: [⟨12 19 0 -22 -42], ⟨0 0 1 2 3]]

POTE generator: ~5/4 = 383.2660

Optimal GPV sequence: 12, 60e, 72

Badness: 0.022235

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 441/440

Mapping: [⟨12 19 0 -22 -42 -67], ⟨0 0 1 2 3 4]]

POTE generator: ~5/4 = 383.9628

Optimal GPV sequence: 12f, 72, 84, 156, 228f, 300cf

Badness: 0.021852

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 225/224, 289/288, 351/350, 441/440

Mapping: [⟨12 19 0 -22 -42 -67 49], ⟨0 0 1 2 3 4 0]]

POTE generator: ~5/4 = 383.7500

Optimal GPV sequence: 12f, 72, 84, 156g, 228fg

Badness: 0.017131

#### Comptone

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 441/440, 1001/1000

Mapping: [⟨12 19 0 -22 -42 100], ⟨0 0 1 2 3 -2]]

POTE generator: ~5/4 = 382.6116

Optimal GPV sequence: 12, 60e, 72, 204cdef, 276cdef

Badness: 0.025144

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 289/288, 325/324, 441/440

Mapping: [⟨12 19 0 -22 -42 100 49], ⟨0 0 1 2 3 -2 0]]

POTE generator: ~5/4 = 382.5968

Optimal GPV sequence: 12, 60e, 72, 132deg, 204cdefg

Badness: 0.016361

## Catler

In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of 12EDO. Catler can also be characterized as the 12&24 temperament. 36EDO or 48EDO are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7, 7/5, and most importantly, 64/63.

Subgroup: 2.3.5.7

Comma list: 81/80, 128/125

Mapping: [⟨12 19 28 0], ⟨0 0 0 1]]

POTE generator: ~64/63 = 26.790

Optimal GPV sequence: 12, 24, 36, 48c

Badness: 0.050297

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 128/125

POTE generator: ~64/63 = 22.723

Mapping: [⟨12 19 28 0 -26], ⟨0 0 0 1 2]]

Optimal GPV sequence: 12, 36e, 48c, 108ccd

Badness: 0.058213

### Catlat

Subgroup: 2.3.5.7.11

Comma list: 81/80, 128/125, 540/539

POTE generator: ~64/63 = 27.864

Mapping: [⟨12 19 28 0 109], ⟨0 0 0 1 -2]]

Optimal GPV sequence: 36, 48c, 84c

Badness: 0.081909

### Catcall

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 128/125

POTE generator: ~36/35 = 32.776

Mapping: [⟨12 19 28 0 8], ⟨0 0 0 1 1]]

Optimal GPV sequence: 12, 24, 36, 72ce

Badness: 0.034478

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 81/80, 105/104

POTE generator: ~36/35 = 37.232

Mapping: [⟨12 19 28 0 8 11], ⟨0 0 0 1 1 1]]

Optimal GPV sequence: 12f, 24, 36f, 60cf

Badness: 0.028363

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 56/55, 66/65, 81/80, 105/104

POTE generator: ~36/35 = 39.777

Mapping: [⟨12 19 28 0 8 11 49], ⟨0 0 0 1 1 1 0]]

Optimal GPV sequence: 12f, 24, 36f, 60cf

Badness: 0.023246

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95

POTE generator: ~36/35 = 40.165

Mapping: [⟨12 19 28 0 8 11 49 51], ⟨0 0 0 1 1 1 0 0]]

Optimal GPV sequence: 12f, 24, 36f, 60cf

Badness: 0.018985

#### Duodecic

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 81/80, 91/90, 128/125

POTE generator: ~36/35 = 37.688

Mapping: [⟨12 19 28 0 8 78], ⟨0 0 0 1 1 -1]]

Optimal GPV sequence: 12, 24, 36, 60c

Badness: 0.038307

##### 17-limit

Subgroup: 2.3.5.7.11.13.17 Comma list: 51/50, 56/55, 81/80, 91/90, 128/125

POTE generator: ~36/35 = 38.097

Mapping: [⟨12 19 28 0 8 78 49], ⟨0 0 0 1 1 -1 0]]

Optimal GPV sequence: 12, 24, 36, 60c

Badness: 0.027487

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95

POTE generator: ~36/35 = 38.080

Mapping: [⟨12 19 28 0 8 78 49 51], ⟨0 0 0 1 1 -1 0 0]]

Optimal GPV sequence: 12, 24, 36, 60c

Badness: 0.020939

## Duodecim

*See also: Jubilismic clan #Duodecim*

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 64/63

Mapping: [⟨12 19 28 34 0], ⟨0 0 0 0 1]]

POTE generator: ~45/44 = 34.977

Badness: 0.030536

## Hours

The hours temperament has a period of 1/24 octave and tempers out the cataharry comma (19683/19600) and the mirwomo comma (33075/32768). The name "hours" was so named for the following reasons - the period is 1/24 octave, and there are 24 hours per a day.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 33075/32768

Mapping: [⟨24 38 0 123], ⟨0 0 1 -1]]

Wedgie: ⟨⟨0 24 -24 38 -38 -123]]

POTE generator: ~5/4 = 384.033

Optimal GPV sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd

Badness: 0.116091

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 9801/9800

Mapping: [⟨24 38 0 123 83], ⟨0 0 1 -1 0]]

POTE generator: ~5/4 = 384.054

Optimal GPV sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde

Badness: 0.036248

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 385/384

Mapping: [⟨24 38 0 123 83 33], ⟨0 0 1 -1 0 1]]

POTE generator: ~5/4 = 384.652

Optimal GPV sequence: 24, 48f, 72, 168df, 240dff

Badness: 0.026931

## Decades

The decades temperament has a period of 1/36 octave and tempers out the gamelisma (1029/1024) and the stearnsma (118098/117649). The name "decades" was so named for the following reasons - the period is 1/36 octave, and there are 36 decades (*ten days*) per a year (12 months × 3 decades per a month).

Subgroup: 2.3.5.7

Comma list: 1029/1024, 118098/117649

Mapping: [⟨36 57 0 101], ⟨0 0 1 0]]

Wedgie: ⟨⟨0 36 0 57 0 -101]]

POTE generator: ~5/4 = 384.764

Optimal GPV sequence: 36, 72, 252, 324bd, 396bd

Badness: 0.108016

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1029/1024, 4000/3993

Mapping: [⟨36 57 0 101 41], ⟨0 0 1 0 1]]

POTE generator: ~5/4 = 384.150

Optimal GPV sequence: 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd

Badness: 0.043088

## Omicronbeta

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 441/440, 4375/4356

Mapping: [⟨72 114 167 202 249 266], ⟨0 0 0 0 0 1]]

POTE generator: ~13/8 = 837.814

Optimal GPV sequence: 72, 144, 216c, 288cdf, 504bcdef

Badness: 0.029956