Pythagorean family

The Pythagorean 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 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

Pythagorean

Comma list: 531441/524288

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

Mapping: [⟨12 19 0], ⟨0 0 1]

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; 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 the either the 5 or 7-limit, 240edo is an excellent tuning, with 81/80 coming in at 15 cents exactly. 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.

Comma list: 225/224, 250047/250000

POTE generator: ~5/4 = 383.775 or ~81/80 = 16.225

Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]

11-limit

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

POTE generator: ~5/4 = 383.266 or ~81/80 = 16.734

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

13-limit

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

POTE generator: ~5/4 = 383.963 or ~81/80 = 16.037

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

Badness: 0.0219

Comptone

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

POTE generator: ~5/4 = 382.612 or ~81/80 = 17.388

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

Badness: 0.0251

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.

Comma list: 81/80, 128/125

POTE generator: ~64/63 = 26.790

Mapping: [⟨12 19 28 0], ⟨0 0 0 1]]

11-limit

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]]

Badness: 0.0582

Catlat

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]]

Badness: 0.0819

Catcall

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]]

Badness: 0.0345

13-limit

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]]

Badness: 0.0284

Duodecic

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]]

Badness: 0.0383

17-limit

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]]

Badness: 0.0275

19-limit

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]]

Badness: 0.0209

Duodecim

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

POTE generator: ~45/44 = 34.977

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

Omicronbeta

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

POTE generator: ~13/8 = 837.814

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

Badness: 0.0300

Hours

Commas: 19683/19600, 33075/32768

POTE generator: ~225/224 = 2.100

Map: [<24 38 0 123 83|, <0 0 1 -1 0|]

Wedgie: <0 24 -24 38 -38 -123|

EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd

Badness: 0.1161

11-limit

Commas: 243/242, 385/384, 9801/9800

POTE generator: ~225/224 = 2.161

Map: [<24 38 0 123 83|, <0 0 1 -1 0|]

EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde

Badness: 0.0362

13-limit

Commas: 243/242, 351/350, 364/363, 385/384

POTE generator: ~225/224 = 3.955

Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|]

EDOs: 24, 48f, 72, 168df, 240df

Badness: 0.0269