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.
POTE generator: 15.116
Map: [<12 19 0|, <0 0 1|]
EDOs: 12, 72, 84, 156, 240, 396
Compton temperament[edit]
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.
Commas: 225/224, 250047/250000
POTE generator: ~5/4 = 383.775 (16.225)
Map: [<12 19 0 -22|, <0 0 1 2|]
EDOs: 12, 60, 72, 228, 300c, 372bc, 444bc
11-limit[edit]
Commas: 225/224, 441/440, 4375/4356
POTE generator: ~5/4 = 383.266 (16.734)
Map: [<12 19 0 -22 -42|, <0 0 1 2 3|]
EDOs: 12, 60e, 72
13-limit[edit]
Commas: 225/224, 441/440, 351/350, 364/363
POTE generator: ~5/4 = 383.963 (16.037)
Map: [<12 19 0 -22 -42 -67|, <0 0 1 2 3 4|]
EDOs: 72, 228f, 300cf
Badness: 0.0219
Comptone[edit]
Commas: 225/224, 441/440, 325/324, 1001/1000
POTE generator: ~5/4 = 382.612 (17.388)
Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|]
EDOs: 12, 60e, 72, 204cdef, 276cdef
Badness: 0.0251
Catler temperament[edit]
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, and 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7 or 7/5 are possible generators.
Commas: 81/80, 128/125
POTE generator: 26.790
Map: [<12 19 28 0|, <0 0 0 1|]
11-limit[edit]
Commas: 81/80, 99/98, 128/125
POTE generator: ~36/35 = 22.723
Map: [<12 19 28 0 -26|, <0 0 0 1 2|]
EDOs: 12, 48c, 108cd
Badness: 0.0582
Catlat[edit]
Commas: 81/80, 128/125, 540/539
POTE generator: ~36/35 = 27.864
Map: [<12 19 28 0 109|, <0 0 0 1 -2|]
EDOs: 36, 48c, 84c
Badness: 0.0819
Catcall[edit]
Commas: 56/55, 81/80, 128/125
POTE generator: ~36/35 = 32.776
Map: [<12 19 28 0 8|, <0 0 0 1 1|]
EDOs: 12, 24, 36, 72ce
Badness: 0.0345
13-limit[edit]
Commas: 56/55, 66/65, 81/80, 105/104
POTE generator: ~36/35 = 37.232
Map: [<12 19 28 0 8 11|, <0 0 0 1 1 1|]
EDOs: 12f, 24, 36f, 60cf
Badness: 0.0284
Omicronbeta temperament[edit]
Commas: 225/224, 243/242, 441/440, 4375/4356
Generator: ~13/8 = 837.814
Map: [<72 114 167 202 249 266|, <0 0 0 0 0 1|]
EDOs: 72, 144, 216c, 288cdf, 504bcdef
Badness: 0.0300
Hours[edit]
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[edit]
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[edit]
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