Archytas clan
The archytas clan tempers out the Archytas comma, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to 81/80 in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of 22edo. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, and 2430/2401 gives quasisuper. Pajara is discussed under diaschismic family, dominant under meantone family, augene under augmented family and porcupine under porcupine family. The rest are considered below.
Mother
Commas: 16/15, 21/20
POTE generator: 721.569
Map: [<1 0 4 6|, <0 1 -1 -2|]
Wedgie: <<1 -1 -2 -4 -6 -2||
EDOs: 5, 148, 153
Badness: 0.0242
Blacksmith
Commas: 28/27, 49/48
POTE generator: ~5/4 = 392.767
Map: [<5 8 0 14|, <0 0 1 0|]
Wedgie: <<0 5 0 8 0 -14||
EDOs: 5, 10, 15, 40b, 55b
Badness: 0.0256
11-limit
Commas: 28/27, 49/48, 55/54
POTE generator: ~5/4 = 394.948
Map: [<5 8 0 14 29|, <0 0 1 0 -1|]
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde
Badness: 0.0246
13-limit
Commas: 28/27, 40/39, 49/48, 55/54
POTE generator: ~5/4 = 391.0367
Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]
EDOs: 5, 10, 15, 25e, 40bef
Badness: 0.0205
Farrier
Commas: 28/27, 49/48, 77/75
POTE generator: ~5/4 = 398.070
Map: [<5 8 0 14 -6|, <0 0 1 0 2|]
EDOs: 5e, 15
Badness: 0.0292
13-limit
Commas: 28/27, 40/39, 49/48, 66/65
POTE generator: ~5/4 = 396.812
Map: [<5 8 0 14 -6 7|, <0 0 1 0 2 1|]
EDOs: 5e, 10e, 15
Badness: 0.0223
Ferrum
Commas: 28/27, 35/33, 49/48
POTE generator: ~5/4 = 374.763
Map: [<5 8 0 14 6|, <0 0 1 0 1|]
EDOs: 10
Badness: 0.0309
Blackwood
Blackwood (5-limit)
Comma: 256/243
POTE generator: 399.594
Map: [<5 8 0|, <0 0 1|]
EDOs: 5, 10, 15
Badness: 0.0638
Superpyth
Commas: 64/63, 245/243
POTE generator: 710.291
Map: [<1 0 -12 6|, <0 1 9 -2|]
Wedgie: <<1 9 -2 12 -6 -30||
EDOs: 5, 17, 22, 27, 49
Badness: 0.0323
11-limit
Commas: 64/63, 100/99, 245/243
POTE generator: 710.175
Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
EDOs: 22, 49
Badness: 0.0250
13-limit
Commas: 64/63, 78/77, 91/90, 100/99
POTE generator: ~3/2 = 710.479
Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
EDOs: 22, 27e, 49, 76bcde
Badness: 0.0247
Suprapyth
Commas: 55/54, 64/63, 99/98
POTE generator: ~3/2 = 709.495
Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
EDOs: 5, 7, 12, 17, 22
Badness: 0.0328
13-limit
Commas: 55/54, 64/63, 65/63, 364/363
POTE generator: ~3/2 = 708.703
Map: [<1 0 -12 6 13 18|, <0 1 9 -2 -6 -9|]
EDOs: 17, 22, 83cdf
Badness: 0.0363
Beatles
Comma: 524288/492075
POTE generator: ~512/405 = 355.930
Map: [<1 1 5|,<0 2 -9|]
EDOs: 10, 17c, 27, 64b, 91bc, 118bc
Badness: 0.3585
7-limit
Commas: 64/63, 686/675
POTE generator: ~49/40 = 355.904
Map: [<1 1 5 4|,<0 2 -9 -4|]
Wedgie: <<2 -9 -4 -19 -12 16||
EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd
Badness: 0.0459
Music: Beatles Improv by Herman Miller
11-limit
Commas: 64/63, 100/99, 686/675
POTE generator: ~49/40 = 356.140
Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]
EDOs: 27e, 37, 64be, 91bcde
Badness: 0.0456
13-limit
Commas: 64/63, 91/90, 100/99, 169/168
POTE generator: ~16/13 = 356.229
Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|]
EDOs: 27e, 37, 64be
Badness: 0.0302
Ringo
Commas: 56/55, 64/63, 540/539
POTE generator: ~11/9 = 355.419
Map: [<1 1 5 4 2|,<0 2 -9 -4 5|]
EDOs: 10, 17c, 27e
Badness: 0.0329
13-limit
Commas: 56/55, 64/63, 78/77, 91/90
POTE generator: ~11/9 = 355.456
Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|]
EDOs: 10, 17c, 27e
Badness: 0.0226
Schism
Commas: 64/63, 360/343
POTE generator: ~3/2 = 701.556
Map: [<1 0 15 6|, <0 1 -8 -2|]
Wedgie: <<1 -8 -2 -15 -6 18||
EDOs: 12, 41d, 53d
Badness: 0.0566
11-limit
Commas: 45/44, 64/63, 99/98
POTE generator ~3/2 = 702.136
Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
EDOs: 12, 29de, 41de
Badness: 0.0375
Passion
5-limit
Comma: 262144/253125
POTE generator: ~16/15 = 98.670
Map: [<1 2 2|, <0 -5 4|]
EDOs: 11, 12, 49, 61, 73
Badness: 0.1686
Passive
Commas: 225/224, 256/245
POTE generator: ~16/15 = 98.809
Map: [<1 2 2 3|, <0 -5 4 -2|]
EDOs: 1, 11, 12, 49d
Badness: 0.0751
7-limit
Commas: 64/63, 3125/3087
POTE generator: ~16/15 = 98.153
Map: [<1 2 2 2|, <0 -5 4 10|]
Wedgie: <<5 -4 -10 -18 -30 -12||
Generators: 2, 16/15
EDOs: 12, 37, 49, 110bcd
Badness: 0.0623
11-limit
Commas: 64/63, 100/99, 1375/1372
POTE generator: ~16/15 = 98.019
Map: [<1 2 2 2 2|, <0 -5 4 10 18|]
EDOs: 12, 37, 49
Badness: 0.0408
13-limit
Commas: 64/63 100/99 196/195 275/273
POTE generator: ~16/15 = 97.910
Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|]
EDOs: 12f, 37, 49f
Badness: 0.0309
Fervor
Comma: 67108864/61509375
POTE generator: ~64/45 = 577.705
Map: [<1 4 -2|, <0 -5 9|]
EDOs: 25, 27
Badness: 0.8526
7-limit
Commas: 64/63, 9604/9375
POTE generator: ~7/5 = 577.777
Map: [<1 4 -2 -2|, <0 -5 9 10|]
Wedgie: <<5 -9 -10 -26 -30 2||
EDOs: 25, 27
Badness: 0.1085
11-limit
Commas: 56/55, 64/63, 1350/1331
POTE generator: ~7/5 = 577.850
Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|]
EDOs: 25e, 27e
Badness: 0.0521
13-limit
Commas: 56/55, 64/63, 78/77, 507/500
POTE generator: ~7/5 = 578.060
Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|]
EDOs: 27e
Badness: 0.0397
Quasisuper
Commas: 64/63, 2430/2401
POTE generator: 708.328
Map: [<1 0 23 6|, <0 1 -13 -2|]
Wedgie: <<1 -13 -2 -23 -2 -6 32||
EDOs: 22, 61
Badness: 0.0638
Quasisupra
Commas: 64/63, 99/98, 121/120
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament supra, with the quasisuper mapping of 5 thrown in (rather than the superpyth mapping of 5, which results in suprapyth).
POTE generator: ~3/2 = 708.205
Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]
EDOs: 17c, 22, 27c, 39d, 61d
Badness: 0.0322
13-limit
Commas: 64/63, 78/77, 91/90, 121/120
POTE generator: ~3/2 = 708.004
Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|]
EDOs: 17c, 22, 39d, 61df, 100bcdf
Badness: 0.0302
Quasisoup
Commas: 55/54, 64/63, 2430/2401
POTE generator: ~3/2 = 709.021
Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|]
EDOs: 22
Badness: 0.0835
Progress
Comma: 32768/30375
POTE generator: ~64/45 = 561.264
Map: [<1 0 5|, <0 3 -5|]
EDOs: 4, 13, 15, 32c, 47bc, 62bc
Badness: 0.2461
7-limit
Commas: 64/63, 392/375
POTE generator: ~7/5 = 562.122
Map: [<1 0 5 6|, <0 3 -5 -6|]
Wedgie: <<3 -5 -6 -15 -18 0||
EDOs: 13, 15, 32c, 79bcc, 111bcc
Badness: 0.0664
11-limit
Commas: 56/55, 64/63, 77/75
POTE generator: ~7/5 = 562.085
Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]
EDOs: 13, 15, 32c, 47bc, 79bcce
Badness: 0.0310
13-limit
Commas: 56/55, 64/63, 66/65, 77/75
POTE generator: ~7/5 = 562.365
Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|]
EDOs: 15, 17c, 32cf
Badness: 0.0262
Progressive
Commas: 26/25, 56/55, 64/63, 77/75
POTE generator: ~7/5 = 563.239
Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|]
EDOs: 15f, 17c, 32c, 49c
Badness: 0.0327