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, 2430/2401 gives quasisuper, and 4375/4374 gives modus.
Discussed under their respective 5-limit families are mother, dominant, augene, porcupine, pajara, catalan, modus, and immunized. The rest are considered below.
Archy
Subgroup: 2.3.7
Comma list: 64/63
Sval mapping: [⟨1 0 6], ⟨0 1 -2]]
Sval mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]
Gencom: [2 3/2; 64/63]
POTE generator: ~3/2 = 709.321
Scales: archy5, archy7, archy12
Supra
Subgroup: 2.3.7.11
Comma list: 64/63, 99/98
Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]
Sval mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]
Gencom: [2 3/2; 64/63 99/98]
POTE generator: ~3/2 = 707.192
Supraphon
Subgroup: 2.3.7.11.13
Comma list: 64/63, 78/77, 99/98
Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]
Sval mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]
Gencom: [2 3/2; 64/63 78/77 99/98]
POTE generator: ~3/2 = 706.137
Suhajira
Subgroup: 2.3.7.11
Comma list: 64/63, 243/242
Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]
Sval mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 0 4 2], ⟨0 1 0 -4 5]]
Gencom: [2 3/2; 64/63 99/98]
POTE generator: ~11/9 = 353.958
Scales: suhajira7, suhajira10, suhajira17
2.3.7.11.13
Subgroup: 2.3.7.11.13
Comma list: 64/63, 78/77, 144/143
Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]
Sval mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 1 0 -4 5 -1]]
Gencom: [2 3/2; 64/63 78/77 99/98]
POTE generator: ~11/9 = 353.775
Scales: suhajira7, suhajira10, suhajira17
Superpyth
Subgroup: 2.3.5.7
Comma list: 64/63, 245/243
Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]
Wedgie: ⟨⟨ 1 9 -2 12 -6 -30 ]]
POTE generator: ~3/2 = 710.291
Badness: 0.0323
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 245/243
Mapping: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]
POTE generator: ~3/2 = 710.175
Badness: 0.0250
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 91/90, 100/99
Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]
POTE generator: ~3/2 = 710.479
Badness: 0.0247
Suprapyth
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 99/98
Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]
POTE generator: ~3/2 = 709.495
Badness: 0.0328
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 65/63, 99/98
Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]
POTE generator: ~3/2 = 708.703
Badness: 0.0363
Quasisuper
Subgroup: 2.3.5.7
Comma list: 64/63, 2430/2401
Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]
Wedgie: ⟨⟨ 1 -13 -2 -23 -2 -6 32 ]]
POTE generator: ~3/2 = 708.328
Badness: 0.0638
Quasisupra
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).
Subgroup: 2.3.5.7.11
Comma list: 64/63, 99/98, 121/120
Mapping: [⟨1 2 -3 2 1], ⟨0 -1 13 2 6]]
POTE generator: ~3/2 = 708.205
Badness: 0.0322
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 91/90, 121/120
Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]
POTE generator: ~3/2 = 708.004
Badness: 0.0302
Quasisoup
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 2430/2401
Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]
POTE generator: ~3/2 = 709.021
Badness: 0.0835
Schism
Subgroup: 2.3.5.7
Comma list: 64/63, 360/343
Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]
Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]
POTE generator: ~3/2 = 701.556
Badness: 0.0566
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
POTE generator: ~3/2 = 702.136
Badness: 0.0375
Blacksmith
5-limit (blackwood)
Subgroup: 2.3.5
Comma list: 256/243
Mapping: [⟨5 8 0], ⟨0 0 1]]
Mapping generators: ~9/8, ~5
POTE generator: ~5/4 = 399.594
Badness: 0.0638
7-limit
Subgroup: 2.3.5.7
Comma list: 28/27, 49/48
Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]
Mapping generators: ~7/6, ~5
Wedgie: ⟨⟨ 0 5 0 8 0 -14 ]]
POTE generator: ~5/4 = 392.767
Badness: 0.0256
11-limit
Subgroup: 2.3.5.7.11
Comma list: 28/27, 49/48, 55/54
Mapping: [⟨5 8 0 14 29], ⟨0 0 1 0 -1]]
POTE generator: ~5/4 = 394.948
Badness: 0.0246
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 49/48, 55/54
Mapping: [⟨5 8 0 14 29 7], ⟨0 0 1 0 -1 1]]
POTE generator: ~5/4 = 391.0367
Badness: 0.0205
Farrier
Subgroup: 2.3.5.7.11
Comma list: 28/27, 49/48, 77/75
Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]
POTE generator: ~5/4 = 398.070
Badness: 0.0292
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 49/48, 66/65
Mapping: [⟨5 8 0 14 -6 7], ⟨0 0 1 0 2 1]]
POTE generator: ~5/4 = 396.812
Badness: 0.0223
Ferrum
Subgroup: 2.3.5.7.11
Comma list: 28/27, 35/33, 49/48
Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]
POTE generator: ~5/4 = 374.763
Badness: 0.0309
Beatles
5-limit
Subgroup: 2.3.5
Comma list: 524288/492075
Mapping: [⟨1 1 5], ⟨0 2 -9]]
POTE generator: ~512/405 = 355.930
Badness: 0.3585
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 686/675
Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]
Wedgie: ⟨⟨ 2 -9 -4 -19 -12 16 ]]
POTE generator: ~49/40 = 355.904
Badness: 0.0459
- Music
- Beatles Improv by Herman Miller
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 686/675
Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]
POTE generator: ~49/40 = 356.140
Badness: 0.0456
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 91/90, 100/99, 169/168
Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]
POTE generator: ~16/13 = 356.229
Badness: 0.0302
Ringo
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63, 540/539
Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]
POTE generator: ~11/9 = 355.419
Badness: 0.0329
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 64/63, 78/77, 91/90
Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]
POTE generator: ~11/9 = 355.456
Badness: 0.0226
Passion
5-limit
Subgroup: 2.3.5
Comma list: 262144/253125
Mapping: [⟨1 2 2], ⟨0 -5 4]]
Mapping generators: 2, 16/15
POTE generator: ~16/15 = 98.670
Badness: 0.1686
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 3125/3087
Mapping: [⟨1 2 2 2], ⟨0 -5 4 10]]
Mapping generators: 2, 16/15
Wedgie: ⟨⟨ 5 -4 -10 -18 -30 -12 ]]
POTE generator: ~16/15 = 98.153
Badness: 0.0623
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 1375/1372
Mapping: [⟨1 2 2 2 2], ⟨0 -5 4 10 18]]
POTE generator: ~16/15 = 98.019
Badness: 0.0408
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 100/99, 196/195, 275/273
Mapping: [⟨1 2 2 2 2 2], ⟨0 -5 4 10 18 21]]
POTE generator: ~16/15 = 97.910
Badness: 0.0309
Fervor
5-limit
Subgroup: 2.3.5
Comma list: 67108864/61509375
Mapping: [⟨1 4 -2], ⟨0 -5 9]]
POTE generator: ~64/45 = 577.705
Badness: 0.8526
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 9604/9375
Mapping: [⟨1 4 -2 -2], ⟨0 -5 9 10]]
Wedgie: ⟨⟨ 5 -9 -10 -26 -30 2 ]]
POTE generator: ~7/5 = 577.777
Badness: 0.1085
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63, 1350/1331
Mapping: [⟨1 4 -2 -2 3], ⟨0 -5 9 10 1]]
POTE generator: ~7/5 = 577.850
Badness: 0.0521
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 64/63, 78/77, 507/500
Mapping: [⟨1 4 -2 -2 3 -4], ⟨0 -5 9 10 1 16]]
POTE generator: ~7/5 = 578.060
Badness: 0.0397
Progress
5-limit
Subgroup: 2.3.5
Comma list: 32768/30375
Mapping: [⟨1 0 5], ⟨0 3 -5]]
POTE generator: ~64/45 = 561.264
Badness: 0.2461
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 392/375
Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]
Wedgie: ⟨⟨ 3 -5 -6 -15 -18 0 ]]
POTE generator: ~7/5 = 562.122
Badness: 0.0664
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63, 77/75
Mapping: [⟨1 0 5 6 4], ⟨0 3 -5 -6 -1]]
POTE generator: ~7/5 = 562.085
Badness: 0.0310
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 64/63, 66/65, 77/75
Mapping: [⟨1 0 5 6 4 0], ⟨0 3 -5 -6 -1 7]]
POTE generator: ~7/5 = 562.365
Badness: 0.0262
Progressive
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 56/55, 64/63, 77/75
Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]
POTE generator: ~7/5 = 563.239
Badness: 0.0327
Sixix
- See also: Dual-fifth temperaments#Dual-3 Sixix
5-limit
Subgroup: 2.3.5
Comma list: 3125/2916
Mapping: [⟨1 3 4], ⟨0 -5 -6]]
POTE generator: ~6/5 = 338.365
Badness: 0.1531
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 3125/2916
Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]
POTE generator: ~6/5 = 337.4419