Schismatic family
The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is [-15 8 1⟩, and flipping that yields ⟨⟨ 1 -8 -15 ]] for the wedgie. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth. In fact, 10 = (4/3)8 × 32805/32768.
Schismatic aka Helmholtz
The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz, schismic or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismatic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
Comma list: 32805/32768
POTE generator: ~3/2 = 701.736
Mapping generators: ~2, ~3
Mapping: [⟨1 0 15], ⟨0 1 -8]]
- valid range: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
- nice range: ~3/2 = [701.711, 701.955]
- strict range: ~3/2 = [701.711, 701.955]
Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc
Badness: 0.00426
Seven-limit extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- Garibaldi adds [25 -14 0 -1⟩,
- Grackle adds [-44 26 0 1⟩,
- Schism adds [6 -2 0 -1⟩,
- Pontiac adds [-59 39 0 -1⟩.
Those all have a fifth as generator.
- Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period.
- Guiron adds [-10 1 0 3⟩, with an 8/7 generator, three of which give the fifth.
- Term adds [-94 54 0 3⟩ with a 1/3 octave period.
- Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.
Temperaments not discussed here include salsa, hemischis and karadeniz.
Garibaldi
Comma list: 225/224, 3125/3087
POTE generator: ~3/2 = 702.085
Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -14 -15 -25 -10 ]]
- 7-odd-limit:
- [[1 0 0 0⟩, [5/3 1/15 0 -1/15⟩, [5/3 -8/15 0 8/15⟩, [5/3 -14/15 0 14/15⟩]
- Eigenmonzos: 2, 7/6
- 9-odd-limit:
- [[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]
- Eigenmonzos: 2, 9/7
- valid range: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- nice range: ~3/2 = [701.711, 702.915]
- strict range: ~3/2 = [701.711, 702.915]
Badness: 0.0216
11-limit
Comma list: 225/224, 385/384, 2200/2187
POTE generator: ~3/2 = 702.157
Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]
Mapping generators: ~2, ~3
Minimax tuning:
- 11-odd-limit:
- [[1 0 0 0 0⟩, [25/16 1/8 0 -1/16 0⟩, [5/2 -1 0 1/2 0⟩, [25/8 -7/4 0 7/8 0⟩, [47/16 23/8 0 -23/16 0⟩]
- Eigenmonzos: 2, 9/7
Badness: 0.0274
13-limit
Comma list: 225/224, 275/273, 325/324, 385/384
POTE generator: ~3/2 = 702.113
Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]
Mapping generators: ~2, ~3
Badness: 0.0207
Andromeda
Comma list: 100/99, 225/224, 245/242
POTE generator: ~3/2 = 702.321
Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]
Mapping generators: ~2, ~3
Badness: 0.0236
13-limit
Comma list: 100/99, 105/104, 196/195, 245/242
POTE generator: ~3/2 = 702.559
Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]
Mapping generators: ~2, ~3
Badness: 0.0207
Helenus
Comma list: 99/98, 176/175, 3125/3087
POTE generator: ~3/2 = 701.725
Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]
Mapping generators: ~2, ~3
Badness: 0.0356
13-limit
Comma list: 99/98, 176/175, 275/273, 847/845
POTE generator: ~3/2 = 701.747
Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]
Mapping generators: ~2, ~3
Badness: 0.0263
Hemigari
Comma list: 121/120, 225/224, 3125/3087
POTE generator: ~63/55 = 248.918
Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]
Mapping generators: ~2, ~110/63
Badness: 0.0507
13-limit
Comma list: 121/120, 169/168, 225/224, 275/273
POTE generator: ~15/13 = 248.918
Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]
Mapping generators: ~2, ~26/15
Badness: 0.027464
Sanjaab
Comma list: 225/224, 1331/1323, 3125/3087
POTE generator: ~11/10 = 165.974
Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]
Mapping generators: ~2, ~11/10
Badness: 0.0580
13-limit
Comma list: 225/224, 275/273, 847/845, 1331/1323
POTE generator: ~11/10 = 165.963
Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]
Mapping generators: ~2, ~11/10
Badness: 0.0338
Guiron
Comma list: 1029/1024, 10976/10935
POTE generator: ~8/7 = 233.930
Mapping: [⟨1 1 7 3], ⟨0 3 -24 -1]]
Mapping generators: ~2, ~8/7
Wedgie: ⟨⟨ 3 -24 -1 -45 -10 65 ]]
- 7- and 9-odd-limit:
- [[1 0 0 0⟩, [15/8 0 -1/8 0⟩, [0 0 1 0⟩, [65/24 0 1/24 0⟩]
- Eigenmonzos: 2, 5/4
Badness: 0.0475
11-limit
Comma list: 385/384, 441/440, 10976/10935
POTE generator: ~8/7 = 233.931
Mapping: [⟨1 1 7 3 -2], ⟨0 3 -24 -1 28]]
Mapping generators: ~2, ~8/7
Minimax tuning:
- [[1 0 0 0 0⟩, [15/8 0 -1/8 0 0⟩, [0 0 1 0 0⟩, [65/24 0 1/24 0 0⟩, [37/6 0 -7/6 0 0⟩]
- Eigenmonzos: 2, 5/4
Badness: 0.0266
13-limit
Comma list: 196/195, 352/351, 385/384, 729/728
POTE generator: ~8/7 = 233.890
Mapping generators: ~2, ~8/7
Mapping: [⟨1 1 7 3 -2 0], ⟨0 3 -24 -1 28 19]]
Badness: 0.0284
Pogo
Comma list: 32805/32768, 118098/117649
POTE generator: ~9/7 = 433.901
Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]
Mapping generators: ~343/243, ~9/7
Badness: 0.0796
11-limit
Comma list: 540/539, 4000/3993, 32805/32768
POTE generator: ~9/7 = 433.911
Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]
Mapping generators: ~99/70, ~9/7
Badness: 0.0319
13-limit
Comma list: 540/539, 729/728, 4000/3993, 4225/4224
POTE generator: ~9/7 = 433.911
Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]
Mapping generators: ~99/70, ~9/7
Badness: 0.0175
Squirrel
Commmas: 686/675, 32805/32768
POTE generator: ~160/147 = 166.140
Map: [<1 2 -1 1|, <0 -3 24 13|]
EDOs: 29, 36, 65
Badness: 0.1747
11-limit
Commas: 245/242, 686/675, 896/891
POTE generator: ~11/10 = 166.097
Map: [<1 2 -1 1 0|, <0 -3 24 13 25|]
EDOs: 29, 36, 65
Badness: 0.0683
13-limit
Commas: 91/90, 169/168, 245/242, 896/891
POTE generator: ~11/10 = 166.054
Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|]
EDOs: 29, 36, 65f, 94df, 159df
Badness: 0.0437
Schism
Commas: 64/63, 360/343
POTE generator: ~3/2 = 701.556
Mapping generator: ~3
Map: [<1 0 15 6|, <0 1 -8 -2|]
Wedgie: <<1 -8 -2 -15 -6 18||
EDOs: 12, 29d, 41d, 53d
Badness: 0.0566
11-limit
Commas: 45/44, 64/63, 99/98
POTE generator ~3/2 = 702.136
Mapping generator: ~3
Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
EDOs: 12, 29de, 41de
Badness: 0.0375
Pontiac
Commas: 32805/32768, 4375/4374
7-limit minimax:
[|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, |113/47 0 -39/47 39/47>]
Eigenmonzos: 2, 7/5
9-limit minimax:
[|1 0 0 0>, |3/2 1/5 -1/10 0>, |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]
Eigenmonzos: 2, 10/9
POTE generator: 701.757
Mapping generator: ~3
Map: [<1 0 15 -59|, <0 1 -8 39|]
Wedgie: <<1 -8 39 -15 59 113||
Edos: 53, 118, 171, 1079, 1250, 1421
Badness: 0.0141
Grackle
Commas: 126/125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: 701.239
Mapping generator: ~3
Map: [<1 0 15 -44|, <0 1 -8 -26|]
Wedgie: <<1 -8 -26 -15 -44 -38||
Badness: 0.0704
Bischismic
Commas: 3136/3125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: 701.592
Mapping generator: ~3
Map: [<2 0 30 69|, <0 1 -8 -20|]
Wedgie: <<2 -16 -40 -30 -69 -48||
Badness: 0.0547
11-limit
Commas: 441/440, 3136/3125, 8019/8000
POTE generator: ~3/2 = 701.612
Mapping generator: ~3
Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|]
EDOs: 12, 118, 130, 248
Badness: 0.0282
13-limit
Commas: 441/440, 729/728, 1001/1000, 3136/3125
POTE generator: ~3/2 = 701.590
Mapping generator: ~3
Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|]
EDOs: 12, 118, 130, 248, 378
Badness: 0.0287
Kleischismic
Commas: 32805/32768, 1500625/1492992
POTE generator: ~36/35 = 50.920
Mapping generator: ~35/24
Map: [<2 1 22 -15|, <0 2 -16 19|]
Wedgie: <<4 -32 38 -60 49 178||
EDOs: 24, 94, 118, 212, 330, 542d, 872cd
Badness: 0.1106
11-limit
Commas: 385/384, 9801/9800, 14641/14580
POTE generator: ~36/35 = 50.918
Mapping generator: ~16/11
Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|]
EDOs: 24, 94, 118, 212, 330e, 542de
Badness: 0.0367
13-limit
Commas: 352/351, 385/384, 729/728, 1575/1573
POTE generator: ~36/35 = 50.938
Mapping generator: ~16/11
Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|]
EDOs: 24, 94, 118, 212f
Badness: 0.0376
Term
Commas: 32805/32768, 250047/250000
7-limit minimax
Eigenmonzos: 2, 6/5
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: ~3/2 = 701.742
Mapping generator: ~3
Map: [<3 0 45 94|, <0 1 -8 -18|]
Wedgie: <<3 -24 -54 -45 -94 -58||
Edos: 12, 171, 1038, 1209, 1380, 1551, 1722
Badness: 0.0200
Sesquiquartififths
Commas: 2401/2400, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: ~448/405 = 175.434
Mapping generator: ~448/405
Map: [<1 1 7 5|, <0 4 -32 -15|]
Wedgie: <<4 -32 -15 -60 -35 55||
Edos: 41, 130, 171, 643, 2100edo
Badness: 0.0112
Sesquart
Commas: 243/242, 441/440, 16384/16335
POTE generator: ~256/231 = 175.406
Mapping generator: ~256/231
Map: [<1 1 7 5 2|, <0 4 -32 -15 10|]
EDOs: 41, 89, 130, 171
Badness: 0.0293
13-limit
Commas: 243/242, 364/363, 441/440, 3584/3575
POTE generator: ~256/231 = 175.409
Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|]
EDOs: 41, 89, 130, 301e, 431e
Badness: 0.0224
Bisesqui
Commas: 2401/2400, 9801/9800, 32805/32768
POTE generator: ~448/405 = 175.435
Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|]
EDOs: 130, 212, 342, 1156, 1498
Badness: 0.0170
Sextilififths
Commas: 32768/32805, 235298/234375
POTE generator: ~21/20 = 83.053
Mapping generator: ~21/20
Map: [<1 2 -1 -1|, <0 -6 48 55|]
EDOs: 29, 101, 130, 289, 419
Badness: 0.1088
11-limit
Commas: 441/440, 4000/3993, 235298/234375
POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20
Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|]
EDOs: 29, 130, 159, 289
Badness: 0.0455
13-limit
Commas: 364/363, 441/440, 676/675, 10985/10976
POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20
Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|]
EDOs: 29, 130, 159, 289
Badness: 0.0253
Tsaharuk
Commas: 32805/32768, 420175/419904
POTE generator: ~243/224 = 140.350
Mapping generator: ~243/224
Map: [<1 1 7 0|, <0 5 -40 24|]
Wedgie: <<5 -40 24 -75 24 168||
EDOs: 17, 60c, 77, 94, 171
Badness: 0.0307
Quanharuk
Commas: 16875/16807, 32805/32768
POTE generator: ~56/45 = 380.355
Mapping generator: ~56/45
Map: [<1 0 15 12|, <0 5 -40 -29|]
Wedgie: <<5 -40 -29 -75 -60 45||
EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd
Badness: 0.0720
11-limit
Commas: 540/539, 1375/1372, 32805/32768
POTE generator: ~56/45 = 380.352
Mapping generator: ~56/45
Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|]
EDOs: 41, 142, 183, 224, 631d, 855d, 1079d
Badness: 0.0315
13-limit
Commas: 540/539, 1375/1372, 4096/4095, 6656/6655
POTE generator: ~56/45 = 380.351
Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|]
EDOs: 41, 142, 183, 224, 631d, 855d
Badness: 0.0214
Octant
Commas: 32805/32768, 2259436291848/2251875390625
POTE generator: ~3/2 = 701.713
Mapping generator: ~3
Map: [<8 0 120 -117|, <0 1 -8 11|]
Wedgie: <<8 -64 88 -120 117 384||
EDOs: 24, 224, 472, 696, 1168
Badness: 0.1572
11-limit
Commas: 9801/9800, 32805/32768, 46656/46585
POTE generator: ~3/2 = 701.713
Mapping generator: ~3
Map: [<8 0 120 -117 15|, <0 1 -8 11 1|]
EDOs: 24, 224, 472, 696, 1168
Badness: 0.0448
13-limit
Commas: 729/728, 1575/1573, 2200/2197, 6656/6655
POTE generator: ~3/2 = 701.725
Mapping generator: ~3
Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|]
EDOs: 24, 224, 472, 696
Badness: 0.0304