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]]

Tuning ranges:

• 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 ]]

Minimax tuning:

• 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

Tuning ranges:

• 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, 3125/3087, 1331/1323

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

Commas: 1029/1024, 10976/10935

Minimax tuning:

7+9 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

POTE generator: ~8/7 = 233.930

Mapping generator: ~8/7

Map: [<1 1 7 3|, <0 3 -24 -1|]

Wedgie: <<3 -24 -1 -45 -10 65||

EDOs: 36, 41, 77, 118, 277d

Badness: 0.0475

11-limit

Commas: 385/384, 441/440, 10976/10935

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

POTE generator: ~8/7 = 233.931

Mapping generator: ~8/7

Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]

Edos: 41, 77, 118, 159, 200, 277d

Badness: 0.0266

13-limit

Commas: 196/195, 352/351, 385/384, 729/728

POTE generator: ~8/7 = 233.890

Mapping generator: ~8/7

Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|]

EDOs: 41, 77, 118

Badness: 0.0284

Pogo

Commas: 32805/32768, 118098/117649

POTE generator: ~9/7 = 433.901

Mapping generator: ~9/7

Map: [<2 1 22 2|, <0 3 -24 5|]

EDOs: 36, 94, 130, 224, 354

Badness: 0.0796

11-limit

Commas: 540/539, 4000/3993, 32805/32768

POTE generator: ~9/7 = 433.911

Mapping generator: ~9/7

Map: [<2 1 22 2 25|, <0 3 -24 5 -25|]

EDOs: 36, 94, 130, 224, 354, 578

Badness: 0.0319

13-limit

Commas: 540/539, 729/728, 4000/3993, 4225/4224

POTE generator: ~9/7 = 433.911

Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|]

EDOs: 36, 94, 130, 224, 354, 578

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||

Edos: 77, 89, 101, 166c, 243c

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||

Edos: 118, 130, 248, 378, 508

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