Schismatic family

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Five limit

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 that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.

The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz 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.

POTE generator: ~3/2 = 701.736

Mapping generator: ~3

Map: [<1 0 15|, <0 1 -8|]

EDOs: 12, 29, 41, 53, 65, 77, 94, 118, 171, 183, 224, 289, 407, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc

Badness: 0.00426

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding [25 -14 0 -1⟩ gives garibaldi, [-44 26 0 1⟩ grackle, [6 -2 0 -1⟩ schism and [-59 39 0 -1⟩ pontiac; these all have a fifth as generator. Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = [-10 1 0 3⟩, with an 8/7 generator, three of which give the fifth, and 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 hemischis.

Garibaldi

Commas: 225/224, 3125/3087

7-limit minimax tuning:

7-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-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

POTE generator: ~3/2 = 702.085

Mapping generator: ~3

Map: [<1 0 15 25|, <0 1 -8 -14|]

Wedgie: <<1 -8 -14 -15 -25 -10||

EDOs: 12, 29, 41, 53, 94, 241c, 335cd, 576cd

Badness: 0.0216

11-limit

Commas: 225/224, 385/384, 2200/2187

Minimax tuning:

[|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

POTE generator: ~3/2 = 702.157

Mapping generator: ~3

Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]

Edos: 41, 53, 94, 135, 229c, 323c, 417ce

Badness: 0.0274

13-limit

Commas: 225/224, 275/273, 325/324, 385/384

POTE generator: ~3/2 = 702.113

Mapping generator: ~3

Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]

EDOs: 12e, 41, 53, 94, 135, 429cdef, 523cdef

Badness: 0.0207

Cassandra

Commas: 100/99, 225/224, 245/242

POTE generator: ~3/2 = 702.321

Mapping generator: ~3

Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]

EDOs: 12, 29, 41, 217ce, 258ce

Badness: 0.0236

13-limit

Commas: 100/99, 105/104, 196/195, 245/242

POTE generator: ~3/2 = 702.559

Mapping generator: ~3

Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]

EDOs: 12f, 29, 41, 152cdf, 193cdf, 234cdf

Badness: 0.0207

Helenus

Commas: 99/98 176/175 3125/3087

POTE generator: ~3/2 = 701.725

Mapping generator: ~3

Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|]

EDOs: 12, 53, 118d, 171de

Badness: 0.0356

13-limit

Commas: 99/98, 176/175, 275/273, 847/845

POTE generator: ~3/2 = 701.747

Mapping generator: ~3

Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|]

EDOs: 53, 118d, 171de

Badness: 0.0263

Hemigari

Commas: 121/120, 225/224, 3125/3087

POTE generator: ~63/55 = 248.918

Mapping generator: ~110/63

Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|]

EDOs: 29, 53, 82e, 135e, 188ce

Badness: 0.0507

13-limit

Commas: 121/120, 169/168, 225/224, 275/273

POTE generator: ~15/13 = 248.918

Mapping generator: ~26/15

Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|]

EDOs: 29, 53, 82e, 135ef, 188cef

Badness: 0.027464

Sanjaab

Commas: 225/224, 3125/3087, 1331/1323

POTE generator: ~11/10 = 165.974

Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|]

EDOs: 29, 65d, 94, 441cde, 535cde, 629cde

Badness: 0.0580

13-limit

Commas: 225/224, 275/273, 847/845, 1331/1323

POTE generator: ~11/10 = 165.963

Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|]

EDOs: 29, 65d, 94

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, 171, 183, 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: 171, 1038, 1209

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: 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

Badness: 0.0293

13-limit

Commas: 243/242, 364/363, 441/440, 105644/105625

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