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: {{#rreplace: 12 17 24 29 36 41 53 65 77 82 89 94 101 106 118 130 135 142 147 154 159 171 183 195 200 207 212 219 224 236 248 253 260 265 272 277 289 301 313 318 325 330 342 354 366 371 378 383 390 395 407 419 424 431 436 443 448 460 472 484 489 496 501 508 513 525 537 542 549 554 561 566 578 590 602 607 614 619 626 631 643 655 660 667 672 679 684 696 708 720 725 732 737 744 749 761 773 778 785 790 797 802 814 826 838 843 850 855 862 867 879 891 896 903 908 915 920 932 944 956 961 968 973 985 997 1009 1014 1021 1026 1033 1038 1050 1062 1067 1074 1079 1086 1091 1103 1115 1127 1132 1139 1144 1151 1156 1168 1180 1185 1192 1197 1204 1209 1221 1233 1245 1250 1257 1262 1269 1274 1286 1298 1303 1310 1315 1322 1327 1339 1351 1363 1368 1375 1380 1387 1392 1404 1416 1421 1428 1433 1440 1445 1457 1469 1481 1486 1493 1498 1505 1510 1522 1534 1539 1546 1551 1558 1563 1575 1587 1599 1604 1611 1616 1628 1640 1652 1657 1664 1669 1676 1681 1693 1705 1710 1717 1722 1729 1734 1746 1758 1770 1775 1782 1787 1794 1799 1811 1823 1828 1835 1840 1847 1852 1864 1876 1888 1893 1900 1905 1912 1917 1929 1941 1946 1953 1958 1965 1970 1982 1994 2006 2011 2018 2023 2030 2035 2047 2059 2064 2071 2076 2083 2088 2100 2112 2124 2129 2136 2141 2148 2153 2165 2177 2182 2189 2194 2206 2218 2230 2242 2247 2259 2271 2283 2295 2300 2312 2324 2336 2348 2353 2365 2377 2389 2401 2418 2430 2442 2454 2471 2483 2495 2536 2548 2589 2601 2654 2707|/(\d+)(\D+)?/|\1\2}}

Customized val EDOs: 3456bc, 4205bc, 4954bc, 5703bc

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.

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: 12, 41, 53, 94, 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, 123c, 217ce, 258ce

Badness: 0.0236

13-limit Cassandra

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: 12, 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 Helenus

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: 27.464

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

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

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, 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: 0.070

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, 289

Badness: 0.0455

13-limit

Commas: 363/363, 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, 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, 26, 43, 60, 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