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.

Subgroup: 2.3.5

Comma list: 32805/32768

Mapping: [⟨1 0 15], ⟨0 1 -8]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 701.736

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: 0.004259

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, guiron, hemischis and karadeniz.

Garibaldi

Main article: Garibaldi temperament

Subgroup: 2.3.5.7

Comma list: 225/224, 3125/3087

Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -8 -14 -15 -25 -10 ]]

POTE generator: ~3/2 = 702.085

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: 0.021644

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2200/2187

Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 702.157

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

Vals: Template:Val list

Badness: 0.027396

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324, 385/384

Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 702.113

Vals: Template:Val list

Badness: 0.020676

Andromeda

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/242

Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 702.321

Vals: Template:Val list

Badness: 0.023556

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195, 245/242

Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 702.559

Vals: Template:Val list

Badness: 0.020749

Helenus

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 3125/3087

Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 701.725

Vals: Template:Val list

Badness: 0.035637

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 275/273, 847/845

Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 701.747

Vals: Template:Val list

Badness: 0.026284

Hemigari

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 3125/3087

Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]

Mapping generators: ~2, ~110/63

POTE generator: ~63/55 = 248.918

Vals: Template:Val list

Badness: 0.050681

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 225/224, 275/273

Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]

Mapping generators: ~2, ~26/15

POTE generator: ~15/13 = 248.918

Vals: Template:Val list

Badness: 0.027464

Sanjaab

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323, 3125/3087

Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]

Mapping generators: ~2, ~11/10

POTE generator: ~11/10 = 165.974

Vals: Template:Val list

Badness: 0.058040

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 847/845, 1331/1323

Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]

Mapping generators: ~2, ~11/10

POTE generator: ~11/10 = 165.963

Vals: Template:Val list

Badness: 0.033849

Schism

See also: Archytas clan § Schism

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 701.556

Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]

Badness: 0.056648

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]]

Mapping generators: ~2, ~3

POTE generator ~3/2 = 702.136

Vals: Template:Val list

Badness: 0.037482

Pontiac

Subgroup: 2.3.5.7

Comma list: 4375/4374, 32805/32768

Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -8 39 -15 59 113 ]]

POTE generator: ~3/2 = 701.757

Minimax tuning:

• 7-odd-limit

[[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-odd-limit

[[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

Tuning ranges:

• valid range: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
• nice range: ~3/2 = [701.711, 701.955]
• strict range: ~3/2 = [701.711, 701.886]

Badness: 0.014133

Grackle

Subgroup: 2.3.5.7

Comma list: 126/125, 32805/32768

Mapping: [⟨1 0 15 -44], ⟨0 1 -8 -26]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -8 -26 -15 -44 -38 ]]

POTE generator: ~3/2 = 701.239

Minimax tuning:

• 7-odd-limit eigenmonzos: 2, 7/6
• 9-odd-limit eigenmonzos: 2, 9/7

Badness: 0.070407

Bischismic

Subgroup: 2.3.5.7

Comma list: 3136/3125, 32805/32768

Mapping: [⟨2 0 30 69], ⟨0 1 -8 -20]]

Mapping generators: ~567/400, ~3

Wedgie: ⟨⟨ 2 -16 -40 -30 -69 -48 ]]

POTE generator: ~3/2 = 701.592

Minimax tuning:

• 7-odd-limit eigenmonzos: 2, 7/6
• 9-odd-limit eigenmonzos: 2, 9/7

Badness: 0.054744

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 8019/8000

Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]

Mapping generators: ~99/70, ~3

POTE generator: ~3/2 = 701.612

Vals: Template:Val list

Badness: 0.028160

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 3136/3125

Mapping generators: ~99/70, ~3

Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]

POTE generator: ~3/2 = 701.590

Vals: Template:Val list

Badness: 0.028722

Kleischismic

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1500625/1492992

Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]

Mapping generators: ~1225/864, ~35/24

Wedgie: ⟨⟨ 4 -32 38 -60 49 178 ]]

POTE generator: ~36/35 = 50.920

Badness: 0.110583

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800, 14641/14580

Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]

Mapping generators: ~99/70, ~16/11

POTE generator: ~36/35 = 50.918

Vals: Template:Val list

Badness: 0.036749

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1575/1573

Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]

Mapping generators: ~99/70, ~16/11

POTE generator: ~36/35 = 50.938

Vals: Template:Val list

Badness: 0.037640

Pogo

Subgroup: 2.3.5.7

Comma list: 32805/32768, 118098/117649

Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]

Mapping generators: ~343/243, ~9/7

Wedgie: ⟨⟨ 6 -48 10 -90 -1 158 ]]

POTE generator: ~9/7 = 433.901

Badness: 0.079635

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 32805/32768

Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]

Mapping generators: ~99/70, ~9/7

POTE generator: ~9/7 = 433.911

Vals: Template:Val list

Badness: 0.031857

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4000/3993, 4225/4224

Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]

Mapping generators: ~99/70, ~9/7

POTE generator: ~9/7 = 433.911

Vals: Template:Val list

Badness: 0.017514

Squirrel

Subgroup: 2.3.5.7

Comma list: 686/675, 32805/32768

Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]

Wedgie: ⟨⟨ 3 -24 -13 -45 -29 37 ]]

POTE generator: ~160/147 = 166.140

Badness: 0.174705

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 686/675, 896/891

Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]

POTE generator: ~11/10 = 166.097

Vals: Template:Val list

Badness: 0.068310

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 245/242, 896/891

Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]

POTE generator: ~11/10 = 166.054

Badness: 0.043750

Term

Subgroup: 2.3.5.7

Comma list: 32805/32768, 250047/250000

Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]

Mapping generators: ~63/50, ~3

Wedgie: ⟨⟨ 3 -24 -54 -45 -94 -58 ]]

POTE generator: ~3/2 = 701.742

Minimax tuning:

• 7-odd-limit eigenmonzos: 2, 6/5
• 9-odd-limit eigenmonzos: 2, 9/7

Badness: 0.019950

Sesquiquartififths

Subgroup: 2.3.5.7

Comma list: 2401/2400, 32805/32768

Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]

Mapping generators: ~2, ~448/405

Wedgie: ⟨⟨ 4 -32 -15 -60 -35 55 ]]

POTE generator: ~448/405 = 175.434

Minimax tuning:

• 7-odd-limit eigenmonzos: 2, 7/6
• 9-odd-limit eigenmonzos: 2, 9/7

Badness: 0.011244

Sesquart

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 16384/16335

Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]

Mapping generators: ~2, ~256/231

POTE generator: ~256/231 = 175.406

Vals: Template:Val list

Badness: 0.029306

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 3584/3575

Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]

POTE generator: ~256/231 = 175.409

Vals: Template:Val list

Badness: 0.022396

Bisesqui

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 32805/32768

Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]

POTE generator: ~448/405 = 175.435

Vals: Template:Val list

Badness: 0.016968

Sextilififths

Subgroup: 2.3.5.7

Comma list: 32768/32805, 235298/234375

Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]

Mapping generators: ~2, ~21/20

Wedgie: ⟨⟨ 6 -48 -55 -90 -104 7 ]]

POTE generator: ~21/20 = 83.053

Badness: 0.108794

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 235298/234375

Mapping: [⟨1 2 -1 -1 0], ⟨0 -6 48 55 50]]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 83.049

Vals: Template:Val list

Badness: 0.045457

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 10985/10976

Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 83.049

Vals: Template:Val list

Badness: 0.025276

Tsaharuk

See also: Tsaharuk

Subgroup: 2.3.5.7

Comma list: 32805/32768, 420175/419904

Mapping: [⟨1 1 7 0], ⟨0 5 -40 24]]

Mapping generators: ~2, ~243/224

Wedgie: ⟨⟨ 5 -40 24 -75 24 168 ]]

POTE generator: ~243/224 = 140.350

Badness: 0.030697

Quanharuk

Subgroup: 2.3.5.7

Comma list: 16875/16807, 32805/32768

Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]

Mapping generators: ~2, ~56/45

Wedgie: ⟨⟨ 5 -40 -29 -75 -60 45 ]]

POTE generator: ~56/45 = 380.355

Badness: 0.071950

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 32805/32768

Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]

Mapping generators: ~2, ~56/45

POTE generator: ~56/45 = 380.352

Vals: Template:Val list

Badness: 0.031549

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 4096/4095

Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]

Mapping generators: ~2, ~56/45

POTE generator: ~56/45 = 380.351

Vals: Template:Val list

Badness: 0.021392

Octant

Subgroup: 2.3.5.7

Comma list: 32805/32768, 2259436291848/2251875390625

Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]

Mapping generators: ~42875/39366, ~3

Wedgie: ⟨⟨ 8 -64 88 -120 117 384 ]]

POTE generator: ~3/2 = 701.713

Badness: 0.157186

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 46656/46585

Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]

Mapping generators: ~12/11, ~3

POTE generator: ~3/2 = 701.713

Vals: Template:Val list

Badness: 0.044778

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655

Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]

Mapping generators: ~12/11, ~3

POTE generator: ~3/2 = 701.725

Vals: Template:Val list

Badness: 0.030425