Schismatic family

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The 5-limit parent comma for the schismatic (or schismic) 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:

  • 5-odd-limit diamond monotone: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
  • 5-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955]
  • 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]

Vals12, 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.

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 discussed elsewhere include salsa, guiron, hemischis and karadeniz. Remarkable subgroup temperaments include nestoria and photia.

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:

[[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 (unchanged intervals): 2, 7/6
[[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 (unchanged intervals): 2, 9/7

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
  • 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 702.915]

Vals12, 29, 41, 53, 94, 241c, 335cd, 576ccd

Badness: 0.021644

Cassandra

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: ~3/2 = [9/16 1/8 0 -1/16
Eigenmonzos (unchanged intervals): 2, 9/7

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]

Vals: 41, 53, 94, 229c, 323c, 417ce

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

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34
Eigenmonzos (unchanged intervals): 2, 14/13

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
  • 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
  • 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]

Vals: 41, 53, 94, 429cdef, 523cdef

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

Minimax tuning:

  • 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20
Eigenmonzos (unchanged intervals): 2, 11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
  • 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 703.448]

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

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

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23
Eigenmonzos (unchanged intervals): 2, 13/9

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
  • 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
  • 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.439, 703.448]

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

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

Minimax tuning:

  • 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32
Eigenmonzos (unchanged intervals): 2, 11/9

Vals: 12, 41e, 53, 118d, 171de

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

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32
Eigenmonzos (unchanged intervals): 2, 11/9

Vals: 12f, 41ef, 53, 118d, 171de

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

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

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

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: 29, 65d, 94

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]]

Vals12, 29d, 41d, 53d

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: 12, 29de, 41de

Badness: 0.037482

Pontiac

Main article: 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:

[[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 (unchanged intervals): 2, 7/5
[[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 (unchanged intervals): 2, 10/9

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
  • 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.886]

Vals53, 118, 171, 1592c, 1763c, 1934c, 2105c, 2276cd, 2447cd, 2618cd, 2789cd, 2960cd, 3131bcd

Badness: 0.014133

Helenoid

The helenoid temperament (53&118) is closely related to the helenus temperament, but with the ragisma rather than the marvel comma tempered out.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 4375/4374

Mapping: [1 0 15 -59 51], 0 1 -8 39 -30]]

POTE generator: ~3/2 = 701.722

Minimax tuning:

  • 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69
Eigenmonzos (unchanged intervals): 2, 14/11

Vals: 53, 118, 289e, 407de

Badness: 0.038863

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 729/728

Mapping: [1 0 15 -59 51 56], 0 1 -8 39 -30 -33]]

POTE generator: ~3/2 = 701.745

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72
Eigenmonzos (unchanged intervals): 2, 14/13

Vals: 53, 118, 171e

Badness: 0.033677

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 729/728

Mapping: [1 0 15 -59 51 56 -91], 0 1 -8 39 -30 -33 60]]

POTE generator: ~3/2 = 701.742

Minimax tuning:

  • 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93
Eigenmonzos (unchanged intervals): 2, 17/13

Vals: 53, 118, 171e, 289ef, 460eef

Badness: 0.028891

Ponta

The ponta temperament (53&171) tempers out the swetisma and the ragisma.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 32805/32768

Mapping: [1 0 15 -59 135], 0 1 -8 39 -83]]

POTE generator: ~3/2 = 701.783

Minimax tuning:

  • 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122
Eigenmonzos (unchanged intervals): 2, 14/11

Vals: 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee

Badness: 0.048692

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2200/2197

Mapping: [1 0 15 -59 135 56], 0 1 -8 39 -83 -33]]

POTE generator: ~3/2 = 701.784

Minimax tuning:

  • 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122
Eigenmonzos (unchanged intervals): 2, 14/11

Vals: 53, 171, 224

Badness: 0.023616

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197

Mapping: [1 0 15 -59 135 56 -91], 0 1 -8 39 -83 -33 60]]

POTE generator: ~3/2 = 701.777

Minimax tuning:

  • 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143
Eigenmonzos (unchanged intervals): 2, 22/17

Vals: 53, 171, 224, 395e, 619eg

Badness: 0.022853

Pontic

The pontic temperament (118&171) tempers out the werckisma and the ragisma.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 32805/32768

Mapping: [1 0 15 -59 -136], 0 1 -8 39 88]]

POTE generator: ~3/2 = 701.724

Minimax tuning:

  • 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88
Eigenmonzos (unchanged intervals): 2, 11/8

Vals: 53e, 118, 289, 407d, 696d

Badness: 0.049573

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 625/624, 729/728, 3584/3575

Mapping: [1 0 15 -59 -136 56], 0 1 -8 39 88 -33]]

POTE generator: ~3/2 = 701.738

Minimax tuning:

  • 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121
Eigenmonzos (unchanged intervals): 2, 13/11

Vals: 53e, 118, 171, 289f, 460ef

Badness: 0.045308

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873

Mapping: [1 0 15 -59 -136 56 -91], 0 1 -8 39 88 -33 60]]

POTE generator: ~3/2 = 701.740

Minimax tuning:

  • 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121
Eigenmonzos (unchanged intervals): 2, 13/11

Vals: 53e, 118, 171, 289f, 460ef

Badness: 0.029618

Bipont

The bipont temperament (118&224) has a period of half octave and tempers out the lehmerisma, 3025/3024 and the kalisma, 9801/9800.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 32805/32768

Mapping: [2 3 6 -1 2], 0 1 -8 39 29]]

POTE generator: ~3/2 = 701.757

Vals: 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde

Badness: 0.014629

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 4096/4095

Mapping: [2 3 6 -1 2 13], 0 1 -8 39 29 -33]]

POTE generator: ~3/2 = 701.773

Vals: 106, 118, 224, 566f, 790f

Badness: 0.030172

Counterbipont

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768

Mapping: [2 3 6 -1 2 -6], 0 1 -8 39 29 79]]

POTE generator: ~3/2 = 701.769

Vals: 106f, 118f, 224, 342f, 566, 1356cf, 1922cff

Badness: 0.025547

Quadrapont

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768

Mapping: [4 6 12 -2 4 7], 0 1 -8 39 29 23]]

POTE generator: ~3/2 = 701.756

Vals: 224, 460, 684, 2276cde, 2960cde, 3644bccddee

Badness: 0.021025

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:

Vals12, 53d, 65, 77, 166c, 243c

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:

Vals12, 106d, 118, 130, 248, 378, 508

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]]

POTE generator: ~3/2 = 701.612

Vals: 12, 106de, 118, 130, 248

Badness: 0.028160

13-limit

Subgroup: 2.3.5.7.11.13

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

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

POTE generator: ~3/2 = 701.590

Vals: 12, 106def, 118, 130, 248, 378

Badness: 0.028722

Bischis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 3136/3125

Mapping: [2 3 6 9 12 14], 0 1 -8 -20 -30 -39]]

POTE generator: ~3/2 = 701.565

Vals: 12f, 106deff, 118f, 130

Badness: 0.029321

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

Vals24, 94, 118, 212, 330, 542d, 872cd

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]]

POTE generator: ~36/35 = 50.918

Vals: 24, 94, 118, 212, 330e, 542de

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]]

POTE generator: ~36/35 = 50.938

Vals: 24, 94, 118, 212f

Badness: 0.037640

Kleischis

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1573/1568, 14641/14580

Mapping: [2 1 22 -15 8 -36], 0 2 -16 19 -1 40]]

POTE generator: ~36/35 = 50.9508

Vals: 24f, 94, 118f, 212

Badness: 0.037607

Squirrel

The squirrel temperament (29&36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.

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

Vals29, 36, 65

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: 29, 36, 65

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

Vals: 29, 36, 65f, 94df, 159df

Badness: 0.043750

Tertiaschis

The tertiaschis temperament (94&159) has a ~11/10 generator, sharing the same 2.3.5.11 with #Squirrel, but tempers out 1071785/1062882 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1071875/1062882

Mapping: [1 2 -1 10], 0 -3 24 -52]]

Wedgie⟨⟨3 -24 52 -45 74 188]]

POTE generator: ~192/175 = 166.019

Vals65, 94, 159, 253, 412cd

Badness: 0.211859

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 19712/19683

Mapping: [1 2 -1 10 0], 0 -3 24 -52 25]]

POTE generator: ~11/10 = 166.017

Vals: 65, 94, 159, 253, 412cd

Badness: 0.061336

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 10985/10976

Mapping: [1 2 -1 10 0 12], 0 -3 24 -52 25 -60]]

POTE generator: ~11/10 = 166.016

Vals: 65f, 94, 159, 253, 412cdf, 665ccdef

Badness: 0.036700

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

Mapping: [1 2 -1 10 0 12 -2], 0 -3 24 -52 25 -60 44]]

POTE generator: ~11/10 = 166.012

Vals: 65f, 94, 159, 253

Badness: 0.026504

Countertertiaschis

The countertertiaschis temperament (159&224) has a ~11/10 generator, sharing the same 2.3.5.11 with #Squirrel, but tempers out 244140625/243045684 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 244140625/243045684

Mapping: [1 2 -1 -12], 0 -3 24 107]]

POTE generator: ~625/567 = 166.0621

Vals65d, 159, 224, 383, 607

Badness: 0.188043

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 32805/32768

Mapping: [1 2 -1 -12 0], 0 -3 24 107 25]]

POTE generator: ~11/10 = 166.0628

Vals: 65d, 159, 224, 383, 607

Badness: 0.048943

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976

Mapping: [1 2 -1 -12 0 -10], 0 -3 24 107 25 99]]

POTE generator: ~11/10 = 166.0628

Vals: 65d, 159, 224, 383, 607

Badness: 0.024506

Pogo

The pogo temperament (94&130) splits the period in two to address the difference between #Tertiaschis and #Countertertiaschis.

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

Vals36, 94, 130, 224, 354

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

Badness: 0.031857

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 4096/4095

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

Badness: 0.017514

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:

Vals12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722

Badness: 0.019950

Terminal

The terminal temperament (12&159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 32805/32768

Mapping: [3 0 45 94 134], 0 1 -8 -18 -26]]

POTE generator: ~3/2 = 701.824

Vals: 12, 147de, 159, 330

Badness: 0.059502

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 13720/13689

Mapping: [3 0 45 94 134 168], 0 1 -8 -18 -26 -33]]

POTE generator: ~3/2 = 701.821

Vals: 12f, 147def, 159, 330

Badness: 0.037082

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619

Mapping: [3 0 45 94 134 168 -2], 0 1 -8 -18 -26 -33 3]]

POTE generator: ~3/2 = 701.810

Vals: 12f, 147def, 159, 171, 330

Badness: 0.027073

Terminator

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 137781/137500

Mapping: [3 0 45 94 -137], 0 1 -8 -18 31]]

POTE generator: ~3/2 = 701.685

Vals: 12e, 159e, 171, 183, 354, 537, 891de

Badness: 0.066968

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 31250/31213

Mapping: [3 0 45 94 -137 -103], 0 1 -8 -18 31 24]]

POTE generator: ~3/2 = 701.689

Vals: 171, 183, 354, 891de, 1245dee, 1599ddee

Badness: 0.035487

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095

Mapping: [3 0 45 94 -137 -103 -2], 0 1 -8 -18 31 24 3]]

POTE generator: ~3/2 = 701.688

Vals: 171, 183, 354, 891de, 1245dee, 1599ddee

Badness: 0.020434

Semiterm

The semiterm temperament (12&342, formerly hemiterm) has a period of 1/6 octave and tempers out 9801/9800 (kalisma) and 151263/151250 (odiheim comma).

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 151263/151250

Mapping: [6 0 90 188 287], 0 1 -8 -18 -28]]

Mapping generators: ~55/49, ~3

POTE generator: ~3/2 = 701.7460

Vals: 12, 330e, 342, 1380, 1722, 2064, 2406c

Badness: 0.029438

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375

Mapping: [6 0 90 188 287 355], 0 1 -8 -18 -28 -35]]

POTE tuning: ~3/2 = 701.7256

Vals: 12f, 330eff, 342f, 696f *

* optimal patent val: 354

Badness: 0.044657

Hemiterm

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 32805/32768, 102487/102400

Mapping: [3 0 45 94 8], 0 2 -16 -36 1]]

Mapping generators: ~63/50, ~693/400

POTE generator: ~12/11 = 150.872

Vals: 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce

Badness: 0.020687

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712

Mapping: [3 0 45 94 8 42], 0 2 -16 -36 1 -13]]

POTE generator: ~12/11 = 150.873

Vals: 24d, 159, 183, 342f

Badness: 0.031362

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264

Mapping: [3 0 45 94 8 42 -2], 0 2 -16 -36 1 -13 6]]

POTE generator: ~12/11 = 150.867

Vals: 24d, 159, 183, 342f, 525f, 867ff

Badness: 0.022316

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:

Vals41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd

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: 41, 89, 130, 301e, 431e

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: ~72/65 = 175.409

Vals: 41, 89, 130, 301e, 431e

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: 82e, 130, 212, 342, 1156, 1498, 1840d

Badness: 0.016968

Quintilipyth

The quintilipyth temperament (12&253, formerly quintilischis temperament) slices the pythagorean fourth (4/3) into five semitones and tempers out the compass comma (9765625/9680832, quinruyoyo) in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 9765625/9680832

Mapping: [1 2 -1 -4], 0 -5 40 82]]

Wedgie⟨⟨5 -40 -82 -75 -144 -78]]

POTE generator: ~625/588 = 99.625

Vals12, 253, 265

Badness: 0.253966

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 32805/32768

Mapping: [1 2 -1 -4 -7], 0 -5 40 82 126]]

POTE generator: ~35/33 = 99.616

Vals: 12, 253, 265, 518c, 783cc

Badness: 0.113044

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647

Mapping: [1 2 -1 -4 -7 -9], 0 -5 40 82 126 153]]

POTE generator: ~35/33 = 99.612

Vals: 12f, 253, 518c, 771cc

Badness: 0.069127

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619

Mapping: [1 2 -1 -4 -7 -9 5], 0 -5 40 82 126 153 -11]]

POTE generator: ~18/17 = 99.612

Vals: 12f, 253, 518c, 771cc

Badness: 0.045992

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971

Mapping: [1 2 -1 -4 -7 -9 5 4], 0 -5 40 82 126 153 -11 3]]

POTE generator: ~18/17 = 99.615

Vals: 12f, 253, 265, 518ch

Badness: 0.038155

Quintaschis

The quintaschis temperament (12&289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 (quinzo-alegu) in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 49009212/48828125

Mapping: [1 2 -1 -5], 0 -5 40 94]]

Wedgie⟨⟨5 -40 -94 -75 -163 -106]]

POTE generator: ~200/189 = 99.664

Vals12, 277d, 289

Badness: 0.132890

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 1953125/1951488

Mapping: [1 2 -1 -5 -8], 0 -5 40 94 138]]

POTE generator: ~35/33 = 99.653

Vals: 12, 277d, 289

Badness: 0.111477

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 32805/32768, 109512/109375

Mapping: [1 2 -1 -5 -8 -11], 0 -5 40 94 138 177]]

POTE generator: ~35/33 = 99.658

Vals: 12f, 277df, 289

Badness: 0.074218

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768

Mapping: [1 2 -1 -5 -8 -11 5], 0 -5 40 94 138 177 -11]]

POTE generator: ~18/17 = 99.656

Vals: 12f, 277df, 289

Badness: 0.050571

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859

Mapping: [1 2 -1 -5 -8 -11 5 4], 0 -5 40 94 138 177 -11 3]]

POTE generator: ~18/17 = 99.659

Vals: 12f, 277df, 289

Badness: 0.042120

Quintahelenic

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 8019/8000, 151263/151250

Mapping: [1 2 -1 -5 -9], 0 -5 40 94 150]]

POTE generator: ~200/189 = 99.671

Vals: 12, 289e, 301, 915, 1216ce

Badness: 0.082225

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000

Mapping: [1 2 -1 -5 -9 -11], 0 -5 40 94 150 177]]

POTE generator: ~200/189 = 99.661

Vals: 12f, 289e, 301

Badness: 0.055570

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750

Mapping: [1 2 -1 -5 -9 -11 5], 0 -5 40 94 150 177 -11]]

POTE generator: ~18/17 = 99.665

Vals: 12f, 289e, 301

Badness: 0.040412

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700

Mapping: [1 2 -1 -5 -9 -11 5 4], 0 -5 40 94 150 177 -11 3]]

POTE generator: ~18/17 = 99.668

Vals: 12f, 289e, 301

Badness: 0.036840

Quintahelenoid

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436

Mapping: [1 2 -1 -5 -9 14], 0 -5 40 94 150 -124]]

POTE generator: ~200/189 = 99.672

Vals: 12, 301, 614, 915

Badness: 0.066108

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157

Mapping: [1 2 -1 -5 -9 14 5], 0 -5 40 94 150 -124 -11]]

POTE generator: ~18/17 = 99.671

Vals: 12, 301, 915gg, 1216cegg

Badness: 0.047908

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137

Mapping: [1 2 -1 -5 -9 14 5 4], 0 -5 40 94 150 -124 -11 3]]

POTE generator: ~18/17 = 99.672

Vals: 12, 301, 614gh, 915gghh

Badness: 0.039542

Sextilififths

The sextilififths (130&159, also known as sextilischis) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.

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

Vals29, 72cd, 101, 130, 289, 419

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: 29, 72cde, 101e, 130, 289

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: 29, 72cdef, 101e, 130, 289

Badness: 0.025276

Septiquarschis

The septiquarschis temperament (89&94) splits septimal minor seventh (7/4) into four generators and tempers out 829440/823543 (mynaslender comma, sepru-ayo) and 67108864/66706983 (septiness comma, sasasepru).

Subgroup: 2.3.5.7

Comma list: 32805/32768, 829440/823543

Mapping: [1 3 -9 2], 0 -7 -56 4]]

Wedgie⟨⟨7 56 -4 231 -26 -76]]

POTE generator: ~147/128 = 242.614

Vals89, 94, 183, 460d, 643d, 1103dd

Badness: 0.187047

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 15488/15435, 32805/32768

Mapping: [1 3 -9 2 -2], 0 -7 -56 4 27]]

POTE generator: ~147/128 = 242.616

Vals: 89, 94, 183, 460d, 643d, 826dd

Badness: 0.052002

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1573/1568, 4096/4095

Mapping: [1 3 -9 2 -2 13], 0 -7 -56 4 27 -46]]

POTE generator: ~147/128 = 242.610

Vals: 89, 94, 183, 277, 460d

Badness: 0.035315

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

Vals17, 60c, 77, 94, 171

Badness: 0.030697

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 19712/19683

Mapping: [1 1 7 0 1], 0 5 -40 24 21]]

POTE generator: ~88/81 = 140.365

Vals: 17, 60ce, 77, 94, 171e, 265e, 436ee

Badness: 0.063499

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1331/1323

Mapping: [1 1 7 0 1 3], 0 5 -40 24 21 6]]

POTE generator: ~13/12 = 140.363

Vals: 17, 60ce, 77, 94, 171e, 436ee

Badness: 0.037886

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

Vals41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd

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: 41, 142, 183, 224, 631d, 855d, 1079d

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: 41, 142, 183, 224, 631d, 855d

Badness: 0.021392

Quadrant

The quadrant temperament (12&224) has a period of quarter octave and tempers out the dimcomp comma, 390625/388962. In this temperament, 25/21 is mapped into quarter octave.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 390625/388962

Mapping: [4 0 60 119], 0 1 -8 -17]]

Mapping generators: ~25/21, ~3

Wedgie⟨⟨4 -32 -68 -60 -119 -68]]

POTE generator: ~28/25 = 198.177

Vals12, 200, 212, 224, 436, 660, 1096c

Badness: 0.110242

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 32805/32768

Mapping: [4 0 60 119 185], 0 1 -8 -17 -27]]

POTE generator: ~28/25 = 198.181

Vals: 12, 212, 224, 436, 660

Badness: 0.045738

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647

Mapping: [4 0 60 119 185 224], 0 1 -8 -17 -27 -33]]

POTE generator: ~28/25 = 198.184

Vals: 212, 224, 436, 660

Badness: 0.027243

Septant

The septant temperament (224&301) has a period of 1/7 octave and tempers out the akjaysma, [47 -7 -7 -7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 516560652/514714375

Mapping: [7 11 17 19], 0 1 -8 7]]

Wedgie⟨⟨7 -56 49 -105 58 271]]

POTE generator: ~3/2 = 701.702

Vals77, 147, 224, 301, 525, 826, 1351

Badness: 0.111142

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 24057/24010, 32805/32768

Mapping: [7 11 17 19 23], 0 1 -8 7 13]]

POTE generator: ~3/2 = 701.719

Vals: 77, 147, 224, 301, 525

Badness: 0.044122

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024

Mapping: [7 11 17 19 23 26], 0 1 -8 7 13 -1]]

POTE generator: ~3/2 = 701.724

Vals: 77, 147, 224, 525

Badness: 0.024706

Octant

The octant temperament (224&472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.

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

Vals24, 224, 472, 696, 1168

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: 24, 224, 472, 696, 1168

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: 24, 224, 472, 696

Badness: 0.030425