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.

This defies the tradition of tertian harmony, as the just major triad on C is C-Fb-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-vE-G.

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

Optimal tuning (POTE): ~2 = 1\1, ~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] (1/8-comma to untempered)
  • 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]

Optimal ET sequence12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc

Badness: 0.004259

Overview to 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.
  • Hemischis adds [-34 25 0 -2 and has a hemififth generator.
  • 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

Remarkable subgroup temperaments include

Garibaldi

Garibaldi tempers out the garischisma, equating the septimal comma with both the syntonic comma and the Pythagorean comma. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its S-expression-based comma list is {S8/S9, S15}.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~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]
eigenmonzo (unchanged-interval) basis: 2.7/3
[[1 0 0 0, [25/16 1/8 0 -1/16, [5/2 -1 0 1/2, [25/8 -7/4 0 7/8]
eigenmonzo (unchanged-interval) basis: 2.9/7

Tuning ranges:

Optimal ET sequence12, 29, 41, 53, 94, 241c, 335cd, 576ccd

Badness: 0.021644

Cassandra

Cassandra is one of the best extension of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157

Minimax tuning:

  • 11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16
eigenmonzo (unchanged-interval) basis: 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]

Optimal ET sequence41, 53, 94, 229c, 323c, 417cce

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34
eigenmonzo (unchanged-interval) basis: 2.13/7

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]

Optimal ET sequence41, 53, 94, 429ccdeef, 523ccdeef

Badness: 0.020676

Cassie

Subgroup: 2.3.5.7.11.13.17

Comma list: 120/119, 154/153, 225/224, 273/272, 325/324

Mapping: [1 0 15 25 -33 -28 -7], 0 1 -8 -14 23 20 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092

Optimal ET sequence41, 53, 94g

Badness: 0.023270

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272

Mapping: [1 0 15 25 -33 -28 -7 9], 0 1 -8 -14 23 20 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079

Optimal ET sequence41, 53, 94g

Badness: 0.018189

Cassandric

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 275/273, 325/324, 375/374, 385/384

Mapping: [1 0 15 25 -33 -28 77], 0 1 -8 -14 23 20 -46]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097

Optimal ET sequence41g, 53, 94, 241ce, 335cde

Badness: 0.023167

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374

Mapping: [1 0 15 25 -33 -28 77 9], 0 1 -8 -14 23 20 -46 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.098

Optimal ET sequence41g, 53, 94, 241ceh, 335cdehh

Badness: 0.017635

Cassander

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 225/224, 275/273, 325/324, 385/384

Mapping: [1 0 15 25 -33 -28 -72], 0 1 -8 -14 23 20 48]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144

Optimal ET sequence41, 53g, 94

Badness: 0.022454

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324

Mapping: [1 0 15 25 -33 -28 -72 9], 0 1 -8 -14 23 20 48 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135

Optimal ET sequence41, 53g, 94

Badness: 0.017576

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321

Minimax tuning:

  • 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20
eigenmonzo (unchanged-interval) basis: 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]

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.559

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23
eigenmonzo (unchanged-interval) basis: 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]

Optimal ET sequence12f, 29, 41, 152cdf, 193cdf, 234cdf

Badness: 0.020749

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 189/187, 196/195

Mapping: [1 0 15 25 32 37 -7], 0 1 -8 -14 -18 -21 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312

Optimal ET sequence12f, 29, 41

Badness: 0.023406

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195

Mapping: [1 0 15 25 32 37 -7 9], 0 1 -8 -14 -18 -21 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357

Optimal ET sequence12f, 29, 41

Badness: 0.019154

Schisicosiennic

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 154/153, 170/169, 196/195

Mapping: [1 0 15 25 32 37 58], 0 1 -8 -14 -18 -21 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725

Optimal ET sequence12fg, 29g, 41, 70cd, 111cd

Badness: 0.021758

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189

Mapping: [1 0 15 25 32 37 58 9], 0 1 -8 -14 -18 -21 -34 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753

Optimal ET sequence12fg, 29g, 41, 70cd, 111cdh, 181ccddh

Badness: 0.017902

Schisicosiennoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 85/84, 100/99, 105/104, 119/117, 221/220

Mapping: [1 0 15 25 32 37 12], 0 1 -8 -14 -18 -21 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717

Optimal ET sequence12f, 29g, 41g, 70cdgg

Badness: 0.020895

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152

Mapping: [1 0 15 25 32 37 12 9], 0 1 -8 -14 -18 -21 -5 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716

Optimal ET sequence12f, 29g, 41g, 70cdgg

Badness: 0.016773

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725

Minimax tuning:

  • 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32
eigenmonzo (unchanged-interval) basis: 2.11/9

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32
eigenmonzo (unchanged-interval) basis: 2.11/9

Optimal ET sequence12f, 41ef, 53, 118d, 171de

Badness: 0.026284

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 99/98, 120/119, 176/175, 275/273, 442/441

Mapping: [1 0 15 25 51 56 -7], 0 1 -8 -14 -30 -33 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680

Optimal ET sequence12f, 41ef, 53, 65d, 118dg

Badness: 0.023732

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245

Mapping: [1 0 15 25 51 56 -7 9], 0 1 -8 -14 -30 -33 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705

Optimal ET sequence12f, 41ef, 53, 65d, 118dg

Badness: 0.019411

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

Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)

Optimal ET sequence29, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)

Optimal ET sequence29, 53, 82e, 135ef, 188cef

Badness: 0.027464

Karadeniz

See also: Turkish maqam music temperaments #Karadeniz temperament

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3125/3087

Mapping: [1 1 7 11 2], 0 2 -16 -28 5]]

mapping generators: ~2, ~11/9

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994

Optimal ET sequence41, 106, 147

Badness: 0.041562

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 640/637

Mapping: [1 1 7 11 2 -8], 0 2 -16 -28 5 40]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014

Optimal ET sequence41, 106, 147

Badness: 0.042564

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

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974

Optimal ET sequence29, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963

Optimal ET sequence29, 65d, 94

Badness: 0.033849

Schism

See also: Archytas clan #Schism

Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb).

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping[1 0 15 6], 0 1 -8 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.556

Wedgie⟨⟨1 -8 -2 -15 -6 18]]

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136

Optimal ET sequence12, 29de, 41de

Badness: 0.037482

Pontiac

Pontiac tempers out the ragisma, rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^3A).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 32805/32768

Mapping[1 0 15 -59], 0 1 -8 39]]

Wedgie⟨⟨1 -8 39 -15 59 113]]

Optimal tuning (POTE): ~2 = 1\1, ~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]
eigenmonzo (unchanged-interval) basis: 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]
eigenmonzo (unchanged-interval) basis: 2.9/5

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]

Optimal ET sequence53, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.722

Minimax tuning:

  • 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69
eigenmonzo (unchanged-interval) basis: 2.11/7

Optimal ET sequence53, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.745

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72
eigenmonzo (unchanged-interval) basis: 2.13/7

Optimal ET sequence53, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.742

Minimax tuning:

  • 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93
eigenmonzo (unchanged-interval) basis: 2.17/13

Optimal ET sequence53, 118, 171e, 289ef, 460eef

Badness: 0.028891

Helena

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 3146/3125

Mapping: [1 0 15 -59 51 -28], 0 1 -8 39 -30 20]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740

Optimal ET sequence53, 118f, 171ef

Badness: 0.036281

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125

Mapping: [1 0 15 -59 51 -28 -91], 0 1 -8 39 -30 20 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.730

Optimal ET sequence53, 118f, 171ef, 289eff

Badness: 0.030688

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625

Mapping: [1 0 15 -59 51 -28 -91 9], 0 1 -8 39 -30 20 60 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.729

Optimal ET sequence53, 118f, 171ef, 289effh

Badness: 0.021892

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.783

Minimax tuning:

  • 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122
eigenmonzo (unchanged-interval) basis: 2.11/7

Optimal ET sequence53, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.784

Minimax tuning:

  • 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122
Eigenmonzo (unchanged-interval) basis: 2.11/7

Optimal ET sequence53, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.777

Minimax tuning:

  • 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143
Eigenmonzo (unchanged-interval) basis: 2.17/11

Optimal ET sequence53, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.724

Minimax tuning:

  • 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88
eigenmonzo (unchanged-interval) basis: 2.11

Optimal ET sequence53e, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.738

Minimax tuning:

  • 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121
eigenmonzo (unchanged-interval) basis: 2.13/11

Optimal ET sequence53e, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740

Minimax tuning:

  • 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121
Eigenmonzo (unchanged-interval) basis: 2.13/11

Optimal ET sequence53e, 118, 171, 289f, 460ef

Badness: 0.029618

Pontoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 4375/4374, 32805/32768

Mapping: [1 0 15 -59 -136 -215], 0 1 -8 39 88 138]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735

Optimal ET sequence53ef, 118f, 171, 289, 460e, 749def

Badness: 0.050188

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768

Mapping: [1 0 15 -59 -136 -215 -91], 0 1 -8 39 88 138 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735

Optimal ET sequence53ef, 118f, 171, 289, 460e, 749defg

Badness: 0.029383

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 0 30 -118 -85], 0 1 -8 39 29]]

mapping generators: ~99/70, ~3

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757

Optimal ET sequence106, 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 0 30 -118 -85 112], 0 1 -8 39 29 -33]]

Mapping generators: ~99/70, ~3

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.773

Optimal ET sequence106, 118, 224, 566f, 790f

Badness: 0.030172

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873

Mapping: [2 0 30 -118 -85 112 -182], 0 1 -8 39 29 -33 60]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.765

Optimal ET sequence106g, 118, 224, 342, 566f

Badness: 0.027051

Counterbipont

Subgroup: 2.3.5.7.11.13

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

Mapping: [2 0 30 -118 -85 -243], 0 1 -8 39 29 79]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.769

Optimal ET sequence106f, 118f, 224, 342f, 566, 1356cf, 1922cff

Badness: 0.025547

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768

Mapping: [2 0 30 -118 -85 -243 -182], 0 1 -8 39 29 79 60]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.764

Optimal ET sequence106fg, 118f, 224, 342f, 566, 908fg, 1474cffgg

Badness: 0.025251

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864

Mapping: [2 0 30 -118 -85 -243 -182 -169], 0 1 -8 39 29 79 60 56]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.761

Optimal ET sequence106fgh, 118f, 224, 342f, 566h, 908fgh

Badness: 0.022267

Quadrapont

Subgroup: 2.3.5.7.11.13

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

Mapping: [4 0 60 -236 -170 -131], 0 1 -8 39 29 23]]

mapping generators: ~208/175, ~3

Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756

Optimal ET sequence224, 460, 684, 2276cde, 2960cde, 3644bccddee

Badness: 0.021025

Grackle

Grackle tempers out [-44 26 0 1. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.239

Minimax tuning:

  • 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
  • 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence12, 53d, 65, 77, 166c, 243c

Badness: 0.070407

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 32805/32768

Mapping: [1 0 15 44 70], 0 1 -8 -26 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.172

Optimal ET sequence12, 53dee, 65e, 77, 89, 166c, 255c

Badness: 0.048887

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 5445/5408

Mapping: [1 0 15 44 70 75], 0 1 -8 -26 -42 -45]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.226

Optimal ET sequence12f, 53deeff, 65ef, 77, 166cf, 243cf

Badness: 0.037859

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873

Mapping: [1 0 15 44 70 75 -7], 0 1 -8 -26 -42 -45 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.206

Optimal ET sequence12f, 53deeff, 65ef, 77, 89f, 166cf

Badness: 0.029864

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323

Mapping: [1 0 15 44 70 75 -7 9], 0 1 -8 -26 -42 -45 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217

Optimal ET sequence12f, 53deeff, 65ef, 77, 166cf

Badness: 0.023096

Grackloid

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 729/728, 1287/1280

Mapping: [1 0 15 44 70 -47], 0 1 -8 -26 -42 32]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217

Optimal ET sequence12, 53deef, 65e, 77, 166c

Badness: 0.048511

Grack

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 896/891

Mapping: [1 0 15 44 51], 0 1 -8 -26 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.401

Optimal ET sequence12, 53d, 65, 77e, 142de

Badness: 0.055908

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 245/242, 832/825

Mapping: [1 0 15 44 51 75], 0 1 -8 -26 -30 -45]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.348

Optimal ET sequence12f, 53dff, 65f, 77e

Badness: 0.044458

Catahelenic

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 352/351

Mapping: [1 0 15 44 51 56], 0 1 -8 -26 -30 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.529

Optimal ET sequence12f, 53df, 65

Badness: 0.048524

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]]

Optimal tuning (CTE): ~567/400 = 1\2, ~3/2 = 701.5899

Minimax tuning:

  • 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
  • 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence12, 106d, 118, 130, 248, 378

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]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.6077

Optimal ET sequence12, 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]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.5949

Optimal ET sequence12, 106def, 118, 130, 248, 378

Badness: 0.028722

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125

Mapping: [2 0 30 69 102 -75 5], 0 1 -8 -20 -30 26 1]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.5959

Optimal ET sequence12, 106def, 118, 130, 248g

Badness: 0.029340

Bischis

Subgroup: 2.3.5.7.11.13

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

Mapping: [2 0 30 69 102 131], 0 1 -8 -20 -30 -39]]

Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 701.5708

Optimal ET sequence12f, 106deff, 118f, 130

Badness: 0.029321

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125

Mapping: [2 0 30 69 102 131 5], 0 1 -8 -20 -30 -39 1]]

Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 701.5717

Optimal ET sequence12f, 106deff, 118f, 130, 248fg

Badness: 0.026894

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]]

Optimal tuning (POTE): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)

Optimal ET sequence24, 70c, 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]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)

Optimal ET sequence24, 70c, 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]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)

Optimal ET sequence24, 70c, 94, 118, 212f

Badness: 0.037640

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 385/384, 561/560

Mapping: [2 1 22 -15 8 15 6], 0 2 -16 19 -1 -7 2]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)

Optimal ET sequence24, 70c, 94, 118, 212fg

Badness: 0.025615

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]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)

Optimal ET sequence24f, 70cf, 94, 118f, 212

Badness: 0.037607

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580

Mapping: [2 1 22 -15 8 -36 6], 0 2 -16 19 -1 40 2]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)

Optimal ET sequence24f, 70cf, 94, 118f, 212g

Badness: 0.024734

Hemischis

Subgroup: 2.3.5.7

Comma list: 6144/6125, 19683/19600

Mapping[1 0 15 -17], 0 2 -16 25]]

mapping generators: ~2, ~140/81

Wedgie⟨⟨2 -16 25 -30 34 103]]

Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.797

Optimal ET sequence24, 53, 130, 183, 313

Badness: 0.045817

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 8019/8000

Mapping: [1 0 15 -17 51], 0 2 -16 25 -60]]

Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801

Optimal ET sequence24e, 53, 130, 183, 313

Badness: 0.036289

13-limit

Its S-expression-based comma list is {S12/S14, S13/S15 = S26, S27, S64(, S65)}. Tempering S13, S15 or S25 leads to 53edo (through Catakleismic) while tempering S12/S13, S13/S14, S14/S15 or S49 (thus leading to S12 = S13 = S14 = S15) leads to 130edo.

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 4096/4095

Mapping: [1 0 15 -17 51 14], 0 2 -16 25 -60 -13]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801

Optimal ET sequence24e, 53, 130, 183, 313

Badness: 0.020816

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

Mapping: [1 0 15 -17 51 14 -49], 0 2 -16 25 -60 -13 67]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810

Optimal ET sequence53, 130, 183, 679df

Badness: 0.021073

Music

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]]

Optimal tuning (POTE): ~2 = 1\1, ~160/147 = 166.140

Optimal ET sequence29, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097

Optimal ET sequence29, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054

Optimal ET sequence29, 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 subgroup 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]]

Optimal tuning (POTE): ~2 = 1\1, ~192/175 = 166.019

Optimal ET sequence65, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017

Optimal ET sequence65, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016

Optimal ET sequence65f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012

Optimal ET sequence65f, 94, 159, 253

Badness: 0.026504

Countertertiaschis

The countertertiaschis temperament (159 & 224) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup 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]]

Optimal tuning (POTE): ~2 = 1\1, ~625/567 = 166.0621

Optimal ET sequence65d, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628

Optimal ET sequence65d, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628

Optimal ET sequence65d, 159, 224, 383, 607

Badness: 0.024506

Pogo

See also: Stearnsmic clan #Pogo

The pogo temperament (94 & 130) splits the period in two to address the difference between #Tertiaschis and #Countertertiaschis. The schismic tempering of the fifth is just about right for tempering out the stearnsma.

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]]

Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.901

Optimal ET sequence36, 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]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911

Optimal ET sequence36, 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]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911

Optimal ET sequence36, 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]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.742

Minimax tuning:

Optimal ET sequence12, 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 are 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]]

Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685

Optimal ET sequence12e, 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]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689

Optimal ET sequence171, 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]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688

Optimal ET sequence171, 183, 354, 891de, 1245dee, 1599ddee

Badness: 0.020434

Semiterm

The semiterm temperament (12 & 342) 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

Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256

Optimal ET sequence12f, 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

Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)

Optimal ET sequence24d, 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]]

Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)

Optimal ET sequence24d, 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]]

Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)

Optimal ET sequence24d, 159, 183, 342f, 525f, 867ff

Badness: 0.022316

Altinex

Subgroup: 2.3.5.7

Comma list: 32805/32768, 367653125/362797056

Mapping[3 0 45 -32], 0 2 -16 17]]

mapping generators: ~1536/1225, ~34300/19683

Optimal tuning (CTE): ~1536/1225 = 1\3, ~34300/19683 = 950.9654

Optimal ET sequence24, …, 111c, 135, 159, 612ccdd, 771ccdd

Badness: 0.422026

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 14700/14641, 19712/19683

Mapping: [3 0 45 -32 8], 0 2 -16 17 1]]

Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9658

Optimal ET sequence: 24, …, 111c, 135, 159, 612ccdd, 771ccdd

Badness: 0.101224

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384, 676/675, 19712/19683

Mapping: [3 0 45 -32 8 42], 0 2 -16 17 1 -13]]

Optimal tuning (CTE): ~44/35 = 1\3, ~26/15 = 950.9360

Optimal ET sequence: 24, …, 111cf, 135f, 159

Badness: 0.054894

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]]

Optimal tuning (POTE): ~2 = 1\1, ~448/405 = 175.434

Minimax tuning:

Optimal ET sequence41, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406

Optimal ET sequence: 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]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409

Optimal ET sequence: 41, 89, 130, 301e, 431e

Badness: 0.022396

Sesquartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575

Mapping: [1 1 7 5 2 -2 -6], 0 4 -32 -15 10 39 69]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.424

Optimal ET sequence: 41, 89g, 130, 171, 301e

Badness: 0.023126

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: [1 1 7 5 2 -2 -6 6], 0 4 -32 -15 10 39 69 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.419

Optimal ET sequence: 41, 89g, 130, 171, 301eh

Badness: 0.020466

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: [1 1 7 5 2 -2 -6 6 -6], 0 4 -32 -15 10 39 69 -12 72]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.412

Optimal ET sequence: 41i, 89gi, 130, 171, 301eh

Badness: 0.019043

Heartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 256/255, 273/272, 364/363, 441/440

Mapping: [1 1 7 5 2 -2 0], 0 4 -32 -15 10 39 28]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.386

Optimal ET sequence: 41, 89, 130g

Badness: 0.028443

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440

Mapping: [1 1 7 5 2 -2 0 6], 0 4 -32 -15 10 39 28 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.380

Optimal ET sequence: 41, 89, 130g

Badness: 0.023059

Hearty

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625

Mapping: [1 1 7 5 2 -2 13], 0 4 -32 -15 10 39 -61]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.377

Optimal ET sequence: 41g, 89, 130, 609ceefgg

Badness: 0.030680

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455

Mapping: [1 1 7 5 2 -2 13 6], 0 4 -32 -15 10 39 -61 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.377

Optimal ET sequence: 41g, 89, 130, 609ceefggh

Badness: 0.022816

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440

Mapping: [1 1 7 5 2 -2 13 6 13], 0 4 -32 -15 10 39 -61 -12 -58]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.376

Optimal ET sequence: 41g, 89, 130, 609ceefggh

Badness: 0.019121

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]]

mapping generators: ~99/70, ~448/405

Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435

Optimal ET sequence82e, 130, 212, 342, 1156, 1498, 1840d

Badness: 0.016968

Quintilipyth

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

Optimal tuning (POTE): ~2 = 1\1, ~625/588 = 99.625

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615

Optimal ET sequence12f, 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 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]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.664

Optimal ET sequence12, …, 289, 301, 590, 891, 1192

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]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653

Optimal ET sequence12, …, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658

Optimal ET sequence12f, …, 277dff, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656

Optimal ET sequence12f, 277dff, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671

Optimal ET sequence12, …, 289e, 301, 915

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]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661

Optimal ET sequence12f, …, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.668

Optimal ET sequence12f, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.672

Optimal ET sequence12, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.671

Optimal ET sequence12, 301

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]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.672

Optimal ET sequence12, 301

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: 32805/32768, 235298/234375

Mapping[1 2 -1 -1], 0 -6 48 55]]

mapping generators: ~2, ~21/20

Wedgie⟨⟨6 -48 -55 -90 -104 7]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.053

Optimal ET sequence29, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049

Optimal ET sequence29, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049

Optimal ET sequence29, 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) and 67108864/66706983 (septiness comma).

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]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.614

Optimal ET sequence89, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616

Optimal ET sequence89, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610

Optimal ET sequence89, 94, 183, 277, 460d

Badness: 0.035315

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]]

Optimal tuning (POTE): ~2 = 1\1, ~243/224 = 140.350

Optimal ET sequence17, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365

Optimal ET sequence17, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363

Optimal ET sequence17, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.355

Optimal ET sequence41, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352

Optimal ET sequence41, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351

Optimal ET sequence41, 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]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8234

Optimal ET sequence212, 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]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176

Optimal ET sequence212, 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]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158

Optimal ET sequence212, 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 0 105 -56], 0 1 -8 7]]

mapping generators: ~8575/7776, ~3

Wedgie⟨⟨7 -56 49 -105 58 271]]

Optimal tuning (POTE): ~8575/7776 = 1\7, ~3/2 = 701.702

Optimal ET sequence77, 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 0 105 -56 -120], 0 1 -8 7 13]]

Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.719

Optimal ET sequence77, 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 0 105 -56 -120 37], 0 1 -8 7 13 -1]]

Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.724

Optimal ET sequence77, 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]]

Optimal tuning (POTE): ~42875/39366 = 1\8, ~3/2 = 701.713

Optimal ET sequence24, 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]]

Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.713

Optimal ET sequence24, 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]]

Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.725

Optimal ET sequence24, 224, 472, 696

Badness: 0.030425

Nonant

The nonant temperament (36 & 135) has a period of 1/9 octave and tempers out the septimal ennealimma, [-11 -9 0 9.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 40353607/40310784

Mapping[9 0 135 11], 0 1 -8 1]]

mapping generators: ~2592/2401, ~3

Optimal tuning (CTE): ~2592/2401 = 1\9, ~3/2 = 701.7232

Optimal ET sequence36, 99c, 135, 171

Badness: 0.069896

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 42875/42592

Mapping: [9 0 135 11 131], 0 1 -8 1 -7]]

Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.8398

Optimal ET sequence: 36, 99c, 135, 171, 477ce, 648cee

Badness: 0.126910

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 16807/16731

Mapping: [9 0 135 11 131 -38], 0 1 -8 1 -7 5]]

Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.7998

Optimal ET sequence: 36, 99cf, 135, 171

Badness: 0.076195

Tridecafifths

Tridecafifths divides the perfect 3/2 into 13 quartertones.

Subgroup: 2.3.5.7

Comma list: 32805/32768, [-14 -1 -9 13

Mapping[1 1 7 6], 0 13 -104 -71]]

mapping generators: ~2, ~1323/1280

Optimal tuning (CTE): ~2 = 1\1, ~1323/1280 = 53.9741

Optimal ET sequence89, 200, 289

Badness: 0.432580

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 55296000/55240493

Mapping: [1 1 7 6 4], 0 13 -104 -71 -12]]

Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744

Optimal ET sequence: 89, 200, 289

Badness: 0.127820