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

• 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]

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

[[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

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]

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]

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

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

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

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

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

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

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]

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]

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

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

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

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

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

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

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

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

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

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

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)

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)

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

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

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

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

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). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

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

• CTE: ~2 = 1\1, ~3/2 = 702.2696
• POTE: ~2 = 1\1, ~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]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 703.3833
• POTE: ~2 = 1\1, ~3/2 = 702.136

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

[[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
• 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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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)

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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