Schismatic family

(Redirected from Schismic)

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]

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

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]

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]

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

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

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

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

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

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

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]

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]

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

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

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

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

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

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

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

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

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

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

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)

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)

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

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

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

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

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]]

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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