Schismatic family
The 5-limit parent comma for the schismatic (or schismic) family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is [-15 8 1⟩, and flipping that yields ⟨⟨ 1 -8 -15 ]] for the wedgie. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth. In fact, 10 = (4/3)8 × 32805/32768.
Schismatic aka Helmholtz
The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz, schismic or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53EDO is a possible tuning for schismatic, but you need 118EDO if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
Subgroup: 2.3.5
Comma list: 32805/32768
Mapping: [⟨1 0 15], ⟨0 1 -8]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.736
- 5-odd-limit diamond monotone: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
- 5-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955]
- 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]
Badness: 0.004259
Seven-limit extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- Garibaldi adds [25 -14 0 -1⟩,
- Grackle adds [-44 26 0 1⟩,
- Schism adds [6 -2 0 -1⟩,
- Pontiac adds [-59 39 0 -1⟩.
Those all have a fifth as generator.
- Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period.
- Guiron adds [-10 1 0 3⟩, with an 8/7 generator, three of which give the fifth.
- Term adds [-94 54 0 3⟩ with a 1/3 octave period.
- Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.
Temperaments discussed elsewhere include salsa, guiron and hemischis. Remarkable subgroup temperaments include nestoria and photia.
Garibaldi
Subgroup: 2.3.5.7
Comma list: 225/224, 3125/3087
Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -14 -15 -25 -10 ]]
POTE generator: ~3/2 = 702.085
- 7-odd-limit: ~3/2 = [2/3 1/15 0 -1/15⟩
- [[1 0 0 0⟩, [5/3 1/15 0 -1/15⟩, [5/3 -8/15 0 8/15⟩, [5/3 -14/15 0 14/15⟩]
- Eigenmonzos (unchanged intervals): 2, 7/6
- 9-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩
- [[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]
- Eigenmonzos (unchanged intervals): 2, 9/7
- 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
- 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 702.915]
Badness: 0.021644
Cassandra
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 2200/2187
Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.157
Minimax tuning:
- 11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩
- Eigenmonzos (unchanged intervals): 2, 9/7
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
- 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
- 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
Optimal GPV sequence: Template:Val list
Badness: 0.027396
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 325/324, 385/384
Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]
POTE generator: ~3/2 = 702.113
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34⟩
- Eigenmonzos (unchanged intervals): 2, 14/13
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
- 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
- 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
- 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
Optimal GPV sequence: Template:Val list
Badness: 0.020676
Andromeda
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 245/242
Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.321
Minimax tuning:
- 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20⟩
- Eigenmonzos (unchanged intervals): 2, 11/9
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
- 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 703.448]
Optimal GPV sequence: Template:Val list
Badness: 0.023556
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 196/195, 245/242
Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]
POTE generator: ~3/2 = 702.559
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23⟩
- Eigenmonzos (unchanged intervals): 2, 13/9
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
- 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
- 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
- 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.439, 703.448]
Optimal GPV sequence: Template:Val list
Badness: 0.020749
Helenus
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 3125/3087
Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.725
Minimax tuning:
- 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
- Eigenmonzos (unchanged intervals): 2, 11/9
Optimal GPV sequence: Template:Val list
Badness: 0.035637
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 176/175, 275/273, 847/845
Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]
POTE generator: ~3/2 = 701.747
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
- Eigenmonzos (unchanged intervals): 2, 11/9
Optimal GPV sequence: Template:Val list
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]
POTE generator: ~3/2 = 701.680
Optimal GPV sequence: Template:Val list
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]
POTE generator: ~3/2 = 701.705
Optimal GPV sequence: Template:Val list
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
POTE generator: ~63/55 = 248.918
Optimal GPV sequence: Template:Val list
Badness: 0.050681
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 225/224, 275/273
Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]
Mapping generators: ~2, ~26/15
POTE generator: ~15/13 = 248.918
Optimal GPV sequence: Template:Val list
Badness: 0.027464
Karadeniz
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]]
POTE generator: ~11/9 = 350.994
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~11/9 = 351.014
Optimal GPV sequence: Template:Val list
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
POTE generator: ~11/10 = 165.974
Optimal GPV sequence: Template:Val list
Badness: 0.058040
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 847/845, 1331/1323
Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]
Mapping generators: ~2, ~11/10
POTE generator: ~11/10 = 165.963
Optimal GPV sequence: Template:Val list
Badness: 0.033849
Schism
Subgroup: 2.3.5.7
Comma list: 64/63, 360/343
Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.556
Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]
Badness: 0.056648
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
Mapping generators: ~2, ~3
POTE generator ~3/2 = 702.136
Optimal GPV sequence: Template:Val list
Badness: 0.037482
Pontiac
Subgroup: 2.3.5.7
Comma list: 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 39 -15 59 113 ]]
POTE generator: ~3/2 = 701.757
- 7-odd-limit: ~3/2 = [27/47 0 -1/47 1/47⟩
- [[1 0 0 0⟩, [74/47 0 -1/47 1/47⟩, [113/47 0 8/47 -8/47⟩, [113/47 0 -39/47 39/47⟩]
- Eigenmonzos (unchanged intervals): 2, 7/5
- 9-odd-limit: ~3/2 = [1/2 1/5 -1/10⟩
- [[1 0 0 0⟩, [3/2 1/5 -1/10 0⟩, [3 -8/5 4/5 0⟩, [-1/2 39/5 -39/10 0⟩]
- Eigenmonzos (unchanged intervals): 2, 10/9
- 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]]
POTE generator: ~3/2 = 701.722
Minimax tuning:
- 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69⟩
- Eigenmonzos (unchanged intervals): 2, 14/11
Optimal GPV sequence: Template:Val list
Badness: 0.038863
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 625/624, 729/728
Mapping: [⟨1 0 15 -59 51 56], ⟨0 1 -8 39 -30 -33]]
POTE generator: ~3/2 = 701.745
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72⟩
- Eigenmonzos (unchanged intervals): 2, 14/13
Optimal GPV sequence: Template:Val list
Badness: 0.033677
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 385/384, 561/560, 625/624, 729/728
Mapping: [⟨1 0 15 -59 51 56 -91], ⟨0 1 -8 39 -30 -33 60]]
POTE generator: ~3/2 = 701.742
Minimax tuning:
- 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93⟩
- Eigenmonzos (unchanged intervals): 2, 17/13
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.740
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.730
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.729
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.783
Minimax tuning:
- 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
- Eigenmonzos (unchanged intervals): 2, 14/11
Optimal GPV sequence: Template:Val list
Badness: 0.048692
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 2200/2197
Mapping: [⟨1 0 15 -59 135 56], ⟨0 1 -8 39 -83 -33]]
POTE generator: ~3/2 = 701.784
Minimax tuning:
- 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
- Eigenmonzos (unchanged intervals): 2, 14/11
Optimal GPV sequence: Template:Val list
Badness: 0.023616
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197
Mapping: [⟨1 0 15 -59 135 56 -91], ⟨0 1 -8 39 -83 -33 60]]
POTE generator: ~3/2 = 701.777
Minimax tuning:
- 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143⟩
- Eigenmonzos (unchanged intervals): 2, 22/17
Optimal GPV sequence: Template:Val list
Badness: 0.022853
Pontic
The pontic temperament (118&171) tempers out the werckisma and the ragisma.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 -136], ⟨0 1 -8 39 88]]
POTE generator: ~3/2 = 701.724
Minimax tuning:
- 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88⟩
- Eigenmonzos (unchanged intervals): 2, 11/8
Optimal GPV sequence: Template:Val list
Badness: 0.049573
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 625/624, 729/728, 3584/3575
Mapping: [⟨1 0 15 -59 -136 56], ⟨0 1 -8 39 88 -33]]
POTE generator: ~3/2 = 701.738
Minimax tuning:
- 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
- Eigenmonzos (unchanged intervals): 2, 13/11
Optimal GPV sequence: Template:Val list
Badness: 0.045308
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873
Mapping: [⟨1 0 15 -59 -136 56 -91], ⟨0 1 -8 39 88 -33 60]]
POTE generator: ~3/2 = 701.740
Minimax tuning:
- 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
- Eigenmonzos (unchanged intervals): 2, 13/11
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.735
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.735
Optimal GPV sequence: Template:Val list
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
POTE generator: ~3/2 = 701.757
Optimal GPV sequence: Template:Val list
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
POTE generator: ~3/2 = 701.773
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.765
Optimal GPV sequence: Template:Val list
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]]
Mapping generators: ~99/70, ~3
POTE generator: ~3/2 = 701.769
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.764
Optimal GPV sequence: Template:Val list
Badness: 0.025251
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
POTE generator: ~3/2 = 701.756
Optimal GPV sequence: Template:Val list
Badness: 0.021025
Grackle
Subgroup: 2.3.5.7
Comma list: 126/125, 32805/32768
Mapping: [⟨1 0 15 44], ⟨0 1 -8 -26]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -26 -15 -44 -38 ]]
POTE generator: ~3/2 = 701.239
- 7-odd-limit eigenmonzos (unchanged intervals): 2, 7/6
- 9-odd-limit eigenmonzos (unchanged intervals): 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]]
POTE generator: ~3/2 = 701.172
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.226
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.206
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.217
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.217
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.401
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.348
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.529
Optimal GPV sequence: Template:Val list
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 ]]
POTE generator: ~3/2 = 701.592
- 7-odd-limit eigenmonzos (unchanged intervals): 2, 7/6
- 9-odd-limit eigenmonzos (unchanged intervals): 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]]
POTE generator: ~3/2 = 701.612
Optimal GPV sequence: Template:Val list
Badness: 0.028160
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]
POTE generator: ~3/2 = 701.590
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.600
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.565
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.595
Optimal GPV sequence: Template:Val list
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 ]]
POTE generator: ~36/35 = 50.920
Badness: 0.110583
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 9801/9800, 14641/14580
Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]
POTE generator: ~36/35 = 50.918
Optimal GPV sequence: Template:Val list
Badness: 0.036749
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1575/1573
Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]
POTE generator: ~36/35 = 50.938
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~36/35 = 50.942
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~36/35 = 50.951
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~36/35 = 50.948
Optimal GPV sequence: Template:Val list
Badness: 0.024734
Squirrel
The squirrel temperament (29&36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
Subgroup: 2.3.5.7
Comma list: 686/675, 32805/32768
Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]
Wedgie: ⟨⟨ 3 -24 -13 -45 -29 37 ]]
POTE generator: ~160/147 = 166.140
Badness: 0.174705
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/242, 686/675, 896/891
Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]
POTE generator: ~11/10 = 166.097
Optimal GPV sequence: Template:Val list
Badness: 0.068310
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 245/242, 896/891
Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]
POTE generator: ~11/10 = 166.054
Optimal GPV sequence: Template:Val list
Badness: 0.043750
Tertiaschis
The tertiaschis temperament (94&159) has a ~11/10 generator, sharing the same 2.3.5.11 with #Squirrel, but tempers out 1071785/1062882 for prime 7.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1071875/1062882
Mapping: [⟨1 2 -1 10], ⟨0 -3 24 -52]]
Wedgie: ⟨⟨ 3 -24 52 -45 74 188 ]]
POTE generator: ~192/175 = 166.019
Badness: 0.211859
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 4000/3993, 19712/19683
Mapping: [⟨1 2 -1 10 0], ⟨0 -3 24 -52 25]]
POTE generator: ~11/10 = 166.017
Optimal GPV sequence: Template:Val list
Badness: 0.061336
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1575/1573, 10985/10976
Mapping: [⟨1 2 -1 10 0 12], ⟨0 -3 24 -52 25 -60]]
POTE generator: ~11/10 = 166.016
Optimal GPV sequence: Template:Val list
Badness: 0.036700
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
Mapping: [⟨1 2 -1 10 0 12 -2], ⟨0 -3 24 -52 25 -60 44]]
POTE generator: ~11/10 = 166.012
Optimal GPV sequence: Template:Val list
Badness: 0.026504
Countertertiaschis
The countertertiaschis temperament (159&224) has a ~11/10 generator, sharing the same 2.3.5.11 with #Squirrel, but tempers out 244140625/243045684 for prime 7.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 244140625/243045684
Mapping: [⟨1 2 -1 -12], ⟨0 -3 24 107]]
POTE generator: ~625/567 = 166.0621
Badness: 0.188043
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 32805/32768
Mapping: [⟨1 2 -1 -12 0], ⟨0 -3 24 107 25]]
POTE generator: ~11/10 = 166.0628
Optimal GPV sequence: Template:Val list
Badness: 0.048943
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Mapping: [⟨1 2 -1 -12 0 -10], ⟨0 -3 24 107 25 99]]
POTE generator: ~11/10 = 166.0628
Optimal GPV sequence: Template:Val list
Badness: 0.024506
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 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 ]]
POTE generator: ~9/7 = 433.901
Badness: 0.079635
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 32805/32768
Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]
Mapping generators: ~99/70, ~9/7
POTE generator: ~9/7 = 433.911
Optimal GPV sequence: Template:Val list
Badness: 0.031857
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]
Mapping generators: ~99/70, ~9/7
POTE generator: ~9/7 = 433.911
Optimal GPV sequence: Template:Val list
Badness: 0.017514
Term
Subgroup: 2.3.5.7
Comma list: 32805/32768, 250047/250000
Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]
Mapping generators: ~63/50, ~3
Wedgie: ⟨⟨ 3 -24 -54 -45 -94 -58 ]]
POTE generator: ~3/2 = 701.742
- 7-odd-limit eigenmonzos (unchanged intervals): 2, 6/5
- 9-odd-limit eigenmonzos (unchanged intervals): 2, 9/7
Badness: 0.019950
Terminal
The terminal temperament (12&159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 32805/32768
Mapping: [⟨3 0 45 94 134], ⟨0 1 -8 -18 -26]]
POTE generator: ~3/2 = 701.824
Optimal GPV sequence: Template:Val list
Badness: 0.059502
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 13720/13689
Mapping: [⟨3 0 45 94 134 168], ⟨0 1 -8 -18 -26 -33]]
POTE generator: ~3/2 = 701.821
Optimal GPV sequence: Template:Val list
Badness: 0.037082
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
Mapping: [⟨3 0 45 94 134 168 -2], ⟨0 1 -8 -18 -26 -33 3]]
POTE generator: ~3/2 = 701.810
Optimal GPV sequence: Template:Val list
Badness: 0.027073
Terminator
Subgroup: 2.3.5.7.11
Comma list: 540/539, 32805/32768, 137781/137500
Mapping: [⟨3 0 45 94 -137], ⟨0 1 -8 -18 31]]
POTE generator: ~3/2 = 701.685
Optimal GPV sequence: Template:Val list
Badness: 0.066968
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 31250/31213
Mapping: [⟨3 0 45 94 -137 -103], ⟨0 1 -8 -18 31 24]]
POTE generator: ~3/2 = 701.689
Optimal GPV sequence: Template:Val list
Badness: 0.035487
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
Mapping: [⟨3 0 45 94 -137 -103 -2], ⟨0 1 -8 -18 31 24 3]]
POTE generator: ~3/2 = 701.688
Optimal GPV sequence: Template:Val list
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
POTE generator: ~3/2 = 701.7460
Optimal GPV sequence: Template:Val list
Badness: 0.029438
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
Mapping: [⟨6 0 90 188 287 355], ⟨0 1 -8 -18 -28 -35]]
POTE tuning: ~3/2 = 701.7256
Optimal GPV sequence: Template:Val list *
* optimal patent val: 354
Badness: 0.044657
Hemiterm
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 32805/32768, 102487/102400
Mapping: [⟨3 0 45 94 8], ⟨0 2 -16 -36 1]]
Mapping generators: ~63/50, ~693/400
POTE generator: ~12/11 = 150.872
Optimal GPV sequence: Template:Val list
Badness: 0.020687
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
Mapping: [⟨3 0 45 94 8 42], ⟨0 2 -16 -36 1 -13]]
POTE generator: ~12/11 = 150.873
Optimal GPV sequence: Template:Val list
Badness: 0.031362
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
Mapping: [⟨3 0 45 94 8 42 -2], ⟨0 2 -16 -36 1 -13 6]]
POTE generator: ~12/11 = 150.867
Optimal GPV sequence: Template:Val list
Badness: 0.022316
Sesquiquartififths
Subgroup: 2.3.5.7
Comma list: 2401/2400, 32805/32768
Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]
Mapping generators: ~2, ~448/405
Wedgie: ⟨⟨ 4 -32 -15 -60 -35 55 ]]
POTE generator: ~448/405 = 175.434
- 7-odd-limit eigenmonzos (unchanged intervals): 2, 7/6
- 9-odd-limit eigenmonzos (unchanged intervals): 2, 9/7
Badness: 0.011244
Sesquart
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 16384/16335
Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]
Mapping generators: ~2, ~256/231
POTE generator: ~256/231 = 175.406
Optimal GPV sequence: Template:Val list
Badness: 0.029306
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 3584/3575
Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]
POTE generator: ~72/65 = 175.409
Optimal GPV sequence: Template:Val list
Badness: 0.022396
Bisesqui
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 32805/32768
Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]
POTE generator: ~448/405 = 175.435
Optimal GPV sequence: Template:Val list
Badness: 0.016968
Quintilipyth
The quintilipyth temperament (12&253, formerly quintilischis temperament) slices the pythagorean fourth (4/3) into five semitones and tempers out the compass comma (9765625/9680832, quinruyoyo) in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 9765625/9680832
Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]
Wedgie: ⟨⟨ 5 -40 -82 -75 -144 -78 ]]
POTE generator: ~625/588 = 99.625
Badness: 0.253966
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4375/4356, 32805/32768
Mapping: [⟨1 2 -1 -4 -7], ⟨0 -5 40 82 126]]
POTE generator: ~35/33 = 99.616
Optimal GPV sequence: Template:Val list
Badness: 0.113044
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Mapping: [⟨1 2 -1 -4 -7 -9], ⟨0 -5 40 82 126 153]]
POTE generator: ~35/33 = 99.612
Optimal GPV sequence: Template:Val list
Badness: 0.069127
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Mapping: [⟨1 2 -1 -4 -7 -9 5], ⟨0 -5 40 82 126 153 -11]]
POTE generator: ~18/17 = 99.612
Optimal GPV sequence: Template:Val list
Badness: 0.045992
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Mapping: [⟨1 2 -1 -4 -7 -9 5 4], ⟨0 -5 40 82 126 153 -11 3]]
POTE generator: ~18/17 = 99.615
Optimal GPV sequence: Template:Val list
Badness: 0.038155
Quintaschis
The quintaschis temperament (12&289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 (quinzo-alegu) in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 49009212/48828125
Mapping: [⟨1 2 -1 -5], ⟨0 -5 40 94]]
Wedgie: ⟨⟨ 5 -40 -94 -75 -163 -106 ]]
POTE generator: ~200/189 = 99.664
Badness: 0.132890
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 32805/32768, 1953125/1951488
Mapping: [⟨1 2 -1 -5 -8], ⟨0 -5 40 94 138]]
POTE generator: ~35/33 = 99.653
Optimal GPV sequence: Template:Val list
Badness: 0.111477
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Mapping: [⟨1 2 -1 -5 -8 -11], ⟨0 -5 40 94 138 177]]
POTE generator: ~35/33 = 99.658
Optimal GPV sequence: Template:Val list
Badness: 0.074218
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
Mapping: [⟨1 2 -1 -5 -8 -11 5], ⟨0 -5 40 94 138 177 -11]]
POTE generator: ~18/17 = 99.656
Optimal GPV sequence: Template:Val list
Badness: 0.050571
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Mapping: [⟨1 2 -1 -5 -8 -11 5 4], ⟨0 -5 40 94 138 177 -11 3]]
POTE generator: ~18/17 = 99.659
Optimal GPV sequence: Template:Val list
Badness: 0.042120
Quintahelenic
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 8019/8000, 151263/151250
Mapping: [⟨1 2 -1 -5 -9], ⟨0 -5 40 94 150]]
POTE generator: ~200/189 = 99.671
Optimal GPV sequence: Template:Val list
Badness: 0.082225
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
Mapping: [⟨1 2 -1 -5 -9 -11], ⟨0 -5 40 94 150 177]]
POTE generator: ~200/189 = 99.661
Optimal GPV sequence: Template:Val list
Badness: 0.055570
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
Mapping: [⟨1 2 -1 -5 -9 -11 5], ⟨0 -5 40 94 150 177 -11]]
POTE generator: ~18/17 = 99.665
Optimal GPV sequence: Template:Val list
Badness: 0.040412
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
Mapping: [⟨1 2 -1 -5 -9 -11 5 4], ⟨0 -5 40 94 150 177 -11 3]]
POTE generator: ~18/17 = 99.668
Optimal GPV sequence: Template:Val list
Badness: 0.036840
Quintahelenoid
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
Mapping: [⟨1 2 -1 -5 -9 14], ⟨0 -5 40 94 150 -124]]
POTE generator: ~200/189 = 99.672
Optimal GPV sequence: Template:Val list
Badness: 0.066108
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Mapping: [⟨1 2 -1 -5 -9 14 5], ⟨0 -5 40 94 150 -124 -11]]
POTE generator: ~18/17 = 99.671
Optimal GPV sequence: Template:Val list
Badness: 0.047908
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Mapping: [⟨1 2 -1 -5 -9 14 5 4], ⟨0 -5 40 94 150 -124 -11 3]]
POTE generator: ~18/17 = 99.672
Optimal GPV sequence: Template:Val list
Badness: 0.039542
Sextilififths
The sextilififths (130&159, also known as sextilischis) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
Subgroup: 2.3.5.7
Comma list: 32768/32805, 235298/234375
Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]
Mapping generators: ~2, ~21/20
Wedgie: ⟨⟨ 6 -48 -55 -90 -104 7 ]]
POTE generator: ~21/20 = 83.053
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]]
POTE generator: ~21/20 = 83.049
Optimal GPV sequence: Template:Val list
Badness: 0.045457
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 10985/10976
Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]
POTE generator: ~21/20 = 83.049
Optimal GPV sequence: Template:Val list
Badness: 0.025276
Septiquarschis
The septiquarschis temperament (89&94) splits septimal minor seventh (7/4) into four generators and tempers out 829440/823543 (mynaslender comma, sepru-ayo) and 67108864/66706983 (septiness comma, sasasepru).
Subgroup: 2.3.5.7
Comma list: 32805/32768, 829440/823543
Mapping: [⟨1 3 -9 2], ⟨0 -7 -56 4]]
Wedgie: ⟨⟨ 7 56 -4 231 -26 -76 ]]
POTE generator: ~147/128 = 242.614
Badness: 0.187047
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 15488/15435, 32805/32768
Mapping: [⟨1 3 -9 2 -2], ⟨0 -7 -56 4 27]]
POTE generator: ~147/128 = 242.616
Optimal GPV sequence: Template:Val list
Badness: 0.052002
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Mapping: [⟨1 3 -9 2 -2 13], ⟨0 -7 -56 4 27 -46]]
POTE generator: ~147/128 = 242.610
Optimal GPV sequence: Template:Val list
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 ]]
POTE generator: ~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]]
POTE generator: ~88/81 = 140.365
Optimal GPV sequence: Template:Val list
Badness: 0.063499
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1331/1323
Mapping: [⟨1 1 7 0 1 3], ⟨0 5 -40 24 21 6]]
POTE generator: ~13/12 = 140.363
Optimal GPV sequence: Template:Val list
Badness: 0.037886
Quanharuk
Subgroup: 2.3.5.7
Comma list: 16875/16807, 32805/32768
Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]
Mapping generators: ~2, ~56/45
Wedgie: ⟨⟨ 5 -40 -29 -75 -60 45 ]]
POTE generator: ~56/45 = 380.355
Badness: 0.071950
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 32805/32768
Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]
Mapping generators: ~2, ~56/45
POTE generator: ~56/45 = 380.352
Optimal GPV sequence: Template:Val list
Badness: 0.031549
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]
Mapping generators: ~2, ~56/45
POTE generator: ~56/45 = 380.351
Optimal GPV sequence: Template:Val list
Badness: 0.021392
Quadrant
The quadrant temperament (12&224) has a period of quarter octave and tempers out the dimcomp comma, 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 390625/388962
Mapping: [⟨4 0 60 119], ⟨0 1 -8 -17]]
Mapping generators: ~25/21, ~3
Wedgie: ⟨⟨ 4 -32 -68 -60 -119 -68 ]]
POTE generator: ~25/21 = 301.823
Badness: 0.110242
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 6250/6237, 32805/32768
Mapping: [⟨4 0 60 119 185], ⟨0 1 -8 -17 -27]]
POTE generator: ~25/21 = 301.819
Optimal GPV sequence: Template:Val list
Badness: 0.045738
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
Mapping: [⟨4 0 60 119 185 224], ⟨0 1 -8 -17 -27 -33]]
POTE generator: ~25/21 = 301.816
Optimal GPV sequence: Template:Val list
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 ]]
POTE generator: ~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]]
POTE generator: ~3/2 = 701.719
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~3/2 = 701.724
Optimal GPV sequence: Template:Val list
Badness: 0.024706
Octant
The octant temperament (224&472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 2259436291848/2251875390625
Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]
Mapping generators: ~42875/39366, ~3
Wedgie: ⟨⟨ 8 -64 88 -120 117 384 ]]
POTE generator: ~3/2 = 701.713
Badness: 0.157186
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 46656/46585
Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]
Mapping generators: ~12/11, ~3
POTE generator: ~3/2 = 701.713
Optimal GPV sequence: Template:Val list
Badness: 0.044778
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]
Mapping generators: ~12/11, ~3
POTE generator: ~3/2 = 701.725
Optimal GPV sequence: Template:Val list
Badness: 0.030425