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.
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]
Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc
Badness: 0.004259
Overview to extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- #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.
- #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
- Main article: Garibaldi temperament
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.
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
- 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⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/3
- 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⟩]
- eigenmonzo (unchanged-interval) basis: 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]
Optimal ET sequence: 12, 29, 41, 53, 94, 241c, 335cd, 576ccd
Badness: 0.021644
Cassandra
Cassandra is one of the best extension of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 2200/2187
Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157
Minimax tuning:
- 11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩
- eigenmonzo (unchanged-interval) basis: 2.9/7
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
- 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
- 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
Optimal ET sequence: 41, 53, 94, 229c, 323c, 417cce
Badness: 0.027396
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 325/324, 385/384
Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34⟩
- Eigenmonzo (unchanged-interval) basis: 2.13/7
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
- 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
- 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
- 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
Optimal ET sequence: 41, 53, 94, 429ccdeef, 523ccdeef
Badness: 0.020676
Cassie
Subgroup: 2.3.5.7.11.13.17
Comma list: 120/119, 154/153, 225/224, 273/272, 325/324
Mapping: [⟨1 0 15 25 -33 -28 -7], ⟨0 1 -8 -14 23 20 7]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092
Optimal ET sequence: 41, 53, 94g
Badness: 0.023270
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272
Mapping: [⟨1 0 15 25 -33 -28 -7 9], ⟨0 1 -8 -14 23 20 7 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079
Optimal ET sequence: 41, 53, 94g
Badness: 0.018189
Cassandric
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 275/273, 325/324, 375/374, 385/384
Mapping: [⟨1 0 15 25 -33 -28 77], ⟨0 1 -8 -14 23 20 -46]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097
Optimal ET sequence: 41g, 53, 94, 241ce, 335cde
Badness: 0.023167
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374
Mapping: [⟨1 0 15 25 -33 -28 77 9], ⟨0 1 -8 -14 23 20 -46 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.098
Optimal ET sequence: 41g, 53, 94, 241ceh, 335cdehh
Badness: 0.017635
Cassander
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 225/224, 275/273, 325/324, 385/384
Mapping: [⟨1 0 15 25 -33 -28 -72], ⟨0 1 -8 -14 23 20 48]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144
Optimal ET sequence: 41, 53g, 94
Badness: 0.022454
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324
Mapping: [⟨1 0 15 25 -33 -28 -72 9], ⟨0 1 -8 -14 23 20 48 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135
Optimal ET sequence: 41, 53g, 94
Badness: 0.017576
Andromeda
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 245/242
Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321
Minimax tuning:
- 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20⟩
- eigenmonzo (unchanged-interval) basis: 2.11/9
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
- 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 703.448]
Optimal ET sequence: 12, 29, 41, 217ce, 258ce
Badness: 0.023556
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 196/195, 245/242
Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]
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]
- 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.439, 703.448]
Optimal ET sequence: 12f, 29, 41, 152cdf, 193cdf, 234cdf
Badness: 0.020749
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 120/119, 189/187, 196/195
Mapping: [⟨1 0 15 25 32 37 -7], ⟨0 1 -8 -14 -18 -21 7]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312
Optimal ET sequence: 12f, 29, 41
Badness: 0.023406
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195
Mapping: [⟨1 0 15 25 32 37 -7 9], ⟨0 1 -8 -14 -18 -21 7 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357
Optimal ET sequence: 12f, 29, 41
Badness: 0.019154
Schisicosiennic
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 154/153, 170/169, 196/195
Mapping: [⟨1 0 15 25 32 37 58], ⟨0 1 -8 -14 -18 -21 -34]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725
Optimal ET sequence: 12fg, 29g, 41, 70cd, 111cd
Badness: 0.021758
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189
Mapping: [⟨1 0 15 25 32 37 58 9], ⟨0 1 -8 -14 -18 -21 -34 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753
Optimal ET sequence: 12fg, 29g, 41, 70cd, 111cdh, 181ccddh
Badness: 0.017902
Schisicosiennoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 85/84, 100/99, 105/104, 119/117, 221/220
Mapping: [⟨1 0 15 25 32 37 12], ⟨0 1 -8 -14 -18 -21 -5]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717
Optimal ET sequence: 12f, 29g, 41g, 70cdgg
Badness: 0.020895
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152
Mapping: [⟨1 0 15 25 32 37 12 9], ⟨0 1 -8 -14 -18 -21 -5 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716
Optimal ET sequence: 12f, 29g, 41g, 70cdgg
Badness: 0.016773
Helenus
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 3125/3087
Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725
Minimax tuning:
- 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
- eigenmonzo (unchanged-interval) basis: 2.11/9
Optimal ET sequence: 12, 41e, 53, 118d, 171de
Badness: 0.035637
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 176/175, 275/273, 847/845
Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
- Eigenmonzo (unchanged-interval) basis: 2.11/9
Optimal ET sequence: 12f, 41ef, 53, 118d, 171de
Badness: 0.026284
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 99/98, 120/119, 176/175, 275/273, 442/441
Mapping: [⟨1 0 15 25 51 56 -7], ⟨0 1 -8 -14 -30 -33 7]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680
Optimal ET sequence: 12f, 41ef, 53, 65d, 118dg
Badness: 0.023732
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245
Mapping: [⟨1 0 15 25 51 56 -7 9], ⟨0 1 -8 -14 -30 -33 7 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705
Optimal ET sequence: 12f, 41ef, 53, 65d, 118dg
Badness: 0.019411
Hemigari
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 3125/3087
Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]
- mapping generators: ~2, ~110/63
Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)
Optimal ET sequence: 29, 53, 82e, 135e, 188ce
Badness: 0.050681
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 225/224, 275/273
Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)
Optimal ET sequence: 29, 53, 82e, 135ef, 188cef
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]]
- mapping generators: ~2, ~11/9
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994
Optimal ET sequence: 41, 106, 147
Badness: 0.041562
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 325/324, 640/637
Mapping: [⟨1 1 7 11 2 -8], ⟨0 2 -16 -28 5 40]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
Optimal ET sequence: 41, 106, 147
Badness: 0.042564
Sanjaab
Subgroup: 2.3.5.7.11
Comma list: 225/224, 1331/1323, 3125/3087
Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]
- mapping generators: ~2, ~11/10
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974
Optimal ET sequence: 29, 65d, 94, 441cde, 535cde, 629cde
Badness: 0.058040
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 847/845, 1331/1323
Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963
Optimal ET sequence: 29, 65d, 94
Badness: 0.033849
Schism
- See also: Archytas clan #Schism
Schism is a low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb).
Subgroup: 2.3.5.7
Comma list: 64/63, 360/343
Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.556
Wedgie: ⟨⟨1 -8 -2 -15 -6 18]]
Optimal ET sequence: 12, 29d, 41d, 53d
Badness: 0.056648
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136
Optimal ET sequence: 12, 29de, 41de
Badness: 0.037482
Pontiac
- Main article: 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
- 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⟩]
- eigenmonzo (unchanged-interval) basis: 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⟩]
- 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]
Optimal ET sequence: 53, 118, 171, 1592c, 1763c, 1934c, 2105c, 2276cd, 2447cd, 2618cd, 2789cd, 2960cd, 3131bcd
Badness: 0.014133
Helenoid
The helenoid temperament (53 & 118) is closely related to the helenus temperament, but with the ragisma rather than the marvel comma tempered out.
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3388/3375, 4375/4374
Mapping: [⟨1 0 15 -59 51], ⟨0 1 -8 39 -30]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.722
Minimax tuning:
- 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69⟩
- eigenmonzo (unchanged-interval) basis: 2.11/7
Optimal ET sequence: 53, 118, 289e, 407de
Badness: 0.038863
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 625/624, 729/728
Mapping: [⟨1 0 15 -59 51 56], ⟨0 1 -8 39 -30 -33]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.745
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72⟩
- eigenmonzo (unchanged-interval) basis: 2.13/7
Optimal ET sequence: 53, 118, 171e
Badness: 0.033677
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 385/384, 561/560, 625/624, 729/728
Mapping: [⟨1 0 15 -59 51 56 -91], ⟨0 1 -8 39 -30 -33 60]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.742
Minimax tuning:
- 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93⟩
- eigenmonzo (unchanged-interval) basis: 2.17/13
Optimal ET sequence: 53, 118, 171e, 289ef, 460eef
Badness: 0.028891
Helena
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 385/384, 3146/3125
Mapping: [⟨1 0 15 -59 51 -28], ⟨0 1 -8 39 -30 20]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Optimal ET sequence: 53, 118f, 171ef
Badness: 0.036281
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125
Mapping: [⟨1 0 15 -59 51 -28 -91], ⟨0 1 -8 39 -30 20 60]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.730
Optimal ET sequence: 53, 118f, 171ef, 289eff
Badness: 0.030688
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625
Mapping: [⟨1 0 15 -59 51 -28 -91 9], ⟨0 1 -8 39 -30 20 60 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.729
Optimal ET sequence: 53, 118f, 171ef, 289effh
Badness: 0.021892
Ponta
The ponta temperament (53 & 171) tempers out the swetisma and the ragisma.
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 135], ⟨0 1 -8 39 -83]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.783
Minimax tuning:
- 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
- eigenmonzo (unchanged-interval) basis: 2.11/7
Optimal ET sequence: 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee
Badness: 0.048692
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 2200/2197
Mapping: [⟨1 0 15 -59 135 56], ⟨0 1 -8 39 -83 -33]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.784
Minimax tuning:
- 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
- Eigenmonzo (unchanged-interval) basis: 2.11/7
Optimal ET sequence: 53, 171, 224
Badness: 0.023616
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197
Mapping: [⟨1 0 15 -59 135 56 -91], ⟨0 1 -8 39 -83 -33 60]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.777
Minimax tuning:
- 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143⟩
- Eigenmonzo (unchanged-interval) basis: 2.17/11
Optimal ET sequence: 53, 171, 224, 395e, 619eg
Badness: 0.022853
Pontic
The pontic temperament (118 & 171) tempers out the werckisma and the ragisma.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 -136], ⟨0 1 -8 39 88]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.724
Minimax tuning:
- 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88⟩
- eigenmonzo (unchanged-interval) basis: 2.11
Optimal ET sequence: 53e, 118, 289, 407d, 696d
Badness: 0.049573
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 625/624, 729/728, 3584/3575
Mapping: [⟨1 0 15 -59 -136 56], ⟨0 1 -8 39 88 -33]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.738
Minimax tuning:
- 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
- eigenmonzo (unchanged-interval) basis: 2.13/11
Optimal ET sequence: 53e, 118, 171, 289f, 460ef
Badness: 0.045308
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873
Mapping: [⟨1 0 15 -59 -136 56 -91], ⟨0 1 -8 39 88 -33 60]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Minimax tuning:
- 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
- Eigenmonzo (unchanged-interval) basis: 2.13/11
Optimal ET sequence: 53e, 118, 171, 289f, 460ef
Badness: 0.029618
Pontoid
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 -136 -215], ⟨0 1 -8 39 88 138]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Optimal ET sequence: 53ef, 118f, 171, 289, 460e, 749def
Badness: 0.050188
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768
Mapping: [⟨1 0 15 -59 -136 -215 -91], ⟨0 1 -8 39 88 138 60]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Optimal ET sequence: 53ef, 118f, 171, 289, 460e, 749defg
Badness: 0.029383
Bipont
The bipont temperament (118 & 224) has a period of half octave and tempers out the lehmerisma (3025/3024) and the kalisma (9801/9800).
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 32805/32768
Mapping: [⟨2 0 30 -118 -85], ⟨0 1 -8 39 29]]
- mapping generators: ~99/70, ~3
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757
Optimal ET sequence: 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde
Badness: 0.014629
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1575/1573, 4096/4095
Mapping: [⟨2 0 30 -118 -85 112], ⟨0 1 -8 39 29 -33]]
Mapping generators: ~99/70, ~3
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.773
Optimal ET sequence: 106, 118, 224, 566f, 790f
Badness: 0.030172
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873
Mapping: [⟨2 0 30 -118 -85 112 -182], ⟨0 1 -8 39 29 -33 60]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.765
Optimal ET sequence: 106g, 118, 224, 342, 566f
Badness: 0.027051
Counterbipont
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
Mapping: [⟨2 0 30 -118 -85 -243], ⟨0 1 -8 39 29 79]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.769
Optimal ET sequence: 106f, 118f, 224, 342f, 566, 1356cf, 1922cff
Badness: 0.025547
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768
Mapping: [⟨2 0 30 -118 -85 -243 -182], ⟨0 1 -8 39 29 79 60]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.764
Optimal ET sequence: 106fg, 118f, 224, 342f, 566, 908fg, 1474cffgg
Badness: 0.025251
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864
Mapping: [⟨2 0 30 -118 -85 -243 -182 -169], ⟨0 1 -8 39 29 79 60 56]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.761
Optimal ET sequence: 106fgh, 118f, 224, 342f, 566h, 908fgh
Badness: 0.022267
Quadrapont
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768
Mapping: [⟨4 0 60 -236 -170 -131], ⟨0 1 -8 39 29 23]]
- mapping generators: ~208/175, ~3
Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756
Optimal ET sequence: 224, 460, 684, 2276cde, 2960cde, 3644bccddee
Badness: 0.021025
Grackle
Grackle tempers out [-44 26 0 1⟩. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
Subgroup: 2.3.5.7
Comma list: 126/125, 32805/32768
Mapping: [⟨1 0 15 44], ⟨0 1 -8 -26]]
- mapping generators: ~2, ~3
Wedgie: ⟨⟨1 -8 -26 -15 -44 -38]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.239
- 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 12, 53d, 65, 77, 166c, 243c
Badness: 0.070407
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 32805/32768
Mapping: [⟨1 0 15 44 70], ⟨0 1 -8 -26 -42]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.172
Optimal ET sequence: 12, 53dee, 65e, 77, 89, 166c, 255c
Badness: 0.048887
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 196/195, 5445/5408
Mapping: [⟨1 0 15 44 70 75], ⟨0 1 -8 -26 -42 -45]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.226
Optimal ET sequence: 12f, 53deeff, 65ef, 77, 166cf, 243cf
Badness: 0.037859
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873
Mapping: [⟨1 0 15 44 70 75 -7], ⟨0 1 -8 -26 -42 -45 7]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.206
Optimal ET sequence: 12f, 53deeff, 65ef, 77, 89f, 166cf
Badness: 0.029864
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323
Mapping: [⟨1 0 15 44 70 75 -7 9], ⟨0 1 -8 -26 -42 -45 7 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Optimal ET sequence: 12f, 53deeff, 65ef, 77, 166cf
Badness: 0.023096
Grackloid
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 729/728, 1287/1280
Mapping: [⟨1 0 15 44 70 -47], ⟨0 1 -8 -26 -42 32]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Optimal ET sequence: 12, 53deef, 65e, 77, 166c
Badness: 0.048511
Grack
Subgroup: 2.3.5.7.11
Comma list: 126/125, 245/242, 896/891
Mapping: [⟨1 0 15 44 51], ⟨0 1 -8 -26 -30]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.401
Optimal ET sequence: 12, 53d, 65, 77e, 142de
Badness: 0.055908
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 245/242, 832/825
Mapping: [⟨1 0 15 44 51 75], ⟨0 1 -8 -26 -30 -45]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.348
Optimal ET sequence: 12f, 53dff, 65f, 77e
Badness: 0.044458
Catahelenic
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 126/125, 245/242, 352/351
Mapping: [⟨1 0 15 44 51 56], ⟨0 1 -8 -26 -30 -33]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.529
Optimal ET sequence: 12f, 53df, 65
Badness: 0.048524
Bischismic
Subgroup: 2.3.5.7
Comma list: 3136/3125, 32805/32768
Mapping: [⟨2 0 30 69], ⟨0 1 -8 -20]]
- mapping generators: ~567/400, ~3
Wedgie: ⟨⟨2 -16 -40 -30 -69 -48]]
Optimal tuning (POTE): ~567/400 = 1\2, ~3/2 = 701.592
- 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 12, 106d, 118, 130, 248, 378, 508
Badness: 0.054744
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 8019/8000
Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.612
Optimal ET sequence: 12, 106de, 118, 130, 248
Badness: 0.028160
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.590
Optimal ET sequence: 12, 106def, 118, 130, 248, 378
Badness: 0.028722
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
Mapping: [⟨2 0 30 69 102 -75 5], ⟨0 1 -8 -20 -30 26 1]]
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.600
Optimal ET sequence: 12, 106def, 118, 130, 248g
Badness: 0.029340
Bischis
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 441/440, 3136/3125
Mapping: [⟨2 0 30 69 102 131], ⟨0 1 -8 -20 -30 -39]]
Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.565
Optimal ET sequence: 12f, 106deff, 118f, 130
Badness: 0.029321
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
Mapping: [⟨2 0 30 69 102 131 5], ⟨0 1 -8 -20 -30 -39 1]]
Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.595
Optimal ET sequence: 12f, 106deff, 118f, 130, 248fg, 378fgg
Badness: 0.026894
Kleischismic
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1500625/1492992
Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]
- mapping generators: ~1225/864, ~35/24
Wedgie: ⟨⟨4 -32 38 -60 49 178]]
Optimal tuning (POTE): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)
Optimal ET sequence: 24, 70c, 94, 118, 212, 330, 542d, 872cd
Badness: 0.110583
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 9801/9800, 14641/14580
Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)
Optimal ET sequence: 24, 70c, 94, 118, 212, 330e, 542de
Badness: 0.036749
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1575/1573
Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)
Optimal ET sequence: 24, 70c, 94, 118, 212f
Badness: 0.037640
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
Mapping: [⟨2 1 22 -15 8 15 6], ⟨0 2 -16 19 -1 -7 2]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)
Optimal ET sequence: 24, 70c, 94, 118, 212fg
Badness: 0.025615
Kleischis
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1573/1568, 14641/14580
Mapping: [⟨2 1 22 -15 8 -36], ⟨0 2 -16 19 -1 40]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)
Optimal ET sequence: 24f, 70cf, 94, 118f, 212
Badness: 0.037607
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
Mapping: [⟨2 1 22 -15 8 -36 6], ⟨0 2 -16 19 -1 40 2]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)
Optimal ET sequence: 24f, 70cf, 94, 118f, 212g
Badness: 0.024734
Hemischis
Subgroup: 2.3.5.7
Comma list: 6144/6125, 19683/19600
Mapping: [⟨1 0 15 -17], ⟨0 2 -16 25]]
- mapping generators: ~2, ~140/81
Wedgie: ⟨⟨2 -16 25 -30 34 103]]
Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.797
Optimal ET sequence: 24, 53, 130, 183, 313
Badness: 0.045817
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 5632/5625, 8019/8000
Mapping: [⟨1 0 15 -17 51], ⟨0 2 -16 25 -60]]
Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801
Optimal ET sequence: 24e, 53, 130, 183, 313
Badness: 0.036289
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 4096/4095
Mapping: [⟨1 0 15 -17 51 14], ⟨0 2 -16 25 -60 -13]]
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801
Optimal ET sequence: 24e, 53, 130, 183, 313
Badness: 0.020816
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
Mapping: [⟨1 0 15 -17 51 14 -49], ⟨0 2 -16 25 -60 -13 67]]
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810
Optimal ET sequence: 53, 130, 183, 679df
Badness: 0.021073
Squirrel
The squirrel temperament (29 & 36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
Subgroup: 2.3.5.7
Comma list: 686/675, 32805/32768
Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]
Wedgie: ⟨⟨3 -24 -13 -45 -29 37]]
Optimal tuning (POTE): ~2 = 1\1, ~160/147 = 166.140
Optimal ET sequence: 29, 36, 65
Badness: 0.174705
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/242, 686/675, 896/891
Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097
Optimal ET sequence: 29, 36, 65
Badness: 0.068310
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 245/242, 896/891
Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054
Optimal ET sequence: 29, 36, 65f, 94df, 159df
Badness: 0.043750
Tertiaschis
The tertiaschis temperament (94 & 159) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with #Squirrel, but tempers out 1071785/1062882 for prime 7.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1071875/1062882
Mapping: [⟨1 2 -1 10], ⟨0 -3 24 -52]]
Wedgie: ⟨⟨3 -24 52 -45 74 188]]
Optimal tuning (POTE): ~2 = 1\1, ~192/175 = 166.019
Optimal ET sequence: 65, 94, 159, 253, 412cd
Badness: 0.211859
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 4000/3993, 19712/19683
Mapping: [⟨1 2 -1 10 0], ⟨0 -3 24 -52 25]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017
Optimal ET sequence: 65, 94, 159, 253, 412cd
Badness: 0.061336
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1575/1573, 10985/10976
Mapping: [⟨1 2 -1 10 0 12], ⟨0 -3 24 -52 25 -60]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016
Optimal ET sequence: 65f, 94, 159, 253, 412cdf, 665ccdef
Badness: 0.036700
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
Mapping: [⟨1 2 -1 10 0 12 -2], ⟨0 -3 24 -52 25 -60 44]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012
Optimal ET sequence: 65f, 94, 159, 253
Badness: 0.026504
Countertertiaschis
The countertertiaschis temperament (159 & 224) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with #Squirrel, but tempers out 244140625/243045684 for prime 7.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 244140625/243045684
Mapping: [⟨1 2 -1 -12], ⟨0 -3 24 107]]
Optimal tuning (POTE): ~2 = 1\1, ~625/567 = 166.0621
Optimal ET sequence: 65d, 159, 224, 383, 607
Badness: 0.188043
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 32805/32768
Mapping: [⟨1 2 -1 -12 0], ⟨0 -3 24 107 25]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Optimal ET sequence: 65d, 159, 224, 383, 607
Badness: 0.048943
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Mapping: [⟨1 2 -1 -12 0 -10], ⟨0 -3 24 107 25 99]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Optimal ET sequence: 65d, 159, 224, 383, 607
Badness: 0.024506
Pogo
- See also: Stearnsmic clan #Pogo
The pogo temperament (94 & 130) splits the period in two to address the difference between #Tertiaschis and #Countertertiaschis. The schismic tempering of the fifth is just about right for tempering out the stearnsma.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 118098/117649
Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]
- mapping generators: ~343/243, ~9/7
Wedgie: ⟨⟨6 -48 10 -90 -1 158]]
Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.901
Optimal ET sequence: 36, 94, 130, 224, 354
Badness: 0.079635
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 32805/32768
Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness: 0.031857
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal ET sequence: 36, 94, 130, 224, 354, 578
Badness: 0.017514
Term
Subgroup: 2.3.5.7
Comma list: 32805/32768, 250047/250000
Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]
- mapping generators: ~63/50, ~3
Wedgie: ⟨⟨3 -24 -54 -45 -94 -58]]
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.742
- 7-odd-limit eigenmonzo (unchanged-interval) basis): 2.5/3
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722
Badness: 0.019950
Terminal
The terminal temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 32805/32768
Mapping: [⟨3 0 45 94 134], ⟨0 1 -8 -18 -26]]
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824
Optimal ET sequence: 12, 147de, 159, 330
Badness: 0.059502
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 13720/13689
Mapping: [⟨3 0 45 94 134 168], ⟨0 1 -8 -18 -26 -33]]
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821
Optimal ET sequence: 12f, 147def, 159, 330
Badness: 0.037082
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
Mapping: [⟨3 0 45 94 134 168 -2], ⟨0 1 -8 -18 -26 -33 3]]
Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810
Optimal ET sequence: 12f, 147def, 159, 171, 330
Badness: 0.027073
Terminator
Subgroup: 2.3.5.7.11
Comma list: 540/539, 32805/32768, 137781/137500
Mapping: [⟨3 0 45 94 -137], ⟨0 1 -8 -18 31]]
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685
Optimal ET sequence: 12e, 159e, 171, 183, 354, 537, 891de
Badness: 0.066968
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 31250/31213
Mapping: [⟨3 0 45 94 -137 -103], ⟨0 1 -8 -18 31 24]]
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689
Optimal ET sequence: 171, 183, 354, 891de, 1245dee, 1599ddee
Badness: 0.035487
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
Mapping: [⟨3 0 45 94 -137 -103 -2], ⟨0 1 -8 -18 31 24 3]]
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688
Optimal ET sequence: 171, 183, 354, 891de, 1245dee, 1599ddee
Badness: 0.020434
Semiterm
The semiterm temperament (12 & 342) has a period of 1/6 octave and tempers out 9801/9800 (kalisma) and 151263/151250 (odiheim comma).
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 151263/151250
Mapping: [⟨6 0 90 188 287], ⟨0 1 -8 -18 -28]]
- mapping generators: ~55/49, ~3
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460
Optimal ET sequence: 12, 330e, 342, 1380, 1722, 2064, 2406c
Badness: 0.029438
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
Mapping: [⟨6 0 90 188 287 355], ⟨0 1 -8 -18 -28 -35]]
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256
Optimal ET sequence: 12f, 330eff, 342f, 696f *
* optimal patent val: 354
Badness: 0.044657
Hemiterm
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 32805/32768, 102487/102400
Mapping: [⟨3 0 45 94 8], ⟨0 2 -16 -36 1]]
- mapping generators: ~63/50, ~693/400
Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)
Optimal ET sequence: 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce
Badness: 0.020687
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
Mapping: [⟨3 0 45 94 8 42], ⟨0 2 -16 -36 1 -13]]
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)
Optimal ET sequence: 24d, 159, 183, 342f
Badness: 0.031362
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
Mapping: [⟨3 0 45 94 8 42 -2], ⟨0 2 -16 -36 1 -13 6]]
Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)
Optimal ET sequence: 24d, 159, 183, 342f, 525f, 867ff
Badness: 0.022316
Altinex
Subgroup: 2.3.5.7
Comma list: 32805/32768, 367653125/362797056
Mapping: [⟨3 0 45 -32], ⟨0 2 -16 17]]
- mapping generators: ~1536/1225, ~34300/19683
Optimal tuning (CTE): ~1536/1225 = 1\3, ~34300/19683 = 950.9654
Optimal ET sequence: 24, 135, 159, 612ccdd
Badness: 0.422
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 = 950.9629
Optimal ET sequence: 24, 135, 159
Badness: 0.101
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
- 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
- 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd
Badness: 0.011244
Sesquart
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 16384/16335
Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]
Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
Optimal ET sequence: 41, 89, 130, 301e, 431e
Badness: 0.029306
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 3584/3575
Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409
Optimal ET sequence: 41, 89, 130, 301e, 431e
Badness: 0.022396
Bisesqui
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 32805/32768
Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]
- mapping generators: ~99/70, ~448/405
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435
Optimal ET sequence: 82e, 130, 212, 342, 1156, 1498, 1840d
Badness: 0.016968
Quintilipyth
The quintilipyth temperament (12 & 253, formerly quintilischis) slices the pythagorean fourth (4/3) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 9765625/9680832
Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]
Wedgie: ⟨⟨5 -40 -82 -75 -144 -78]]
Optimal tuning (POTE): ~2 = 1\1, ~625/588 = 99.625
Optimal ET sequence: 12, 253, 265
Badness: 0.253966
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4375/4356, 32805/32768
Mapping: [⟨1 2 -1 -4 -7], ⟨0 -5 40 82 126]]
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616
Optimal ET sequence: 12, 253, 265, 518c, 783cc
Badness: 0.113044
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Mapping: [⟨1 2 -1 -4 -7 -9], ⟨0 -5 40 82 126 153]]
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612
Optimal ET sequence: 12f, 253, 518c, 771cc
Badness: 0.069127
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Mapping: [⟨1 2 -1 -4 -7 -9 5], ⟨0 -5 40 82 126 153 -11]]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612
Optimal ET sequence: 12f, 253, 518c, 771cc
Badness: 0.045992
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Mapping: [⟨1 2 -1 -4 -7 -9 5 4], ⟨0 -5 40 82 126 153 -11 3]]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615
Optimal ET sequence: 12f, 253, 265, 518ch
Badness: 0.038155
Quintaschis
The quintaschis temperament (12 & 289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 49009212/48828125
Mapping: [⟨1 2 -1 -5], ⟨0 -5 40 94]]
Wedgie: ⟨⟨5 -40 -94 -75 -163 -106]]
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.664
Optimal ET sequence: 12, …, 289, 301, 590, 891, 1192
Badness: 0.132890
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 32805/32768, 1953125/1951488
Mapping: [⟨1 2 -1 -5 -8], ⟨0 -5 40 94 138]]
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653
Optimal ET sequence: 12, …, 277d, 289
Badness: 0.111477
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Mapping: [⟨1 2 -1 -5 -8 -11], ⟨0 -5 40 94 138 177]]
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658
Optimal ET sequence: 12f, …, 277dff, 289
Badness: 0.074218
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
Mapping: [⟨1 2 -1 -5 -8 -11 5], ⟨0 -5 40 94 138 177 -11]]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656
Optimal ET sequence: 12f, 277dff, 289
Badness: 0.050571
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Mapping: [⟨1 2 -1 -5 -8 -11 5 4], ⟨0 -5 40 94 138 177 -11 3]]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659
Badness: 0.042120
Quintahelenic
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 8019/8000, 151263/151250
Mapping: [⟨1 2 -1 -5 -9], ⟨0 -5 40 94 150]]
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671
Optimal ET sequence: 12, …, 289e, 301, 915
Badness: 0.082225
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
Mapping: [⟨1 2 -1 -5 -9 -11], ⟨0 -5 40 94 150 177]]
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661
Optimal ET sequence: 12f, …, 289e, 301
Badness: 0.055570
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
Mapping: [⟨1 2 -1 -5 -9 -11 5], ⟨0 -5 40 94 150 177 -11]]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665
Optimal ET sequence: 12f, 289e, 301
Badness: 0.040412
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
Mapping: [⟨1 2 -1 -5 -9 -11 5 4], ⟨0 -5 40 94 150 177 -11 3]]
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
Optimal ET sequence: 12, 301, 614, 915
Badness: 0.066108
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Mapping: [⟨1 2 -1 -5 -9 14 5], ⟨0 -5 40 94 150 -124 -11]]
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
Optimal ET sequence: 29, 72cd, 101, 130, 289, 419
Badness: 0.108794
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4000/3993, 235298/234375
Mapping: [⟨1 2 -1 -1 0], ⟨0 -6 48 55 50]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
Optimal ET sequence: 29, 72cde, 101e, 130, 289
Badness: 0.045457
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 10985/10976
Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
Optimal ET sequence: 29, 72cdef, 101e, 130, 289
Badness: 0.025276
Septiquarschis
The septiquarschis temperament (89 & 94) splits septimal minor seventh (7/4) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
Subgroup: 2.3.5.7
Comma list: 32805/32768, 829440/823543
Mapping: [⟨1 3 -9 2], ⟨0 -7 -56 4]]
Wedgie: ⟨⟨7 56 -4 231 -26 -76]]
Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.614
Optimal ET sequence: 89, 94, 183, 460d, 643d, 1103dd
Badness: 0.187047
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 15488/15435, 32805/32768
Mapping: [⟨1 3 -9 2 -2], ⟨0 -7 -56 4 27]]
Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616
Optimal ET sequence: 89, 94, 183, 460d, 643d, 826dd
Badness: 0.052002
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Mapping: [⟨1 3 -9 2 -2 13], ⟨0 -7 -56 4 27 -46]]
Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610
Optimal ET sequence: 89, 94, 183, 277, 460d
Badness: 0.035315
Tsaharuk
- Main article: Tsaharuk
Subgroup: 2.3.5.7
Comma list: 32805/32768, 420175/419904
Mapping: [⟨1 1 7 0], ⟨0 5 -40 24]]
- mapping generators: ~2, ~243/224
Wedgie: ⟨⟨5 -40 24 -75 24 168]]
Optimal tuning (POTE): ~2 = 1\1, ~243/224 = 140.350
Optimal ET sequence: 17, 60c, 77, 94, 171
Badness: 0.030697
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1331/1323, 19712/19683
Mapping: [⟨1 1 7 0 1], ⟨0 5 -40 24 21]]
Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365
Optimal ET sequence: 17, 60ce, 77, 94, 171e, 265e, 436ee
Badness: 0.063499
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1331/1323
Mapping: [⟨1 1 7 0 1 3], ⟨0 5 -40 24 21 6]]
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363
Optimal ET sequence: 17, 60ce, 77, 94, 171e, 436ee
Badness: 0.037886
Quanharuk
Subgroup: 2.3.5.7
Comma list: 16875/16807, 32805/32768
Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]
- mapping generators: ~2, ~56/45
Wedgie: ⟨⟨5 -40 -29 -75 -60 45]]
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.355
Optimal ET sequence: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd
Badness: 0.071950
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 32805/32768
Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
Optimal ET sequence: 41, 142, 183, 224, 631d, 855d, 1079d
Badness: 0.031549
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
Optimal ET sequence: 41, 142, 183, 224, 631d, 855d
Badness: 0.021392
Quadrant
The quadrant temperament (12 & 224) has a period of quarter octave and tempers out the dimcomp comma, 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 390625/388962
Mapping: [⟨4 0 60 119], ⟨0 1 -8 -17]]
- mapping generators: ~25/21, ~3
Wedgie: ⟨⟨4 -32 -68 -60 -119 -68]]
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8234
Optimal ET sequence: 212, 224, 436, 660, 1096c
Badness: 0.110242
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 6250/6237, 32805/32768
Mapping: [⟨4 0 60 119 185], ⟨0 1 -8 -17 -27]]
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176
Optimal ET sequence: 212, 224, 436, 660
Badness: 0.045738
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
Mapping: [⟨4 0 60 119 185 224], ⟨0 1 -8 -17 -27 -33]]
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158
Optimal ET sequence: 212, 224, 436, 660
Badness: 0.027243
Septant
The septant temperament (224 & 301) has a period of 1/7 octave and tempers out the akjaysma, [47 -7 -7 -7⟩.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 516560652/514714375
Mapping: [⟨7 0 105 -56], ⟨0 1 -8 7]]
- mapping generators: ~8575/7776, ~3
Wedgie: ⟨⟨7 -56 49 -105 58 271]]
Optimal tuning (POTE): ~8575/7776 = 1\7, ~3/2 = 701.702
Optimal ET sequence: 77, 147, 224, 301, 525, 826, 1351
Badness: 0.111142
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 24057/24010, 32805/32768
Mapping: [⟨7 0 105 -56 -120], ⟨0 1 -8 7 13]]
Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.719
Optimal ET sequence: 77, 147, 224, 301, 525
Badness: 0.044122
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
Mapping: [⟨7 0 105 -56 -120 37], ⟨0 1 -8 7 13 -1]]
Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.724
Optimal ET sequence: 77, 147, 224, 525
Badness: 0.024706
Octant
The octant temperament (224 & 472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 2259436291848/2251875390625
Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]
- mapping generators: ~42875/39366, ~3
Wedgie: ⟨⟨8 -64 88 -120 117 384]]
Optimal tuning (POTE): ~42875/39366 = 1\8, ~3/2 = 701.713
Optimal ET sequence: 24, 224, 472, 696, 1168
Badness: 0.157186
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 46656/46585
Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.713
Optimal ET sequence: 24, 224, 472, 696, 1168
Badness: 0.044778
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.725
Optimal ET sequence: 24, 224, 472, 696
Badness: 0.030425
Tridecafifths
Tridecafifths divides the perfect 3/2 into 13 quartertones.
Subgroup: 2.3.5.7
Comma list: 32805/32768, [-14 -1 -9 13⟩
Mapping: [⟨1 1 7 6], ⟨0 13 -104 -71]]
- mapping generators: ~2, ~1323/1280
Optimal tuning (CTE): ~2 = 1\1, ~1323/1280 = 53.9741
Optimal ET sequence: 89, 200, 289
Badness: 0.433
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: 0.128