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
The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in Subgroup extensions.
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
- 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]
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]
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]
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]
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]
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
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]]
Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]
Optimal ET sequence: 5c, 7c, 12
Badness: 0.056648
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 703.3833
- POTE: ~2 = 1\1, ~3/2 = 702.136
Optimal ET sequence: 5c, 7ce, 12, 29de
Badness: 0.037482
Pontiac
Pontiac tempers out the ragisma, rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^3A).
Subgroup: 2.3.5.7
Comma list: 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]
Wedgie: ⟨⟨ 1 -8 39 -15 59 113 ]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.757
- 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 (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
Optimal ET sequence: 12, 106d, 118, 130, 248, 378
Badness: 0.054744
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 8019/8000
Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.6077
Optimal ET 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 (CTE): ~99/70 = 1\2, ~3/2 = 701.5949
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 (CTE): ~99/70 = 1\2, ~3/2 = 701.5959
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 (CTE): ~55/39 = 1\2, ~3/2 = 701.5708
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 (CTE): ~55/39 = 1\2, ~3/2 = 701.5717
Optimal ET sequence: 12f, 106deff, 118f, 130, 248fg
Badness: 0.026894
Kleischismic
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1500625/1492992
Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]
- mapping generators: ~1225/864, ~35/24
Wedgie: ⟨⟨ 4 -32 38 -60 49 178 ]]
Optimal tuning (POTE): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)
Optimal ET 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
Its S-expression-based comma list is {S12/S14, S13/S15 = S26, S27, S64(, S65)}. Tempering S13, S15 or S25 leads to 53edo (through Catakleismic) while tempering S12/S13, S13/S14, S14/S15 or S49 (thus leading to S12 = S13 = S14 = S15) leads to 130edo.
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 4096/4095
Mapping: [⟨1 0 15 -17 51 14], ⟨0 2 -16 25 -60 -13]]
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801
Optimal ET 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
- Music
Squirrel
The squirrel temperament (29 & 36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
Subgroup: 2.3.5.7
Comma list: 686/675, 32805/32768
Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]
Wedgie: ⟨⟨ 3 -24 -13 -45 -29 37 ]]
Optimal tuning (POTE): ~2 = 1\1, ~160/147 = 166.140
Optimal ET 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
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, …, 111c, 135, 159, 612ccdd, 771ccdd
Badness: 0.422026
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 14700/14641, 19712/19683
Mapping: [⟨3 0 45 -32 8], ⟨0 2 -16 17 1]]
Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9658
Optimal ET sequence: 24, …, 111c, 135, 159, 612ccdd, 771ccdd
Badness: 0.101224
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 385/384, 676/675, 19712/19683
Mapping: [⟨3 0 45 -32 8 42], ⟨0 2 -16 17 1 -13]]
Optimal tuning (CTE): ~44/35 = 1\3, ~26/15 = 950.9360
Optimal ET sequence: 24, …, 111cf, 135f, 159
Badness: 0.054894
Sesquiquartififths
Subgroup: 2.3.5.7
Comma list: 2401/2400, 32805/32768
Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]
- mapping generators: ~2, ~448/405
Wedgie: ⟨⟨ 4 -32 -15 -60 -35 55 ]]
Optimal tuning (POTE): ~2 = 1\1, ~448/405 = 175.434
- 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
Sesquartia
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
Mapping: [⟨1 1 7 5 2 -2 -6], ⟨0 4 -32 -15 10 39 69]]
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.424
Optimal ET sequence: 41, 89g, 130, 171, 301e
Badness: 0.023126
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
Mapping: [⟨1 1 7 5 2 -2 -6 6], ⟨0 4 -32 -15 10 39 69 -12]]
Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.419
Optimal ET sequence: 41, 89g, 130, 171, 301eh
Badness: 0.020466
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
Mapping: [⟨1 1 7 5 2 -2 -6 6 -6], ⟨0 4 -32 -15 10 39 69 -12 72]]
Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.412
Optimal ET sequence: 41i, 89gi, 130, 171, 301eh
Badness: 0.019043
Heartia
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
Mapping: [⟨1 1 7 5 2 -2 0], ⟨0 4 -32 -15 10 39 28]]
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.386
Optimal ET sequence: 41, 89, 130g
Badness: 0.028443
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
Mapping: [⟨1 1 7 5 2 -2 0 6], ⟨0 4 -32 -15 10 39 28 -12]]
Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.380
Optimal ET sequence: 41, 89, 130g
Badness: 0.023059
Hearty
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
Mapping: [⟨1 1 7 5 2 -2 13], ⟨0 4 -32 -15 10 39 -61]]
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.377
Optimal ET sequence: 41g, 89, 130, 609ceefgg
Badness: 0.030680
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
Mapping: [⟨1 1 7 5 2 -2 13 6], ⟨0 4 -32 -15 10 39 -61 -12]]
Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.377
Optimal ET sequence: 41g, 89, 130, 609ceefggh
Badness: 0.022816
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
Mapping: [⟨1 1 7 5 2 -2 13 6 13], ⟨0 4 -32 -15 10 39 -61 -12 -58]]
Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.376
Optimal ET sequence: 41g, 89, 130, 609ceefggh
Badness: 0.019121
Bisesqui
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 32805/32768
Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]
- mapping generators: ~99/70, ~448/405
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435
Optimal ET 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
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
Nonant
The nonant temperament (36 & 135) has a period of 1/9 octave and tempers out the septimal ennealimma, [-11 -9 0 9⟩.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 40353607/40310784
Mapping: [⟨9 0 135 11], ⟨0 1 -8 1]]
- mapping generators: ~2592/2401, ~3
Optimal tuning (CTE): ~2592/2401 = 1\9, ~3/2 = 701.7232
Optimal ET sequence: 36, 99c, 135, 171
Badness: 0.069896
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 32805/32768, 42875/42592
Mapping: [⟨9 0 135 11 131], ⟨0 1 -8 1 -7]]
Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.8398
Optimal ET sequence: 36, 99c, 135, 171, 477ce, 648cee
Badness: 0.126910
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 16807/16731
Mapping: [⟨9 0 135 11 131 -38], ⟨0 1 -8 1 -7 5]]
Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.7998
Optimal ET sequence: 36, 99cf, 135, 171
Badness: 0.076195
Tridecafifths
Tridecafifths divides the perfect 3/2 into 13 quartertones.
Subgroup: 2.3.5.7
Comma list: 32805/32768, [-14 -1 -9 13⟩
Mapping: [⟨1 1 7 6], ⟨0 13 -104 -71]]
- mapping generators: ~2, ~1323/1280
Optimal tuning (CTE): ~2 = 1\1, ~1323/1280 = 53.9741
Optimal ET sequence: 89, 200, 289
Badness: 0.432580
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 32805/32768, 55296000/55240493
Mapping: [⟨1 1 7 6 4], ⟨0 13 -104 -71 -12]]
Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744
Optimal ET sequence: 89, 200, 289
Badness: 0.127820
Subgroup extensions
Photia (2.3.5.17)
Subgroup: 2.3.5.17
Comma list: 256/255, 1458/1445
Sval mapping: [⟨1 0 15 -7], ⟨0 1 -8 7]]
Gencom mapping: [⟨1 0 15 0 0 0 -7], ⟨0 1 -8 0 0 0 7]]
- gencom: [2 3; 256/255 1458/1445]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.491
Optimal ET sequence: 12, 41, 53, 65
RMS error: 0.4842 cents
2.3.5.17.19
Subgroup: 2.3.5.17.19
Comma list: 171/170, 256/255, 324/323
Sval mapping: [⟨1 0 15 -7 9], ⟨0 1 -8 7 -3]]
Gencom mapping: [⟨1 0 15 0 0 0 -7 9], ⟨0 1 -8 0 0 0 7 -3]]
- gencom: [2 3; 171/170 256/255 324/323]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.470
Optimal ET sequence: 12, 41, 53, 65
RMS error: 0.5374 cents
Nestoria (2.3.5.19)
- See also: No-elevens subgroup temperaments #Garibaldia and #Pontia
The S-expression-based comma list of this temperament is {S16/S18, S19 (, S15/S20)}.
Subgroup: 2.3.5.19
Comma list: 361/360, 513/512
Sval mapping: [⟨1 0 15 9], ⟨0 1 -8 -3]]
- mapping generators: ~2, ~3
Gencom mapping: [⟨1 0 15 0 0 0 0 9], ⟨0 1 -8 0 0 0 0 -3]]
- gencom: [2 3; 361/360 513/512]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.746
Optimal ET sequence: 12, 29, 41, 53, 118, 171
RMS error: 0.1763 cents
Taylor (2.3.5.13)
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.
Subgroup: 2.3.5.13
Comma list: 676/675, 32805/32768
Sval mapping: [⟨1 0 15 14], ⟨0 2 -16 -13]]
- mapping generators: ~2, ~26/15
Gencom mapping: [⟨1 0 15 0 0 14], ⟨0 2 -16 0 0 -13]]
- gencom: [2 26/15; 676/675 32805/32768]
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.855
Optimal ET sequence: 24, 53, 130, 183, 236
RMS error: 0.1485 cents
Quintilischis (2.3.5.17)
- For full 17- and 19-limit extensions, see #Quintilipyth or #Quintaschis.
Subgroup: 2.3.5.17
Comma list: 32805/32768, 1419857/1417176
Sval mapping: [⟨1 2 -1 5], ⟨0 -5 40 -11]]
- mapping generators: ~2, ~18/17
Gencom mapping: [⟨1 2 -1 0 0 0 5], ⟨0 -5 40 0 0 0 -11]]
- gencom: [2 18/17; 32805/32768 1419857/1417176]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.649
Optimal ET sequence: 12, 253, 265, 277, 289
RMS error: 0.0719 cents
2.3.5.17.19
Subgroup: 2.3.5.17.19
Comma list: 4624/4617, 6144/6137, 6885/6859
Sval mapping: [⟨1 2 -1 5 4], ⟨0 -5 40 -11 3]]
Gencom mapping: [⟨1 2 -1 0 0 0 5 4], ⟨0 -5 40 0 0 0 -11 3]]
- gencom: [2 18/17; 4624/4617 6144/6137 6885/6859]
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.652
Optimal ET sequence: 12, 253, 265, 277, 289
RMS error: 0.1636 cents