Garischismic family
The garischismic family of rank-3 temperaments tempers out the garischisma, [25 -14 0 -1⟩ = 33554432/33480783. The head of this family is garischismic, which is generated by a perfect fifth and an independent generator for 5/4. Two apotomes i.e. 14 fifths octave-reduced make a septimal major second (8/7). Equivalently stated, the harmonic seventh (7/4) is found at the double diminished octave (C-Cbb).
The best extension to the 11-limit identifies the 11/8 at +23 fifths. This is also the mapping used in cassandra, so we call it cassaschismic. An alternative, supported by andromeda, is androschismic.
Garischismic
Subgroup: 2.3.5.7
Comma list: 33554432/33480783
Mapping: [⟨1 0 0 25], ⟨0 1 0 -14], ⟨0 0 1 0]]
- mapping generators: ~2, ~3, ~5
Optimal ET sequence: 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd
Badness: 1.31 × 10-3
Cassaschismic
Subgroup: 2.3.5.7.11
Comma list: 19712/19683, 41503/41472
Mapping: [⟨1 0 0 25 -33], ⟨0 1 0 -14 23], ⟨0 0 1 0 0]]
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 1121
Badness: 1.41 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4096/4095, 19712/19683
Mapping: [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 702.2289, ~5/4 = 386.2869
- CWE: ~2 = 1\1, ~3/2 = 702.2303, ~5/4 = 386.3031
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 2283b
Badness: 0.872 × 10-3
2.3.5.7.11.13.19 subgroup
Subgroup: 2.3.5.7.11.13.19
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079
Sval mapping: [⟨1 0 0 25 -33 -13 -6], ⟨0 1 0 -14 23 12 5], ⟨0 0 1 0 0 -1 1]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2293, ~5/4 = 386.3021
Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851
Badness: 0.496 × 10-3
Androschismic
Subgroup: 2.3.5.7.11
Comma list: 151263/151250, 200704/200475
Mapping: [⟨1 0 0 25 62], ⟨0 1 0 -14 -34], ⟨0 0 1 0 -2]]
- CTE: ~2 = 1\1, ~3/2 = 702.2308, ~5/4 = 386.3806
- CWE: ~2 = 1\1, ~3/2 = 702.2178, ~5/4 = 386.5048
Optimal ET sequence: 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee
Badness: 1.64 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 43904/43875, 154880/154791
Mapping: [⟨1 0 0 25 62 82], ⟨0 1 0 -14 -34 -43], ⟨0 0 1 0 -2 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 702.2305, ~5/4 = 386.3748
- CWE: ~2 = 1\1, ~3/2 = 702.2174, ~5/4 = 386.4968
Optimal ET sequence: 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff
Badness: 1.01 × 10-3