Garischismic clan
The garischismic clan of temperaments tempers out the garischisma, [25 -14 0 -1⟩ = 33554432/33480783. The head of this clan is gary, which is generated by a perfect fifth. 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 second comma of the comma list determines which full 7-limit family member we are looking at. Garibaldi adds the schisma, or equivalently 225/224 and finds 5/4 at the diminished fourth. Cotoneum adds 10976/10935 and finds 5/4 at the septuple diminished octave. These are generated by the fifth as is gary.
Newt adds 2401/2400, slicing the fifth in two. Sextile adds 250047/250000 with a 1/3-octave period. Trident adds 6144/6125, slicing the twelfth in three. Satin adds 2100875/2097152, slicing the fourth in three. Vulture adds 4375/4374, slicing the twelfth in four. World calendar adds 390625/388962 with a 1/4-octave period as well as a bisect generator. Quintagar adds 3136/3125, slicing the fourth in five. Paramity adds 65625/65536, slicing the eleventh in five.
- Garibaldi → Schismatic family (+225/224)
- Newt → Breedsmic temperaments (+2401/2400)
- Sextile → Landscape microtemperaments (+250047/250000)
- Satin → Canousmic temperaments (+2100875/2097152)
- Trident → Tricot family (+6144/6125)
- Vulture → Vulture family (+4375/4374)
- Quintagar → Quindromeda family (+3136/3125)
- Paramity → Amity family (+65625/65536)
- Garistearn → Stearnsmic clan (+118098/117649)
Gary
Subgroup: 2.3.7
Comma list: 33554432/33480783
Sval mapping: [⟨1 0 25], ⟨0 1 -14]]
- sval mapping generators: ~2, ~3
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2079
Optimal ET sequence: 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd
Badness: 0.0135
2.3.7.11 subgroup
Subgroup: 2.3.7.11
Comma list: 19712/19683, 41503/41472
Sval mapping: [⟨1 0 25 -33], ⟨0 1 -14 23]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292
Optimal ET sequence: 12e, 41, 94, 135, 716, 851, 986, 1121, 1256
Badness: 0.00731
Cotoneum
The cotoneum temperament tempers out 10976/10935 (hemimage comma), and 823543/819200 (quince comma) in addition to the garischisma. This temperament can be described as 41 & 217, and is supported by 176-, 217-, and 258edo. 5/4 is found at the septuple diminished octave (C-Cbbbbbbb) or equivalently at the perfect fourth minus four Pyth. commas. It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
Subgroup: 2.3.5.7
Comma list: 10976/10935, 823543/819200
Mapping: [⟨1 0 80 25], ⟨0 1 -49 -14]]
Wedgie: ⟨⟨ 1 -49 -14 -80 -25 105 ]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.317
Optimal ET sequence: 41, 135c, 176, 217, 258, 475
Badness: 0.105632
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 10976/10935, 16384/16335
Mapping: [⟨1 0 80 25 -33], ⟨0 1 -49 -14 23]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303
Optimal ET sequence: 41, 135c, 176, 217
Badness: 0.050966
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Mapping: [⟨1 0 80 25 -33 -93], ⟨0 1 -49 -14 23 61]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306
Optimal ET sequence: 41, 176, 217
Badness: 0.036951
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
Mapping: [⟨1 0 80 25 -33 -93 -137], ⟨0 1 -49 -14 23 61 89]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307
Optimal ET sequence: 41, 176, 217
Badness: 0.029495
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Mapping: [⟨1 0 80 25 -33 -93 -137 74], ⟨0 1 -49 -14 23 61 89 -44]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308
Optimal ET sequence: 41, 176, 217
Badness: 0.021811
World calendar
World calendar tempers out the dimcomp comma and the garischisma, and can be described as the 12 & 364 temperament. The name derives from a certain calendar layout by the same name.
Subgroup: 2.3.5.7
Comma list: 390625/388962, 33554432/33480783
Mapping: [⟨4 1 -44 86], ⟨0 2 -13 -28]]
- mapping generators: ~25/21, ~91125/57344
Optimal tuning (POTE): ~25/21 = 1\4, ~91125/57344 = 801.0947
Optimal ET sequence: 12, …, 352, 364
Badness: 0.292
2.3.5.7.17 subgroup
Subgroup: 2.3.5.7.17
Comma list: 2025/2023, 24576/24565, 390625/388962
Sval mapping: [⟨4 1 -44 86 3], ⟨0 2 -13 -28 5]]
Optimal tuning (POTE): ~25/21 = 1\4, ~27/17 = 801.0908
Optimal ET sequence: 12, …, 352, 364
Badness: 0.0743
2.3.5.7.17.19 subgroup
Subgroup: 2.3.5.7.17.19
Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000
Sval mapping: [⟨4 1 -44 86 3 25], ⟨0 2 -13 -28 5 -3]]
Optimal tuning (POTE): ~25/21 = 1\4, ~27/17 = 801.0945
Optimal ET sequence: 12, …, 352, 364
Badness: 0.0378