Garischismic clan

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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.

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

Main article: 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

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