Hemimage temperaments

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This is a collection of temperaments tempering out the hemimage comma, [5 -7 -1 3 = 10976/10935. These include commatic, chromat, degrees, subfourth, bisupermajor and cotoneum, considered below, as well as the following discussed elsewhere:

Commatic

The commatic temperament has a period of half octave and a generator of 20.4 cents. It is so named because the generator is a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.


Subgroup: 2.3.5.7

Comma list: 10976/10935, 50421/50000

Mapping: [2 3 4 5], 0 5 19 18]]

Wedgie⟨⟨10 38 36 37 29 -23]]

POTE generator: ~81/80 = 20.377

Vals58, 118, 294, 412d, 530d

Badness: 0.084317

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [2 3 4 5 6], 0 5 19 18 27]]

POTE generator: ~81/80 = 20.390

Vals: 58, 118, 294, 412d

Badness: 0.030461

Chromat

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an amity extension with third-octave period.


Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [3 4 5 6], 0 5 13 16]]

Wedgie⟨⟨15 39 48 27 34 2]]

POTE generator: ~28/27 = 60.528

Vals39d, 60, 99, 258, 357, 456

Badness: 0.057499

Degrees

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.


Subgroup: 2.3.5.7

Comma list: 10976/10935, 390625/388962

Mapping: [20 0 -17 -39], 0 1 2 3]]

Wedgie⟨⟨20 40 60 17 39 27]]

POTE generator: ~3/2 = 703.015

Vals60, 80, 140, 640b, 780b, 920b

Badness: 0.106471

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [20 0 -17 -39 -26], 0 1 2 3 3]]

POTE generator: ~3/2 = 703.231

Vals: 60e, 80, 140, 360, 500be, 860bde

Badness: 0.046770

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: [20 0 -17 -39 -26 74], 0 1 2 3 3 0]]

POTE generator: ~3/2 = 703.080

Vals: 60e, 80, 140, 500be, 640be, 780be

Badness: 0.032718

Subfourth

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65536/64827

Mapping: [1 0 17 4], 0 4 -37 -3]]

Wedgie⟨⟨4 -37 -3 -68 -16 97]]

POTE generator: ~21/16 = 475.991

Vals58, 121, 179, 300bd, 479bcd

Badness: 0.140722

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 12005/11979

Mapping: [1 0 17 4 11], 0 4 -37 -3 -19]]

POTE generator: ~21/16 = 475.995

Vals: 58, 121, 179e, 300bde

Badness: 0.045323

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: [1 0 17 4 11 16], 0 4 -37 -3 -19 -31]]

POTE generator: ~21/16 = 475.996

Vals: 58, 121, 179ef, 300bdef

Badness: 0.023800

Bisupermajor

See also: Very high accuracy temperaments #Kwazy

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65625/65536

Mapping: [2 1 6 1], 0 8 -5 17]]

Wedgie⟨⟨16 -10 34 -53 9 107]]

POTE generator: ~192/175 = 162.806

Vals22, 74d, 96d, 118, 140, 258, 398, 656d

Badness: 0.065492

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: [2 1 6 1 8], 0 8 -5 17 -4]]

POTE generators: ~11/10 = 162.773

Vals: 22, 74d, 96d, 118, 258e, 376de

Badness: 0.032080

Cotoneum

Main article: Cotoneum

The cotoneum temperament (41&217, named after the Latin for "quince") tempers out the quince comma, 823543/819200 and the garischisma, 33554432/33480783. This temperament is supported by 41, 176, 217, and 258 EDOs, and 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 2 -18 -3], 0 -1 49 14]]

Wedgie⟨⟨1 -49 -14 -80 -25 105]]

POTE generator: ~3/2 = 702.317

Vals41, 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 2 -18 -3 13], 0 -1 49 14 -23]]

POTE generator: ~3/2 = 702.303

Vals: 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 2 -18 -3 13 29], 0 -1 49 14 -23 -61]]

POTE generator: ~3/2 = 702.306

Vals: 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 2 -18 -3 13 29 41], 0 -1 49 14 -23 -61 -89]]

POTE generator: ~3/2 = 702.307

Vals: 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 2 -18 -3 13 29 41 -14], 0 -1 49 14 -23 -61 -89 44]]

POTE generator: ~3/2 = 702.308

Vals: 41, 176, 217

Badness: 0.021811

Squarschmidt

A generator for the squarschimidt temperament is the fourth root of 5/2, (5/2)1/4, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.


Subgroup: 2.3.5

Comma: [61 4 -29

Mapping: [1 -8 1], 0 29 4]]

POTE generator: ~98304/78125 = 396.621

Vals118, 593, 711, 829, 947

Badness: 0.218314

7-limit

Subgroup: 2.3.5.7

Comma list: 10976/10935, 29360128/29296875

Mapping: [1 -8 1 -20], 0 29 4 69]]

Wedgie⟨⟨29 4 69 -61 28 149]]

POTE generator: ~1125/896 = 396.643

Vals118, 239, 357, 596, 1549bd

Badness: 0.132821

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 10976/10935

Mapping: [1 -8 1 -20 -21], 0 29 4 69 74]]

POTE generator: ~44/35 = 396.644

Vals: 118, 239, 357, 596

Badness: 0.038186