Hemimage temperaments
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:
- quasisuper, {64/63, 2430/2401} → Archytas clan #Quasisuper
- liese, {81/80, 686/675} → Meantone family #Liese
- unicorn, {126/125, 10976/10935} → Unicorn family #Septimal unicorn
- magic, {225/224, 245/243} → Magic family #Magic
- guiron, {1029/1024, 10976/10935} → Gamelismic clan #Guiron
- echidna, {1728/1715, 2048/2025} → Diaschismic family #Echidna
- hemififths, {2401/2400, 5120/5103} → Breedsmic temperaments #Hemififths
- dodecacot, {3125/3087, 10976/10935} → Tetracot family #Dodecacot
- parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments #Parakleismic
- pluto, {4000/3969, 10976/10935} → Mirkwai clan #Pluto
- hendecatonic, {6144/6125, 10976/10935} → Porwell temperaments #Hendecatonic
- marfifths, {10976/10935, 15625/15552} → Kleismic family #Marfifths
- yarman, {10976/10935, 244140625/243045684} → Turkish maqam music temperaments #Yarman
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
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: Template:Val list
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
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
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: Template:Val list
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: Template:Val list
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
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: Template:Val list
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: Template:Val list
Badness: 0.023800
Bisupermajor
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
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: Template:Val list
Badness: 0.032080
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
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: Template:Val list
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: Template:Val list
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: Template:Val list
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: Template:Val list
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
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
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: Template:Val list
Badness: 0.038186