Hemimage temperaments
This is a collection of rank-2 temperaments tempering out the hemimage comma, [5 -7 -1 3⟩ = 10976/10935. These include commatic, chromat, degrees, subfourth, and bisupermajor, considered below, as well as the following discussed elsewhere:
- Quasisuper (+64/63) → Archytas clan
- Liese (+81/80) → Meantone family
- Unicorn (+126/125) → Unicorn family
- Magic (+225/224 or 245/243) → Magic family
- Guiron (+1029/1024) → Gamelismic clan
- Echidna (+1728/1715 or 2048/2025) → Diaschismic family
- Hemififths (+2401/2400 or 5120/5103) → Breedsmic temperaments
- Dodecacot (+3125/3087) → Tetracot family
- Parakleismic (+3136/3125 or 4375/4374) → Ragismic microtemperaments
- Pluto (+4000/3969) → Mirkwai clan
- Hendecatonic (+6144/6125) → Porwell temperaments
- Marfifths (+15625/15552) → Kleismic family
- Cotoneum (+33554432/33480783) → Garischismic clan
- Yarman I (+244140625/243045684) → Turkish maqam music temperaments
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]]
- mapping generators: ~63/50, ~28/27
Optimal tuning (POTE): ~63/50 = 1\3, ~28/27 = 60.528
Optimal ET sequence: 39d, 60, 99, 258, 357, 456
Badness: 0.057499
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 10976/10935
Mapping: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
Optimal ET sequence: 60e, 99e, 159, 258, 417d
Badness: 0.050379
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 10976/10935
Mapping: [⟨3 4 5 6 6 4], ⟨0 5 13 16 29 47]]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428
Optimal ET sequence: 99ef, 159, 258, 417d
Badness: 0.046006
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
Mapping: [⟨3 4 5 6 6 4 10], ⟨0 5 13 16 29 47 15]]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438
Optimal ET sequence: 99ef, 159, 258, 417dg
Badness: 0.031678
Catachrome
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
Mapping: [⟨3 4 5 6 6 12], ⟨0 5 13 16 29 -6]]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378
Optimal ET sequence: 60e, 99e, 159
Badness: 0.043844
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
Mapping: [⟨3 4 5 6 6 12 10], ⟨0 5 13 16 29 -6 15]]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377
Optimal ET sequence: 60e, 99e, 159
Badness: 0.030218
Chromic
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 729/728, 1875/1859
Mapping: [⟨3 4 5 6 6 9], ⟨0 5 13 16 29 14]]
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456
Optimal ET sequence: 60e, 99ef, 159f, 258ff
Badness: 0.049857
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
Mapping: [⟨3 4 5 6 6 9 10], ⟨0 5 13 16 29 14 15]]
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459
Optimal ET sequence: 60e, 99ef, 159f, 258ff
Badness: 0.031043
Hemichromat
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 10976/10935, 102487/102400
Mapping: [⟨3 4 5 6 10], ⟨0 10 26 32 5]]
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
Optimal ET sequence: 39d, 120cd, 159, 198, 357, 912b
Badness: 0.0672
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
Mapping: [⟨3 4 5 6 10 8], ⟨0 10 26 32 5 41]]
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
Optimal ET sequence: 39df, 120cdff, 159, 198, 357, 912b
Badness: 0.0334
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]]
- mapping generators: ~1225/864, ~192/175
Wedgie: ⟨⟨16 -10 34 -53 9 107]]
Optimal tuning (POTE): ~1225/864 = 1\2, ~192/175 = 162.806
Optimal ET sequence: 22, 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]]
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de
Badness: 0.032080
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]]
- mapping generators: ~567/400, ~81/80
Wedgie: ⟨⟨10 38 36 37 29 -23]]
Optimal tuning (POTE): ~567/400 = 1\2, ~81/80 = 20.377
Optimal ET sequence: 58, 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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390
Optimal ET sequence: 58, 118, 294, 412d
Badness: 0.030461
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 729/728, 1001/1000
Mapping: [⟨2 3 4 5 6 7], ⟨0 5 19 18 27 12]]
Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427
Optimal ET sequence: 58, 118, 176f
Badness: 0.026336
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Mapping: [⟨2 3 4 5 6 7 8], ⟨0 5 19 18 27 12 5]]
Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
Optimal ET sequence: 58, 118, 294ffg, 412dffgg
Badness: 0.022396
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]]
- mapping generators: ~28/27, ~3
Wedgie: ⟨⟨20 40 60 17 39 27]]
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.015
Optimal ET sequence: 60, 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]]
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080
Optimal ET sequence: 60e, 80, 140, 500be, 640be, 780be
Badness: 0.032718
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 list: [61 4 -29⟩
Mapping: [⟨1 -8 1], ⟨0 29 4]]
- mapping generators: ~2, ~98304/78125
Optimal tuning (POTE): ~2 = 1\1, ~98304/78125 = 396.621
Optimal ET sequence: 118, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~1125/896 = 396.643
Optimal ET sequence: 118, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
Optimal ET sequence: 118, 239, 357, 596
Badness: 0.038186
Subfourth
Subgroup: 2.3.5.7
Comma list: 10976/10935, 65536/64827
Mapping: [⟨1 0 17 4], ⟨0 4 -37 -3]]
- mapping generators: ~2, ~21/16
Wedgie: ⟨⟨4 -37 -3 -68 -16 97]]
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.991
Optimal ET sequence: 58, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996
Optimal ET sequence: 58, 121, 179ef, 300bdef
Badness: 0.023800