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) → Quartonic family
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: 0.067173
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: 0.033420
Bisupermajor
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.
An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with 23/20, 6\20 = 3\10 with 69/56, 7\20 with 23/18, etc. By observing that 1\20 works as 30/29~29/28~28/27, with 29/28 being especially accurate, and by equating 29/22 with 2\5 = 240 ¢, we get a uniquely elegant extension to the 29-limit which tempers out (33/25)/(29/22) = 726/725, S28 = 784/783 and S29 = 841/840. An edo as large as 220 supports it by patent val, though it does not appear in the optimal ET sequence, and 80edo and 140edo are both much more recommendable tunings.
By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.
By looking at the mapping, we observe an 80-note mos scale is ideal, so that 80edo is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of 20edo by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in 140edo.
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 (~126/125 = 16.985)
Optimal ET sequence: 20cd, 60, 80, 140, 640b, 780b
Badness: 0.106471
Badness (Dirichlet): 2.694
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 (~100/99 = 16.769)
Optimal ET sequence: 20cd, 60e, 80, 140, 360
Badness: 0.046770
Badness (Dirichlet): 1.546
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 (~100/99 = 16.920)
Optimal ET sequence: 20cde, 60e, 80, 140
Badness: 0.032718
Badness (Dirichlet): 1.352
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
Mapping: [⟨20 0 -17 -39 -26 74 50], ⟨0 1 2 3 3 0 1]]
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
Optimal ET sequence: 20cde, 60e, 80, 140
Badness (Dirichlet): 1.171
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475
Mapping: [⟨20 0 -17 -39 -26 74 50 85], ⟨0 1 2 3 3 0 1 0]]
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
Optimal ET sequence: 20cde, 60e, 80, 140
Badness (Dirichlet): 1.273
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
Mapping: [⟨20 0 -17 -39 -26 74 50 85 27], ⟨0 1 2 3 3 0 1 0 2]]
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)
Optimal ET sequence: 20cdei, 60e, 80, 140
Badness (Dirichlet): 1.209
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2], ⟨0 1 2 3 3 0 1 0 2 3]]
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)
Optimal ET sequence: 20cdeij, 60e, 80, 140
Badness (Dirichlet): 1.134
no-31's 37-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9], ⟨0 1 2 3 3 0 1 0 2 3 3]]
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)
Optimal ET sequence: 20cdeijl, 60el, 80, 140
Badness (Dirichlet): 1.127
no-31's 41-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870
Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9 12], ⟨0 1 2 3 3 0 1 0 2 3 3 3]]
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
Optimal ET sequence: 20cdeijlm, 60el, 80, 140
Badness (Dirichlet): 1.100
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