Hemimage temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
This is a collection of rank-2 temperaments tempering out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, 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) → Octagar temperaments
- Hendecatonic (+6144/6125) → Porwell temperaments
- Marfifths (+15625/15552) → Kleismic family
- Subfourth (+65536/64827) → Buzzardsmic clan
- 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]]
- mapping generators: ~63/50, ~28/27
- WE: ~63/50 = 399.9549 ¢, ~28/27 = 60.5216 ¢
- error map: ⟨-0.135 +0.473 +0.241 -0.751]
- CWE: ~63/50 = 400.0000 ¢, ~28/27 = 60.5162 ¢
- error map: ⟨0.000 +0.626 +0.397 -0.567]
Optimal ET sequence: 39d, 60, 99, 258, 357, 456
Badness (Sintel): 1.46
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 tunings:
- WE: ~44/35 = 400.0359 ¢, ~28/27 = 60.4357 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢
Optimal ET sequence: 60e, 99e, 159, 258
Badness (Sintel): 1.67
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 tunings:
- WE: ~44/35 = 400.0382 ¢, ~28/27 = 60.4342 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4331 ¢
Optimal ET sequence: 60eff, 99ef, 159, 258, 417d
Badness (Sintel): 1.90
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 tunings:
- WE: ~44/35 = 399.9982 ¢, ~28/27 = 60.4374 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢
Optimal ET sequence: 99ef, 159, 258, 417dg
Badness (Sintel): 1.61
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 tunings:
- WE: ~44/35 = 400.1386 ¢, ~28/27 = 60.3986 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3929 ¢
Optimal ET sequence: 60e, 99e, 159
Badness (Sintel): 1.81
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 tunings:
- WE: ~44/35 = 400.1115 ¢, ~28/27 = 60.3935 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3893 ¢
Optimal ET sequence: 60e, 99e, 159
Badness (Sintel): 1.54
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 tunings:
- WE: ~44/35 = 399.9082 ¢, ~28/27 = 60.4425 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4380 ¢
Optimal ET sequence: 60e, 99ef, 159f
Badness (Sintel): 2.06
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 tunings:
- WE: ~44/35 = 399.8948 ¢, ~28/27 = 60.4435 ¢
- CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4385 ¢
Optimal ET sequence: 60e, 99ef, 159f
Badness (Sintel): 1.58
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 tunings:
- WE: ~63/50 = 399.9750 ¢, ~55/54 = 30.2568 ¢
- CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2561 ¢
Optimal ET sequence: 39d, 120cd, 159, 198, 357, 912b
Badness (Sintel): 2.22
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 tunings:
- WE: ~63/50 = 399.9741 ¢, ~55/54 = 30.2584 ¢
- CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2577 ¢
Optimal ET sequence: 39df, 120cdff, 159, 198, 357, 912b
Badness (Sintel): 1.38
Bisupermajor
- For the 5-limit version, see 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
- WE: ~1225/864 = 600.0294 ¢, ~192/175 = 162.8141 ¢
- error map: ⟨+0.059 +0.587 -0.208 -0.957]
- CWE: ~1225/864 = 600.0000 ¢, ~192/175 = 162.8082 ¢
- error map: ⟨0.000 +0.510 -0.355 -1.087]
Optimal ET sequence: 22, 74d, 96d, 118, 140, 258, 398, 656d
Badness (Sintel): 1.66
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 tunings:
- WE: ~99/70 = 600.1224 ¢, ~11/10 = 162.8065 ¢
- CWE: ~99/70 = 600.0000 ¢, ~11/10 = 162.7788 ¢
Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de, 634dee
Badness (Sintel): 1.06
Bicommatic
Used to be known simply as the commatic temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, 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
- WE: ~567/400 = 600.0497 ¢, ~81/80 = 20.3790 ¢
- error map: ⟨+0.099 +0.089 +1.085 -1.756]
- CWE: ~567/400 = 600.0000 ¢, ~81/80 = 20.3837 ¢
- error map: ⟨0.000 -0.037 +0.976 -1.920]
Optimal ET sequence: 58, 118, 294, 412d
Badness (Sintel): 2.13
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 tunings:
- WE: ~99/70 = 600.0401 ¢, ~81/80 = 20.3913 ¢
- CWE: ~99/70 = 600.0000 ¢, ~81/80 = 20.3948 ¢
Optimal ET sequence: 58, 118, 294, 412d
Badness (Sintel): 1.01
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 tunings:
- WE: ~99/70 = 599.8514 ¢, ~66/65 = 20.4215 ¢
- CWE: ~99/70 = 600.0000 ¢, ~66/65 = 20.4093 ¢
Optimal ET sequence: 58, 118, 176f
Badness (Sintel): 1.09
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 tunings:
- WE: ~17/12 = 600.0257 ¢, ~66/65 = 20.3789 ¢
- CWE: ~17/12 = 600.0000 ¢, ~66/65 = 20.3804 ¢
Badness (Sintel): 1.14
Degrees
- This page is about the regular temperament. For scale degrees, see degree.
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
- WE: ~28/27 = 59.9922 ¢, ~3/2 = 702.9233 ¢ (~126/125 = 16.9828 ¢)
- error map: ⟨-0.157 +0.812 -0.647 -0.220]
- CWE: ~28/27 = 60.0000 ¢, ~3/2 = 702.9324 ¢ (~126/125 = 17.0676 ¢)
- error map: ⟨0.000 +0.977 -0.449 -0.029]
Optimal ET sequence: 60, 80, 140, 640b, 780b
Badness (Sintel): 2.69
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 tunings:
- WE: ~28/27 = 59.9929 ¢, ~3/2 = 703.1478 ¢ (~100/99 = 16.7666 ¢)
- CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1556 ¢ (~100/99 = 16.8444 ¢)
Optimal ET sequence: 60e, 80, 140, 360
Badness (Sintel): 1.55
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 tunings:
- WE: ~28/27 = 59.9996 ¢, ~3/2 = 703.0749 ¢ (~100/99 = 16.9197 ¢)
- CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.0770 ¢ (~100/99 = 16.9230 ¢)
Optimal ET sequence: 60e, 80, 140
Badness (Sintel): 1.35
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 tunings:
- WE: ~28/27 = 60.0058 ¢, ~3/2 = 703.0364 ¢ (~100/99 = 17.0335 ¢)
- CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.0061 ¢ (~100/99 = 16.9939 ¢)
Optimal ET sequence: 60e, 80, 140
Badness (Sintel): 1.17
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 tunings:
- WE: ~28/27 = 59.9961 ¢, ~3/2 = 703.1523 ¢ (~100/99 = 16.8015 ¢)
- CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1777 ¢ (~100/99 = 16.8223 ¢)
Optimal ET sequence: 60e, 80, 140
Badness (Sintel): 1.27
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 tunings:
- WE: ~28/27 = 59.9990 ¢, ~3/2 = 703.1804 ¢ (~100/99 = 16.8074 ¢)
- CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1870 ¢ (~100/99 = 16.8130 ¢)
Optimal ET sequence: 60e, 80, 140
Badness (Sintel): 1.21
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 tunings:
- WE: ~29/28 = 59.9990 ¢, ~3/2 = 703.1829 ¢ (~100/99 = 16.8055 ¢)
- CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.1891 ¢ (~100/99 = 16.8109 ¢)
Optimal ET sequence: 60e, 80, 140
Badness (Sintel): 1.13
2.3.5.7.11.13.17.19.23.29.37 subgroup
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 tunings:
- WE: ~29/28 = 60.0001 ¢, ~3/2 = 703.2183 ¢ (~100/99 = 16.7827 ¢)
- CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.2178 ¢ (~100/99 = 16.7822 ¢)
Optimal ET sequence: 60el, 80, 140
Badness (Sintel): 1.13
2.3.5.7.11.13.17.19.23.29.37.41 subgroup
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 tunings:
- WE: ~29/28 = 59.9998 ¢, ~3/2 = 703.2088 ¢ (~100/99 = 16.7882 ¢)
- CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.2104 ¢ (~100/99 = 16.7896 ¢)
Optimal ET sequence: 60el, 80, 140
Badness (Sintel): 1.10
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, it 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
- WE: ~2 = 1199.9653 ¢, ~98304/78125 = 396.6094 ¢
- error map: ⟨-0.099 +0.543 +0.029 -0.719]
- CWE: ~2 = 1200.0000 ¢, ~98304/78125 = 396.6201 ¢
- error map: ⟨0.000 +0.653 +0.253 -0.552]
Optimal ET sequence: 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc
Badness (Sintel): 5.12
7-limit
Subgroup: 2.3.5.7
Comma list: 10976/10935, 29360128/29296875
Mapping: [⟨1 -8 1 -20], ⟨0 29 4 69]]
- WE: ~2 = 1199.9006 ¢, ~1125/896 = 396.6104 ¢
- error map: ⟨-0.099 +0.543 +0.029 -0.719]
- CWE: ~2 = 1200.0000 ¢, ~1125/896 = 396.6417 ¢
- error map: ⟨0.000 +0.653 +0.253 -0.552]
Optimal ET sequence: 118, 239, 357, 596
Badness (Sintel): 3.36
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 tunings:
- WE: ~2 = 1199.9005 ¢, ~44/35 = 396.6107 ¢
- CWE: ~2 = 1200.0000 ¢, ~44/35 = 396.6419 ¢
Optimal ET sequence: 118, 239, 357, 596
Badness (Sintel): 1.26