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{{Technical data page}}
Hemimage temperaments temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935.
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 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 #Quasisuper|Archytas clan]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


=Commatic=
== 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 family|amity extension]] with third-octave period.


==7-limit==
[[Subgroup]]: 2.3.5.7
Commas: 10976/10935, 50421/50000


POTE generator: ~81/80 = 20.377
[[Comma list]]: 10976/10935, 235298/234375


Map: [<2 3 4 5|, <0 5 19 18|]
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}


Wedgie: <<10 38 36 37 29 -23||
: mapping generators: ~63/50, ~28/27


EDOs: 58, 118, 294, 412d, 530d
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528


Badness: 0.0843
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


==11-limit==
[[Badness]]: 0.057499
Commas: 441/440, 3388/3375, 8019/8000


POTE generator: ~81/80 = 20.390
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<2 3 4 5 6|, <0 5 19 18 27|]
Comma list: 441/440, 4375/4356, 10976/10935


EDOs: 58, 118, 294, 412d
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


Badness: 0.0305
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430


= Chromat =
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
{{see also|Amity family}}


Commas: 10976/10935, 235298/234375
Badness: 0.050379


POTE generator: ~28/27 = 60.528
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<3 4 5 6|, <0 5 13 16|]
Comma list: 364/363, 441/440, 625/624, 10976/10935


Wedgie: <<15 39 48 27 34 2||
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


EDOs: 60, 99, 258, 357, 456
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428


Badness: 0.0575
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}


=Degrees=
Badness: 0.046006
Commas: 10976/10935, 390625/388962


POTE generator: ~3/2 = 703.015
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [<20 0 -17 -39|, <0 1 2 3|]
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


Wedgie: <<20 40 60 17 39 27||
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


EDOs: 60, 80, 140, 640b, 780b, 920b
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438


Badness: 0.1065
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}


==11-limit==
Badness: 0.031678
Commas: 1331/1323, 1375/1372, 2200/2187


POTE generator: ~3/2 = 703.231
==== Catachrome ====
Subgroup: 2.3.5.7.11.13


Map: [<20 0 -17 -39 -26|, <0 1 2 3 3|]
Comma list: 325/324, 441/440, 1001/1000, 10976/10935


EDOs: 80, 140, 360, 500be, 860bde
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


Badness: 0.0468
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378


==13-limit==
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
Commas: 325/324, 352/351, 1001/1000, 1331/1323


POTE generator: ~3/2 = 703.080
Badness: 0.043844


Map: [<20 0 -17 -39 -26 74|, <0 1 2 3 3 0|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 60e, 80, 140, 500be, 640be, 780be
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


Badness: 0.0327
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


=Subfourth=
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377
Commas: 10976/10935, 65536/64827


POTE generator: ~21/16 = 475.991
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Map: [<1 0 17 4|, <0 4 -37 -3|]
Badness: 0.030218


EDOs: 58, 121, 179, 300bd, 479bcd
==== Chromic ====
Subgroup: 2.3.5.7.11.13


Badness: 0.1407
Comma list: 196/195, 352/351, 729/728, 1875/1859


==11-limit==
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
Commas: 540/539, 896/891, 12005/11979


POTE generator: ~21/16 = 475.995
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456


Map: [<1 0 17 4 11|, <0 4 -37 -3 -19|]
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


EDOs: 58, 121, 179e, 300bde
Badness: 0.049857


Badness: 0.0453
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
Commas: 352/351, 364/363, 540/539, 676/675


POTE generator: ~21/16 = 475.996
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


Map: [<1 0 17 4 11 16|, <0 4 -37 -3 -19 -31|]
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459


EDOs: 58, 121, 179ef, 300bdef
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


Badness: 0.0238
Badness: 0.031043


[[Category:Theory]]
=== Hemichromat ===
[[Category:Temperament]]
Subgroup: 2.3.5.7.11
[[Category:Hemimage]]
 
Comma list: 3025/3024, 10976/10935, 102487/102400
 
Mapping: {{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|legend=1| 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: {{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|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
 
Badness: 0.033420
 
== Bisupermajor ==
{{See also| Very high accuracy temperaments #Kwazy }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 65625/65536
 
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
 
: mapping generators: ~1225/864, ~192/175
 
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
 
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
 
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}
 
Badness: 0.032080
 
== 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|legend=1| 2 3 4 5 | 0 5 19 18 }}
 
: mapping generators: ~567/400, ~81/80
 
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 58, 118, 294ffg, 412dffgg }}
 
Badness: 0.022396
 
== Degrees ==
{{ See also | 20th-octave temperaments }}
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{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|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|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
 
: mapping generators: ~28/27, ~3
 
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)
 
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
 
[[Badness]]: 0.106471
 
Badness (Sintel): 2.694
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1331/1323, 1375/1372, 2200/2187
 
Mapping: {{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|legend=1| 20cd, 60e, 80, 140, 360 }}
 
Badness: 0.046770
 
Badness (Sintel): 1.546
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 1001/1000, 1331/1323
 
Mapping: {{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|legend=1| 20cde, 60e, 80, 140 }}
 
Badness: 0.032718
 
Badness (Sintel): 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: {{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|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 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: {{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|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 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: {{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|legend=1| 20cdei, 60e, 80, 140 }}
 
Badness (Sintel): 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: {{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|legend=1| 20cdeij, 60e, 80, 140 }}
 
Badness (Sintel): 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: {{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|legend=1| 20cdeijl, 60el, 80, 140 }}
 
Badness (Sintel): 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: {{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|legend=1| 20cdeijlm, 60el, 80, 140 }}
 
Badness (Sintel): 1.100
 
== Squarschmidt ==
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 61 4 -29 }}
 
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
 
: mapping generators: ~2, ~98304/78125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
 
[[Badness]]: 0.218314
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 29360128/29296875
 
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
 
Badness: 0.038186
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"