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'''Hemimage temperaments''' temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935.  
{{Technical data page}}
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]]


Discussed elsewhere are:
== Chromat ==
* [[Archytas clan #quasisuper|Quasisuper]]
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.
* [[Meantone family #Liese|Liese]]
* [[Unicorn family #Alicorn|Alicorn]]
* [[Magic family #magic|Magic]]
* [[Schismatic family #Guiron|Guiron]]
* [[Diaschismic family #Echidna|Echidna]]
* [[Breedsmic temperaments #hemififths|Hemififths]]
* [[Ragismic microtemperaments #Parakleismic|Parakleismic]]
* [[Mirkwai clan #Pluto|Pluto]]
* [[Porwell temperaments #Hendecatonic|Hendecatonic]]
* [[Turkish maqam music temperaments|Yarman]]


= Commatic =
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 10976/10935, 50421/50000
[[Comma list]]: 10976/10935, 235298/234375


Mapping: [<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


POTE generator: ~81/80 = 20.377
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528


EDOs: 58, 118, 294, 412d, 530d
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


Badness: 0.0843
[[Badness]]: 0.057499


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4375/4356, 10976/10935
 
Mapping: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 60e, 99ef, 159f, 258ff }}
 
Badness: 0.031043
 
=== Hemichromat ===
Subgroup: 2.3.5.7.11
 
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
Subgroup: 2.3.5.7.11


Comma list: 441/440, 3388/3375, 8019/8000
Comma list: 441/440, 3388/3375, 8019/8000


Mapping: [<2 3 4 5 6|, <0 5 19 18 27|]
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390


POTE generator: ~81/80 = 20.390
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


EDOs: 58, 118, 294, 412d
Badness: 0.030461


Badness: 0.0305
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


= Chromat =
Comma list: 196/195, 352/351, 729/728, 1001/1000
{{see also|Amity family}}


Subgroup: 2.3.5.7
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


Comma list: 10976/10935, 235298/234375
Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427


Mapping: [<3 4 5 6|, <0 5 13 16|]
{{Optimal ET sequence|legend=1| 58, 118, 176f }}


Wedgie: <<15 39 48 27 34 2||
Badness: 0.026336


POTE generator: ~28/27 = 60.528
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 60, 99, 258, 357, 456
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


Badness: 0.0575
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


= Degrees =
Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
Subgroup: 2.3.5.7


Comma list: 10976/10935, 390625/388962
{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}


Mapping: [<20 0 -17 -39|, <0 1 2 3|]
Badness: 0.022396


Wedgie: <<20 40 60 17 39 27||
== 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.


POTE generator: ~3/2 = 703.015
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.


EDOs: 60, 80, 140, 640b, 780b, 920b
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.


Badness: 0.1065
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]].


== 11-limit ==
[[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
Subgroup: 2.3.5.7.11


Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 1331/1323, 1375/1372, 2200/2187


Mapping: [<20 0 -17 -39 -26|, <0 1 2 3 3|]
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)


POTE generator: ~3/2 = 703.231
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}


EDOs: 80, 140, 360, 500be, 860bde
Badness: 0.046770


Badness: 0.0468
Badness (Sintel): 1.546


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


Mapping: [<20 0 -17 -39 -26 74|, <0 1 2 3 3 0|]
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)


POTE generator: ~3/2 = 703.080
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}


EDOs: 60e, 80, 140, 500be, 640be, 780be
Badness (Sintel): 1.209


Badness: 0.0327
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


= Subfourth =
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
Subgroup: 2.3.5.7


Comma list: 10976/10935, 65536/64827
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


Mapping: [<1 0 17 4|, <0 4 -37 -3|]
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)


POTE generator: ~21/16 = 475.991
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}


EDOs: 58, 121, 179, 300bd, 479bcd
Badness (Sintel): 1.134


Badness: 0.1407
=== no-31's 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37


== 11-limit ==
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
Subgroup: 2.3.5.7.11
 
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: 540/539, 896/891, 12005/11979
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: [<1 0 17 4 11|, <0 4 -37 -3 -19|]
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 }}


POTE generator: ~21/16 = 475.995
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207


EDOs: 58, 121, 179e, 300bde
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}


Badness: 0.0453
Badness (Sintel): 1.100


== 13-limit ==
== Squarschmidt ==
Subgroup: 2.3.5.7.11.13
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.


Comma list: 352/351, 364/363, 540/539, 676/675
[[Subgroup]]: 2.3.5


Mapping: [&lt;1 0 17 4 11 16|, &lt;0 4 -37 -3 -19 -31|]
[[Comma list]]: {{monzo| 61 4 -29 }}


POTE generator: ~21/16 = 475.996
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}


EDOs: 58, 121, 179ef, 300bdef
: mapping generators: ~2, ~98304/78125


Badness: 0.0238
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621


= Bisupermajor =
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
{{see also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
[[Badness]]: 0.218314


Comma list: 10976/10935, 65625/65536
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Mapping: [<2 1 6 1|, <0 8 -5 17|]
[[Comma list]]: 10976/10935, 29360128/29296875


POTE generator: ~192/175 = 162.8061
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


EDOs: 22, 74d, 96d, 118, 140, 258, 398
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643


Badness: 0.0655
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


== 11-limit ==
[[Badness]]: 0.132821


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 3025/3024, 5632/5625, 10976/10935


Mapping: [<2 1 6 1 8|, <0 8 -5 17 -4|]
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


POTE generators: ~11/10 = 162.7733
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644


EDOs: 22, 74d, 96d, 118, 258e, 376de
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


Badness: 0.0321
Badness: 0.038186


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"