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This is a collection of temperaments tempering out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include commatic, chromat, degrees, subfourth, bisupermajor and cotoneum, considered below, as well as the following discussed elsewhere:  
{{Technical data page}}
* ''[[quasisuper]]'', {64/63, 2430/2401} → [[Archytas clan #Quasisuper]]
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:  
* ''[[liese]]'', {81/80, 686/675} → [[Meantone family #Liese]]
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* [[magic]], {225/224, 245/243} → [[Magic family #Magic]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[guiron]]'', {1029/1024, 10976/10935} → [[Gamelismic clan #Guiron]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* [[hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[dodecacot]]'', {3125/3087, 10976/10935} → [[Tetracot family #Dodecacot]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* ''[[pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic]]
* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[yarman]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman]]
* ''[[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 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.
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


{{Multival|legend=1| 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


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


[[Badness]]: 0.084317
[[Badness]]: 0.057499


=== 11-limit ===
=== 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, 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


Mapping: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}


POTE generator: ~81/80 = 20.390
Badness: 0.050379


Vals: {{Val list| 58, 118, 294, 412d }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.030461
Comma list: 364/363, 441/440, 625/624, 10976/10935


== Chromat ==
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
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.


Subgroup: 2.3.5.7
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428


[[Comma list]]: 10976/10935, 235298/234375
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
Badness: 0.046006


{{Multival|legend=1| 15 39 48 27 34 2 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~28/27 = 60.528
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


[[Badness]]: 0.057499
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438


== Degrees ==
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
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
Badness: 0.031678


[[Comma list]]: 10976/10935, 390625/388962
==== Catachrome ====
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
Comma list: 325/324, 441/440, 1001/1000, 10976/10935


{{Multival|legend=1| 20 40 60 17 39 27 }}
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


[[POTE generator]]: ~3/2 = 703.015
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


[[Badness]]: 0.106471
Badness: 0.043844


=== 11-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


POTE generator: ~3/2 = 703.231
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377


Vals: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Badness: 0.046770
Badness: 0.030218


=== 13-limit ===
==== Chromic ====
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: 196/195, 352/351, 729/728, 1875/1859
 
Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]


POTE generator: ~3/2 = 703.080
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


Vals: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456


Badness: 0.032718
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


== Subfourth ==
Badness: 0.049857
Subgroup: 2.3.5.7


[[Comma list]]: 10976/10935, 65536/64827
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


[[POTE generator]]: ~21/16 = 475.991
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


[[Badness]]: 0.140722
Badness: 0.031043


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


Comma list: 540/539, 896/891, 12005/11979
Comma list: 3025/3024, 10976/10935, 102487/102400


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


POTE generator: ~21/16 = 475.995
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511


Vals: {{Val list| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.045323
Badness: 0.067173


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


Comma list: 352/351, 364/363, 540/539, 676/675
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


POTE generator: ~21/16 = 475.996
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527


Vals: {{Val list| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}


Badness: 0.023800
Badness: 0.033420


== Bisupermajor ==
== Bisupermajor ==
{{see also| Very high accuracy temperaments #Kwazy }}
{{See also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 65625/65536
[[Comma list]]: 10976/10935, 65625/65536


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
: mapping generators: ~1225/864, ~192/175


[[POTE generator]]: ~192/175 = 162.8061
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]]: 0.065492
[[Badness]]: 0.065492
Line 167: Line 173:
Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 385/384, 3388/3375, 9801/9800


Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}


POTE generators: ~11/10 = 162.7733
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773


Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}


Badness: 0.032080
Badness: 0.032080


== Cotoneum ==
== Bicommatic ==
{{Main| Cotoneum }}
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


The ''cotoneum'' temperament (41&217, named after the Latin for "[[Wikipedia:quince|quince]]") tempers out the [[Quince clan|quince comma]], 823543/819200 and the [[garischisma]], 33554432/33480783. This temperament is supported by [[41edo|41]], [[176edo|176]], [[217edo|217]], and [[258edo|258]] EDOs, and can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
== 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.  


Subgroup: 2.3.5.7
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.


[[Comma list]]: 10976/10935, 823543/819200
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.


[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]
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]].


{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~3/2 = 702.317
[[Comma list]]: 10976/10935, 390625/388962


[[Minimax tuning]]:
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
* 7-odd-limit: ~3/2 = {{Monzo| 3/5 1/50 -1/50 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 8/5 1/50 -1/50 0 }}, {{Monzo| 8/5 -49/50 49/50 0 }}, {{Monzo| 13/5 -7/25 7/25 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 6/5
* 9-odd-limit: ~3/2 = {{Monzo| 29/51 2/51 -1/51 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 80/51 2/51 -1/51 0 }}, {{Monzo| 160/51 -98/51 49/51 0 }}, {{Monzo| 155/51 -28/51 14/51 0 }}]
: Eigenmonzos (unchanged intervals): 2, 10/9


[[Tuning ranges]]:
: mapping generators: ~28/27, ~3
* 7-odd-limit [[diamond monotone]]: ~3/2 = [701.5385, 702.8571] (38\65 to 41\70)
* 9-odd-limit diamond monotone: ~3/2 = [701.8868, 702.8571] (31\53 to 41\70)
* [[Diamond tradeoff]] range: ~3/2 = [701.9550, 702.3575]
* Diamond monotone and tradeoff: ~3/2 = [701.9550, 702.3575]


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)


[[Badness]]: 0.105632
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
 
[[Badness]]: 0.106471
 
Badness (Sintel): 2.694


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


Comma list: 441/440, 10976/10935, 16384/16335
Comma list: 1331/1323, 1375/1372, 2200/2187


Mapping: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


POTE generator: ~3/2 = 702.303
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)


Minimax tuning:
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}
* 11-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}
: Eigenmonzos (unchanged intervals): 2, 11/10


Tuning ranges:
Badness: 0.046770
* Diamond monotone range: ~3/2 = [702.1277, 702.4390] (55\94 to 24\41)
* Diamond tradeoff range: ~3/2 = [701.9550, 702.3575]
* Diamond monotone and tradeoff: ~3/2 = [702.1277, 702.3575]
 
Vals: {{Val list| 41, 135c, 176, 217 }}


Badness: 0.050966
Badness (Sintel): 1.546


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


Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


POTE generator: ~3/2 = 702.306
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)


Minimax tuning:
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
* 13-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}
: Eigenmonzos (unchanged intervals): 2, 11/10
* 15-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
: Eigenmonzos (unchanged intervals): 2, 15/11


Tuning ranges:
Badness: 0.032718
* Diamond monotone range: ~3/2 = [702.2222, 702.4390] (79\135 to 24\41)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3575]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3693]
* 13-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2222, 702.3575]
* 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2222, 702.3693]
 
Vals: {{Val list| 41, 176, 217 }}


Badness: 0.036951
Badness (Sintel): 1.352


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
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)


Mapping: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}


POTE generator: ~3/2 = 702.307
Badness (Sintel): 1.209


Minimax tuning:
=== 29-limit ===
* 17-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
Subgroup: 2.3.5.7.11.13.17.19.23.29
: Eigenmonzos (unchanged intervals): 2, 15/11


Tuning ranges:
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
* Diamond monotone range: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
* Diamond tradeoff range: ~3/2 = [701.9550, 702.3693]
* Diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3693]


Vals: {{Val list| 41, 176, 217 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


Badness: 0.029495
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)


=== 19-limit ===
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}


Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Badness (Sintel): 1.127


Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
=== no-31's 41-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41


POTE generator: ~3/2 = 702.308
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870


Minimax tuning:  
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 }}
* 19- and 21-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
: Eigenmonzos (unchanged intervals): 2, 15/11


Tuning ranges:
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
* Diamond monotone range: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
* Diamond tradeoff range: ~3/2 = [701.9550, 702.3771]
* Diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3771]


Vals: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}


Badness: 0.021811
Badness (Sintel): 1.100


== Squarschmidt ==
== 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 temperament
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
[[Subgroup]]: 2.3.5


[[Comma]]: {{monzo| 61 4 -29 }}
[[Comma list]]: {{monzo| 61 4 -29 }}


[[Mapping]]: [{{val| 1 -8 1 }}, {{val| 0 29 4 }}]
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}


[[POTE generator]]: ~98304/78125 = 396.6208
: mapping generators: ~2, ~98304/78125


{{Val list|legend=1| 118, 593, 711, 829, 947 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}


[[Badness]]: 0.218314
[[Badness]]: 0.218314


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 29360128/29296875
[[Comma list]]: 10976/10935, 29360128/29296875


[[Mapping]]: [{{val| 1 -8 1 -20 }}, {{val| 0 29 4 69 }}]
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
 
{{Multival|legend=1| 29 4 69 -61 28 149 }}


[[POTE generator]]: ~1125/896 = 396.643
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643


{{Val list|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


[[Badness]]: 0.132821
[[Badness]]: 0.132821
Line 335: Line 406:
Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 3025/3024, 5632/5625, 10976/10935


Mapping: [{{val| 1 -8 1 -20 -21 }}, {{val| 0 29 4 69 74 }}]
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


POTE generator: ~44/35 = 396.644
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644


Vals: {{Val list| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


Badness: 0.038186
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"