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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).
* ''[[liese]]'', {81/80, 686/675} → [[Meantone family #Liese]]
* ''[[unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn]]
* [[magic]], {225/224, 245/243} → [[Magic family #Magic]]
* ''[[guiron]]'', {1029/1024, 10976/10935} → [[Gamelismic clan #Guiron]]
* ''[[echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna]]
* [[hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths]]
* ''[[dodecacot]]'', {3125/3087, 10976/10935} → [[Tetracot family #Dodecacot]]
* [[parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic]]
* ''[[pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto]]
* ''[[hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic]]
* ''[[marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths]]
* ''[[yarman]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman]]


== Commatic ==
Temperaments discussed elsewhere are:
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.
* ''[[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) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Hendecatonic (temperament)|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]]


Subgroup: 2.3.5.7
Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].  


[[Comma list]]: 10976/10935, 50421/50000
== 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.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
[[Comma list]]: 10976/10935, 235298/234375


{{Multival|legend=1| 10 38 36 37 29 -23 }}
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27


[[POTE generator]]: ~81/80 = 20.377
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}


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


[[Badness]]: 0.084317
[[Badness]] (Sintel): 1.46


=== 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 tunings:
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


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


POTE generator: ~81/80 = 20.390
Badness (Sintel): 1.67


Optimal GPV sequence: {{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 tunings:  
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}


[[Comma list]]: 10976/10935, 235298/234375
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
Badness (Sintel): 1.90


{{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 tunings:  
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


== Degrees ==
{{Optimal ET sequence|legend=0| 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 (Sintel): 1.61


[[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 tunings:  
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


[[Badness]]: 0.106471
Badness (Sintel): 1.81


=== 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 tunings:  
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}


Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness: 0.046770
Badness (Sintel): 1.54


=== 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 }}]
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


POTE generator: ~3/2 = 703.080
Optimal tunings:  
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}


Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Badness: 0.032718
Badness (Sintel): 2.06


== Subfourth ==
===== 17-limit =====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17
 
[[Comma list]]: 10976/10935, 65536/64827


[[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 tunings:  
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


[[Badness]]: 0.140722
Badness (Sintel): 1.58


=== 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 tunings:  
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}


Optimal GPV sequence: {{Val list| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.045323
Badness (Sintel): 2.22


=== 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 tunings:  
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}


Optimal GPV sequence: {{Val list| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}


Badness: 0.023800
Badness (Sintel): 1.38


== Bisupermajor ==
== Bisupermajor ==
{{see also| Very high accuracy temperaments #Kwazy }}
: ''For the 5-limit version, see [[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 }}
: mapping generators: ~1225/864, ~192/175


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}
: [[error map]]: {{val| +0.059 +0.587 -0.208 -0.957 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}
: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}


[[POTE generator]]: ~192/175 = 162.806
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
[[Badness]] (Sintel): 1.66


[[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 tunings:
* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=1| 2 3 4 5 | 0 5 19 18 }}
: mapping generators: ~567/400, ~81/80
 
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}
: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}
: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}
 
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
 
[[Badness]] (Sintel): 2.13


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


Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 441/440, 3388/3375, 8019/8000
 
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
{{Optimal ET sequence|legend=0| 58, 118 }}


POTE generators: ~11/10 = 162.773
Badness (Sintel): 1.14


Optimal GPV sequence: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
== Degrees ==
{{About|the regular temperament|scale degrees|degree}}
{{See also| 20th-octave temperaments }}


Badness: 0.032080
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.  


== Cotoneum ==
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.
{{Main| Cotoneum }}


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.
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.


Subgroup: 2.3.5.7
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]].


[[Comma list]]: 10976/10935, 823543/819200
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]
[[Comma list]]: 10976/10935, 390625/388962


{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
: mapping generators: ~28/27, ~3


[[POTE generator]]: ~3/2 = 702.317
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})
: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}
* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})
: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


[[Badness]]: 0.105632
[[Badness]] (Sintel): 2.69


=== 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 tunings:  
* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})


Optimal GPV sequence: {{Val list| 41, 135c, 176, 217 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


Badness: 0.050966
Badness (Sintel): 1.55


=== 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 tunings:  
* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness: 0.036951
Badness (Sintel): 1.35


=== 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: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


POTE generator: ~3/2 = 702.307
Optimal tunings:  
* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness: 0.029495
Badness (Sintel): 1.17


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
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 tunings:
* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})
 
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~3/2 = 703.2183{{c}} (~100/99 = 16.7827{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2178{{c}} (~100/99 = 16.7822{{c}})
 
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
Badness (Sintel): 1.13


POTE generator: ~3/2 = 702.308
=== 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


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870


Badness: 0.021811
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 tunings:
* WE: ~29/28 = 59.9998{{c}}, ~3/2 = 703.2088{{c}} (~100/99 = 16.7882{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2104{{c}} (~100/99 = 16.7896{{c}})
 
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}
 
Badness (Sintel): 1.10


== 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 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''
 
Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma]]: {{monzo| 61 4 -29 }}
[[Comma list]]: 10976/10935, 29360128/29296875


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


[[POTE generator]]: ~98304/78125 = 396.621
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9006{{c}}, ~1125/896 = 396.6104{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Val list|legend=1| 118, 593, 711, 829, 947 }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


[[Badness]]: 0.218314
[[Badness]] (Sintel): 3.36


=== 7-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 10976/10935, 29360128/29296875
Comma list: 3025/3024, 5632/5625, 10976/10935
 
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
 
Optimal tunings:
* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}
 
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}
 
Badness (Sintel): 1.26
 
== Leapmonth ==
Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 -8 1 -20 }}, {{val| 0 29 4 69 }}]
[[Comma list]]: 10976/10935, 51200/50421


{{Multival|legend=1| 29 4 69 -61 28 149 }}
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}


[[POTE generator]]: ~1125/896 = 396.643
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}}
: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
: error map: {{val| 0.000 +2.977 +1.093 +5.150 }}


{{Val list|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}


[[Badness]]: 0.132821
[[Badness]] (Sintel): 4.79


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


Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 540/539, 896/891, 1331/1323
 
Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}
 
Optimal tunings:
* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}
 
Badness (Sintel): 1.88
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 352/351, 364/363, 540/539


Mapping: [{{val| 1 -8 1 -20 -21 }}, {{val| 0 29 4 69 74 }}]
Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}


POTE generator: ~44/35 = 396.644
Optimal tunings:  
* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}


Optimal GPV sequence: {{Val list| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}


Badness: 0.038186
Badness (Sintel): 1.53


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