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Hemimage temperaments: Difference between revisions

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+ leapmonth (63 & 80)
- chromat (more properly addressed in amity family). Sort links
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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic 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]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat|Amity family]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[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]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].  
Considered below are degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].  
 
== 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
 
[[Comma list]]: 10976/10935, 235298/234375
 
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{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}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
 
Optimal tunings:
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
 
Optimal tunings:
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
 
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 ET sequence|legend=0| 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: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
 
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 ET sequence|legend=0| 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: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
 
Optimal tunings:
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
 
Badness (Sintel): 1.58
 
=== 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 tunings:
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
 
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 ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
 
Badness (Sintel): 1.38


== Bisupermajor ==
== Bisupermajor ==
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