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Canousmic temperaments: Difference between revisions

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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Godzilla]] (+49/48 or 81/80) → [[Slendro clan #Godzilla|Slendro clan]]
* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Satin]]'' (+2100875/2097152) → [[Garischismic clan #Satin|Garischismic clan]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* ''[[Amicable]]'' (+2401/2400) → [[Breedsmic temperaments #Amicable|Breedsmic temperaments]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]
* ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]
Considered below are satin and superlimmal.
== Satin ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Satin]].''
Satin tempers out the [[rainy comma]], the canousma, and the [[garischisma]], and may be described as the {{nowrap| 94 & 217 }} temperament. It uses [[~]][[11/10]] as a generator, three of which gives a [[4/3|perfect fourth]], tempering out [[4000/3993]] in the 11-limit and onwards. Its [[ploidacot]] is omega-tricot.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2100875/2097152, 4802000/4782969
{{Mapping|legend=1| 1 2 12 -3 | 0 -3 -70 42 }}
: mapping generators: ~2, ~8575/7776
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0198{{c}}, ~8575/7776 = 165.9161{{c}}
: [[error map]]: {{val| +0.020 +0.336 -0.200 -0.411 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8575/7776 = 165.9133{{c}}
: error map: {{val| 0.000 +0.305 -0.241 -0.469 }}
{{Optimal ET sequence|legend=1| 94, 217, 311, 839, 1150 }}
[[Badness]] (Sintel): 4.99
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4000/3993, 19712/19683, 41503/41472
Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}
Optimal tunings:
* WE: ~2 = 1199.9931{{c}}, ~11/10 = 165.9145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9155{{c}}
{{Optimal ET sequence|legend=0| 94, 217, 311 }}
Badness (Sintel): 1.92
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}
Optimal tunings:
* WE: ~2 = 1199.9607{{c}}, ~11/10 = 165.9085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9141{{c}}
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}
Badness (Sintel): 1.25
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}
Optimal tunings:
* WE: ~2 = 1199.9843{{c}}, ~11/10 = 165.9110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9132{{c}}
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}
Badness (Sintel): 1.02
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}
Optimal tunings:
* WE: ~2 = 1199.9875{{c}}, ~11/10 = 165.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9129{{c}}
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}
Badness (Sintel): 0.881
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}
Optimal tunings:
* WE: ~2 = 1199.9745{{c}}, ~11/10 = 165.9103{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9140{{c}}
{{Optimal ET sequence|legend=0| 94, 217, 311 }}


Badness (Sintel): 0.871
Considered below is superlimmal.


== Superlimmal ==
== Superlimmal ==
Superlimmal is essentially an 80-form, and may be described as the {{nowrap| 80 & 311 }} temperament. It uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. Note that in the data that follow, the generator is its [[octave complement]], [[~]][[50/27]], so that 57 of them [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]].  
Superlimmal is essentially an 80-form, and may be described as the {{nowrap| 80 & 311 }} temperament. It uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. Note that in the data that follow, the generator is its [[octave complement]], [[~]][[50/27]], so that 57 of them [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]].  


Superlimmal gets all the primes up to [[29/1|29]] reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple [[comma basis]]. It can also be extended to include prime [[37/1|37]] by tempering out ([[27/25]])/([[40/37]]) = [[1000/999]], where 40/37 is notably the mediant of [[27/25]] and [[13/12]], which could be interpreted as an explanation of the sharpened limma.  
Superlimmal gets all the primes up to [[29/1|29]] reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple [[comma basis]]. It can also be extended to include prime [[37/1|37]] by mapping it to 87 generator steps, tempering out ([[27/25]])/([[40/37]]) = [[1000/999]]. Since 40/37 is the mediant of [[27/25]] and [[13/12]], this extension further consolidates the sharpened limma.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Sintel): 1.09
Badness (Sintel): 1.09
=== No-31's 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 -72 | 0 57 86 135 67 -24 17 60 22 99 87 }}
Optimal tuning:
* WE: ~2 = 1199.9549{{c}}, ~37/20 = 1064.9140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~37/20 = 1064.9538{{c}}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
Badness (Sintel): 1.06


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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