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Canousmic temperaments: Difference between revisions

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These are rank-2 temperaments that temper out the [[canousma]], 4802000/4782969 = {{monzo|4 -14 3 4}}. For the rank-3 temperament, see [[Canou family]].  
{{Technical data page}}
This is a collection of rank-2 temperaments that temper out the [[canousma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969). For the rank-3 temperament, see [[Canou family]].  


Note that 4802000/4782969 = 2×([[10/9]])<sup>3</sup>/([[9/7]])<sup>4</sup>, these intervals tend to have lower complexity.
Temperaments discussed elsewhere are:
* [[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]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[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]]
* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]


Temperaments not dicussed here include [[Meantone family #Godzilla|godzilla]] (a trivial case) and these:
Considered below is superlimmal.
* [[Sycamore family #Betic|Betic]]
* [[Breedsmic temperaments #Amicable|Amicable]]
* [[Ragismic microtemperaments #Parakleismic|Parakleismic]]
* [[Hemifamity temperaments #Septiquarter|Septiquarter]]
* [[Schismatic family #Kleischismic|Kleischismic]]
* [[Garischismic temperaments #Satin|Satin]]


[[Category:Temperament]]
== Superlimmal ==
[[Category:Canou]]
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 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
 
[[Comma list]]: 4802000/4782969, 52734375/52706752
 
{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}
: mapping generators: ~2, ~50/27
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9770{{c}}, ~50/27 = 1064.9332{{c}}
: [[error map]]: {{val| -0.023 +0.365 -0.356 -0.152 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9533{{c}}
: error map: {{val| 0.000 +0.386 -0.326 -0.124 }}
 
{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}
 
[[Badness]] (Sintel): 6.39
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 1479016/1476225
 
Mapping: {{mapping| 1 -49 -74 -117 -56 | 0 57 86 135 67 }}
 
Optimal tuning:
* WE: ~2 = 1199.9235{{c}}, ~50/27 = 1064.8866{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311, 1013e, 1324be }}
 
Badness (Sintel): 2.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
 
Mapping: {{mapping| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }}
 
Optimal tuning:
* WE: ~2 = 1199.8904{{c}}, ~50/27 = 1064.8582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9547{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311, 702, 1013e }}
 
Badness (Sintel): 1.61
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
 
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }}
 
Optimal tuning:
* WE: ~2 = 1199.9634{{c}}, ~50/27 = 1064.9213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
 
Badness (Sintel): 1.53
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499
 
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 | 0 57 86 135 67 -24 17 60 }}
 
Optimal tuning:
* WE: ~2 = 1199.9800{{c}}, ~50/27 = 1064.9358{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9535{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
 
Badness (Sintel): 1.24
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495
 
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }}
 
Optimal tuning:
* WE: ~2 = 1199.9546{{c}}, ~50/27 = 1064.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9539{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
 
Badness (Sintel): 1.16
 
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495
 
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 | 0 57 86 135 67 -24 17 60 22 99 }}
 
Optimal tuning:
* WE: ~2 = 1199.9430{{c}}, ~50/27 = 1064.9035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9538{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
 
Badness (Sintel): 1.09
 
[[Category:Temperament collections]]
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
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