Canousmic temperaments: Difference between revisions

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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]]

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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

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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:Canousmic temperaments| ]]

[[Category:Rank 2]]

@@ Line 4: / Line 4: @@
 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]]
@@ Line 22: / Line 23: @@
 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
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 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:Canousmic temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]