Canousmic temperaments: Difference between revisions

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]].  

* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]

* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]

* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]

* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]

* ''[[Turkey (temperament)|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]

* ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]

Considered below is 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 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.  

{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9770{{c}}, ~50/27 = 1064.9332{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9533{{c}}

: error map: {{val| 0.000 +0.386 -0.326 -0.124 }}

[[Badness]] (Sintel): 6.39

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

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

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

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

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

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