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Canousmic temperaments: Difference between revisions

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


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
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== Satin ==
== Satin ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Satin]].''
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Satin]].''


The satin temperament (94 & 217) uses [[11/10]] as a generator, three of them gives [[4/3]], and tempers out both the [[rainy comma]] and the canousma.
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
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 2 12 -3 | 0 -3 -70 42 }}
{{Mapping|legend=1| 1 2 12 -3 | 0 -3 -70 42 }}
: mapping generators: ~2, ~8575/7776


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8575/7776 = 165.913
[[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 }}
{{Optimal ET sequence|legend=1| 94, 217, 311, 839, 1150 }}


[[Badness]]: 0.197207
[[Badness]] (Sintel): 4.99


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}
Mapping: {{mapping| 1 2 12 -3 13 | 0 -3 -70 42 -69 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.915
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=1| 94, 217, 311 }}
{{Optimal ET sequence|legend=0| 94, 217, 311 }}


Badness: 0.057972
Badness (Sintel): 1.92


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}
Mapping: {{mapping| 1 2 12 -3 13 -1 | 0 -3 -70 42 -69 34 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914
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=1| 94, 217, 311, 839e, 1150e }}
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}


Badness: 0.030316
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}
Mapping: {{mapping| 1 2 12 -3 13 -1 11 | 0 -3 -70 42 -69 34 -50 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913
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=1| 94, 217, 311, 839e, 1150eg }}
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}


Badness: 0.020007
Badness (Sintel): 1.02


=== 19-limit ===
=== 19-limit ===
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Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 | 0 -3 -70 42 -69 34 -50 -85 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913
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=1| 94, 217, 311, 839e, 1150eg }}
{{Optimal ET sequence|legend=0| 94, 217, 311, 839e }}


Badness: 0.014479
Badness (Sintel): 0.881


=== 23-limit ===
=== 23-limit ===
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Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}
Mapping: {{mapping| 1 2 12 -3 13 -1 11 16 16 | 0 -3 -70 42 -69 34 -50 -85 -83 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914
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=1| 94, 217, 311, 839ei, 1150egi }}
{{Optimal ET sequence|legend=0| 94, 217, 311 }}


Badness: 0.012158
Badness (Sintel): 0.871


== Superlimmal ==
== Superlimmal ==
The superlimmal temperament (80 & 311) uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. It gets all the primes up to 29 reasonably covered, but still acceptable just as a 13-limit microtemperament, judging from its [[comma basis]]. While the [[mos scale]] may not be the most effective approach, the 80-tone mos is presumably the place to start if it is used. It can also be extended to prime 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 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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 4802000/4782969, 52734375/52706752
[[Comma list]]: 4802000/4782969, 52734375/52706752


{{Mapping|legend=1| 1 8 12 18 | 0 -57 -86 -135 }}
{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}
: mapping generators: ~2, ~50/27


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 135.0464
[[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 }}
{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}


[[Badness]]: 0.252387
[[Badness]] (Sintel): 6.39


=== 11-limit ===
=== 11-limit ===
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Comma list: 3025/3024, 4000/3993, 1479016/1476225
Comma list: 3025/3024, 4000/3993, 1479016/1476225


Mapping: {{mapping| 1 8 12 18 11 | 0 -57 -86 -135 -67 }}
Mapping: {{mapping| 1 -49 -74 -117 -56 | 0 57 86 135 67 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0455
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=1| 80, 231, 311, 1013e, 1324be }}
{{Optimal ET sequence|legend=0| 80, 231, 311, 1013e, 1324be }}


Badness: 0.060667
Badness (Sintel): 2.01


=== 13-limit ===
=== 13-limit ===
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Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455


Mapping: {{mapping| 1 8 12 18 11 1 | 0 -57 -86 -135 -67 24 }}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0446
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=1| 80, 231, 311, 702, 1013e }}
{{Optimal ET sequence|legend=0| 80, 231, 311, 702, 1013e }}


Badness: 0.039017
Badness (Sintel): 1.61


=== 17-limit ===
=== 17-limit ===
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Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224


Mapping: {{mapping| 1 8 12 18 11 1 6 | 0 -57 -86 -135 -67 24 -17 }}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0462
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=1| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.030077
Badness (Sintel): 1.53


=== 19-limit ===
=== 19-limit ===
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Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499


Mapping: {{mapping| 1 8 12 18 11 1 6 11 | 0 -57 -86 -135 -67 24 -17 -60 }}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 | 0 57 86 135 67 -24 17 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0464
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=1| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.020460
Badness (Sintel): 1.24


=== 23-limit ===
=== 23-limit ===
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Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495


Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 | 0 -57 -86 -135 -67 24 -17 -60 -22 }}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0458
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=1| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.016146
Badness (Sintel): 1.16


=== 29-limit ===
=== 29-limit ===
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Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495


Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 16 | 0 -57 -86 -135 -67 24 -17 -60 -22 -99 }}
Mapping: {{mapping| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 | 0 57 86 135 67 -24 17 60 22 99 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460
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=1| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.013054
Badness (Sintel): 1.09


=== No-31's 37-limit ===
=== No-31's 37-limit ===
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Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274
Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274


Mapping: {{mapping| 1 8 12 18 11 1 6 11 7 16 15 | 0 -57 -86 -135 -67 24 -17 -60 -22 -99 -87 }}
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 (POTE): ~2 = 1\1, ~27/25 = 135.0460
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=1| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.010901
Badness (Sintel): 1.06


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Canousmic_temperaments"