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Canou family: Difference between revisions

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The '''canou family''' of rank-3 temperaments tempers out the [[canousma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}, a 7-limit comma measuring about 6.9 cents.
The '''canou family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[canousma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}, a 7-limit comma measuring about 6.9 [[cent]]s.


== Canou ==
== Canou ==
{{Main| Canou temperament }}
{{Main| Canou temperament }}


The canou temperament features a period of an octave and generators of [[3/2]] and [[81/70]]. The 81/70-generator is about 255 cents. Two of them interestingly make [[980/729]] at about 510 cents, an audibly off perfect fourth. Three make [[14/9]]; four make [[9/5]]. It therefore also features splitting the septimal diesis, [[49/48]], into three equal parts, making two distinct [[interseptimal]] intervals related to the 35th harmonic.  
The canou temperament features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The 81/70-generator is about 255 cents. Two of them make [[980/729]] at about 510 cents, an audibly off perfect fourth. Three make [[14/9]]; four make [[9/5]]. It therefore also features splitting the septimal diesis, [[49/48]], into three equal parts, making two distinct [[interseptimal interval]]s related to the 35th harmonic.  


For tunings, a basic option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament.  
For tunings, a basic option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament.  


It has a neat extension to the 2.3.5.7.17.19 subgroup with virtually no additional errors. The [[comma basis]] is {1216/1215, 1225/1224, 1445/1444}. Otherwise, 11- and 13-limit extensions are somewhat less ideal.  
It has a neat extension to the 2.3.5.7.17.19 [[subgroup]] with virtually no additional errors. The [[comma basis]] is {1216/1215, 1225/1224, 1445/1444}. Otherwise, 11- and 13-limit extensions are somewhat less ideal.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 14: Line 14:
[[Comma list]]: [[4802000/4782969]]
[[Comma list]]: [[4802000/4782969]]


[[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 -4 3 }}]
{{Mapping|legend=1| 1 0 0 -1 | 0 1 2 2 | 0 0 -4 3 }}
 
: mapping generators: ~2, ~3, ~81/70


Lattice basis:  
Lattice basis:  
Line 20: Line 22:
: Angle (3/2, 81/70) = 73.88 deg
: Angle (3/2, 81/70) = 73.88 deg


Optimal tuning ([[CTE]]): ~3/2 = 702.3175, ~81/70 = 254.6220
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.3175, ~81/70 = 254.6220


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5.7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7
* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5


{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}
{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}
Line 39: Line 41:
Comma list: 1225/1224, 295936/295245
Comma list: 1225/1224, 295936/295245


Mapping: [{{val| 1 0 0 -1 -5 }}, {{val| 0 1 2 2 6 }}, {{val| 0 0 -4 3 -2 }}]
Mapping: {{mapping| 1 0 0 -1 -5 | 0 1 2 2 6 | 0 0 -4 3 -2 }}


Optimal tuning (CTE): ~3/2 = 702.3458, ~81/70 = 254.6233
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3458, ~81/70 = 254.6233


{{Optimal ET sequence|legend=1| 94, 99, 193, 217, 292, 311, 410, 1131, 1541b }}
{{Optimal ET sequence|legend=1| 94, 99, 193, 217, 292, 311, 410, 1131, 1541b }}
Line 52: Line 54:
Comma list: 1216/1215, 1225/1224, 1445/1444
Comma list: 1216/1215, 1225/1224, 1445/1444


Mapping: [{{val| 1 0 0 -1 -5 -6 }}, {{val| 0 1 2 2 6 7 }}, {{val| 0 0 -4 3 -2 -4 }}]
Mapping: {{mapping| 1 0 0 -1 -5 -6 | 0 1 2 2 6 7 | 0 0 -4 3 -2 -4 }}


Optimal tuning (CTE): ~3/2 = 702.3233, ~81/70 = 254.6279
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3233, ~81/70 = 254.6279


{{Optimal ET sequence|legend=1| 94, 99, 118, 193, 217, 292h, 311, 410, 721 }}
{{Optimal ET sequence|legend=1| 94, 99, 118, 193, 217, 292h, 311, 410, 721 }}
Line 67: Line 69:
[[Comma list]]: 19712/19683, 42875/42768
[[Comma list]]: 19712/19683, 42875/42768


[[Mapping]]: [{{val| 1 0 0 -1 -7 }}, {{val| 0 1 2 2 7 }}, {{val| 0 0 -4 3 -3 }}]
{{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 0 0 -4 3 -3 }}


Optimal tuning ([[CTE]]): ~3/2 = 702.2115, ~81/70 = 254.6215
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2115, ~81/70 = 254.6215


{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
Line 82: Line 84:
Comma list: 2080/2079, 19712/19683, 42875/42768
Comma list: 2080/2079, 19712/19683, 42875/42768


Mapping: [{{val| 1 0 0 -1 -7 -13 }}, {{val| 0 1 2 2 7 10 }}, {{val| 0 0 -4 3 -3 4 }}]
Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 0 0 -4 3 -3 4 }}


Optimal tuning (CTE): ~3/2 = 702.2075, ~81/70 = 254.6183
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2075, ~81/70 = 254.6183


{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
Line 95: Line 97:
Comma list: 595/594, 833/832, 1156/1155, 19712/19683
Comma list: 595/594, 833/832, 1156/1155, 19712/19683


Mapping: [{{val| 1 0 0 -1 -7 -13 -5 }}, {{val| 0 1 2 2 7 10 6 }}, {{val| 0 0 -4 3 -3 4 -2 }}]
Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}


Optimal tuning (CTE): ~3/2 = 702.2296, ~51/44 = 254.6012
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2296, ~51/44 = 254.6012


{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
Line 108: Line 110:
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215


Mapping: [{{val| 1 0 0 -1 -7 -13 -5 -6 }}, {{val| 0 1 2 2 7 10 6 7 }}, {{val| 0 0 -4 3 -3 4 -2 -4 }}]
Mapping: {{mapping|| 1 0 0 -1 -7 -13 -5 -6 | 0 1 2 2 7 10 6 7 | 0 0 -4 3 -3 4 -2 -4 }}


Optimal tuning (CTE): ~3/2 = 702.2355, ~22/19 = 254.5930
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2355, ~22/19 = 254.5930


{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
Line 123: Line 125:
[[Comma list]]: 896/891, 472392/471625
[[Comma list]]: 896/891, 472392/471625


[[Mapping]]: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}


Optimal tuning ([[CTE]]): ~3/2 = 702.8093, ~64/55 = 254.3378
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.8093, ~64/55 = 254.3378


{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}
Line 136: Line 138:
Comma list: 352/351, 364/363, 472392/471625
Comma list: 352/351, 364/363, 472392/471625


Mapping: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]
Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}


Optimal tuning (CTE): ~3/2 = 703.6228, ~64/55 = 254.3447
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.6228, ~64/55 = 254.3447


{{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}
{{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}
Line 149: Line 151:
The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9-11/9-11/9-11/10.  
The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9-11/9-11/9-11/10.  


Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to any of them.  
Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to all of them.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 155: Line 157:
[[Comma list]]: 9801/9800, 14641/14580
[[Comma list]]: 9801/9800, 14641/14580


[[Mapping]]: [{{val| 2 0 0 -2 1 }}, {{val| 0 1 2 2 2 }}, {{val| 0 0 -4 3 -1 }}]
{{Mapping|legend=1| 2 0 0 -2 1 | 0 1 2 2 2 | 0 0 -4 3 -1 }}


Mapping generators: ~99/70, ~3, ~81/70
: mapping generators: ~99/70, ~3, ~81/70


Optimal tuning ([[CTE]]): ~3/2 = 702.4262, ~81/70 = 254.6191
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.4262, ~81/70 = 254.6191


{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
Line 170: Line 172:
Comma list: 1716/1715, 2080/2079, 14641/14580
Comma list: 1716/1715, 2080/2079, 14641/14580


Mapping: [{{val| 2 0 0 -2 1 -11 }}, {{val| 0 1 2 2 2 5 }}, {{val| 0 0 -4 3 -1 6 }}]
Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}


Optimal tuning (CTE): ~3/2 = 702.4802, ~81/70 = 254.6526
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.4802, ~81/70 = 254.6526


{{Optimal ET sequence|legend=1| 80f, 94, 118f, 198, 410 }}
{{Optimal ET sequence|legend=1| 80f, 94, 118f, 198, 410 }}
Line 183: Line 185:
Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580
Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580


Mapping: [{{val| 2 0 0 -2 1 -11 -10 }}, {{val| 0 1 2 2 2 5 6 }}, {{val| 0 0 -4 3 -1 6 -2 }}]
Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}


Optimal tuning (CTE): ~3/2 = 702.4415, ~81/70 = 254.6663
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.4415, ~81/70 = 254.6663


{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}
{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}
Line 196: Line 198:
Comma list: 715/714, 1089/1088, 1216/1215, 1225/1224, 1445/1444
Comma list: 715/714, 1089/1088, 1216/1215, 1225/1224, 1445/1444


Mapping: [{{val| 2 0 0 -2 1 -11 -10 -12 }}, {{val| 0 1 2 2 2 5 6 7 }}, {{val| 0 0 -4 3 -1 6 -2 -4 }}]
Mapping: {{mapping| 2 0 0 -2 1 -11 -10 -12 | 0 1 2 2 2 5 6 7 | 0 0 -4 3 -1 6 -2 -4 }}


Optimal tuning (CTE): ~3/2 = 702.4030, ~81/70 = 254.6870
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.4030, ~81/70 = 254.6870


{{Optimal ET sequence|legend=1| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
{{Optimal ET sequence|legend=1| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
Line 211: Line 213:
Comma list: 352/351, 9801/9800, 14641/14580
Comma list: 352/351, 9801/9800, 14641/14580


Mapping: [{{val| 2 0 0 -2 1 11 }}, {{val| 0 1 2 2 2 -1 }}, {{val| 0 0 -4 3 -1 -1 }}]
Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}


Optimal tuning (CTE): ~3/2 = 702.5374, ~81/70 = 254.6819
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.5374, ~81/70 = 254.6819


{{Optimal ET sequence|legend=1| 80, 94, 118, 174d, 198, 490f }}
{{Optimal ET sequence|legend=1| 80, 94, 118, 174d, 198, 490f }}
Line 228: Line 230:
Comma list: 351/350, 364/363, 11011/10935
Comma list: 351/350, 364/363, 11011/10935


Mapping: [{{val| 2 0 0 -2 1 0 }}, {{val| 0 1 2 2 2 3 }}, {{val| 0 0 -4 3 -1 -5 }}]
Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}


Optimal tuning (CTE): ~3/2 = 702.7417, ~15/13 = 254.3382
Optimal tuning (CTE): ~3/2 = 702.7417, ~15/13 = 254.3382
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Canou_family"