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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.
{{Technical data page}}
The '''canou family''' of [[rank-3 temperament|rank-3]] [[regular temperament|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 [[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.  
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. Three make [[14/9]]; four make [[9/5]]. It therefore splits the large 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.
A basic tuning 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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 22: Line 21:
: Angle (3/2, 81/70) = 73.88 deg
: Angle (3/2, 81/70) = 73.88 deg


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.3175, ~81/70 = 254.6220
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
: [[error map]]: {{val| 0.0000 +0.3625 -0.1667 -0.3249 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237
: error map: {{val| 0.0000 +0.3904 -0.1175 -0.2640 }}


[[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|eigenmonzo (unchanged-interval) basis]]: 2.5.7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) 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|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) 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 }}


[[Badness]]: 1.122 × 10<sup>-3</sup>
[[Badness]] (Smith): 1.122 × 10<sup>-3</sup>


[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5


=== 2.3.5.7.17 subgroup ===
== Undecimal canou ==
Subgroup: 2.3.5.7.17
The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]], which suggests the [[wilschisma]] should also be tempered out in the 13-limit.  
 
Comma list: 1225/1224, 295936/295245
 
Mapping: {{mapping| 1 0 0 -1 -5 | 0 1 2 2 6 | 0 0 -4 3 -2 }}
 
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 }}
 
Badness: 0.775 × 10<sup>-3</sup>
 
=== 2.3.5.7.17.19 subgroup ===
Subgroup: 2.3.5.7.17.19
 
Comma list: 1216/1215, 1225/1224, 1445/1444
 
Mapping: {{mapping| 1 0 0 -1 -5 -6 | 0 1 2 2 6 7 | 0 0 -4 3 -2 -4 }}
 
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 }}


Badness: 0.548 × 10<sup>-3</sup>
Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]]. From a commatic point of view, notice the other 11-limit comma, [[42875/42768]], is {{nowrap| S34 × S35<sup>2</sup> }}, suggesting tempering out [[595/594]] (S34 × S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. 311edo remains an excellent tuning in all the limits.  
 
== Synca ==
Synca, for symbiotic canou, adds the [[symbiotic comma]] and the [[wilschisma]] to the comma list.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 71: Line 50:
{{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 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]]): ~2 = 1\1, ~3/2 = 702.2115, ~81/70 = 254.6215
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
: [[error map]]: {{val| 0.0000 +0.2565 -0.3768 -0.5383 +0.2980 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186
: error map: {{val| 0.0000 +0.2279 -0.4221 -0.6043 +0.1069 }}


{{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 }}


[[Badness]]: 2.04 × 10<sup>-3</sup>
[[Badness]] (Smith): 2.04 × 10<sup>-3</sup>


[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5
[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5
Line 86: Line 69:
Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 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): ~2 = 1\1, ~3/2 = 702.2075, ~81/70 = 254.6183
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
* CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222


{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}


Badness: 2.56 × 10<sup>-3</sup>
Badness (Smith): 2.56 × 10<sup>-3</sup>


=== 17-limit ===
=== 17-limit ===
Line 99: Line 84:
Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 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): ~2 = 1\1, ~3/2 = 702.2296, ~51/44 = 254.6012
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
* CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066


{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}


Badness: 1.49 × 10<sup>-3</sup>
Badness (Smith): 1.49 × 10<sup>-3</sup>


=== 19-limit ===
=== 19-limit ===
Line 110: Line 97:
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: {{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 }}
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 tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
* CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983
 
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
 
Badness (Smith): 1.00 × 10<sup>-3</sup>
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2355, ~22/19 = 254.5930
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155


{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 -6 4 | 0 1 2 2 7 10 6 7 1 | 0 0 -4 3 -3 4 -2 -4 -5 }}


Badness: 1.00 × 10<sup>-3</sup>
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222
* CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223
 
{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}
 
Badness (Smith): 0.948 × 10<sup>-3</sup>


== Canta ==
== Canta ==
Line 127: Line 131:
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 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]]): ~2 = 1\1, ~3/2 = 702.8093, ~64/55 = 254.3378
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
: [[error map]]: {{val| 0.0000 +0.8543 +1.9537 -0.1940 +6.0769 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492
: error map: {{val| 0.0000 +1.5699 +2.5393 +1.8714 +5.2799 }}


{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


[[Badness]]: 4.523 × 10<sup>-3</sup>
[[Badness]] (Smith): 4.52 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 140: Line 148:
Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 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): ~2 = 1\1, ~3/2 = 703.6228, ~64/55 = 254.3447
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
* CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887


{{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef }}


Badness: 4.781 × 10<sup>-3</sup>
Badness (Smith): 4.78 × 10<sup>-3</sup>


== Semicanou ==
== Semicanou ==
Semicanou adds [[9801/9800]], the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing 99/70~140/99. Note that 99/70 = (81/70)(11/9), this extension is more than natural.  
Semicanou adds [[9801/9800]], the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing 99/70~140/99. Note that {{nowrap| 99/70 {{=}} (81/70)(11/9) }}, this extension is more than natural.  


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 about 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, and 11/10.  
 
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 161: Line 169:
: mapping generators: ~99/70, ~3, ~81/70
: mapping generators: ~99/70, ~3, ~81/70


[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.4262, ~81/70 = 254.6191
[[Optimal tuning]]s:
* [[CTE]]: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
: [[error map]]: {{val| 0.0000 +0.4712 +0.0625 -0.1163 -1.0846 }}
* [[CWE]]: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179
: error map: {{val| 0.0000 +0.4498 +0.0245 -0.1627 -1.1262 }}


{{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 }}


[[Badness]]: 2.197 × 10<sup>-3</sup>
[[Badness]] (Smith): 2.20 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 174: Line 186:
Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 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): ~99/70 = 1\2, ~3/2 = 702.4802, ~81/70 = 254.6526
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
* CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511


{{Optimal ET sequence|legend=1| 80f, 94, 118f, 198, 410 }}
{{Optimal ET sequence|legend=0| 80f, 94, 118f, 198, 410 }}


Badness: 2.974 × 10<sup>-3</sup>
Badness (Smith): 2.97 × 10<sup>-3</sup>


<!-- debatable canonicity
==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 187: Line 202:
Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 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): ~99/70 = 1\2, ~3/2 = 702.4415, ~81/70 = 254.6663
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663


{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}
{{Optimal ET sequence|legend=0| 94, 118f, 198g, 212g, 292, 410 }}


Badness: 2.421 × 10<sup>-3</sup>
Badness (Smith): 2.42 × 10<sup>-3</sup>


==== 19-limit ====
==== 19-limit ====
Line 200: Line 216:
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 }}
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): ~99/70 = 1\2, ~3/2 = 702.4030, ~81/70 = 254.6870
Optimal tunings:
 
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870
{{Optimal ET sequence|legend=1| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}


Badness: 2.177 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}


Badness (Smith): 2.18 × 10<sup>-3</sup>
-->
=== Semicanoumint ===
=== Semicanoumint ===
This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.  
This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.  
Line 215: Line 232:
Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 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): ~99/70 = 1\2, ~3/2 = 702.5374, ~81/70 = 254.6819
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
* CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704


{{Optimal ET sequence|legend=1| 80, 94, 118, 174d, 198, 490f }}
{{Optimal ET sequence|legend=0| 80, 94, 118, 174d, 198, 490f }}


Badness: 2.701 × 10<sup>-3</sup>
Badness (Smith): 2.70 × 10<sup>-3</sup>


=== Semicanouwolf ===
=== Semicanouwolf ===
Line 232: Line 251:
Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 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 tunings:
* CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
* CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396


{{Optimal ET sequence|legend=1| 80, 104c, 118f, 198f, 420cff }}
{{Optimal ET sequence|legend=0| 80, 104c, 118f, 198f, 420cff }}


Badness: 3.511 × 10<sup>-3</sup>
Badness (Smith): 3.51 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Canou_family"