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'''Canou''' is a rank-3 temperament that tempers out the [[canousma]], 4802000/4782969 = {{monzo|4 -14 3 4}}, a 7-limit comma measuring about 6.9 cents.
{{Technical data page}}
The '''canou family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[canousma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s.


The 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 a [[980/729]] at about 510 cents, an audibly off perfect fourth. Three of them make a [[14/9]]; four of them make a [[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.
== Canou ==
{{Main| Canou }}


Decent amount of harmonic resources are provided by a simple 9-note scale. [[Flora Canou]] commented:
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 wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic.


:''— It sounds somewhat like a Phrygian scale but the abundance of small intervals of [[28/27]] makes it melodically active.''
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]], associating it with the [[amicable]] temperament.  


14- and 19-note scales are also possible. See [[canou scales]] for more information.  
[[Subgroup]]: 2.3.5.7


For tunings, a basic option would be [[80edo]]. Others such as [[94edo]], [[99edo]] and [[118edo]] are more accurate; [[19edo]] (perferably with stretched octaves) also provides a good trivial case, whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[Amicable|amicable temperament]].
[[Comma list]]: 4802000/4782969


Comma: 4802000/4782969
{{Mapping|legend=1| 1 0 0 -1 | 0 1 2 2 | 0 0 -4 3 }}
: mapping generators: ~2, ~3, ~81/70


Map: [<1 0 0 -1|, <0 1 2 2|, <0 0 -4 3|]
Lattice basis:  
: 3/2 length = 0.8110, 81/70 length = 0.5135
: Angle (3/2, 81/70) = 73.88 deg


Wedgie: <<<4 -3 -14 -4 |||
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9597{{c}}, ~3/2 = 702.3492{{c}}, ~81/70 = 254.6168{{c}}
: [[error map]]: {{val| -0.040 +0.354 -0.163 -0.317 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3455{{c}}, ~81/70 = 254.6237{{c}}
: error map: {{val| 0.000 +0.390 -0.118 -0.264 }}


POTE generators: ~3/2 = 702.3728, ~81/70 = 254.6253
[[Minimax tuning]]:
* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
: [[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
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5


EDOs: {{EDOs|75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b}}
{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}


Badness: 0.001122
[[Badness]] (Sintel): 4.95


== Semicanou ==
[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
 
== Undecimal canou ==
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]].
 
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]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests 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.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 19712/19683, 42875/42768
 
{{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 0 0 -4 3 -3 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0568{{c}}, ~3/2 = 702.2009{{c}}, ~81/70 = 254.6291{{c}}
: [[error map]]: {{val| +0.057 +0.303 -0.314 -0.480 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1829{{c}}, ~81/70 = 254.6186{{c}}
: error map: {{val| 0.0000 +0.228 -0.422 -0.604 +0.107 }}
 
{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
 
[[Badness]] (Sintel): 2.45
 
[[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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 19712/19683, 42875/42768
 
Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 0 0 -4 3 -3 4 }}
 
Optimal tunings:
* WE: ~2 = 1200.0501{{c}}, ~3/2 = 702.2100{{c}}, ~81/70 = 254.6345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1889{{c}}, ~81/70 = 254.6222{{c}}
 
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
 
Badness (Sintel): 2.39
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 833/832, 1156/1155, 19712/19683
 
Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}
 
Optimal tunings:
* WE: ~2 = 1200.0630{{c}}, ~3/2 = 702.2317{{c}}, ~51/44 = 254.6224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2055{{c}}, ~51/44 = 254.6066{{c}}
 
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
 
Badness (Sintel): 1.41
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1200.0624{{c}}, ~3/2 = 702.2377{{c}}, ~22/19 = 254.6139{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2117{{c}}, ~22/19 = 254.5983{{c}}
 
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
 
Badness (Sintel): 1.03
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1200.0004{{c}}, ~3/2 = 702.2361{{c}}, ~22/19 = 254.6225{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~22/19 = 254.6223{{c}}
 
{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}


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. Note that 99/70 = (81/70)×(11/9), this extension is more than natural.  
Badness (Sintel): 1.09


The other comma necessary to define it is 14641/14580], 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.  
== Canta ==
By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as {{nowrap| 75e & 80 & 99e }}, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)⋅(364/363), named ''gentcanta'' in earlier materials.  


Still 80edo can be used as a tuning. Other options include 94edo, 118edo, and [[104edo]] in 104c val.  
[[Subgroup]]: 2.3.5.7.11


Commas: 9801/9800, 14641/14580
[[Comma list]]: 896/891, 472392/471625


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


POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0708{{c}}, ~3/2 = 703.1969{{c}}, ~64/55 = 254.4161{{c}}
: [[error map]]: {{val| -0.929 +0.313 +0.557 -0.113 +1.820 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.5249{{c}}, ~64/55 = 254.5492{{c}}
: error map: {{val| 0.000 +1.570 +2.539 +1.871 +5.280 }}


EDOs: {{EDOs|80, 94, 118, 198, 212, 292, 330e, 410}}
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


Badness: 0.002197
[[Badness]] (Sintel): 5.43


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 472392/471625
Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}
Optimal tunings:
* WE: ~2 = 1199.0093{{c}}, ~3/2 = 703.2884{{c}}, ~64/55 = 254.4219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8323{{c}}, ~64/55 = 254.5887{{c}}
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}
Badness (Sintel): 4.47
== Semicanou ==
Semicanou adds 9801/9800, the [[kalisma]], to the comma list, and may be described as {{nowrap| 80 & 94 & 118 }}. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that {{nowrap| 99/70 {{=}} (81/70)⋅(11/9) }}.
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.
[[Subgroup]]: 2.3.5.7.11


This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit.
[[Comma list]]: 9801/9800, 14641/14580


Commas: 352/351, 9801/9800, 14641/14580
{{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


Map: [<2 0 0 -2 1 11|, <0 1 2 2 2 -1|, <0 0 4 -3 1 1|]
[[Optimal tuning]]s:  
* [[WE]]: ~99/70 = 600.0142{{c}}, ~3/2 = 702.4017{{c}}, ~81/70 = 254.6228{{c}}
: [[error map]]: {{val| +0.028 +0.475 +0.055 -0.126 -1.066 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.4048{{c}}, ~81/70 = 254.6179{{c}}
: error map: {{val| 0.0000 +0.450 +0.024 -0.163 -1.126 }}


POTE generators: ~3/2 = 702.8788, ~81/70 = 254.6664 or ~11/9 = 345.3336
{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}


EDOs: {{EDOs|80, 94, 118, 174d, 198}}
[[Badness]] (Sintel): 2.64


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