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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.
{{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.


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


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 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.  


For tunings, a basic option would be [[99edo]]. Others such as [[80edo]], [[94edo]], and [[118edo]] are possible; [[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]] 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]], associating it with the [[amicable]] temperament.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[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 18: Line 20:
: Angle (3/2, 81/70) = 73.88 deg
: Angle (3/2, 81/70) = 73.88 deg


[[POTE generator]]s: ~3/2 = 702.3728, ~81/70 = 254.6253
[[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 }}


[[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]]s: 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]]s: 2, 7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5


{{Val list|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]] (Sintel): 4.95


[[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


=== Overview to extensions ===
== Undecimal canou ==
Canou 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.
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]].  


== Synca ==
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.  
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


[[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 }}


[[POTE generator]]s: ~3/2 = 702.2549, ~81/70 = 254.6291
[[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 }}


{{Val list|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}


[[Badness]]: 2.042 × 10<sup>-3</sup>
[[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
[[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 57: Line 66:
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 }}


POTE generators: ~3/2 = 702.1807, ~81/70 = 254.6239
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 GPV sequence: {{val list| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}


Badness: 2.555 × 10<sup>-3</sup>
Badness (Sintel): 2.39


== Canta ==
=== 17-limit ===
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 75e & 80 & 99e, and 80edo makes the optimal.  
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 595/594, 833/832, 1156/1155, 19712/19683


Comma list: 896/891, 472392/471625
Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}


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


POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}


{{Val list|legend=1| 75e, 80, 99e, 179e }}
Badness (Sintel): 1.41


Badness: 4.523 × 10<sup>-3</sup>
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
=== Cantawolf ===
This extension was named ''canta'' in the earlier materials. It adds [[351/350]], the ratwolfsma, to the comma list. Since 351/350 = (81/70)/(15/13). The 81/70-generator simultaneously represents 15/13, adding a lot of fun to the scale. Again 80edo makes the optimal.
 
Subgroup: 2.3.5.7.11


Comma list: 351/350, 832/825, 13013/12960
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215


Mapping: [<1 0 0 -1 6 0|, <0 1 2 2 -2 3|, <0 0 4 -3 -3 5|]
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 }}


POTE generators: ~3/2 = 703.8423, ~15/13 = 254.3605
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 GPV sequence: {{val list| 75ef, 80, 99e, 104c, 179e, 184c, 203ce }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}


Badness: 3.470 × 10<sup>-3</sup>
Badness (Sintel): 1.03


=== Cantamint ===
=== 23-limit ===
This extension was named ''gentcanta'' in the earlier materials. It adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal.  
Subgroup: 2.3.5.7.11.13.17.19.23


Subgroup: 2.3.5.7.11.13
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155


Comma list: 352/351, 364/363, 472392/471625
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 }}


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


POTE generators: ~3/2 = 703.8695, ~64/55 = 254.6321
{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}


Optimal GPV sequence: {{val list| 75e, 80, 99ef, 179ef }}
Badness (Sintel): 1.09


Badness: 4.781 × 10<sup>-3</sup>
== 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.  
== 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. Note that 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.  


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


Subgroup: 2.3.5.7.11
[[Comma list]]: 896/891, 472392/471625


[[Comma list]]: 9801/9800, 14641/14580
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}


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


Mapping generators: ~99/70, ~3, ~81/70
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


[[POTE generator]]s: ~3/2 = 702.3850, ~81/70 = 254.6168
[[Badness]] (Sintel): 5.43
 
{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
 
[[Badness]]: 2.197 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 14641/14580
Comma list: 352/351, 364/363, 472392/471625


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


POTE generators: ~3/2 = 702.5046, ~81/70 = 254.6501
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 GPV sequence: {{val list| 80f, 94, 118f, 198, 410 }}
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}


Badness: 2.974 × 10<sup>-3</sup>
Badness (Sintel): 4.47


==== 17-limit ====
== Semicanou ==
Subgroup: 2.3.5.7.11.13.17
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) }}.  


Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580
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.


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 }}]
[[Subgroup]]: 2.3.5.7.11


POTE generators: ~3/2 = 702.4241, ~81/70 = 254.6672
[[Comma list]]: 9801/9800, 14641/14580
 
Optimal GPV sequence: {{val list| 94, 118f, 198g, 212g, 292, 410 }}
 
Badness: 2.421 × 10<sup>-3</sup>
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}]
 
POTE generators: ~3/2 = 702.3551, ~81/70 = 254.6866
 
Optimal GPV sequence: {{val list| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
 
Badness: 2.177 × 10<sup>-3</sup>
 
=== 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.
 
Subgroup: 2.3.5.7.11.13
 
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 }}]
 
POTE generators: ~3/2 = 702.8788, ~81/70 = 254.6664 or ~11/9 = 345.3336
 
Optimal GPV sequence: {{val list| 80, 94, 118, 174d, 198, 490f }}
 
Badness: 2.701 × 10<sup>-3</sup>
 
=== Semicanouwolf ===
This extension was named ''gentsemicanou'' in the earlier materials. It adds [[351/350]], the ratwolfsma, as wells as [[364/363]], the gentle comma, to the comma list. Since 351/350 = (81/70)/(15/13), the 81/70-generator simultaneously represents 15/13, adding a lot of fun to the scale.
 
Not supported by many patent vals, 80edo easily makes the optimal. Yet 104edo in 104c val and 118edo in 118f val are worth mentioning, and the temperament may be described as 80 & 104c & 118f.
 
Subgroup: 2.3.5.7.11.13
 
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|legend=1| 2 0 0 -2 1 | 0 1 2 2 2 | 0 0 -4 3 -1 }}
: mapping generators: ~99/70, ~3, ~81/70


POTE generators: ~3/2 = 702.7876, ~15/13 = 254.3411 or ~11/9 = 345.6789
[[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 }}


Optimal GPV sequence: {{val list| 80, 104c, 118f, 198f, 420cff }}
{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}


Badness: 3.511 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.64


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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