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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 '''canthartic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[cathartic comma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s.


= Canou =
== Cathartic ==
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 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.
[[File:Lattice Orthocanousmic.png|thumb|Lattice for cathartic.]]
[[File:Lattice Orthocanousmic Rearranged.png|thumb|Ditto, but re-arranged into hemitwelfths.]]
{{See also| Cathartic scales }}


Decent amount of harmonic resources are provided by a simple 9-note scale. [[Flora Canou]] commented:
Cathartic (formerly ''canou'') 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.  
9-, 14- and 19-note scales are highly characteristic for the temperament, with the abundance of [[28/27]] subminor seconds making it melodically active.  


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]] temperament.  
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4802000/4782969
[[Comma list]]: 4802000/4782969


[[POTE generator]]s: ~3/2 = 702.3728, ~81/70 = 254.6253
{{Mapping|legend=1| 1 0 0 -1 | 0 1 2 2 | 0 0 -4 3 }}
: mapping generators: ~2, ~3, ~81/70


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


{{Multival|legend=1|rank=3| 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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7-odd-limit: 3 +c/14, 5 and 7 just
* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
: Eigenmonzos: 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
: Eigenmonzos: 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 }}
 
[[Badness]] (Sintel): 4.95
 
[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5


Lattice basis:
== Undecimal cathartic ==
: 3/2 length = 0.8110, 81/70 length = 0.5135
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]].  
: Angle (3/2, 81/70) = 73.88 deg
 
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]]. These extensions add little additional error, and [[311edo]] remains an excellent tuning in all cases.
 
[[Subgroup]]: 2.3.5.7.11


{{Val list|legend=1| 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b }}
[[Comma list]]: 19712/19683, 42875/42768


[[Badness]]: 1.122 × 10<sup>-3</sup>
{{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 0 0 -4 3 -3 }}


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


= Semicanou =
{{Optimal ET sequence|legend=1| 24, …, 75e, 94, 99e, 118, 193, 212, 311, 740, 1051d }}


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): 2.45


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.
[[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


Still 80edo can be used as a tuning. Other options include 94edo, 118edo, and [[104edo]] in 104c val.  
=== 2.3.5.7.11.17 subgroup ===
Subgroup: 2.3.5.7.11.17


Comma list: 9801/9800, 14641/14580
Comma list: 595/594, 1156/1155, 19712/19683


POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 | 0 1 2 2 7 6 | 0 0 -4 3 -3 -2 }}


Mapping: [{{val| 2 0 0 -2 1 }}, {{val| 0 1 2 2 2 }}, {{val| 0 0 4 -3 1 }}]
Optimal tunings:  
* WE: ~2 = 1200.0485{{c}}, ~3/2 = 702.2481{{c}}, ~51/44 = 254.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2325{{c}}, ~51/44 = 254.6253{{c}}


{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg }}


Badness: 2.197 × 10<sup>-3</sup>
Badness (Sintel): 1.01


== 13-limit ==
=== 2.3.5.7.11.17.19 subgroup ===
Subgroup: 2.3.5.7.11.17.19


This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit.
Comma list: 595/594, 969/968, 1156/1155, 1216/1215


Comma list: 352/351, 9801/9800, 14641/14580
Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 -6 | 0 1 2 2 7 6 7 | 0 0 -4 3 -3 -2 -4 }}


POTE generators: ~3/2 = 702.8788, ~81/70 = 254.6664 or ~11/9 = 345.3336
Optimal tunings:  
* WE: ~2 = 1200.0444{{c}}, ~3/2 = 702.2569{{c}}, ~22/19 = 254.6305{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2421{{c}}, ~22/19 = 254.6224{{c}}


Mapping: [{{val| 2 0 0 -2 1 11 }}, {{val| 0 1 2 2 2 -1 }}, {{val| 0 0 4 -3 1 1 }}]
{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh }}


{{Val list|legend=1| 80, 94, 118, 174d, 198, 490f }}
Badness (Sintel): 0.641


Badness: 2.701 × 10<sup>-3</sup>
=== 2.3.5.7.11.17.19.23 subgroup ===
Subgroup: 2.3.5.7.11.17.19.23


== Gentsemicanou ==
Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155


This 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.
Mapping: {{mapping| 1 0 0 -1 -7 -5 -6 4 | 0 1 2 2 7 6 7 1 | 0 0 -4 3 -3 -2 -4 -5 }}


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.  
Optimal tunings:
* WE: ~2 = 1199.9778{{c}}, ~3/2 = 702.2794{{c}}, ~22/19 = 254.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2883{{c}}, ~22/19 = 254.6625{{c}}


Comma list: 351/350, 364/363, 11011/10935
{{Optimal ET sequence|legend=0| 24, 75e, 94, 99e, 118, 193, 217, 311 }}


POTE generators: ~3/2 = 702.7876, ~15/13 = 254.3411 or ~11/9 = 345.6789
Badness (Sintel): 0.703


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


{{Val list|legend=1| 80, 104c, 118f, 198f, 420cff }}
[[Subgroup]]: 2.3.5.7.11


Badness: 3.511 × 10<sup>-3</sup>
[[Comma list]]: 896/891, 472392/471625


= Canta =
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}
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.


Comma list: 896/891, 472392/471625
[[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 }}


POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


Mapping: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
[[Badness]] (Sintel): 5.43


{{Val list|legend=1| 75e, 80, 99e, 179e }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 4.523 × 10<sup>-3</sup>
Comma list: 352/351, 364/363, 472392/471625


== 13-limit ==
Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}
This 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.


Comma list: 351/350, 832/825, 13013/12960
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}}


POTE generators: ~3/2 = 703.8423, ~15/13 = 254.3605
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}


Mapping: [<1 0 0 -1 6 0|, <0 1 2 2 -2 3|, <0 0 4 -3 -3 5|]
Badness (Sintel): 4.47


{{Val list|legend=1| 75ef, 80, 99e, 104c, 179e, 184c, 203ce }}
== Semicathart ==
Semicathart (formerly ''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) }}.


Badness: 3.470 × 10<sup>-3</sup>
The other comma necessary to define it is 14641/14580, the [[semicathartisma]], 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.


== Gentcanta ==
[[Subgroup]]: 2.3.5.7.11
This 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.  


Comma list: 352/351, 364/363, 472392/471625
[[Comma list]]: 9801/9800, 14641/14580


POTE generators: ~3/2 = 703.8695, ~64/55 = 254.6321
{{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: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]
[[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 }}


{{Val list|legend=1| 75e, 80, 99ef, 179ef }}
{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}


Badness: 4.781 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.64


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