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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 [[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.
[[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


{{Multival|legend=1|rank=3| 4 -3 -14 -4 }}
Lattice basis:
: 3/2 length = 0.8110, 81/70 length = 0.5135
: 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
: 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
 
Lattice basis:
: 3/2 length = 0.8110, 81/70 length = 0.5135
: Angle (3/2, 81/70) = 73.88 deg


{{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


= Synca =
== Undecimal cathartic ==
Synca, for symbiotic canou, adds the [[symbiotic comma]] to the comma list.  
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]].
 
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
[[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 }}
{{Optimal ET sequence|legend=1| 24, …, 75e, 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


= Semicanou =
=== 2.3.5.7.11.17 subgroup ===
Subgroup: 2.3.5.7.11.17


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.
Comma list: 595/594, 1156/1155, 19712/19683


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.
Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 | 0 1 2 2 7 6 | 0 0 -4 3 -3 -2 }}


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


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg }}


Comma list: 9801/9800, 14641/14580
Badness (Sintel): 1.01


POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
=== 2.3.5.7.11.17.19 subgroup ===
Subgroup: 2.3.5.7.11.17.19


Mapping: [{{val| 2 0 0 -2 1 }}, {{val| 0 1 2 2 2 }}, {{val| 0 0 4 -3 1 }}]
Comma list: 595/594, 969/968, 1156/1155, 1216/1215


{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
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 }}


Badness: 2.197 × 10<sup>-3</sup>
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}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 0.641


This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit.  
=== 2.3.5.7.11.17.19.23 subgroup ===
Subgroup: 2.3.5.7.11.17.19.23


Comma list: 352/351, 9801/9800, 14641/14580
Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155


POTE generators: ~3/2 = 702.8788, ~81/70 = 254.6664 or ~11/9 = 345.3336
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 }}


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


{{Val list|legend=1| 80, 94, 118, 174d, 198, 490f }}
{{Optimal ET sequence|legend=0| 24, 75e, 94, 99e, 118, 193, 217, 311 }}


Badness: 2.701 × 10<sup>-3</sup>
Badness (Sintel): 0.703


== Gentsemicanou ==
== 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.


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.  
[[Subgroup]]: 2.3.5.7.11


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.
[[Comma list]]: 896/891, 472392/471625


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}


Comma list: 351/350, 364/363, 11011/10935
[[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 = 702.7876, ~15/13 = 254.3411 or ~11/9 = 345.6789
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


Mapping: [{{val| 2 0 0 -2 1 0 }}, {{val| 0 1 2 2 2 3 }}, {{val| 0 0 4 -3 1 5 }}]
[[Badness]] (Sintel): 5.43


{{Val list|legend=1| 80, 104c, 118f, 198f, 420cff }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 3.511 × 10<sup>-3</sup>
Comma list: 352/351, 364/363, 472392/471625
 
= 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 75e & 80 & 99e, and 80edo makes the optimal.
 
Subgroup: 2.3.5.7
 
Comma list: 896/891, 472392/471625
 
POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133
 
Mapping: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
 
{{Val list|legend=1| 75e, 80, 99e, 179e }}
 
Badness: 4.523 × 10<sup>-3</sup>
 
== 13-limit ==
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.
 
Subgroup: 2.3.5.7.11


Comma list: 351/350, 832/825, 13013/12960
Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}


POTE generators: ~3/2 = 703.8423, ~15/13 = 254.3605
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}}


Mapping: [<1 0 0 -1 6 0|, <0 1 2 2 -2 3|, <0 0 4 -3 -3 5|]
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}


{{Val list|legend=1| 75ef, 80, 99e, 104c, 179e, 184c, 203ce }}
Badness (Sintel): 4.47


Badness: 3.470 × 10<sup>-3</sup>
== 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) }}.


== Gentcanta ==
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.  
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.  


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11


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

Latest revision as of 14:30, 10 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The canthartic family of rank-3 temperaments tempers out the cathartic comma (monzo: [4 -14 3 4⟩, ratio: 4802000/4782969), a 7-limit comma measuring about 6.9 cents.

Cathartic

Lattice for cathartic.
Ditto, but re-arranged into hemitwelfths.
See also: Cathartic scales

Cathartic (formerly canou) features a period of an octave and generators 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 intervals related to the 35th harmonic.

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.

9-, 14- and 19-note scales are highly characteristic for the temperament, with the abundance of 28/27 subminor seconds making it melodically active.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969

Mapping: [⟨1 0 0 -1], ⟨0 1 2 2], ⟨0 0 -4 3]]

mapping generators: ~2, ~3, ~81/70

Lattice basis:

3/2 length = 0.8110, 81/70 length = 0.5135
Angle (3/2, 81/70) = 73.88 deg

Optimal tunings:

  • WE: ~2 = 1199.9597 ¢, ~3/2 = 702.3492 ¢, ~81/70 = 254.6168 ¢
error map: ⟨-0.040 +0.354 -0.163 -0.317]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3455 ¢, ~81/70 = 254.6237 ¢
error map: ⟨0.000 +0.390 -0.118 -0.264]

Minimax tuning:

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
unchanged-interval (eigenmonzo) basis: 2.7/5

Optimal ET sequence: 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

Undecimal cathartic

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.

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⋅S352), 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

Comma list: 19712/19683, 42875/42768

Mapping: [⟨1 0 0 -1 -7], ⟨0 1 2 2 7], ⟨0 0 -4 3 -3]]

Optimal tunings:

  • WE: ~2 = 1200.0568 ¢, ~3/2 = 702.2009 ¢, ~81/70 = 254.6291 ¢
error map: ⟨+0.057 +0.303 -0.314 -0.480 +0.201]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1829 ¢, ~81/70 = 254.6186 ¢
error map: ⟨0.0000 +0.228 -0.422 -0.604 +0.107]

Optimal ET sequence: 24, …, 75e, 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

2.3.5.7.11.17 subgroup

Subgroup: 2.3.5.7.11.17

Comma list: 595/594, 1156/1155, 19712/19683

Subgroup-val mapping: [⟨1 0 0 -1 -7 -5], ⟨0 1 2 2 7 6], ⟨0 0 -4 3 -3 -2]]

Optimal tunings:

  • WE: ~2 = 1200.0485 ¢, ~3/2 = 702.2481 ¢, ~51/44 = 254.6343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2325 ¢, ~51/44 = 254.6253 ¢

Optimal ET sequence: 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg

Badness (Sintel): 1.01

2.3.5.7.11.17.19 subgroup

Subgroup: 2.3.5.7.11.17.19

Comma list: 595/594, 969/968, 1156/1155, 1216/1215

Subgroup-val mapping: [⟨1 0 0 -1 -7 -5 -6], ⟨0 1 2 2 7 6 7], ⟨0 0 -4 3 -3 -2 -4]]

Optimal tunings:

  • WE: ~2 = 1200.0444 ¢, ~3/2 = 702.2569 ¢, ~22/19 = 254.6305 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2421 ¢, ~22/19 = 254.6224 ¢

Optimal ET sequence: 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh

Badness (Sintel): 0.641

2.3.5.7.11.17.19.23 subgroup

Subgroup: 2.3.5.7.11.17.19.23

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

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]]

Optimal tunings:

  • WE: ~2 = 1199.9778 ¢, ~3/2 = 702.2794 ¢, ~22/19 = 254.6572 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2883 ¢, ~22/19 = 254.6625 ¢

Optimal ET sequence: 24, 75e, 94, 99e, 118, 193, 217, 311

Badness (Sintel): 0.703

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 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.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 472392/471625

Mapping: [⟨1 0 0 -1 6], ⟨0 1 2 2 -2], ⟨0 0 4 -3 -3]]

Optimal tunings:

  • WE: ~2 = 1199.0708 ¢, ~3/2 = 703.1969 ¢, ~64/55 = 254.4161 ¢
error map: ⟨-0.929 +0.313 +0.557 -0.113 +1.820]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.5249 ¢, ~64/55 = 254.5492 ¢
error map: ⟨0.000 +1.570 +2.539 +1.871 +5.280]

Optimal ET sequence: 75e, 80, 99e, 179e, 457bcddeeee

Badness (Sintel): 5.43

13-limit

Subgroup: 2.3.5.7.11.13

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

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 ¢, ~3/2 = 703.2884 ¢, ~64/55 = 254.4219 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8323 ¢, ~64/55 = 254.5887 ¢

Optimal ET sequence: 75e, 80, 99ef, 179ef, 462bccddeeeff

Badness (Sintel): 4.47

Semicathart

Semicathart (formerly 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. This takes advantage of the fact that 99/70 = (81/70)⋅(11/9).

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.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 14641/14580

Mapping: [⟨2 0 0 -2 1], ⟨0 1 2 2 2], ⟨0 0 -4 3 -1]]

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

Optimal tunings:

  • WE: ~99/70 = 600.0142 ¢, ~3/2 = 702.4017 ¢, ~81/70 = 254.6228 ¢
error map: ⟨+0.028 +0.475 +0.055 -0.126 -1.066]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.4048 ¢, ~81/70 = 254.6179 ¢
error map: ⟨0.0000 +0.450 +0.024 -0.163 -1.126]

Optimal ET sequence: 80, 94, 118, 198, 212, 292, 330e, 410

Badness (Sintel): 2.64

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