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


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.  
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.  


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.  
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
[[Subgroup]]: 2.3.5.7
Line 36: Line 40:
[[Badness]] (Sintel): 4.95
[[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]].


== Undecimal canou ==
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.  
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
[[Subgroup]]: 2.3.5.7.11
Line 55: Line 57:
: error map: {{val| 0.0000 +0.228 -0.422 -0.604 +0.107 }}
: 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 }}
{{Optimal ET sequence|legend=1| 24, …, 75e, 94, 99e, 118, 193, 212, 311, 740, 1051d }}


[[Badness]] (Sintel): 2.45
[[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
=== 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
Comma list: 595/594, 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 }}
{{Mapping|legend=2| 1 0 0 -1 -7 -5 | 0 1 2 2 7 6 | 0 0 -4 3 -3 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0630{{c}}, ~3/2 = 702.2317{{c}}, ~51/44 = 254.6224{{c}}
* WE: ~2 = 1200.0485{{c}}, ~3/2 = 702.2481{{c}}, ~51/44 = 254.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2055{{c}}, ~51/44 = 254.6066{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2325{{c}}, ~51/44 = 254.6253{{c}}


{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg }}


Badness (Sintel): 1.41
Badness (Sintel): 1.01


=== 19-limit ===
=== 2.3.5.7.11.17.19 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.17.19


Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215
Comma list: 595/594, 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|legend=2| 1 0 0 -1 -7 -5 -6 | 0 1 2 2 7 6 7 | 0 0 -4 3 -3 -2 -4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0624{{c}}, ~3/2 = 702.2377{{c}}, ~22/19 = 254.6139{{c}}
* WE: ~2 = 1200.0444{{c}}, ~3/2 = 702.2569{{c}}, ~22/19 = 254.6305{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2117{{c}}, ~22/19 = 254.5983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2421{{c}}, ~22/19 = 254.6224{{c}}


{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh }}


Badness (Sintel): 1.03
Badness (Sintel): 0.641


=== 23-limit ===
=== 2.3.5.7.11.17.19.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.17.19.23


Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155
Comma list: 595/594, 760/759, 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 }}
{{Mapping|legend=0| 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:  
Optimal tunings:  
* WE: ~2 = 1200.0004{{c}}, ~3/2 = 702.2361{{c}}, ~22/19 = 254.6225{{c}}
* WE: ~2 = 1199.9778{{c}}, ~3/2 = 702.2794{{c}}, ~22/19 = 254.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~22/19 = 254.6223{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2883{{c}}, ~22/19 = 254.6625{{c}}


{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}
{{Optimal ET sequence|legend=0| 24, 75e, 94, 99e, 118, 193, 217, 311 }}


Badness (Sintel): 1.09
Badness (Sintel): 0.703


== Canta ==
== Canta ==
Line 145: Line 130:
Comma list: 352/351, 364/363, 472392/471625
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 }}
{{Mapping|legend=0| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 155: Line 140:
Badness (Sintel): 4.47
Badness (Sintel): 4.47


== Semicanou ==
== Semicathart ==
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) }}.  
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) }}.  


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.  
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
[[Subgroup]]: 2.3.5.7.11
Line 177: Line 162:
[[Badness]] (Sintel): 2.64
[[Badness]] (Sintel): 2.64


[[Category:Cathartic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Canou family| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]

Latest revision as of 17:44, 30 September 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

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

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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