Canou family: Difference between revisions

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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]], 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]], 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.  
 
It 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.  


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


Optimal tuning ([[CTE]]): ~3/2 = 702.3175, ~81/70 = 254.6220
[[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 basis]]: 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 basis]]: 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


=== 2.3.5.7.17 subgroup ===
== Undecimal canou ==
Subgroup: 2.3.5.7.17
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]].  
 
Comma list: 1225/1224, 295936/295245
 
Mapping: [{{val| 1 0 0 -1 -5 }}, {{val| 0 1 2 2 6 }}, {{val| 0 0 -4 3 -2 }}]
 
Optimal tuning (CTE): ~3/2 = 702.3458, ~81/70 = 254.6233
 
Optimal GPV sequence: {{val list| 94, 99, 193, 217, 292, 311, 410, 1131, 1541b }}
 
Badness: 0.775 × 10<sup>-3</sup>
 
=== 2.3.5.7.17.19 subgroup ===
Subgroup: 2.3.5.7.17.19
 
Comma list: 1216/1215, 1225/1224, 1445/1444
 
Mapping: [{{val| 1 0 0 -1 -5 -6 }}, {{val| 0 1 2 2 6 7 }}, {{val| 0 0 -4 3 -2 -4 }}]
 
Optimal tuning (CTE): ~3/2 = 702.3233, ~81/70 = 254.6279


Optimal GPV sequence: {{val list| 94, 99, 118, 193, 217, 292h, 311, 410, 721 }}
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.  
 
Badness: 0.548 × 10<sup>-3</sup>
 
== Synca ==
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
Line 67: Line 47:
[[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 }}


Optimal tuning ([[CTE]]): ~3/2 = 702.2115, ~81/70 = 254.6215
[[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 82: 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 }}


Optimal tuning (CTE): ~3/2 = 702.2075, ~81/70 = 254.6183
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. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.  
Subgroup: 2.3.5.7.11.13.17


[[Subgroup]]: 2.3.5.7.11
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}}


Optimal tuning ([[CTE]]): ~3/2 = 702.8093, ~64/55 = 254.3378
{{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


=== 13-limit ===
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 364/363, 472392/471625
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: [{{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 = 1200.0624{{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 tuning (CTE): ~3/2 = 703.6228, ~64/55 = 254.3447
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}


Optimal GPV sequence: {{val list| 75e, 80, 99ef, 179ef }}
Badness (Sintel): 1.03


Badness: 4.781 × 10<sup>-3</sup>
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


== Semicanou ==
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155
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. 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.
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 }}


Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to any of them.  
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}}


[[Subgroup]]: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}


[[Comma list]]: 9801/9800, 14641/14580
Badness (Sintel): 1.09


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


Mapping generators: ~99/70, ~3, ~81/70
[[Subgroup]]: 2.3.5.7.11


Optimal tuning ([[CTE]]): ~3/2 = 702.4262, ~81/70 = 254.6191
[[Comma list]]: 896/891, 472392/471625


{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}


[[Badness]]: 2.197 × 10<sup>-3</sup>
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 14641/14580
[[Badness]] (Sintel): 5.43
 
Mapping: [{{val| 2 0 0 -2 1 -11 }}, {{val| 0 1 2 2 2 5 }}, {{val| 0 0 -4 3 -1 6 }}]
 
Optimal tuning (CTE): ~3/2 = 702.4802, ~81/70 = 254.6526
 
Optimal GPV sequence: {{val list| 80f, 94, 118f, 198, 410 }}
 
Badness: 2.974 × 10<sup>-3</sup>
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580
 
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 }}]
 
Optimal tuning (CTE): ~3/2 = 702.4415, ~81/70 = 254.6663
 
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 }}]
 
Optimal tuning (CTE): ~3/2 = 702.4030, ~81/70 = 254.6870
 
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.


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


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


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


Optimal tuning (CTE): ~3/2 = 702.5374, ~81/70 = 254.6819
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| 80, 94, 118, 174d, 198, 490f }}
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}


Badness: 2.701 × 10<sup>-3</sup>
Badness (Sintel): 4.47


=== Semicanouwolf ===
== Semicanou ==
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.  
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) }}.  


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.  
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: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11


Comma list: 351/350, 364/363, 11011/10935
[[Comma list]]: 9801/9800, 14641/14580


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


Optimal tuning (CTE): ~3/2 = 702.7417, ~15/13 = 254.3382
[[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]]

Latest revision as of 06:42, 9 June 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 canou family of rank-3 temperaments tempers out the canousma (monzo: [4 -14 3 4⟩, ratio: 4802000/4782969), a 7-limit comma measuring about 6.9 cents.

Canou

Main article: 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 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.

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 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⋅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. 311edo remains an excellent tuning in all the limits.

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: 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: [⟨1 0 0 -1 -7 -13], ⟨0 1 2 2 7 10], ⟨0 0 -4 3 -3 4]]

Optimal tunings:

  • WE: ~2 = 1200.0501 ¢, ~3/2 = 702.2100 ¢, ~81/70 = 254.6345 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1889 ¢, ~81/70 = 254.6222 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.2317 ¢, ~51/44 = 254.6224 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2055 ¢, ~51/44 = 254.6066 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.2377 ¢, ~22/19 = 254.6139 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2117 ¢, ~22/19 = 254.5983 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.2361 ¢, ~22/19 = 254.6225 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2359 ¢, ~22/19 = 254.6223 ¢

Optimal ET sequence: 94, 193f, 212gh, 217, 311

Badness (Sintel): 1.09

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

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 99/70~140/99. This takes advantage of the fact that 99/70 = (81/70)⋅(11/9).

The other comma necessary to define it is 14641/14580, the 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

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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