Canou family: Difference between revisions

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The '''canou family''' of [[~~Rank~~-3 temperament|rank-3]] [[temperament]]s [[~~Tempering~~ out|tempers out]] the [[canousma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}, a 7-limit comma measuring about 6.9 [[cent]]s.

The '''canou family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperament]]s [[tempering out|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.

== Canou ==

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~~. Two of them 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 interval]]s 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. Three make [[14/9]]; four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, making two distinct [[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]], relating it to 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

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

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.3175, ~81/70 = 254.6220

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220

[[Minimax tuning]]:

* [[7-odd-limit]]: 3 +c/14, 5 and 7 just

: [[~~Eigenmonzo~~ basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7

: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7

* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just

: [[~~Eigenmonzo~~ basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5

: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5

{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}

[[Badness]]: 1.122 × 10-3

[[Badness]] (Smith): 1.122 × 10-3

[[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: {{mapping| 1 0 0 -1 -5 | 0 1 2 2 6 | 0 0 -4 3 -2 }}~~

~~Optimal tuning (CTE~~): ~~~2 = 1\1~~, ~3/~~2 = 702~~.~~3458~~, ~81/70 ~~= 254.6233~~

~~{{Optimal ET sequence|legend=1| 94, 99, 193, 217, 292, 311, 410, 1131, 1541b }}~~

~~Badness: 0.775 × 10-3<~~/~~sup>~~

~~=== 2.3.5.7.17.19 subgroup ===~~

~~Subgroup: 2.3.5.7.17.~~19

~~Comma list: 1216/1215~~, ~~1225/1224~~, ~~1445~~/~~1444~~

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

~~Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3233, ~81/70 = 254~~.~~6279~~

~~{{Optimal ET sequence|legend=1| 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]]. From a commatic point of view, notice the other 11-limit comma, [[42875/42768]], is {{nowrap| S34 × S352 }}, suggesting 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~~ × 10-3

~~== Synca ==~~

~~Synca~~, ~~for symbiotic canou~~, ~~adds the~~ [[~~symbiotic comma~~]] and the [[~~wilschisma~~]] to the ~~comma list~~.

[[Subgroup]]: 2.3.5.7.11

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.2115, ~81/70 = 254.6215

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215

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

[[Badness]]: 2.04 × 10-3

[[Badness]] (Smith): 2.04 × 10-3

[[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

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Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 0 0 -4 3 -3 4 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.2075, ~81/70 = 254.6183

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183

{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}

{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}

Badness: 2.56 × 10-3

Badness (Smith): 2.56 × 10-3

=== 17-limit ===

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Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.2296, ~51/44 = 254.6012

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012

{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}

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

Badness: 1.49 × 10-3

Badness (Smith): 1.49 × 10-3

=== 19-limit ===

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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 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.2355, ~22/19 = 254.5930

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930

{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}

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

Badness: 1.00 × 10-3

Badness (Smith): 1.00 × 10-3

== Canta ==

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.8093, ~64/55 = 254.3378

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378

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Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 703.6228, ~64/55 = 254.3447

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447

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== 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. Note that 99/70 = (81/70)(11/9), this extension is more than natural.

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 {{nowrap| 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.

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.

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

[[Subgroup]]: 2.3.5.7.11

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

[[Optimal tuning]] ([[CTE]]): ~99/70 = ~~1\2~~, ~3/2 = 702.4262, ~81/70 = 254.6191

[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191

{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}

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Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}

Optimal tuning (CTE): ~99/70 = ~~1\2~~, ~3/2 = 702.4802, ~81/70 = 254.6526

Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526

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Badness: 2.974 × 10-3

<!-- debatable canonicity

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}

Optimal tuning (CTE): ~99/70 = ~~1\2~~, ~3/2 = 702.4415, ~81/70 = 254.6663

Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663

{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}

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Mapping: {{mapping| 2 0 0 -2 1 -11 -10 -12 | 0 1 2 2 2 5 6 7 | 0 0 -4 3 -1 6 -2 -4 }}

Optimal tuning (CTE): ~99/70 = ~~1\2~~, ~3/2 = 702.4030, ~81/70 = 254.6870

Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870

{{Optimal ET sequence|legend=1| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}

Badness: 2.177 × 10-3

-->

=== 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.

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Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}

Optimal tuning (CTE): ~99/70 = ~~1\2~~, ~3/2 = 702.5374, ~81/70 = 254.6819

Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819

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Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}

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

Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382

@@ Line 1: / Line 1: @@
-The '''canou family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[canousma]], 4802000/4782969 = {{monzo| 4 -14 3 4 }}, a 7-limit comma measuring about 6.9 [[cent]]s.
+The '''canou family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperament]]s [[tempering out|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.
 == Canou ==
 {{Main| Canou temperament }}
-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. Two of them 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 interval]]s 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. Three make [[14/9]]; four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, making two distinct [[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]], relating it to 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
@@ Line 22: / Line 20: @@
 : Angle (3/2, 81/70) = 73.88 deg
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.3175, ~81/70 = 254.6220
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
 [[Minimax tuning]]:
 * [[7-odd-limit]]: 3 +c/14, 5 and 7 just
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7
+: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7
 * [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
 {{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]] (Smith): 1.122 × 10<sup>-3</sup>
 [[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: {{mapping| 1 0 0 -1 -5 | 0 1 2 2 6 | 0 0 -4 3 -2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3458, ~81/70 = 254.6233
-{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 0 -1 -5 -6 | 0 1 2 2 6 7 | 0 0 -4 3 -2 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3233, ~81/70 = 254.6279
-{{Optimal ET sequence|legend=1| 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]]. From a commatic point of view, notice the other 11-limit comma, [[42875/42768]], is {{nowrap| S34 × S35<sup>2</sup> }}, suggesting 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
@@ Line 71: / Line 45: @@
 {{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 0 0 -4 3 -3 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2115, ~81/70 = 254.6215
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
 {{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
-[[Badness]]: 2.04 × 10<sup>-3</sup>
+[[Badness]] (Smith): 2.04 × 10<sup>-3</sup>
 [[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 86: / Line 60: @@
 Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 0 0 -4 3 -3 4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2075, ~81/70 = 254.6183
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
-{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
+{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
-Badness: 2.56 × 10<sup>-3</sup>
+Badness (Smith): 2.56 × 10<sup>-3</sup>
 === 17-limit ===
@@ Line 99: / Line 73: @@
 Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2296, ~51/44 = 254.6012
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
-{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
+{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
-Badness: 1.49 × 10<sup>-3</sup>
+Badness (Smith): 1.49 × 10<sup>-3</sup>
 === 19-limit ===
@@ Line 112: / Line 86: @@
 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 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2355, ~22/19 = 254.5930
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
-{{Optimal ET sequence|legend=1| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
+{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
-Badness: 1.00 × 10<sup>-3</sup>
+Badness (Smith): 1.00 × 10<sup>-3</sup>
 == Canta ==
@@ Line 127: / Line 101: @@
 {{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.8093, ~64/55 = 254.3378
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
 {{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}
@@ Line 140: / Line 114: @@
 Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.6228, ~64/55 = 254.3447
+Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
 {{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}
@@ Line 147: / Line 121: @@
 == 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. Note that 99/70 = (81/70)(11/9), this extension is more than natural.
+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 {{nowrap| 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.
+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.
-Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to all of them.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 161: / Line 133: @@
 : mapping generators: ~99/70, ~3, ~81/70
-[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.4262, ~81/70 = 254.6191
+[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
 {{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
@@ Line 174: / Line 146: @@
 Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}
-Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.4802, ~81/70 = 254.6526
+Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
 {{Optimal ET sequence|legend=1| 80f, 94, 118f, 198, 410 }}
@@ Line 180: / Line 152: @@
 Badness: 2.974 × 10<sup>-3</sup>
+<!-- debatable canonicity
 ==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 187: / Line 160: @@
 Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}
-Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.4415, ~81/70 = 254.6663
+Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663
 {{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}
@@ Line 200: / Line 173: @@
 Mapping: {{mapping| 2 0 0 -2 1 -11 -10 -12 | 0 1 2 2 2 5 6 7 | 0 0 -4 3 -1 6 -2 -4 }}
-Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.4030, ~81/70 = 254.6870
+Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870
 {{Optimal ET sequence|legend=1| 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.
@@ Line 215: / Line 188: @@
 Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}
-Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.5374, ~81/70 = 254.6819
+Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
 {{Optimal ET sequence|legend=1| 80, 94, 118, 174d, 198, 490f }}
@@ Line 232: / Line 205: @@
 Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}
-Optimal tuning (CTE): ~3/2 = 702.7417, ~15/13 = 254.3382
+Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
 {{Optimal ET sequence|legend=1| 80, 104c, 118f, 198f, 420cff }}