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.
{{Technical data page}}
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 ==
== Canou ==
{{Main| Canou temperament }}
{{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
[[Subgroup]]: 2.3.5.7
Line 22: Line 21:
: Angle (3/2, 81/70) = 73.88 deg
: Angle (3/2, 81/70) = 73.88 deg


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.3175, ~81/70 = 254.6220
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
: [[error map]]: {{val| 0.0000 +0.3625 -0.1667 -0.3249 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237
: error map: {{val| 0.0000 +0.3904 -0.1175 -0.2640 }}


[[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|eigenmonzo (unchanged-interval) 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|eigenmonzo (unchanged-interval) basis]]: 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 }}
{{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
[[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 }}


Badness: 0.548 × 10<sup>-3</sup>
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.  
 
== 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 71: Line 50:
{{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 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]]): ~2 = 1\1, ~3/2 = 702.2115, ~81/70 = 254.6215
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
: [[error map]]: {{val| 0.0000 +0.2565 -0.3768 -0.5383 +0.2980 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186
: error map: {{val| 0.0000 +0.2279 -0.4221 -0.6043 +0.1069 }}


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


[[Badness]]: 2.04 × 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
[[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 69:
Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 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): ~2 = 1\1, ~3/2 = 702.2075, ~81/70 = 254.6183
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
* CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222


{{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 ===
=== 17-limit ===
Line 99: Line 84:
Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}
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 tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
* CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066


{{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 ===
=== 19-limit ===
Line 110: Line 97:
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215
Comma list: 595/594, 833/832, 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: {{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:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
* CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983
 
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
 
Badness (Smith): 1.00 × 10<sup>-3</sup>
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2355, ~22/19 = 254.5930
Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155


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


Badness: 1.00 × 10<sup>-3</sup>
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222
* CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223
 
{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}
 
Badness (Smith): 0.948 × 10<sup>-3</sup>


== Canta ==
== Canta ==
Line 127: Line 131:
{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}
{{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]]s:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
: [[error map]]: {{val| 0.0000 +0.8543 +1.9537 -0.1940 +6.0769 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492
: error map: {{val| 0.0000 +1.5699 +2.5393 +1.8714 +5.2799 }}


{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


[[Badness]]: 4.523 × 10<sup>-3</sup>
[[Badness]] (Smith): 4.52 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 140: Line 148:
Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}
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 tunings:
* CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
* CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887


{{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef }}


Badness: 4.781 × 10<sup>-3</sup>
Badness (Smith): 4.78 × 10<sup>-3</sup>


== Semicanou ==
== 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
[[Subgroup]]: 2.3.5.7.11
Line 161: Line 169:
: mapping generators: ~99/70, ~3, ~81/70
: mapping generators: ~99/70, ~3, ~81/70


[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.4262, ~81/70 = 254.6191
[[Optimal tuning]]s:
* [[CTE]]: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
: [[error map]]: {{val| 0.0000 +0.4712 +0.0625 -0.1163 -1.0846 }}
* [[CWE]]: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179
: error map: {{val| 0.0000 +0.4498 +0.0245 -0.1627 -1.1262 }}


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


[[Badness]]: 2.197 × 10<sup>-3</sup>
[[Badness]] (Smith): 2.20 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 174: Line 186:
Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}
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 tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
* CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511


{{Optimal ET sequence|legend=1| 80f, 94, 118f, 198, 410 }}
{{Optimal ET sequence|legend=0| 80f, 94, 118f, 198, 410 }}


Badness: 2.974 × 10<sup>-3</sup>
Badness (Smith): 2.97 × 10<sup>-3</sup>


<!-- debatable canonicity
==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 187: Line 202:
Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}
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 tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663


{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}
{{Optimal ET sequence|legend=0| 94, 118f, 198g, 212g, 292, 410 }}


Badness: 2.421 × 10<sup>-3</sup>
Badness (Smith): 2.42 × 10<sup>-3</sup>


==== 19-limit ====
==== 19-limit ====
Line 200: Line 216:
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 }}
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 tunings:
 
* 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>
{{Optimal ET sequence|legend=0| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}


Badness (Smith): 2.18 × 10<sup>-3</sup>
-->
=== Semicanoumint ===
=== 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.  
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 232:
Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}
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 tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
* CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704


{{Optimal ET sequence|legend=1| 80, 94, 118, 174d, 198, 490f }}
{{Optimal ET sequence|legend=0| 80, 94, 118, 174d, 198, 490f }}


Badness: 2.701 × 10<sup>-3</sup>
Badness (Smith): 2.70 × 10<sup>-3</sup>


=== Semicanouwolf ===
=== Semicanouwolf ===
Line 232: Line 251:
Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}
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 tunings:
* CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
* CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396


{{Optimal ET sequence|legend=1| 80, 104c, 118f, 198f, 420cff }}
{{Optimal ET sequence|legend=0| 80, 104c, 118f, 198f, 420cff }}


Badness: 3.511 × 10<sup>-3</sup>
Badness (Smith): 3.51 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:35, 24 June 2025

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, 4802000/4782969 ([4 -14 3 4⟩), a 7-limit comma measuring about 6.9 cents.

Canou

Main article: Canou temperament

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. 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 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, relating it to 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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
error map: ⟨0.0000 +0.3625 -0.1667 -0.3249]
  • CWE: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237
error map: ⟨0.0000 +0.3904 -0.1175 -0.2640]

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 (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

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. From a commatic point of view, notice the other 11-limit comma, 42875/42768, is 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.

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
error map: ⟨0.0000 +0.2565 -0.3768 -0.5383 +0.2980]
  • CWE: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186
error map: ⟨0.0000 +0.2279 -0.4221 -0.6043 +0.1069]

Optimal ET sequence: 94, 99e, 118, 193, 212, 311, 740, 1051d

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

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
  • CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222

Optimal ET sequence: 94, 118f, 193f, 212, 217, 311, 740, 1051d

Badness (Smith): 2.56 × 10-3

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
  • CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066

Optimal ET sequence: 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg

Badness (Smith): 1.49 × 10-3

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
  • CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983

Optimal ET sequence: 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh

Badness (Smith): 1.00 × 10-3

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222
  • CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223

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

Badness (Smith): 0.948 × 10-3

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
error map: ⟨0.0000 +0.8543 +1.9537 -0.1940 +6.0769]
  • CWE: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492
error map: ⟨0.0000 +1.5699 +2.5393 +1.8714 +5.2799]

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

Badness (Smith): 4.52 × 10-3

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
  • CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887

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

Badness (Smith): 4.78 × 10-3

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.

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:

  • CTE: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
error map: ⟨0.0000 +0.4712 +0.0625 -0.1163 -1.0846]
  • CWE: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179
error map: ⟨0.0000 +0.4498 +0.0245 -0.1627 -1.1262]

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

Badness (Smith): 2.20 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 14641/14580

Mapping: [⟨2 0 0 -2 1 -11], ⟨0 1 2 2 2 5], ⟨0 0 -4 3 -1 6]]

Optimal tunings:

  • CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
  • CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511

Optimal ET sequence: 80f, 94, 118f, 198, 410

Badness (Smith): 2.97 × 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.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 9801/9800, 14641/14580

Mapping: [⟨2 0 0 -2 1 11], ⟨0 1 2 2 2 -1], ⟨0 0 -4 3 -1 -1]]

Optimal tunings:

  • CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
  • CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704

Optimal ET sequence: 80, 94, 118, 174d, 198, 490f

Badness (Smith): 2.70 × 10-3

Semicanouwolf

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.

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.

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 11011/10935

Mapping: [⟨2 0 0 -2 1 0], ⟨0 1 2 2 2 3], ⟨0 0 -4 3 -1 -5]]

Optimal tunings:

  • CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
  • CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396

Optimal ET sequence: 80, 104c, 118f, 198f, 420cff

Badness (Smith): 3.51 × 10-3

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