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Canou family: Difference between revisions

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The 19712/19683 extension is now canon. I need to think about 17-limit semicanou so they are commented out now
 
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{{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.
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.


Line 20: Line 21:
: Angle (3/2, 81/70) = 73.88 deg
: Angle (3/2, 81/70) = 73.88 deg


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
Line 45: 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 = 1200.0000, ~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 }}
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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 = 1200.0000, ~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=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
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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 }}
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 = 1200.0000, ~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=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
Line 84: 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 tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
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 }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}


Badness (Smith): 1.00 × 10<sup>-3</sup>
Badness (Smith): 1.00 × 10<sup>-3</sup>
=== 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: {{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|legend=0| 94, 193f, 212gh, 217, 311 }}
Badness (Smith): 0.948 × 10<sup>-3</sup>


== Canta ==
== Canta ==
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{{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 = 1200.0000, ~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 ===
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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 = 1200.0000, ~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 ==
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: mapping generators: ~99/70, ~3, ~81/70
: mapping generators: ~99/70, ~3, ~81/70


[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000, ~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 ===
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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 = 600.0000, ~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
<!-- debatable canonicity
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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 }}
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 = 600.0000, ~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 ====
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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 }}
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 = 600.0000, ~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 }}
{{Optimal ET sequence|legend=0| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}


Badness: 2.177 × 10<sup>-3</sup>
Badness (Smith): 2.18 × 10<sup>-3</sup>
-->
-->
=== Semicanoumint ===
=== Semicanoumint ===
Line 188: 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 = 600.0000, ~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 205: 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): ~99/70 = 600.0000, ~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]]
Retrieved from "https://en.xen.wiki/w/Canou_family"