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. | 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. | ||
== Canou | == Canou == | ||
{{Main| 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. 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. | |||
For tunings, a basic option would be [[99edo]]. Others such as [[80edo]], [[94edo]], and [[118edo]] are possible; [[19edo]] (perferably with stretched octaves) also provides a good trivial case, whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament. | For tunings, a basic option would be [[99edo]]. Others such as [[80edo]], [[94edo]], and [[118edo]] are possible; [[19edo]] (perferably with stretched octaves) also provides a good trivial case, whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament. | ||
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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]]: [{{val| 1 0 0 -1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 -4 3 }}] | ||
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[[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]]s: 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]]s: 2, 7/5 | ||
Lattice basis: | Lattice basis: | ||
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[[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 | ||
== Synca | == Synca == | ||
Synca, for symbiotic canou, adds the [[symbiotic comma]] to the comma list. | Synca, for symbiotic canou, adds the [[symbiotic comma]] to the comma list. | ||
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[[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 | ||
== 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. 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. Note that 99/70 = (81/70)×(11/9), this extension is more than natural. | ||
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Badness: 2.197 × 10<sup>-3</sup> | Badness: 2.197 × 10<sup>-3</sup> | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 2.701 × 10<sup>-3</sup> | Badness: 2.701 × 10<sup>-3</sup> | ||
=== Gentsemicanou | === Gentsemicanou === | ||
This 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. | This 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. | ||
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Badness: 3.511 × 10<sup>-3</sup> | Badness: 3.511 × 10<sup>-3</sup> | ||
== Canta | == 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. | 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. | ||
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Badness: 4.523 × 10<sup>-3</sup> | Badness: 4.523 × 10<sup>-3</sup> | ||
=== 13-limit | === 13-limit === | ||
This adds [[351/350]], the ratwolfsma, 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. Again 80edo makes the optimal. | This adds [[351/350]], the ratwolfsma, 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. Again 80edo makes the optimal. | ||
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Badness: 3.470 × 10<sup>-3</sup> | Badness: 3.470 × 10<sup>-3</sup> | ||
=== Gentcanta | === Gentcanta === | ||
This adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal. | This adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal. | ||