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 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 a [[980/729]] at about 510 cents, an audibly off perfect fourth. Three of them make a [[14/9]]; four of them make a [[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. | = 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 a [[980/729]] at about 510 cents, an audibly off perfect fourth. Three of them make a [[14/9]]; four of them make a [[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. | |||
Decent amount of harmonic resources are provided by a simple 9-note scale. [[Flora Canou]] commented: | Decent amount of harmonic resources are provided by a simple 9-note scale. [[Flora Canou]] commented: | ||
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Badness: 1.122 × 10<sup>-3</sup> | Badness: 1.122 × 10<sup>-3</sup> | ||
= 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 == | |||
This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit. | This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit. | ||
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Badness: 2.701 × 10<sup>-3</sup> | Badness: 2.701 × 10<sup>-3</sup> | ||
== 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 = | |||
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 == | |||
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 == | |||
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. | ||
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[[Category:Planar temperament]] | [[Category:Planar temperament]] | ||
[[Category:Canou]] | [[Category:Canou]] | ||
[[Category:Rank 3]] | |||