Cathartic family: Difference between revisions
→Undecimal cathartic: rework into a no-13 temp |
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== Cathartic == | == Cathartic == | ||
{{ | [[File:Lattice Orthocanousmic.png|thumb|Lattice for cathartic.]] | ||
[[File:Lattice Orthocanousmic Rearranged.png|thumb|Ditto, but re-arranged into hemitwelfths.]] | |||
{{See also| Cathartic scales }} | |||
Cathartic (formerly ''canou'') features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic. | Cathartic (formerly ''canou'') features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s 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]], associating it with 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]], associating it with the [[amicable]] temperament. | ||
9-, 14- and 19-note scales are highly characteristic for the temperament, with the abundance of [[28/27]] subminor seconds making it melodically active. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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[[Badness]] (Sintel): 4.95 | [[Badness]] (Sintel): 4.95 | ||
== Undecimal cathartic == | == Undecimal cathartic == | ||
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]]. | 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]]. | ||
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]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests 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 | 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]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests 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]]. These extensions add little additional error, and [[311edo]] remains an excellent tuning in all cases. | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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[[Badness]] (Sintel): 2.45 | [[Badness]] (Sintel): 2.45 | ||
=== 2.3.5.7.11.17 subgroup === | === 2.3.5.7.11.17 subgroup === | ||
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Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155 | Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155 | ||
{{Mapping|legend=0| 1 0 0 -1 -7 -5 -6 4 | 0 1 2 2 7 6 7 1 | 0 0 -4 3 -3 -2 -4 -5 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 352/351, 364/363, 472392/471625 | Comma list: 352/351, 364/363, 472392/471625 | ||
{{Mapping|legend=0| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
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[[Badness]] (Sintel): 2.64 | [[Badness]] (Sintel): 2.64 | ||
[[Category:Cathartic family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-3 temperaments]] | ||