Cathartic family: Difference between revisions
Switch to Sintel's badness, WE & CWE tunings. - all 13-limit semicanou extensions (not very notable). - redundant category |
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{{Technical data page}} | {{Technical data page}} | ||
The ''' | The '''canthartic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[cathartic comma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s. | ||
== | == Cathartic == | ||
{{Main| Canou }} | {{Main| Canou }} | ||
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. | ||
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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 | ||
== Undecimal | == 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]], which suggests the [[wilschisma]] should also be tempered out in the [[13-limit]]. | 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]]. | ||
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Badness (Sintel): 4.47 | Badness (Sintel): 4.47 | ||
== | == Semicathart == | ||
Semicathart (formerly ''semicanou'') adds 9801/9800, the [[kalisma]], to the comma list, and may be described as {{nowrap| 80 & 94 & 118 }}. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that {{nowrap| 99/70 {{=}} (81/70)⋅(11/9) }}. | |||
The other comma necessary to define it is 14641/14580, the [[ | The other comma necessary to define it is 14641/14580, the [[semicathartisma]], 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 | [[Subgroup]]: 2.3.5.7.11 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Cathartic family| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||