Cathartic family: Difference between revisions
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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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== 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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: error map: {{val| 0.0000 +0.228 -0.422 -0.604 +0.107 }} | : error map: {{val| 0.0000 +0.228 -0.422 -0.604 +0.107 }} | ||
{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }} | {{Optimal ET sequence|legend=1| 24, …, 75e, 94, 99e, 118, 193, 212, 311, 740, 1051d }} | ||
[[Badness]] (Sintel): 2.45 | [[Badness]] (Sintel): 2.45 | ||
| Line 61: | Line 65: | ||
[[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 | ||
=== | === 2.3.5.7.11.17 subgroup === | ||
Subgroup: 2.3.5.7.11.17 | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 595/594 | Comma list: 595/594, 1156/1155, 19712/19683 | ||
Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 | 0 1 2 2 7 6 | 0 0 -4 3 -3 -2 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.0485{{c}}, ~3/2 = 702.2481{{c}}, ~51/44 = 254.6343{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702. | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2325{{c}}, ~51/44 = 254.6253{{c}} | ||
{{Optimal ET sequence|legend=0| 94, | {{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.01 | ||
=== 19 | === 2.3.5.7.11.17.19 subgroup === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.17.19 | ||
Comma list: 595/594 | Comma list: 595/594, 969/968, 1156/1155, 1216/1215 | ||
Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 -6 | 0 1 2 2 7 6 7 | 0 0 -4 3 -3 -2 -4 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.0444{{c}}, ~3/2 = 702.2569{{c}}, ~22/19 = 254.6305{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702. | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2421{{c}}, ~22/19 = 254.6224{{c}} | ||
{{Optimal ET sequence|legend=0| 94, | {{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh }} | ||
Badness (Sintel): | Badness (Sintel): 0.641 | ||
=== 23 | === 2.3.5.7.11.17.19.23 subgroup === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.17.19.23 | ||
Comma list: 595/594, 760/759 | Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155 | ||
Mapping: {{mapping| 1 0 0 -1 -7 | Mapping: {{mapping| 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: | ||
* WE: ~2 = | * WE: ~2 = 1199.9778{{c}}, ~3/2 = 702.2794{{c}}, ~22/19 = 254.6572{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702. | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2883{{c}}, ~22/19 = 254.6625{{c}} | ||
{{Optimal ET sequence|legend=0| 94, | {{Optimal ET sequence|legend=0| 24, 75e, 94, 99e, 118, 193, 217, 311 }} | ||
Badness (Sintel): | Badness (Sintel): 0.703 | ||
== Canta == | == Canta == | ||