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Marvel temperaments: Difference between revisions

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This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]], [[Qintosec family|qintosec]] and [[Pelogic family #Mavila|mavila]].
This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]], [[Qintosec family|qintosec]], [[Pelogic family #Mavila|mavila]] and [[Cloudy clan #Decic|decic]].


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
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Badness: 0.0735
Badness: 0.0735
= Decic =
The ''decic'' temperament (10&amp;50, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit, as well as the [[15/14ths equal temperament|linus comma]], {{monzo|11 -10 -10 10}}. This temperament is supported by [[10edo|10]], [[50edo|50]], and [[60edo|60]] EDOs, and can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.
Subgroup: 2.3.5.7
[[Comma list]]: 225/224, 16807/16384
[[Mapping]]: [{{val| 10 0 39 28 }}, {{val| 0 1 -1 0 }}]
Mapping generators: ~15/14, ~3
{{Multival|legend=1| 10 -10 0 -39 -28 28 }}
[[POTE generator]]: ~3/2 = 698.696
[[Tuning ranges]]:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [693.129, 702.512]
* strict range: ~3/2 = [696.000, 700.000]
{{Val list|legend=1| 10, 50, 60, 110d, 170cdd }}
[[Badness]]: 0.089135
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 3087/3025
Mapping: [{{val| 10 0 39 28 3 }}, {{val| 0 1 -1 0 2 }}]
Mapping generators: ~15/14, ~3
POTE generator: ~3/2 = 696.791
Tuning ranges:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]
Vals: {{Val list| 10, 50, 160bcdd, 210bcddde, 260bbcddde, 310bbcdddde }}
Badness: 0.063900
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 196/195, 2200/2197
Mapping: [{{val| 10 0 39 28 3 37 }}, {{val| 0 1 -1 0 2 0 }}]
Mapping generators: ~14/13, ~3
POTE generator: ~3/2 = 696.993
Tuning ranges:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 702.512]
* strict range: ~3/2 = [696.000, 700.000]
Vals: {{Val list| 10, 50, 160bcdd, 210bcddde, 260bbcddde }}
Badness: 0.036880
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
Mapping: [{{val| 10 0 39 28 3 37 25 }}, {{val| 0 1 -1 0 2 0 1 }}]
Mapping generators: ~14/13, ~3
POTE generator: ~3/2 = 697.085
Tuning ranges:
* valid range: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* nice range: ~3/2 = [689.363, 704.955]
* strict range: ~3/2 = [696.000, 700.000]
Vals: {{Val list| 10, 50, 160bcddg, 210bcdddegg }}
Badness: 0.025064


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
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