Revision as of 08:15, 18 April 2021

This page discusses some of the temperaments tempering out [-5 2 2 -1⟩ = 225/224, the marvel comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august, compton, sharp, escapade, qintosec, mavila and decic.

Since (5/4)² = 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.

The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.

Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

Negri

Main article: Negri

Negri tempers out the negri comma in the 5-limit, 49/48 and 225/224 in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 to the comma list; this will be discussed below under the title of negra.

5-limit

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

Wedgie: ⟨⟨ 4 -3 -14 ]]

POTE generator: ~16/15 = 125.7549

@@ Line 1: / Line 1: @@
-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.
@@ Line 947: / Line 947: @@
 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]]

Marvel temperaments: Difference between revisions

Revision as of 08:15, 18 April 2021

Negri

5-limit

7-limit

Negra

11-limit

13-limit

Negril

13-limit

Negric

13-limit

Negroni

13-limit

Wilsec

13-limit

17-limit

19-limit

Passive

Wizard

11-limit

Lizard

17-limit

19-limit

Gizzard

17-limit

19-limit

Mage

Triton

11-limit

Tritonic

11-limit

13-limit

Tritoni

Merman

11-limit

13-limit

Septimin

11-limit

13-limit

Slender

11-limit

13-limit

Marvo

11-limit

13-limit

Marvolo

11-limit

13-limit

Amavil

5-limit (mabila)

7-limit

11-limit

13-limit

Enneaportent

11-limit

13-limit

Submajor

5-limit

7-limit

11-limit

13-limit

Interpental

13-limit

Alphorn

11-limit

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