Catakleismic: Difference between revisions

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The '''catakleismic''' temperament is one of the best [[7-limit]] ~~extensions~~ of [[hanson]], the 5-limit temperament tempering out the [[15625/15552|kleisma]] (15625/15552), though it is naturally viewed as a 2.3.5.7.13 temperament, first extending hanson to include the harmonic 13 (called ~~'''~~cata~~'''~~), and then to include 7.

The '''catakleismic''' [[regular temperament|temperament]] is one of the best [[7-limit]] [[extension]]s of [[hanson]], the [[5-limit]] temperament [[tempering out]] the [[15625/15552|kleisma]] (15625/15552), though it is naturally viewed as a 2.3.5.7.13-[[subgroup]] temperament, first extending hanson to include the [[harmonic]] [[13/1|13]] (called [[cata]]), and then to include [[7/1|7]].

In addition to the kleisma, catakleismic tempers the [[marvel comma]] (225/224), equating the interval of [[25/24]] (which is already equated to [[26/25]] and [[27/26]] in the 2.3.5.13 subgroup interpretation of kleismic) to [[28/27]]. This forces a flatter interpretation of 25/24, which is found 4 [[6/5]] generators up, and therefore a flatter interpretation of the generator, which confines reasonable catakleismic tunings to the portion of the kleismic tuning spectrum between [[19edo]] and [[34edo]]; in fact, catakleismic is the 19 & 34d temperament in the 7-limit. It can additionally be defined by tempering out the marvel comma and the [[ragisma]] (4375/4374), which finds [[7/6]] at the square of [[27/25]], which is found at the square of 25/24. Therefore the 7th harmonic appears 22 generators up the chain.

In addition to the kleisma, catakleismic tempers out the [[marvel comma]] (225/224), equating the interval of [[25/24]] (which is already equated to [[26/25]] and [[27/26]] in the 2.3.5.13 subgroup interpretation of kleismic) to [[28/27]]. This forces a flatter interpretation of 25/24, which is found four [[6/5]] generators up, and therefore a flatter interpretation of the generator, which confines reasonable catakleismic tunings to the portion of the kleismic tuning spectrum between [[19edo]] and [[34edo]]; in fact, catakleismic is the 19 & 34d temperament in the 7-limit. It can additionally be defined by tempering out the marvel comma and the [[ragisma]] (4375/4374), which finds [[7/6]] at the square of [[27/25]], which is found at the square of 25/24. Therefore the 7th harmonic appears 22 generators up the chain.

Various reasonable extensions exist for harmonic 11. These are ''undecimal catakleismic'', mapping 11 to -21 generator steps, ''cataclysmic'', to +32 steps, ''catalytic'', to +51 steps, and cataleptic, to -2 steps.

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! #

! Cents*

! Approximate ~~Ratios~~

! Approximate ratios

|-

| 0

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{| class="wikitable center-all left-4"

|-

! ~~EDO~~ generator

! Edo generator

! [[~~eigenmonzo~~|~~eigenmonzo~~ (unchanged-interval)]]

! [[Eigenmonzo|Eigenmonzo (unchanged-interval)]]

! ~~generator~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

|

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[[Category:Kleismic family]]

[[Category:Marvel temperaments]]

[[Category:Ragismic microtemperaments]]

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-The '''catakleismic''' temperament is one of the best [[7-limit]] extensions of [[hanson]], the 5-limit temperament tempering out the [[15625/15552|kleisma]] (15625/15552), though it is naturally viewed as a 2.3.5.7.13 temperament, first extending hanson to include the harmonic 13 (called '''cata'''), and then to include 7.
+The '''catakleismic''' [[regular temperament|temperament]] is one of the best [[7-limit]] [[extension]]s of [[hanson]], the [[5-limit]] temperament [[tempering out]] the [[15625/15552|kleisma]] (15625/15552), though it is naturally viewed as a 2.3.5.7.13-[[subgroup]] temperament, first extending hanson to include the [[harmonic]] [[13/1|13]] (called [[cata]]), and then to include [[7/1|7]].
-In addition to the kleisma, catakleismic tempers the [[marvel comma]] (225/224), equating the interval of [[25/24]] (which is already equated to [[26/25]] and [[27/26]] in the 2.3.5.13 subgroup interpretation of kleismic) to [[28/27]]. This forces a flatter interpretation of 25/24, which is found 4 [[6/5]] generators up, and therefore a flatter interpretation of the generator, which confines reasonable catakleismic tunings to the portion of the kleismic tuning spectrum between [[19edo]] and [[34edo]]; in fact, catakleismic is the 19 & 34d temperament in the 7-limit. It can additionally be defined by tempering out the marvel comma and the [[ragisma]] (4375/4374), which finds [[7/6]] at the square of [[27/25]], which is found at the square of 25/24. Therefore the 7th harmonic appears 22 generators up the chain.
+In addition to the kleisma, catakleismic tempers out the [[marvel comma]] (225/224), equating the interval of [[25/24]] (which is already equated to [[26/25]] and [[27/26]] in the 2.3.5.13 subgroup interpretation of kleismic) to [[28/27]]. This forces a flatter interpretation of 25/24, which is found four [[6/5]] generators up, and therefore a flatter interpretation of the generator, which confines reasonable catakleismic tunings to the portion of the kleismic tuning spectrum between [[19edo]] and [[34edo]]; in fact, catakleismic is the 19 & 34d temperament in the 7-limit. It can additionally be defined by tempering out the marvel comma and the [[ragisma]] (4375/4374), which finds [[7/6]] at the square of [[27/25]], which is found at the square of 25/24. Therefore the 7th harmonic appears 22 generators up the chain.
 Various reasonable extensions exist for harmonic 11. These are ''undecimal catakleismic'', mapping 11 to -21 generator steps, ''cataclysmic'', to +32 steps, ''catalytic'', to +51 steps, and cataleptic, to -2 steps.
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 ! #
 ! Cents*
-! Approximate Ratios
+! Approximate ratios
 |-
 | 0
@@ Line 109: / Line 109: @@
 {| class="wikitable center-all left-4"
 |-
-! EDO<br>generator
+! Edo<br>generator
-! [[eigenmonzo|eigenmonzo<br>(unchanged-interval)]]
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
-! generator<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 |
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 [[Category:Kleismic family]]
 [[Category:Marvel temperaments]]
+[[Category:Ragismic microtemperaments]]