Catakleismic: Difference between revisions

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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|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 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]]—or further, between [[19edo]] and [[53edo]], as beyond 53, the [[countercata]] mapping of 7 is more reasonable, with the two meeting at 53edo~~. 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|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]]—or further, between [[19edo]] and [[53edo]], as beyond 53, the [[countercata]] mapping of 7 is more reasonable, with the two meeting at 53edo. 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. Undecimal catakleismic is shown in the tables below; additionally, tempering out [[286/285]] gives us an extension to prime 19 at -18 generator steps.

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

|-

! Edo<br />generator

! Edo<br>generator

! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*

! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*

! Generator (¢)

! Comments

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 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|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 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]]—or further, between [[19edo]] and [[53edo]], as beyond 53, the [[countercata]] mapping of 7 is more reasonable, with the two meeting at 53edo. 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|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]]—or further, between [[19edo]] and [[53edo]], as beyond 53, the [[countercata]] mapping of 7 is more reasonable, with the two meeting at 53edo. 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. Undecimal catakleismic is shown in the tables below; additionally, tempering out [[286/285]] gives us an extension to prime 19 at -18 generator steps.
@@ Line 280: / Line 280: @@
 {| class="wikitable center-all left-4"
 |-
-! Edo<br />generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
 ! Generator (¢)
 ! Comments