Catakleismic: Difference between revisions

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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.

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.

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.

See [[Kleismic family #Catakleismic]] for technical data.

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== Tunings ==

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup prime-optimized tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~6/5 = 316.9026{{c}}

| CSEE: ~6/5 = 316.8354{{c}}

| POEE: ~6/5 = 316.5718{{c}}

|-

! Tenney

| CTE: ~6/5 = 316.8865{{c}}

| CWE: ~6/5 = 316.7939{{c}}

| POTE: ~6/5 = 316.7410{{c}}

|-

! Benedetti, Wilson

| CBE: ~6/5 = 316.8827{{c}}

| CSBE: ~6/5 = 316.7927{{c}}

| POBE: ~6/5 = 316.7673{{c}}

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.11.13.19-subgroup prime-optimized tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~6/5 = 316.7941{{c}}

| CSEE: ~6/5 = 316.7860{{c}}

| POEE: ~6/5 = 316.8002{{c}}

|-

! Tenney

| CTE: ~6/5 = 316.8070{{c}}

| CWE: ~6/5 = 316.7816{{c}}

| POTE: ~6/5 = 316.7778{{c}}

|-

! Benedetti, Wilson

| CBE: ~6/5 = 316.8299{{c}}

| CSBE: ~6/5 = 316.7884{{c}}

| POBE: ~6/5 = 316.7625{{c}}

|}

=== Tuning spectrum ===

This tuning spectrum assumes undecimal catakleismic.

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|-

! Edo generator

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

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

! Generator (¢)

! Comments

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<nowiki />* Besides the octave

~~[[Category:Temperaments]]~~

[[Category:Catakleismic| ]]

[[Category:Rank-2 temperaments]]

[[Category:Kleismic family]]

[[Category:Marvel temperaments]]

[[Category:Ragismic microtemperaments]]

@@ Line 3: / Line 3: @@
 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]]&mdash;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.
-Various reasonable extensions exist for harmonic 11. These are ''undecimal catakleismic'', mapping 11 to &minus;21 generator steps, ''cataclysmic'', to +32 steps, ''catalytic'', to +51 steps, and cataleptic, to &minus;2 steps.
+Various reasonable extensions exist for harmonic 11. These are ''undecimal catakleismic'', mapping 11 to &minus;21 generator steps, ''cataclysmic'', to +32 steps, ''catalytic'', to +51 steps, and cataleptic, to &minus;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.
 See [[Kleismic family #Catakleismic]] for technical data.
@@ Line 213: / Line 213: @@
 == Tunings ==
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup prime-optimized tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Equilateral
+| CEE: ~6/5 = 316.9026{{c}}
+| CSEE: ~6/5 = 316.8354{{c}}
+| POEE: ~6/5 = 316.5718{{c}}
+|-
+! Tenney
+| CTE: ~6/5 = 316.8865{{c}}
+| CWE: ~6/5 = 316.7939{{c}}
+| POTE: ~6/5 = 316.7410{{c}}
+|-
+! Benedetti, <br>Wilson
+| CBE: ~6/5 = 316.8827{{c}}
+| CSBE: ~6/5 = 316.7927{{c}}
+| POBE: ~6/5 = 316.7673{{c}}
+|}
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.11.13.19-subgroup prime-optimized tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Equilateral
+| CEE: ~6/5 = 316.7941{{c}}
+| CSEE: ~6/5 = 316.7860{{c}}
+| POEE: ~6/5 = 316.8002{{c}}
+|-
+! Tenney
+| CTE: ~6/5 = 316.8070{{c}}
+| CWE: ~6/5 = 316.7816{{c}}
+| POTE: ~6/5 = 316.7778{{c}}
+|-
+! Benedetti, <br>Wilson
+| CBE: ~6/5 = 316.8299{{c}}
+| CSBE: ~6/5 = 316.7884{{c}}
+| POBE: ~6/5 = 316.7625{{c}}
+|}
 === Tuning spectrum ===
 This tuning spectrum assumes undecimal catakleismic.
@@ Line 218: / Line 270: @@
 |-
 ! Edo<br />generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
 ! Generator (¢)
 ! Comments
@@ Line 374: / Line 426: @@
 <nowiki />* Besides the octave
-[[Category:Temperaments]]
 [[Category:Catakleismic| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
 [[Category:Kleismic family]]
 [[Category:Marvel temperaments]]
 [[Category:Ragismic microtemperaments]]