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]] temperament, first extending hanson to include the [[harmonic]] [[13/1|13]] (called [[cata]]), and then to include [[7/1|7]].

{{Infobox regtemp

| Title = Catakleismic

| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.11.13

| Comma basis = [[225/224]], [[4375/4374]] (7-limit); [[169/168]], [[225/224]], [[325/324]] (2.3.5.7.13); [[169/168]], [[225/224]], [[325/324]], [[385/384]] (13-limit)

| Edo join 1 = 53 | Edo join 2 = 72

| Mapping = 1; 6 5 22 -21 14

| Generators = 6/5 | Generators tuning = 316.7 | Optimization method = CWE

| MOS scales = [[4L 7s]], [[4L 11s]], [[15L 4s]], [[15L 19s]]

| Odd limit 1 = 9 | Mistuning 1 = ??? | Complexity 1 = 34

| Odd limit 2 = 13-limit 21 | Mistuning 2 = ??? | Complexity 2 = 53

}}

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.

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.

See [[Kleismic family #Catakleismic]] for technical data. See [[Catakleismic extensions]] for a discussion on [[11-limit]] extensions.

== Interval chain ==

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Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 53 generator steps, so it vanishes in 53edo, but is tuned to the same size as the syntonic comma in 72edo. 125edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.

== Chords ==

== Chords and harmony ==

~~: ''Main article: [[~~Chords of catakleismic~~]] and [[~~Chords of tridecimal catakleismic~~]]''~~

{{See also| Chords of catakleismic | Chords of tridecimal catakleismic }}

== Scales ==

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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

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

|-

! rowspan="2" |

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

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

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

|-

! rowspan="2" |

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

|-

! Edo generator

! Edo generator

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

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

! Generator (¢)

! Comments

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|

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

== Music ==

; [[Budjarn Lambeth]]

* [https://youtu.be/6Kxq5uxu-c8 ''Improv in CWE Catakleismic11''] (2026)

~~[[Category:Temperaments]]~~

[[Category:Catakleismic| ]]

[[Category:Rank-2 temperaments]]

@@ Line 1: / Line 1: @@
-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]].
+{{Infobox regtemp
+| Title = Catakleismic
+| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.11.13
+| Comma basis = [[225/224]], [[4375/4374]] (7-limit); <br>[[169/168]], [[225/224]], [[325/324]] (2.3.5.7.13); <br>[[169/168]], [[225/224]], [[325/324]], [[385/384]] <br>(13-limit)
+| Edo join 1 = 53 | Edo join 2 = 72
+| Mapping = 1; 6 5 22 -21 14
+| Generators = 6/5 | Generators tuning = 316.7 | Optimization method = CWE
+| MOS scales = [[4L 7s]], [[4L 11s]], [[15L 4s]], [[15L 19s]]
+| Odd limit 1 = 9 | Mistuning 1 = ??? | Complexity 1 = 34
+| Odd limit 2 = 13-limit 21 | Mistuning 2 = ??? | Complexity 2 = 53
+}}
+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]]&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.
+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 &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.
+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.
+See [[Kleismic family #Catakleismic]] for technical data. See [[Catakleismic extensions]] for a discussion on [[11-limit]] extensions.
 == Interval chain ==
@@ Line 202: / Line 213: @@
 Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 53 generator steps, so it vanishes in 53edo, but is tuned to the same size as the syntonic comma in 72edo. 125edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.
-== Chords ==
+== Chords and harmony ==
-: ''Main article: [[Chords of catakleismic]] and [[Chords of tridecimal catakleismic]]''
+{{See also| Chords of catakleismic | Chords of tridecimal catakleismic }}
 == Scales ==
@@ Line 214: / Line 225: @@
 == Tunings ==
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based tunings
 |-
 ! rowspan="2" |
@@ Line 240: / Line 251: @@
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.11.13.19-subgroup prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.11.13.19-subgroup norm-based tunings
 |-
 ! rowspan="2" |
@@ Line 269: / 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
@@ Line 424: / Line 435: @@
 |
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
+== Music ==
+; [[Budjarn Lambeth]]
+* [https://youtu.be/6Kxq5uxu-c8 ''Improv in CWE Catakleismic11''] (2026)
-[[Category:Temperaments]]
 [[Category:Catakleismic| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]