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


== Interval chain ==
== Interval chain ==
In the following table, harmonics 1–21 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
! rowspan="2" | #
! Cents*
! rowspan="2" | Cents*
! Approximate ratios
! colspan="2" | Approximate ratios
|-
! 2.3.5.7.13 subgroup
! add-11 add-19 extension
|-
|-
| 0
| 0
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|-
|-
| 1
| 1
| 316.7
| 316.8
| 6/5
| 6/5
|
|-
|-
| 2
| 2
| 633.5
| 633.6
| 13/9
| 13/9
|
|-
|-
| 3
| 3
| 950.2
| 950.4
| 26/15
| 26/15
| 19/11
|-
|-
| 4
| 4
| 67.0
| 67.2
| 25/24, 26/25, 27/26, 28/27
| 25/24, 26/25, 27/26, 28/27
|
|-
|-
| 5
| 5
| 383.7
| 384.0
| '''5/4'''
| '''5/4'''
|
|-
|-
| 6
| 6
| 700.4
| 700.8
| '''3/2'''
| '''3/2'''
|
|-
|-
| 7
| 7
| 1017.2
| 1017.6
| 9/5
| 9/5
|
|-
|-
| 8
| 8
| 133.9
| 134.4
| 13/12, 14/13, 27/25
| 13/12, 14/13, 27/25
|
|-
|-
| 9
| 9
| 450.7
| 451.1
| 13/10
| 13/10
|
|-
|-
| 10
| 10
| 767.4
| 767.9
| 14/9
| 14/9
|
|-
|-
| 11
| 11
| 1084.1
| 1084.7
| 15/8, 28/15
| 15/8, 28/15
|
|-
|-
| 12
| 12
| 200.9
| 201.5
| 9/8
| '''9/8'''
|
|-
|-
| 13
| 13
| 517.6
| 518.3
| 27/20
| 27/20
|
|-
|-
| 14
| 14
| 834.4
| 835.1
| '''13/8''', 21/13
| '''13/8''', 21/13
|
|-
|-
| 15
| 15
| 1151.1
| 1151.9
| 35/18
| 35/18, 39/20
| 64/33
|-
|-
| 16
| 16
| 267.9
| 268.7
| 7/6
| 7/6
|
|-
|-
| 17
| 17
| 584.6
| 585.5
| 7/5
| 7/5
|
|-
|-
| 18
| 18
| 901.3
| 902.3
| 27/16
| 27/16
| '''32/19'''
|-
|-
| 19
| 19
| 18.1
| 19.1
| 81/80
| 81/80, 91/90, 105/104
| 77/76, 78/77, 96/95, <br>100/99, 133/132, 144/143
|-
| 20
| 335.9
| 39/32
| 40/33
|-
| 21
| 652.7
| 35/24
| '''16/11'''
|-
| 22
| 969.5
| '''7/4'''
|
|-
| 23
| 86.3
| 21/20
| 20/19
|-
| 24
| 403.1
| 63/50
| 24/19
|-
| 25
| 719.8
| 91/60
| 50/33
|-
| 26
| 1036.6
| 91/50
| 20/11
|-
| 27
| 153.4
| 35/32
| 12/11
|-
| 28
| 470.2
| '''21/16'''
|
|-
| 29
| 787.0
| 63/40
| 30/19
|-
| 30
| 1103.8
| 91/48
| 36/19
|-
| 31
| 220.6
| 91/80
| 25/22
|-
| 32
| 537.4
| 117/80
| 15/11, 26/19
|-
| 33
| 854.2
| 49/30
| 18/11
|-
| 34
| 1171.0
| 63/32
| 49/25, 65/33
|}
|}
<nowiki/>* In 2.3.5.7.13 POTE tuning
<nowiki/>* In 2.3.5.7.13-subgroup CWE tuning


=== As a detemperament of 19et ===
=== As a detemperament of 19et ===
Line 113: Line 213:


== Tunings ==
== 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 ===
=== Tuning spectrum ===
This tuning spectrum assumes undecimal catakleismic.  
This tuning spectrum assumes undecimal catakleismic.  
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|-
|-
! Edo<br />generator
! Edo<br />generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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<nowiki />* Besides the octave
<nowiki />* Besides the octave


[[Category:Temperaments]]
[[Category:Catakleismic| ]] <!-- main article -->
[[Category:Catakleismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Kleismic family]]
[[Category:Kleismic family]]
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
Retrieved from "https://en.xen.wiki/w/Catakleismic"