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The '''catakleismic''' temperament is one of the best extensions of hanson, the 5-limit temperament tempering out the [[kleisma]].  
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]].  


Catakleismic 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. 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.  
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. 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>*</nowiki> in 2.3.5.7.13 POTE tuning
<nowiki/>* In 2.3.5.7.13-subgroup CWE tuning
 
=== As a detemperament of 19et ===
[[File: Catakleismic 19et Detempering.png|thumb|Catakleismic as a 72-tone 19et detempering]]
 
Catakleismic is naturally considered as a [[detemperament]] of the [[19edo|19 equal temperament]]. The diagram on the right shows a 72-tone detempered scale, with a generator range of -35 to +36. 72 is the largest number of tones for a mos where intervals in the 19 categories do not overlap. Each category is divided into three or four qualities separated by 19 generator steps, which represent the syntonic comma. Combining this division with the minor and major diatonic qualities of the 19 equal temperament, catakleismic gives us seven or eight qualities for each diatonic category in addition to the four qualities for the categories corresponding to [[interseptimal interval]]s.
 
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 ==
Line 103: Line 212:
* [[Catakleismic34]]
* [[Catakleismic34]]


== Tuning spectrum ==
== 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}}
|}


{| class="wikitable center-all"
=== Tuning spectrum ===
This tuning spectrum assumes undecimal catakleismic.
{| class="wikitable center-all left-4"
|-
|-
! ET<br>generator
! Edo<br />generator
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
! minor<br>third (¢)
! Generator (¢)
! comments
! Comments
|-
|-
|  
|  
| 6/5
| 5/3
| 315.641
| 315.641
|  
|  
|-
|-
| 5\19
| [[19edo|5\19]]
|  
|  
| 315.789
| 315.789
|  
| Lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 14/13
| 13/7
| 316.037
| 316.037
|  
|  
Line 147: Line 309:
|  
|  
|-
|-
| 19\72
| [[72edo|19\72]]
|  
|  
| 316.667
| 316.667
Line 158: Line 320:
|-
|-
|  
|  
| 14/11
| 11/7
| 316.686
| 316.686
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 316.690
| 316.690
|  
|  
Line 177: Line 339:
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
| 52\197
| [[197edo|52\197]]
|  
|  
| 316.751
| 316.751
|  
| 197ef val
|-
|-
|  
|  
| 8/7
| 7/4
| 316.765
| 316.765
| 7-, 9-, 13- and 15-odd-limit minimax
| 7-, 9-, 13- and 15-odd-limit minimax
Line 193: Line 355:
|-
|-
|  
|  
| 10/9
| 9/5
| 316.799
| 316.799
|  
| 1/7-kleisma
|-
|-
| 33\125
| [[125edo|33\125]]
|  
|  
| 316.800
| 316.800
|  
| 125f val
|-
|-
|  
|  
Line 207: Line 369:
|  
|  
|-
|-
| 14\53
| [[53edo|14\53]]
|  
|  
| 316.981
| 316.981
Line 213: Line 375:
|-
|-
|  
|  
| 4/3
| 3/2
| 316.993
| 316.993
| 5-odd-limit minimax
| 5-odd-limit minimax, 1/6-kleisma
|-
|-
|  
|  
| 16/15
| 15/8
| 317.115
| 317.115
|  
| 2/11-kleisma
|-
|-
|  
|  
Line 228: Line 390:
|-
|-
|  
|  
| 16/13
| 13/8
| 317.181
| 317.181
|  
|  
|-
|-
| 23\87
| [[87edo|23\87]]
|  
|  
| 317.241
| 317.241
|  
| 87de val
|-
|-
|  
|  
| 5/4
| 5/4
| 317.263
| 317.263
|  
| 1/5-kleisma
|-
|-
|  
|  
Line 252: Line 414:
|  
|  
|-
|-
| 9\34
| [[34edo|9\34]]
|  
|  
| 317.647
| 317.647
|  
| 34de val, upper bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 18/13
| 13/9
| 318.309
| 318.309
|  
|  
|}
|}
<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]]
Retrieved from "https://en.xen.wiki/w/Catakleismic"