Catakleismic: Difference between revisions

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m Cata is gonna get its own page
"19 & 34d" is contentious (it might give the impression that the optimum is around 34d or 53)
 
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The '''catakleismic''' temperament is one of the best extensions of hanson, the 5-limit temperament tempering out the [[kleisma]].  
{{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]].  


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|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.


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


== 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 and harmony ==
: ''Main article: [[Chords of catakleismic]] and [[Chords of tridecimal catakleismic]]''
{{See also| Chords of catakleismic | Chords of tridecimal catakleismic }}


== Scales ==
== Scales ==
Line 103: Line 223:
* [[Catakleismic34]]
* [[Catakleismic34]]


== Tuning spectrum ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based 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 norm-based 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.
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! EDO<br>generator
! Edo<br>generator
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! generator<br>(¢)
! Generator (¢)
! comments
! Comments
|-
|-
|  
|  
| 6/5
| 5/3
| 315.641
| 315.641
|  
|  
|-
|-
| 5\19
| [[19edo|5\19]]
|  
|  
| 315.789
| 315.789
Line 122: Line 296:
|-
|-
|  
|  
| 14/13
| 13/7
| 316.037
| 316.037
|  
|  
Line 146: Line 320:
|  
|  
|-
|-
| 19\72
| [[72edo|19\72]]
|  
|  
| 316.667
| 316.667
Line 157: Line 331:
|-
|-
|  
|  
| 14/11
| 11/7
| 316.686
| 316.686
|  
|  
|-
|-
|  
|  
| 12/11
| 11/6
| 316.690
| 316.690
|  
|  
Line 176: Line 350:
| 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 192: Line 366:
|-
|-
|  
|  
| 10/9
| 9/5
| 316.799
| 316.799
|  
| 1/7-kleisma
|-
|-
| 33\125
| [[125edo|33\125]]
|  
|  
| 316.800
| 316.800
|  
| 125f val
|-
|-
|  
|  
Line 206: Line 380:
|  
|  
|-
|-
| 14\53
| [[53edo|14\53]]
|  
|  
| 316.981
| 316.981
Line 212: Line 386:
|-
|-
|  
|  
| 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 227: Line 401:
|-
|-
|  
|  
| 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 251: Line 425:
|  
|  
|-
|-
| 9\34
| [[34edo|9\34]]
|  
|  
| 317.647
| 317.647
| Upper bound of 9-odd-limit diamond monotone
| 34de val, upper bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 18/13
| 13/9
| 318.309
| 318.309
|  
|  
|}
|}
<nowiki/>* Besides the octave
== Music ==
; [[Budjarn Lambeth]]
* [https://youtu.be/6Kxq5uxu-c8 ''Improv in CWE Catakleismic11''] (2026)


[[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]]

Latest revision as of 09:42, 16 February 2026

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)
Reduced mapping ⟨1; 6 5 22 -21 14]
ET join 53 & 72
Generators (CWE) ~6/5 = 316.7 ¢
MOS scales 4L 7s, 4L 11s, 15L 4s, 15L 19s
Ploidacot alpha-hexacot
Minimax error 9-odd-limit: ??? ¢;
13-limit 21-odd-limit: ??? ¢
Target scale size 9-odd-limit: 34 notes;
13-limit 21-odd-limit: 53 notes

The catakleismic temperament is one of the best 7-limit extensions of hanson, the 5-limit temperament tempering out the 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 (called cata), and then to include 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. 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 Catakleismic extensions for a discussion on 11-limit extensions.

Interval chain

In the following table, harmonics 1–21 and their inverses are in bold.

# Cents* Approximate ratios
2.3.5.7.13 subgroup add-11 add-19 extension
0 0.0 1/1
1 316.8 6/5
2 633.6 13/9
3 950.4 26/15 19/11
4 67.2 25/24, 26/25, 27/26, 28/27
5 384.0 5/4
6 700.8 3/2
7 1017.6 9/5
8 134.4 13/12, 14/13, 27/25
9 451.1 13/10
10 767.9 14/9
11 1084.7 15/8, 28/15
12 201.5 9/8
13 518.3 27/20
14 835.1 13/8, 21/13
15 1151.9 35/18, 39/20 64/33
16 268.7 7/6
17 585.5 7/5
18 902.3 27/16 32/19
19 19.1 81/80, 91/90, 105/104 77/76, 78/77, 96/95,
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

* In 2.3.5.7.13-subgroup CWE tuning

As a detemperament of 19et

Catakleismic as a 72-tone 19et detempering

Catakleismic is naturally considered as a detemperament of the 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 intervals.

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 and harmony

See also: Chords of catakleismic and Chords of tridecimal catakleismic

Scales

Tunings

2.3.5.7.13-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~6/5 = 316.9026 ¢ CSEE: ~6/5 = 316.8354 ¢ POEE: ~6/5 = 316.5718 ¢
Tenney CTE: ~6/5 = 316.8865 ¢ CWE: ~6/5 = 316.7939 ¢ POTE: ~6/5 = 316.7410 ¢
Benedetti,
Wilson 		CBE: ~6/5 = 316.8827 ¢ CSBE: ~6/5 = 316.7927 ¢ POBE: ~6/5 = 316.7673 ¢
2.3.5.7.11.13.19-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~6/5 = 316.7941 ¢ CSEE: ~6/5 = 316.7860 ¢ POEE: ~6/5 = 316.8002 ¢
Tenney CTE: ~6/5 = 316.8070 ¢ CWE: ~6/5 = 316.7816 ¢ POTE: ~6/5 = 316.7778 ¢
Benedetti,
Wilson 		CBE: ~6/5 = 316.8299 ¢ CSBE: ~6/5 = 316.7884 ¢ POBE: ~6/5 = 316.7625 ¢

Tuning spectrum

This tuning spectrum assumes undecimal catakleismic.

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
5/3 315.641
5\19 315.789 Lower bound of 9-odd-limit diamond monotone
13/7 316.037
15/14 316.414
9/7 316.492
11/8 316.604
7/5 316.618
19\72 316.667
7/6 316.679
11/7 316.686
11/6 316.690
11/10 316.731
11/9 316.745 11-odd-limit minimax
52\197 316.751 197ef val
7/4 316.765 7-, 9-, 13- and 15-odd-limit minimax
15/11 316.780
9/5 316.799 1/7-kleisma
33\125 316.800 125f val
13/11 316.835
14\53 316.981
3/2 316.993 5-odd-limit minimax, 1/6-kleisma
15/8 317.115 2/11-kleisma
13/10 317.135
13/8 317.181
23\87 317.241 87de val
5/4 317.263 1/5-kleisma
13/12 317.322
15/13 317.420
9\34 317.647 34de val, upper bound of 9-odd-limit diamond monotone
13/9 318.309

* Besides the octave

Music

Budjarn Lambeth
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