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{{Redirect|Hanson and cata|the rank-2 temperament family|Kleismic family}}
{{Interwiki
{{Redirect|Hanson and cata|the rank-3 temperament family|Kleismic rank three family}}
| en = Kleismic
| de = Hanson-Kleismisch
}}
{{Infobox regtemp
| Title = Kleismic
| Subgroups = 2.3.5, 2.3.5.13
| Comma basis = [[15625/15552]] (2.3.5); <br>[[325/324]], [[625/624]] (2.3.5.13)
| Edo join 1 = 15 | Edo join 2 = 19
| Mapping = 1; 6 5 14
| Generators = 6/5 | Generators tuning = 317.1 | Optimization method = CWE
| MOS scales = [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], [[4L&nbsp;11s]], [[15L&nbsp;4s]]
| Pergen = (P8, P12/6)
| Color name = Tribiyoti
| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 7
| Odd limit 2 = 2.3.5.13 15 | Mistuning 2 = 2.35 | Complexity 2 = 15
}}
'''Kleismic''', alternatively called '''hanson''' in the [[5-limit]], is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which stacked are equated to the [[3/1|perfect twelfth (3/1)]], and thereby characterized by the vanishing of the [[15625/15552|kleisma]] ([[ratio]]: 15625/15552, {{monzo|legend=1| -6 -5 6 }}).


'''Kleismic''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] of the [[kleismic family]], characterized by the vanishing of the kleisma ( [[15625/15552]]). In the [[5-limit]], it is also known as '''hanson'''. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].  
Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As {{nowrap| 9/8 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S|25}}) and the marveltwin comma [[325/324]] ([[S-expression|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.  


However, another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As 9/8 = (25/24)(26/25)(27/26), it would be illogical not to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering the marveltwin comma (S25 * S26 = [[325/324]]), and the tunbarsma (S25 = [[625/624]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''. As the chain of generators naturally gives us hemitwelfths at only 3 generator steps, this also corresponds directly to an interpretation of these as [[26/15]] (and thus hemifourths as [[15/13]]) by tempering S26 = [[676/675]].
Extensions with prime 7 include [[catakleismic]] (which adds [[225/224]], finding 7 at 22 generators up), [[countercata]] (which adds [[5120/5103]], finding 7 at 31 generators down), [[metakleismic]] (which adds [[179200/177147]], finding 7 at 56 generators up), [[keemun]] (which adds [[49/48]], finding 7 at 3 generators up), anakleismic (which adds [[2240/2187]], finding 7 at 37 generators up), and [[catalan]] (which adds [[64/63]], finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical extension, as it makes an intuitive further equivalence of 25/24~26/25~27/26 to [[28/27]] (by tempering out the [[square superparticular]] comma [[729/728]] ({{S|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, identifying [[64/63]] with [[65/64]] by tempering out [[4096/4095]]. Catakleismic and countercata merge in [[53edo]], as the former finds 7 at 22 generators up while the latter finds it at 31 generators down (22 + 31 = 53).


Extensions to prime 7 include [[catakleismic]], [[countercata]], [[metakleismic]], [[keemun]], and [[catalan]]. Of these, catakleismic can be considered canonical, as it makes a natural further equivalence of 25/24~26/25~27/26 to [[28/27]] and can be defined in the [[7-limit]] by tempering [[225/224]] and [[4375/4374]].
Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).


For technical data, see [[Kleismic family #Hanson]].  
For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].


== Interval chain ==
== Interval chain ==
Line 16: Line 32:
! #
! #
! Cents*
! Cents*
! class="unsortable" | Approximate Ratios
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
Line 28: Line 44:
| 2
| 2
| 634.2
| 634.2
| 13/9
| 13/9, 36/25
|-
|-
| 3
| 3
| 950.3
| 951.3
| 26/15
| 26/15
|-
|-
Line 39: Line 55:
|-
|-
| 5
| 5
| 385.6
| 385.5
| '''5/4'''
| '''5/4'''
|-
|-
| 6
| 6
| 702.7
| 702.6
| '''3/2'''
| '''3/2'''
|-
|-
| 7
| 7
| 1019.8
| 1019.6
| 9/5
| 9/5
|-
|-
| 8
| 8
| 136.9
| 136.7
| 13/12, 27/25
| 13/12, 27/25
|-
|-
| 9
| 9
| 454.0
| 453.8
| 13/10
| 13/10
|-
|-
| 10
| 10
| 771.1
| 770.9
| 25/16
| 25/16, 39/25
|-
|-
| 11
| 11
| 1088.2
| 1088.0
| '''15/8'''
| '''15/8'''
|-
|-
| 12
| 12
| 205.3
| 205.1
| '''9/8'''
| '''9/8'''
|-
|-
| 13
| 13
| 522.4
| 522.2
| 27/20
| 27/20
|-
|-
| 14
| 14
| 839.6
| 839.3
| '''13/8'''
| '''13/8'''
|-
|-
| 15
| 15
| 1156.7
| 1156.4
| 39/20
| 39/20
|-
|-
| 16
| 16
| 273.8
| 273.5
| 75/64
| 75/64
|-
|-
| 17
| 17
| 590.9
| 590.6
| 45/32
| 45/32
|-
|-
| 18
| 18
| 908.0
| 907.7
| 27/16
| 27/16
|-
|-
| 19
| 19
| 25.1
| 24.7
| 65/64, 81/80
| 65/64, 81/80
|}
|}
<nowiki>*</nowiki> in 2.3.5.13-subgroup [[CTE tuning]]
<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced


== Tunings ==
== Tunings ==
[[File:Kleismic.png|thumb|alt=Kleismic.png|A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.]]
=== Optimized tunings ===
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! Weight-skew\Order !! Euclidean
! rowspan="2" |  !! colspan="2" | Euclidean
|-
|-
| Tenney || (2.3.5) CTE: ~6/5 = 317.0308¢
! Constrained !! Destretched
|-
! Tenney
| CTE: ~6/5 = 317.0308{{c}} || POTE: ~6/5 = 317.007{{c}}
|-
! Equilateral
| CEE: ~6/5 = 317.1033{{c}}<br>(11/61-kleisma)
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.13-subgroup norm-based tunings
|-
|-
| Tenney || (2.3.5) POTE: ~6/5 = 317.007¢
! rowspan="2" | !! colspan="2" | Euclidean
|-
|-
| Tenney || (2.3.5.13) CTE: ~6/5 = 317.1110¢
! Constrained !! Destretched
|-
|-
| Tenney || (2.3.5.13) POTE: ~6/5 = 317.0756¢
! Tenney
| CTE: ~6/5 = 317.1110{{c}} || POTE: ~6/5 = 317.0756{{c}}
|}
|}


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
|-
| 3:4:5 (+1 +1) || ~6/5 = 317.1496 || ''g''<sup>6</sup> + 2''g''<sup>5</sup> - 8 = 0 || 1-3-5 equal-beating tuning, close to 8/43-kleisma
| 3:4:5 (+1 +1) || ~6/5 = 317.1496 || ''g''<sup>6</sup> + 2''g''<sup>5</sup> &minus; 8 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 8/43-kleisma
|-
|-
| 4:5:6 (+1 +1) || ~6/5 = 317.9593 || ''g''<sup>6</sup> - 2''g''<sup>5</sup> + 2 = 0 || 1-3-5 equal-beating tuning, close to 2/7-kleisma
| 4:5:6 (+1 +1) || ~6/5 = 317.9593 || ''g''<sup>6</sup> &minus; 2''g''<sup>5</sup> + 2 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 2/7-kleisma
|-
|-
| 10:12:15 (+2 +3) || ~6/5 = 317.6675 || ''g''<sup>6</sup> - 5''g'' + 3 = 0 || Close to 1/4-kleisma
| 10:12:15 (+2 +3) || ~6/5 = 317.6675 || ''g''<sup>6</sup> &minus; 5''g'' + 3 = 0 || Close to 1/4-kleisma
|-
|-
| 9:13:15 (+2 +1) || ~6/5 = 317.5679 || 3''g''<sup>3</sup> + 4''g'' - 10 = 0 || Close to 13/36-marveltwin comma
| 9:13:15 (+2 +1) || ~6/5 = 317.5679 || 3''g''<sup>3</sup> + 4''g'' &minus; 10 = 0 || Close to 13/36-marveltwin comma
|-
|-
| 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''<sup>3</sup> - ''g'' - 4 = 0 || Close to 13/51-marveltwin comma
| 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''<sup>3</sup> &minus; ''g'' &minus; 4 = 0 || Close to 13/51-marveltwin comma
|}
|}
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.0000{{c}}, ~6/5 = 317.1983{{c}}


=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br>Generator
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]*
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 197: Line 230:
| [[3/2]]
| [[3/2]]
| 316.9925
| 316.9925
| 1/6-kleisma
| 1/6-kleisma; 5- and 9-odd-limit minimax tuning
|-
|-
| [[246edo|65\246]]
|
| 317.0732
|  
|  
| [[75/52]]
| 317.0274
| 1/2-tunbarsma
|-
|-
| [[193edo|51\193]]
| [[193edo|51\193]]
Line 213: Line 246:
| 317.1153
| 317.1153
| 2/11-kleisma
| 2/11-kleisma
|-
| [[333edo|88\333]]
|
| 317.1171
|
|-
|-
|  
|  
| [[13/10]]
| [[13/10]]
| 317.1349
| 317.1349
|  
| 13- and 15-odd-limit minimax tuning
|-
|-
| [[140edo|37\140]]
| [[140edo|37\140]]
Line 298: Line 326:
| 318.3673
| 318.3673
| 49f val
| 49f val
|-
|
| [[125/104]]
| 318.4135
| Full tunbarsma
|-
|-
|  
|  
Line 313: Line 336:
| '''320.0000'''
| '''320.0000'''
| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''
| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''
|-
|
| [[65/54]]
| 320.9764
| Full marveltwin comma
|}
|}
<nowiki>*</nowiki> besides the octave
<nowiki/>* Besides the octave
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983


== Scales ==
== Scales ==
* [[Cata7]]
* [[Cata7]] ([[4L 3s]])
* [[Cata11]]
* [[Cata11]] ([[4L 7s]])
* [[Cata15]]
* [[Cata15]] ([[4L 11s]])
* [[Cata19]]
* [[Cata19]] ([[15L 4s]])


== Music ==
== Music ==
Line 340: Line 355:
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm ''11 note chain-of-minor-thirds scale''], by [[David Keenan]]
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm ''11 note chain-of-minor-thirds scale''], by [[David Keenan]]


[[Category:Temperaments]]
[[Category:Kleismic| ]] <!-- Main article -->
[[Category:Hanson]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Cata| ]] <!-- main article -->
[[Category:Kleismic| ]] <!-- main article -->
[[Category:Kleismic family]]
[[Category:Kleismic family]]

Latest revision as of 21:30, 26 March 2026

Kleismic
Subgroups 2.3.5, 2.3.5.13
Comma basis 15625/15552 (2.3.5);
325/324, 625/624 (2.3.5.13)
Reduced mapping ⟨1; 6 5 14]
ET join 15 & 19
Generators (CWE) ~6/5 = 317.1 ¢
MOS scales 3L 1s, 4L 3s, 4L 7s, 4L 11s, 15L 4s
Ploidacot alpha-hexacot
Pergen (P8, P12/6)
Color name Tribiyoti
Minimax error 5-odd-limit: 1.35 ¢;
2.3.5.13 15-odd-limit: 2.35 ¢
Target scale size 5-odd-limit: 7 notes;
2.3.5.13 15-odd-limit: 15 notes

Kleismic, alternatively called hanson in the 5-limit, is a rank-2 temperament and parent of the kleismic family, generated by a classical minor third (6/5), six of which stacked are equated to the perfect twelfth (3/1), and thereby characterized by the vanishing of the kleisma (ratio: 15625/15552, monzo: [-6 -5 6⟩).

Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, 25/24, into a third-tone by equating three of this interval to 9/8. As 9/8 = (27/26)⋅(26/25)⋅(25/24), it is natural to equate 25/24 to 26/25 and 27/26 as well, thereby tempering out the tunbarsma 625/624 (S25) and the marveltwin comma 325/324 (S25⋅S26) respectively, and resulting in a low-complexity but high-accuracy extension to the 2.3.5.13-subgroup sometimes known as cata. From there we can see that 676/675 (S26) is also tempered out, meaning 4/3 is split into two 15/13's and that 3/1 is split into two 26/15's. From 325/324 = (13/9)/(6/5)2 we can see that 13/9 is split into two 6/5's, so that it is equated with 36/25 (giving rise to the other S-expression of 325/324, S10/S12); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

Extensions with prime 7 include catakleismic (which adds 225/224, finding 7 at 22 generators up), countercata (which adds 5120/5103, finding 7 at 31 generators down), metakleismic (which adds 179200/177147, finding 7 at 56 generators up), keemun (which adds 49/48, finding 7 at 3 generators up), anakleismic (which adds 2240/2187, finding 7 at 37 generators up), and catalan (which adds 64/63, finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical extension, as it makes an intuitive further equivalence of 25/24~26/25~27/26 to 28/27 (by tempering out the square superparticular comma 729/728 (S27) in addition to 625/624 and 676/675), and can be defined independently in the 7-limit by tempering out 225/224 and 4375/4374. However, countercata is well-tuned closer to the optimal range of kleismic (between 53edo and 87edo), especially that of 2.3.5.13 cata, and naturally emerges in that context, identifying 64/63 with 65/64 by tempering out 4096/4095. Catakleismic and countercata merge in 53edo, as the former finds 7 at 22 generators up while the latter finds it at 31 generators down (22 + 31 = 53).

Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out 385/384, equating the ~6/5 generator to 77/64. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between just 6/5 (just flat of 19edo) and 77/64 (just sharp of 15edo).

For technical data, see Kleismic family #Kleismic a.k.a. hanson.

Interval chain

In the following table, odd harmonics 1–15 are labeled in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 317.1 6/5
2 634.2 13/9, 36/25
3 951.3 26/15
4 68.4 25/24, 26/25, 27/26
5 385.5 5/4
6 702.6 3/2
7 1019.6 9/5
8 136.7 13/12, 27/25
9 453.8 13/10
10 770.9 25/16, 39/25
11 1088.0 15/8
12 205.1 9/8
13 522.2 27/20
14 839.3 13/8
15 1156.4 39/20
16 273.5 75/64
17 590.6 45/32
18 907.7 27/16
19 24.7 65/64, 81/80

* In 2.3.5.13-subgroup CWE tuning, octave reduced

Tunings

Kleismic.png
A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.

Optimized tunings

5-limit norm-based tunings
Euclidean
Constrained Destretched
Tenney CTE: ~6/5 = 317.0308 ¢ POTE: ~6/5 = 317.007 ¢
Equilateral CEE: ~6/5 = 317.1033 ¢
(11/61-kleisma)
2.3.5.13-subgroup norm-based tunings
Euclidean
Constrained Destretched
Tenney CTE: ~6/5 = 317.1110 ¢ POTE: ~6/5 = 317.0756 ¢
DR and equal-beating tunings
Optimized chord Generator value Polynomial Further notes
3:4:5 (+1 +1) ~6/5 = 317.1496 g6 + 2g5 − 8 = 0 1 – 3 – 5 equal-beating tuning, close to 8/43-kleisma
4:5:6 (+1 +1) ~6/5 = 317.9593 g6 − 2g5 + 2 = 0 1 – 3 – 5 equal-beating tuning, close to 2/7-kleisma
10:12:15 (+2 +3) ~6/5 = 317.6675 g6 − 5g + 3 = 0 Close to 1/4-kleisma
9:13:15 (+2 +1) ~6/5 = 317.5679 3g3 + 4g − 10 = 0 Close to 13/36-marveltwin comma
13:15:18 (+2 +3) ~6/5 = 317.0010 3g3 − g − 4 = 0 Close to 13/51-marveltwin comma

Other tunings

  • DKW (2.3.5): ~2 = 1200.0000 ¢, ~6/5 = 317.1983 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
6/5 315.6413 Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
5\19 315.7895 Lower bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone
27/26 316.3343 1/4-tunbarsma
29\110 316.3636 110ff val
24\91 316.4835 91f val
27/25 316.6547 1/8-kleisma
19\72 316.6667
9/5 316.7995 1/7-kleisma
33\125 316.8000 125f val
26/25 316.9750 1/4-marveltwin comma
14\53 316.9811
3/2 316.9925 1/6-kleisma; 5- and 9-odd-limit minimax tuning
65\246 317.0732
51\193 317.0984
15/8 317.1153 2/11-kleisma
13/10 317.1349 13- and 15-odd-limit minimax tuning
37\140 317.1429
13/8 317.1805
60\227 317.1807
23\87 317.2414
5/4 317.2627 1/5-kleisma, upper bound of 5-odd-limit diamond tradeoff
13/12 317.3216
32\121 317.3554
41\155 317.4194
15/13 317.4197 1/3-marveltwin comma
9\34 317.6471
25/24 317.6681 1/4-kleisma, virtually DR 10:12:15
22\83 318.0723 83f val
13/9 318.3088 1/2-marveltwin comma, upper bound of 2.3.5.13-subgroup 15-odd-limit diamond tradeoff
125/72 318.3437 1/3-kleisma
13\49 318.3673 49f val
625/432 319.6949 1/2-kleisma
4\15 320.0000 Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone

* Besides the octave

Scales

Music

Petr Pařízek
Chris Vaisvil

External links

Retrieved from "https://en.xen.wiki/index.php?title=Kleismic&oldid=226929"