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'''Kleismic''' may refer to:
{{Interwiki
* [[Kleismic rank three family]], for the rank-3 kleismic temperament and its associated family  
| en = Kleismic
* [[Kleismic family]], for the rank-2 hanson temperament and its associated family
| 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 }}).


{{Disambiguation}}
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.


[[Category:Kleismic]]
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).
 
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 #Kleismic a.k.a. hanson]].
 
== Interval chain ==
In the following table, odd harmonics 1–15 are labeled in '''bold'''.
 
{| class="wikitable sortable center-1 right-2"
! #
! Cents*
! class="unsortable" | 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
|}
<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced
 
== 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 ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |  !! colspan="2" | Euclidean
|-
! 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
|-
! rowspan="2" |  !! colspan="2" | Euclidean
|-
! Constrained !! Destretched
|-
! Tenney
| CTE: ~6/5 = 317.1110{{c}} || POTE: ~6/5 = 317.0756{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
! 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> &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> &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> &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'' &minus; 10 = 0 || Close to 13/36-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 ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Comments
|-
|
| [[6/5]]
| 315.6413
| Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
|-
| '''[[19edo|5\19]]'''
|
| '''315.7895'''
| '''Lower bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''
|-
|
| [[27/26]]
| 316.3343
| 1/4-[[625/624|tunbarsma]]
|-
| [[110edo|29\110]]
|
| 316.3636
| 110ff val
|-
| [[91edo|24\91]]
|
| 316.4835
| 91f val
|-
|
| [[27/25]]
| 316.6547
| 1/8-kleisma
|-
| [[72edo|19\72]]
|
| 316.6667
|
|-
|
| [[9/5]]
| 316.7995
| 1/7-kleisma
|-
| [[125edo|33\125]]
|
| 316.8000
| 125f val
|-
|
| [[26/25]]
| 316.9750
| 1/4-[[325/324|marveltwin comma]]
|-
| [[53edo|14\53]]
|
| 316.9811
|
|-
|
| [[3/2]]
| 316.9925
| 1/6-kleisma; 5- and 9-odd-limit minimax tuning
|-
| [[246edo|65\246]]
|
| 317.0732
|
|-
| [[193edo|51\193]]
|
| 317.0984
|
|-
|
| [[15/8]]
| 317.1153
| 2/11-kleisma
|-
|
| [[13/10]]
| 317.1349
| 13- and 15-odd-limit minimax tuning
|-
| [[140edo|37\140]]
|
| 317.1429
|
|-
|
| [[13/8]]
| 317.1805
|
|-
| [[227edo|60\227]]
|
| 317.1807
|
|-
| [[87edo|23\87]]
|
| 317.2414
|
|-
|
| [[5/4]]
| 317.2627
| 1/5-kleisma, upper bound of 5-odd-limit diamond tradeoff
|-
|
| [[13/12]]
| 317.3216
|
|-
| [[121edo|32\121]]
|
| 317.3554
|
|-
| [[155edo|41\155]]
|
| 317.4194
|
|-
|
| [[15/13]]
| 317.4197
| 1/3-marveltwin comma
|-
| [[34edo|9\34]]
|
| 317.6471
|
|-
|
| [[25/24]]
| 317.6681
| 1/4-kleisma, virtually [[Delta-rational chord|DR]] 10:12:15
|-
| [[83edo|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
|-
| [[49edo|13\49]]
|
| 318.3673
| 49f val
|-
|
| [[625/432]]
| 319.6949
| 1/2-kleisma
|-
| '''[[15edo|4\15]]'''
|
| '''320.0000'''
| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''
|}
<nowiki/>* Besides the octave
 
== Scales ==
* [[Cata7]] ([[4L 3s]])
* [[Cata11]] ([[4L 7s]])
* [[Cata15]] ([[4L 11s]])
* [[Cata19]] ([[15L 4s]])
 
== Music ==
; [[Petr Pařízek]]
* [https://web.archive.org/web/20201127013042/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 ''Hanson Improv'']
 
; [[Chris Vaisvil]]
* [http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 ''In Hanson11'']
 
== External links ==
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm ''11 note chain-of-minor-thirds scale''], by [[David Keenan]]
 
[[Category:Kleismic| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Kleismic family]]
Retrieved from "https://en.xen.wiki/w/Kleismic"