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| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
| Mapping = 1; 6 5 14
| Mapping = 1; 6 5 14
| Ploidacot = haploid alpha-hexacot
| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 15
| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 15
| Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34
| Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34
}}
}}
: ''"Kleismic" redirects here. For the temperament families, see [[Kleismic family]] and [[Kleismic rank three family]].''
<br>
 
{{Infobox RT
'''Kleismic''', known in the [[5-limit]] as either '''hanson''' or simply '''kleismic''', is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], characterized by the vanishing of the kleisma ([[15625/15552]]). It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].
| Title = Würschmidt
 
| Subgroups = 2.3.5, 2.3.5.23
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 is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out 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 out S26 = [[676/675]].
| Comma basis = [[393216/390625]] (2.3.5); <br> [[576/575]], [[12167/12150]] (2.3.5.23)
 
| Edo join 1 = 31 | Edo join 2 = 34
Extensions with prime 7 include [[catakleismic]], [[countercata]], [[metakleismic]], [[keemun]], and [[catalan]]. Of these, catakleismic can perhaps be considered the 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 out [[225/224]] and [[4375/4374]].
| Generator = 5/4 | Generator tuning = 387.734 | Optimization method = CTE
 
| MOS scales = [[3L 1s]], [[3L 4s]] ... [[3L 28s]], [[31L 3s]]
For technical data, see [[Kleismic family #Hanson]].
| Mapping = 1; 8 1 14
 
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 19
== Interval chain ==
| Odd limit 2 = (2.3.5.23) 25 | Mistuning 2 = 2.86 | Complexity 2 = 34
In the following table, odd harmonics 1–15 are labeled in '''bold'''.
}}
 
{| class="wikitable sortable center-1 right-2"
! &#35;
! Cents*
! class="unsortable" | Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 317.1
| 6/5
|-
| 2
| 634.2
| 36/25, 13/9
|-
| 3
| 950.3
| 26/15, 45/26
|-
| 4
| 68.4
| 25/24, 26/25, 27/26
|-
| 5
| 385.6
| '''5/4''', 81/65
|-
| 6
| 702.7
| '''3/2'''
|-
| 7
| 1019.8
| 9/5, 65/36
|-
| 8
| 136.9
| 13/12, 27/25
|-
| 9
| 454.0
| 13/10
|-
| 10
| 771.1
| 25/16, 39/25, 81/52
|-
| 11
| 1088.2
| '''15/8'''
|-
| 12
| 205.3
| '''9/8'''
|-
| 13
| 522.4
| 27/20, 65/48
|-
| 14
| 839.6
| '''13/8''', 81/50
|-
| 15
| 1156.7
| 39/20
|-
| 16
| 273.8
| 75/64
|-
| 17
| 590.9
| 45/32
|-
| 18
| 908.0
| 27/16
|-
| 19
| 25.1
| 65/64, 81/80
|}
<nowiki />* In 2.3.5.13-subgroup [[CTE tuning]]
 
== Tunings ==
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings
|-
! Weight-skew\Order !! Euclidean
|-
| Tenney || (2.3.5) CTE: ~6/5 = 317.0308¢
|-
| Tenney || (2.3.5) POTE: ~6/5 = 317.007¢
|-
| Tenney || (2.3.5.13) CTE: ~6/5 = 317.1110¢
|-
| Tenney || (2.3.5.13) POTE: ~6/5 = 317.0756¢
|}
 
{| 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
|}
 
=== 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
|-
|
| [[75/52]]
| 317.0274
| 1/2-tunbarsma
|-
| [[193edo|51\193]]
|
| 317.0984
|
|-
|
| [[15/8]]
| 317.1153
| 2/11-kleisma
|-
| [[333edo|88\333]]
|
| 317.1171
|
|-
|
| [[13/10]]
| 317.1349
|
|-
| [[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
|-
|
| [[125/104]]
| 318.4135
| Full tunbarsma
|-
|
| [[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'''
|-
|
| [[65/54]]
| 320.9764
| Full marveltwin comma
|}
<nowiki />* Besides the octave
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983
 
== Scales ==
* [[Cata7]]
* [[Cata11]]
* [[Cata15]]
* [[Cata19]]
 
== 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:Temperaments]]
[[Category:Hanson]] <!-- Main article -->
[[Category:Cata| ]] <!-- Main article -->
[[Category:Kleismic| ]] <!-- Main article -->
[[Category:Kleismic family]]
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