Xenharmonic Wiki

Kleismic: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,044 edits
m noted minimax tunings
Overthink (talk | contribs)
autopatrolled
2,698 edits
m single link (see talk)
 
(40 intermediate revisions by 9 users not shown)
Line 1: Line 1:
{{Infobox Regtemp
{{Interwiki
| Title = Kleismic; hanson; cata
| en = Kleismic
| de = Hanson-Kleismisch
}}
{{Infobox regtemp
| Title = Kleismic
| Subgroups = 2.3.5, 2.3.5.13
| Subgroups = 2.3.5, 2.3.5.13
| Comma basis = [[15625/15552]] (2.3.5); <br> [[325/324]], [[625/624]] (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
| Edo join 1 = 15 | Edo join 2 = 19
| Generator = 6/5 | Generator tuning = 317.1 | Optimization method = CTE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
| Mapping = 1; 6 5 14
| Mapping = 1; 6 5 14
| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 15
| Generators = 6/5 | Generators tuning = 317.1 | Optimization method = CWE
| Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34
| 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" redirects here. For the temperament families, see [[Kleismic family]] and [[Kleismic rank three family]].''
'''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''', 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)]].  
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.  


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]].
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 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, 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]]. However, countercata exists closer to the truly optimal range of kleismic (between [[53edo]] and [[87edo]]) and tempers out [[4096/4095]] where [[65/64]] and therefore [[64/63]] are close to just.
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]]).


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]], which works well since ~6/5 should be tuned sharp of just, bringing it closer to 77/64, which is in fact just at very close to [[15edo]]'s minor third of 320c.
For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].
 
For technical data, see [[Kleismic family #Hanson]].  


== Interval chain ==
== Interval chain ==
Line 26: Line 30:


{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 right-2"
! &#35;
! #
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
Line 36: Line 40:
| 1
| 1
| 317.1
| 317.1
| 6/5, 65/54
| 6/5
|-
|-
| 2
| 2
Line 43: Line 47:
|-
|-
| 3
| 3
| 950.3
| 951.3
| 26/15, 45/26
| 26/15
|-
|-
| 4
| 4
Line 51: Line 55:
|-
|-
| 5
| 5
| 385.6
| 385.5
| '''5/4''', 81/65
| '''5/4'''
|-
|-
| 6
| 6
| 702.7
| 702.6
| '''3/2'''
| '''3/2'''
|-
|-
| 7
| 7
| 1019.8
| 1019.6
| 9/5, 65/36
| 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, 39/25, 81/52
| 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, 65/48
| 27/20
|-
|-
| 14
| 14
| 839.6
| 839.3
| '''13/8''', 81/50
| '''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 />* 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
|-
|-
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
! rowspan="2" | !! colspan="2" | Euclidean
|-
|-
! Constrained !! Destretched
! Constrained !! Destretched
|-
|-
! Tenney
! Tenney
| (2.3.5) CTE: ~6/5 = 317.0308¢ || (2.3.5) POTE: ~6/5 = 317.007¢
| CTE: ~6/5 = 317.0308{{c}} || POTE: ~6/5 = 317.007{{c}}
|-
|-
! Equilateral
! Equilateral
| (2.3.5) CEE: ~6/5 = 317.1033¢
| CEE: ~6/5 = 317.1033{{c}}<br>(11/61-kleisma)
(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
! Tenney
| (2.3.5.13) CTE: ~6/5 = 317.1110¢ || (2.3.5.13) POTE: ~6/5 = 317.0756¢
| CTE: ~6/5 = 317.1110{{c}} || POTE: ~6/5 = 317.0756{{c}}
|}
|}


Line 147: Line 160:
| 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
| 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 215: Line 232:
| 1/6-kleisma; 5- and 9-odd-limit minimax tuning
| 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 229: Line 246:
| 317.1153
| 317.1153
| 2/11-kleisma
| 2/11-kleisma
|-
| [[333edo|88\333]]
|
| 317.1171
|
|-
|-
|  
|  
Line 314: Line 326:
| 318.3673
| 318.3673
| 49f val
| 49f val
|-
|
| [[125/104]]
| 318.4135
| Full tunbarsma
|-
|-
|  
|  
Line 329: 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 />* Besides the octave
<nowiki/>* Besides the octave
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983


== Scales ==
== Scales ==
Line 345: Line 344:
* [[Cata15]] ([[4L 11s]])
* [[Cata15]] ([[4L 11s]])
* [[Cata19]] ([[15L 4s]])
* [[Cata19]] ([[15L 4s]])
== Images ==
[[File:Kleismic.png|alt=Kleismic.png|600x560px]]
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.


== Music ==
== Music ==
Line 361: 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:Hanson]] <!-- Main article -->
[[Category:Cata| ]] <!-- Main article -->
[[Category:Kleismic| ]] <!-- Main article -->
[[Category:Kleismic| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Kleismic family]]
[[Category:Kleismic family]]
Retrieved from "https://en.xen.wiki/w/Kleismic"