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Diaschismic extensions: Difference between revisions

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{{URWTC}}
{{Breadcrumb|Diaschismic}}
{{Infobox Regtemp
In the [[5-limit]], '''diaschismic''' is a [[regular temperament]] (also known as ''srutal'', though they refer to different extensions in higher limits) defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]]. There are multiple ways to extend diaschismic to primes [[7/1|7]], [[11/1|11]], and [[13/1|13]].
| Title = Srutal; srutal archagall
| Subgroups = 2.3.5, 2.3.5.17
| Comma basis = [[2048/2025]] (2.3.5); <br /> [[136/135]], [[256/255]] (2.3.5.17)
| Edo join 1 = 12 | Edo join 2 = 22
| Generator = 16/15 | Generator tuning = 104.898 | Optimization method = POTE
| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
| Mapping = 2; 1 -2 1
| Pergen = (P8/2, P5)
| Color name = Saguguti
| Odd limit 1 = 5 | Mistuning 1 = 3.259 | Complexity 1 = 12
| Odd limit 2 = (2.3.5.17) 25 | Mistuning 2 = ??? | Complexity 2 = 22
}}


'''Srutal''', known interchangeably as '''diaschismic''' in the [[5-limit]], is a [[regular temperament]] defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and therefore as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]].
== 7-limit extensions ==
The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&amp;46) and diaschismic (46&amp;58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&amp;22) and keen (22&amp;34).
 
=== Srutal ===
Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave.
 
For technical data on 7-limit and higher-limit srutal: see [[Diaschismic family #Srutal]].


{{Tdlink|Diaschismic family #Srutal aka diaschismic}}
=== Diaschismic ===
Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.


== 7-limit extensions ==
For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].
The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&amp;46) and diaschismic (46&amp;58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&amp;22) and keen (22&amp;34).


Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave. Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.
Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order. The extensions to prime [[11/1|11]] and [[13/1|13]] can be characterized by [[parapyth]], which makes sense as the fifth is tuned slightly sharp, and prime 17 is found via srutal archagall.


Both of these can be extended straightforwardly to the [[11-limit|11-]], [[13-limit|13-]], and [[17-limit]] by adding 176/175, 352/351, and 221/220 to the comma list in this order.
=== Pajara ===
[[Pajara]] combines diaschismic with [[archy]], tempering the fifth to about 709 cents. The interval of two stacked fifths is equated to 16/7, and the harmonic seventh [[7/4]] and the just major third [[5/4]] are separated by a perfect semioctave.


For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Diaschismic]].
For technical data on 7-limit and higher-limit pajara, see [[Diaschismic family #Pajara]].


For technical data on 7-limit and higher-limit srutal: see [[Diaschismic family #Srutal]].
=== Keen ===
{{todo|inline=1|complete section}}


== Interval chains ==
== Intervals ==
=== Diaschismic ===
{{todo|cleanup|complete section|inline=1}}
=== Diaschismic (2.3.5.7.17) ===
<div><div style="display: inline-grid; margin-right: 25px;">
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
Line 121: Line 118:
</div>
</div>


== Intervals ==
=== Srutal (17-limit) ===
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height: 1.6;">'''Intervals of srutal {{nowrap|(34d &amp; 46)}}'''</div>
<div style="line-height: 1.6;">'''Intervals of srutal {{nowrap|(34d &amp; 46)}}'''</div>
Line 270: Line 267:
| 21/16
| 21/16
| 7/5
| 7/5
|}
<nowiki />* In 17-limit POTE tuning
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height: 1.6;">'''Intervals of diaschismic {{nowrap|(46 &amp; 58)}}'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
! Generator
! &minus;17
! &minus;16
! &minus;15
! &minus;14
! &minus;13
! &minus;12
|-
! Cents*
| 35.19
| 139.01
| 242.82
| 346.63
| 450.44
| 554.25
|-
! Ratios
|
| 13/12
|
| 11/9
| 22/17, 13/10
| 11/8
|-
! Generator
! &minus;11
! &minus;10
! &minus;9
! &minus;8
! &minus;7
! &minus;6
|-
! Cents*
| 58.07
| 161.88
| 265.69
| 369.50
| 473.32
| 577.13
|-
! Ratios
|
| 11/10
| 7/6
| 21/17, 26/21
| 21/16
| 7/5
|-
! Generator
! &minus;5
! &minus;4
! &minus;3
! &minus;2
! &minus;1
! &minus; 0
|-
! Cents*
| 80.94
| 184.75
| 288.56
| 392.38
| 496.19
| 600.00
|-
! Ratios
| 22/21, 21/20
| 10/9
| 20/17, 13/11
| 5/4
| 4/3
| 24/17, 17/12
|-
! Generator
! 0
! 1
! 2
! 3
! 4
! 5
|-
! Cents*
| 0.00
| 103.81
| 207.62
| 311.44
| 415.25
| 519.06
|-
! Ratios
| 1/1
| 18/17, 17/16, <br />16/15
| 9/8, 17/15
| 6/5
| 14/11
|
|-
! Generator
! 6
! 7
! 8
! 9
! 10
! 11
|-
! Cents*
| 22.87
| 126.68
| 230.50
| 334.31
| 438.12
| 541.93
|-
! Ratios
|
| 15/14, 14/13
| 8/7
| 17/14
| 9/7
| 15/11
|-
! Generator
! 12
! 13
! 14
! 15
! 16
! 17
|-
! Cents*
| 45.75
| 149.56
| 253.37
| 357.18
| 460.99
| 564.81
|-
! Ratios
|
| 12/11
| 15/13
| 16/13
| 17/13
| 18/13
|}
|}
<nowiki />* In 17-limit POTE tuning
<nowiki />* In 17-limit POTE tuning
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* [[Lumatone mapping for diaschismic]]
* [[Lumatone mapping for diaschismic]]


[[Category:Temperaments]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
[[Category:Srutal]]
[[Category:Diaschismic family]]
[[Category:Diaschismic]]
[[Category:Srutal |Srutal ]]
[[Category:Diaschismic |Diaschismic ]]
Retrieved from "https://en.xen.wiki/w/Diaschismic_extensions"