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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
| Odd limit 1 = 5 | Mistuning 1 = ??? | Complexity 1 = 22
| Odd limit 2 = (2.3.5.17) 17 | 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]], 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 37: Line 36:
! 2.3.5.17 subgroup !! Intervals of 7
! 2.3.5.17 subgroup !! Intervals of 7
|-
|-
| &minus;8 || 220.3 || 100/81 || 21/17, 56/45
| &minus;8 || 370.6 || 100/81 || 21/17, 56/45
|-
|-
| &minus;7 || 440.6 || 125/96 || '''21/16''', 112/85
| &minus;7 || 474.2 || 125/96 || '''21/16''', 112/85
|-
|-
| &minus;6 || 660.9 || 25/18 || 7/5
| &minus;6 || 577.9 || 25/18 || 7/5
|-
|-
| &minus;5 || 881.1 || 25/24 || 21/20
| &minus;5 || 81.6 || 25/24 || 21/20
|-
|-
| &minus;4 || 1101.4 || 10/9, 75/68 || 28/25
| &minus;4 || 185.3 || 10/9, 75/68 || 28/25
|-
|-
| &minus;3 || 1321.7 || 20/17, 32/27 || 119/100
| &minus;3 || 289.0 || 20/17, 32/27 || 119/100
|-
|-
| &minus;2 || 1542.0 || '''5/4''', 34/27, 64/51 || 63/50
| &minus;2 || 392.6 || '''5/4''', 34/27, 64/51 || 63/50
|-
|-
| &minus;1 || 1762.2 || 4/3, 45/34 || 168/125
| &minus;1 || 496.3 || 4/3, 45/34 || 168/125
|-
|-
| 0 || 0.0 || '''1/1''' || 126/125
| 0 || 0.0 || '''1/1''' || 126/125
|-
|-
| 1 || 220.3 || 16/15, '''17/16''', 18/17 ||  
| 1 || 103.7 || 16/15, '''17/16''', 18/17 ||  
|-
|-
| 2 || 440.6 || '''9/8''', 17/15 || 125/112
| 2 || 207.4 || '''9/8''', 17/15 || 125/112
|-
|-
| 3 || 660.9 || 6/5, 81/68 || 25/21
| 3 || 311.0 || 6/5, 81/68 || 25/21
|-
|-
| 4 || 881.1 || 32/25, 51/40, 81/64 || 80/63
| 4 || 414.7 || 32/25, 51/40, 81/64 || 80/63
|-
|-
| 5 || 1101.4 || 27/20, 34/25 || 75/56, 85/63
| 5 || 518.4 || 27/20, 34/25 || 75/56, 85/63
|-
|-
| 6 || 1321.7 || 51/50, 81/80 || 85/84
| 6 || 22.1 || 51/50, 81/80 || 85/84
|-
|-
| 7 || 1542.0 || 27/25 || 15/14, 68/63
| 7 || 125.8 || 27/25 || 15/14, 68/63
|-
|-
| 8 || 1762.2 || 144/125 || 8/7
| 8 || 229.5 || 144/125 || 8/7
|}
|}
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
<nowiki />* In 7-limit [[POTE]] tuning
</div>
</div>


Line 82: Line 81:
! 2.3.5.17 subgroup !! Intervals of 7
! 2.3.5.17 subgroup !! Intervals of 7
|-
|-
| &minus;8 || 1762.2 || 125/72 || '''7/4'''
| &minus;8 || 970.6 || 125/72 || '''7/4'''
|-
|-
| &minus;7 || 1542.0 || 50/27 || 28/15, 63/34
| &minus;7 || 1074.2 || 50/27 || 28/15, 63/34
|-
|-
| &minus;6 || 1321.7 || 100/51, 160/81 || 168/85
| &minus;6 || 1177.9 || 100/51, 160/81 || 168/85
|-  
|-  
| &minus;5 || 1101.4 || 40/27, 25/17 || 112/75, 126/85
| &minus;5 || 681.6 || 40/27, 25/17 || 112/75, 126/85
|-
|-
| &minus;4 || 881.1 || '''25/16''', 80/51, 128/81 || 63/40
| &minus;4 || 785.3 || '''25/16''', 80/51, 128/81 || 63/40
|-
|-
| &minus;3 || 660.9 || 5/3, 136/81 || 42/25
| &minus;3 || 889.0 || 5/3, 136/81 || 42/25
|-
|-
| &minus;2 || 440.6 || 16/9, 30/17 || 224/125
| &minus;2 || 992.6 || 16/9, 30/17 || 224/125
|-
|-
| &minus;1 || 220.3 || '''15/8''', 17/9, 32/17 ||
| &minus;1 || 1096.3 || '''15/8''', 17/9, 32/17 ||
|-
|-
| 0 || 0.0 || 17/12, 24/17, 45/32, 64/45 ||
| 0 || 600.0 || 17/12, 24/17, 45/32, 64/45 ||
|-
|-
| 1 || 1762.2 || '''3/2''', 68/45 || 125/84
| 1 || 703.7 || '''3/2''', 68/45 || 125/84
|-
|-
| 2 || 1542.0 || 8/5, 27/17, 51/32 || 100/63
| 2 || 807.4 || 8/5, 27/17, 51/32 || 100/63
|-
|-
| 3 || 1321.7 || 17/10, '''27/16''' || 200/119
| 3 || 911.0 || 17/10, '''27/16''' || 200/119
|-
|-
| 4 || 1101.4 || 9/5, 136/75 || 25/14
| 4 || 1014.7 || 9/5, 136/75 || 25/14
|-
|-
| 5 || 881.1 || 48/25 || 40/21
| 5 || 1118.4 || 48/25 || 40/21
|-
|-
| 6 || 660.9 || 36/25 || 10/7
| 6 || 622.1 || 36/25 || 10/7
|-
|-
| 7 || 440.6 || 192/125 || 32/21, 85/56
| 7 || 725.8 || 192/125 || 32/21, 85/56
|-
|-
| 8 || 220.3 || 81/50 || 34/21, 45/28
| 8 || 829.5 || 81/50 || 34/21, 45/28
|}
|}
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
<nowiki />* In 7-limit [[POTE]] tuning
</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 268: 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
Line 435: Line 281:
* [[Lumatone mapping for diaschismic]]
* [[Lumatone mapping for diaschismic]]


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