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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 ==
== 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).
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.
=== 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.  


Both of these can be extended naturally 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.
For technical data on 7-limit and higher-limit srutal: see [[Diaschismic family #Srutal]].


== Interval chains (WIP) ==
=== 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.


For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].
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.
=== 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 pajara, see [[Diaschismic family #Pajara]].
=== Keen ===
{{todo|inline=1|complete section}}
== Intervals ==
{{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"
|+ style="font-size: 105%;" | First period
|+ style="font-size: 105%;" | First period
|-
|-
! rowspan="3" | &#35; !! rowspan="3" | Cents&#42; !! colspan="3" | Approximate Ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
|-
! rowspan="2" | 2.3.5.17 subgroup !! colspan="2" | Intervals of 7
|-
|-
! Srutal !! Diaschismic
! 2.3.5.17 subgroup !! Intervals of 7
|-
|-
| -8 || 220.3 || 100/81
| &minus;8 || 370.6 || 100/81 || 21/17, 56/45
|-
|-
| -7 || 440.6 || 125/96
| &minus;7 || 474.2 || 125/96 || '''21/16''', 112/85
|-
|-
| -6 || 660.9 || 25/18
| &minus;6 || 577.9 || 25/18 || 7/5
|-
|-
| -5 || 881.1 || 25/24
| &minus;5 || 81.6 || 25/24 || 21/20
|-
|-
| -4 || 1101.4 || 10/9, 75/68
| &minus;4 || 185.3 || 10/9, 75/68 || 28/25
|-
|-
| -3 || 1321.7 || 20/17, 32/27
| &minus;3 || 289.0 || 20/17, 32/27 || 119/100
|-
|-
| -2 || 1542.0 || '''5/4''', 34/27, 64/51
| &minus;2 || 392.6 || '''5/4''', 34/27, 64/51 || 63/50
|-
|-
| -1 || 1762.2 || 4/3, 45/34  
| &minus;1 || 496.3 || 4/3, 45/34 || 168/125
|-
|-
| 0 || 0.0 || '''1/1''' || ||
| 0 || 0.0 || '''1/1''' || 126/125
|-
|-
| 1 || 220.3 || 16/15, '''17/16''', 18/17 || || 200/189
| 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, 150/119
| 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, 120/119, 225/224
| 6 || 22.1 || 51/50, 81/80 || 85/84
|-
|-
| 7 || 1542.0 || 27/25 || || 15/14, 68/63, 128/119
| 7 || 125.8 || 27/25 || 15/14, 68/63
|-
|-
| 8 || 1762.2 || 144/125 || || 8/7, 135/119
| 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 74: Line 77:
|+ style="font-size: 105%;" | Second period
|+ style="font-size: 105%;" | Second period
|-
|-
! rowspan="3" | &#35; !! rowspan="3" | Cents&#42; !! colspan="3" | Approximate Ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
|-
! rowspan="2" | 2.3.5.17 subgroup !! colspan="2" | Intervals of 7
|-
! Srutal !! Diaschismic
|-
|-
| -8 || 1762.2 || 125/72
! 2.3.5.17 subgroup !! Intervals of 7
|-
|-
| -7 || 1542.0 || 50/27
| &minus;8 || 970.6 || 125/72 || '''7/4'''
|-
|-
| -6 || 1321.7 || 100/51, 160/81
| &minus;7 || 1074.2 || 50/27 || 28/15, 63/34
|-
|-
| -5 || 1101.4 || 40/27, 25/17
| &minus;6 || 1177.9 || 100/51, 160/81 || 168/85
|-  
| &minus;5 || 681.6 || 40/27, 25/17 || 112/75, 126/85
|-
|-
| -4 || 881.1 || '''25/16''', 80/51, 128/81
| &minus;4 || 785.3 || '''25/16''', 80/51, 128/81 || 63/40
|-
|-
| -3 || 660.9 || 5/3, 136/81
| &minus;3 || 889.0 || 5/3, 136/81 || 42/25
|-
|-
| -2 || 440.6 || 16/9, 30/17
| &minus;2 || 992.6 || 16/9, 30/17 || 224/125
|-
|-
| -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
| 1 || 703.7 || '''3/2''', 68/45 || 125/84
|-
|-
| 2 || 1542.0 || 8/5, 27/17, 51/32
| 2 || 807.4 || 8/5, 27/17, 51/32 || 100/63
|-
|-
| 3 || 1321.7 || 17/10, '''27/16'''
| 3 || 911.0 || 17/10, '''27/16''' || 200/119
|-
|-
| 4 || 1101.4 || 9/5, 136/75
| 4 || 1014.7 || 9/5, 136/75 || 25/14
|-
|-
| 5 || 881.1 || 48/25
| 5 || 1118.4 || 48/25 || 40/21
|-
|-
| 6 || 660.9 || 36/25
| 6 || 622.1 || 36/25 || 10/7
|-
|-
| 7 || 440.6 || 192/125
| 7 || 725.8 || 192/125 || 32/21, 85/56
|-
|-
| 8 || 220.3 || 81/50
| 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 (34d&amp;46)'''</div>
<div style="line-height: 1.6;">'''Intervals of srutal {{nowrap|(34d &amp; 46)}}'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
! | Generator
! | -17
! | -16
! | -15
! | -14
! | -13
! | -12
|-
! | Cents*
| | 17.73
| | 122.57
| | 227.40
| | 332.24
| | 437.08
| | 541.92
|-
! | Ratios
| |
| | 15/14
| | 8/7
| | 17/14
| | 9/7
| | 15/11
|-
! | Generator
! | -11
! | -10
! | -9
! | -8
! | -7
! | -6
|-
! | Cents*
| | 46.76
| | 151.60
| | 256.44
| | 361.28
| | 466.12
| | 570.96
|-
! | Ratios
| |
| | 12/11
| | 15/13
| | 16/13
| | 17/13
| | 18/13
|-
! | Generator
! | -5
! | -4
! | -3
! | -2
! | -1
! | 0
|-
! | Cents*
| | 75.80
| | 180.64
| | 285.48
| | 390.32
| | 495.16
| | 600.00
|-
! | Ratios
| | 22/21
| | 10/9
| | 20/17, 13/11
| | 5/4
| | 4/3
| | 24/17, 17/12
|-
! | Generator
! | 0
! | 1
! | 2
! | 3
! | 4
! | 5
|-
! | Cents*
| | 0.00
| | 104.84
| | 209.68
| | 314.52
| | 419.36
| | 524.20
|-
! | 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*
| | 29.04
| | 133.88
| | 238.72
| | 343.56
| | 448.40
| | 553.24
|-
! | Ratios
| |
| | 14/13, 13/12
| |
| | 11/9
| | 22/17, 13/10
| | 11/8
|-
! | Generator
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
|-
! | Cents*
| | 58.08
| | 162.92
| | 267.76
| | 372.60
| | 477.43
| | 582.27
|-
! | Ratios
| |
| | 11/10
| | 7/6
| | 21/17
| | 21/16
| | 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 (46&amp;58)'''</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Generator
! Generator
! | -17
! &minus;17
! | -16
! &minus;16
! | -15
! &minus;15
! | -14
! &minus;14
! | -13
! &minus;13
! | -12
! &minus;12
|-
|-
! | Cents*
! Cents*
| | 35.19
| 17.73
| | 139.01
| 122.57
| | 242.82
| 227.40
| | 346.63
| 332.24
| | 450.44
| 437.08
| | 554.25
| 541.92
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 13/12
| 15/14
| |
| 8/7
| | 11/9
| 17/14
| | 22/17, 13/10
| 9/7
| | 11/8
| 15/11
|-
|-
! | Generator
! Generator
! | -11
! &minus;11
! | -10
! &minus;10
! | -9
! &minus;9
! | -8
! &minus;8
! | -7
! &minus;7
! | -6
! &minus;6
|-
|-
! | Cents*
! Cents*
| | 58.07
| 46.76
| | 161.88
| 151.60
| | 265.69
| 256.44
| | 369.50
| 361.28
| | 473.32
| 466.12
| | 577.13
| 570.96
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 11/10
| 12/11
| | 7/6
| 15/13
| | 21/17, 26/21
| 16/13
| | 21/16
| 17/13
| | 7/5
| 18/13
|-
|-
! | Generator
! Generator
! | -5
! &minus;5
! | -4
! &minus;4
! | -3
! &minus;3
! | -2
! &minus;2
! | -1
! &minus;1
! | 0
! 0
|-
|-
! | Cents*
! Cents*
| | 80.94
| 75.80
| | 184.75
| 180.64
| | 288.56
| 285.48
| | 392.38
| 390.32
| | 496.19
| 495.16
| | 600.00
| 600.00
|-
|-
! | Ratios
! Ratios
| | 22/21, 21/20
| 22/21
| | 10/9
| 10/9
| | 20/17, 13/11
| 20/17, 13/11
| | 5/4
| 5/4
| | 4/3
| 4/3
| | 24/17, 17/12
| 24/17, 17/12
|-
|-
! | Generator
! Generator
! | 0
! 0
! | 1
! 1
! | 2
! 2
! | 3
! 3
! | 4
! 4
! | 5
! 5
|-
|-
! | Cents*
! Cents*
| | 0.00
| 0.00
| | 103.81
| 104.84
| | 207.62
| 209.68
| | 311.44
| 314.52
| | 415.25
| 419.36
| | 519.06
| 524.20
|-
|-
! | Ratios
! Ratios
| | 1/1
| 1/1
| | 18/17, 17/16, <br>16/15
| 18/17, 17/16, <br />16/15
| | 9/8, 17/15
| 9/8, 17/15
| | 6/5
| 6/5
| | 14/11
| 14/11
| |  
|  
|-
|-
! | Generator
! Generator
! | 6
! 6
! | 7
! 7
! | 8
! 8
! | 9
! 9
! | 10
! 10
! | 11
! 11
|-
|-
! | Cents*
! Cents*
| | 22.87
| 29.04
| | 126.68
| 133.88
| | 230.50
| 238.72
| | 334.31
| 343.56
| | 438.12
| 448.40
| | 541.93
| 553.24
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 15/14, 14/13
| 14/13, 13/12
| | 8/7
|  
| | 17/14
| 11/9
| | 9/7
| 22/17, 13/10
| | 15/11
| 11/8
|-
|-
! | Generator
! Generator
! | 12
! 12
! | 13
! 13
! | 14
! 14
! | 15
! 15
! | 16
! 16
! | 17
! 17
|-
|-
! | Cents*
! Cents*
| | 45.75
| 58.08
| | 149.56
| 162.92
| | 253.37
| 267.76
| | 357.18
| 372.60
| | 460.99
| 477.43
| | 564.81
| 582.27
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 12/11
| 11/10
| | 15/13
| 7/6
| | 16/13
| 21/17
| | 17/13
| 21/16
| | 18/13
| 7/5
|}
|}
<nowiki />* In 17-limit POTE tuning
<nowiki />* In 17-limit POTE tuning
Line 424: Line 272:


== Scales ==
== Scales ==
* [[Srutal12]] - proper [[10L 2s]]
* [[Srutal12]] &ndash; proper [[10L 2s]]
* [[Srutal22]] - improper [[12L 10s]]
* [[Srutal22]] &ndash; improper [[12L 10s]]
* [[Diaschismic12]] - proper [[10L 2s]]
* [[Diaschismic12]] &ndash; proper [[10L 2s]]
* [[Diaschismic22]] - improper [[12L 10s]]
* [[Diaschismic22]] &ndash; improper [[12L 10s]]
* [[Diaschismic34]] - improper [[12L 22s]]
* [[Diaschismic34]] &ndash; improper [[12L 22s]]


== See also ==
== See also ==
* [[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"