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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]].
 
=== 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 the 5-limit and 2.3.5.17 temperaments: {{Tdlink|Diaschismic family #Srutal aka diaschismic}}
For technical data on 7-limit and higher-limit diaschismic: see [[Diaschismic family #Septimal diaschismic]].


== 7-limit extensions ==
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.
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.
=== 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.


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


For 7-limit and higher-limit diaschismic: {{Tdlink|Diaschismic family #Diaschismic}}
=== Keen ===
For 7-limit and higher-limit srutal: {{Tdlink|Diaschismic family #Srutal}}
{{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 36: 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 81: 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 267: 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 434: 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"