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 ==
== 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 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 srutal: see [[Diaschismic family #Srutal]].


== Interval chains ==
=== 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.
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* !! colspan="2" | Approximate Ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
|-
|-
! rowspan="2" | 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 72: Line 77:
|+ style="font-size: 105%;" | Second period
|+ style="font-size: 105%;" | Second period
|-
|-
! rowspan="3" | &#35; !! rowspan="3" | Cents* !! colspan="2" | Approximate Ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
|-
|-
! rowspan="2" | 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 262: 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 429: 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]]

Latest revision as of 14:41, 28 May 2026

< Diaschismic

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, 11, and 13.

7-limit extensions

The two alternative names for this temperament are assigned to different strong extensions to the 7-limit: srutal (34d&46) and diaschismic (46&58), though there are other mappings that are comparable in complexity and error: pajara (12&22) and keen (22&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 6/5s 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 5/4s 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-, 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 and 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: complete section

Intervals

Todo: cleanup, complete section

Diaschismic (2.3.5.7.17)

First period
# Cents* Approximate Ratios
2.3.5.17 subgroup Intervals of 7
−8 370.6 100/81 21/17, 56/45
−7 474.2 125/96 21/16, 112/85
−6 577.9 25/18 7/5
−5 81.6 25/24 21/20
−4 185.3 10/9, 75/68 28/25
−3 289.0 20/17, 32/27 119/100
−2 392.6 5/4, 34/27, 64/51 63/50
−1 496.3 4/3, 45/34 168/125
0 0.0 1/1 126/125
1 103.7 16/15, 17/16, 18/17
2 207.4 9/8, 17/15 125/112
3 311.0 6/5, 81/68 25/21
4 414.7 32/25, 51/40, 81/64 80/63
5 518.4 27/20, 34/25 75/56, 85/63
6 22.1 51/50, 81/80 85/84
7 125.8 27/25 15/14, 68/63
8 229.5 144/125 8/7

* In 7-limit POTE tuning

Second period
# Cents* Approximate Ratios
2.3.5.17 subgroup Intervals of 7
−8 970.6 125/72 7/4
−7 1074.2 50/27 28/15, 63/34
−6 1177.9 100/51, 160/81 168/85
−5 681.6 40/27, 25/17 112/75, 126/85
−4 785.3 25/16, 80/51, 128/81 63/40
−3 889.0 5/3, 136/81 42/25
−2 992.6 16/9, 30/17 224/125
−1 1096.3 15/8, 17/9, 32/17
0 600.0 17/12, 24/17, 45/32, 64/45
1 703.7 3/2, 68/45 125/84
2 807.4 8/5, 27/17, 51/32 100/63
3 911.0 17/10, 27/16 200/119
4 1014.7 9/5, 136/75 25/14
5 1118.4 48/25 40/21
6 622.1 36/25 10/7
7 725.8 192/125 32/21, 85/56
8 829.5 81/50 34/21, 45/28

* In 7-limit POTE tuning

Srutal (17-limit)

Intervals of srutal (34d & 46)
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,
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

* In 17-limit POTE tuning

Scales

See also

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