Diaschismic extensions: Difference between revisions

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! Srutal !! Diaschismic
! Srutal !! Diaschismic
|-
|-
| -8 || 220.3 || 100/81 || || 21/17, 56/45
| −8 || 220.3 || 100/81 || || 21/17, 56/45
|-
|-
| -7 || 440.6 || 125/96 || || '''21/16''', 112/85
| −7 || 440.6 || 125/96 || || '''21/16''', 112/85
|-
|-
| -6 || 660.9 || 25/18 || || 7/5
| −6 || 660.9 || 25/18 || || 7/5
|-
|-
| -5 || 881.1 || 25/24 || || 21/20
| −5 || 881.1 || 25/24 || || 21/20
|-
|-
| -4 || 1101.4 || 10/9, 75/68 || || 28/25
| −4 || 1101.4 || 10/9, 75/68 || || 28/25
|-
|-
| -3 || 1321.7 || 20/17, 32/27 || || 119/100
| −3 || 1321.7 || 20/17, 32/27 || || 119/100
|-
|-
| -2 || 1542.0 || '''5/4''', 34/27, 64/51 || || 63/50
| −2 || 1542.0 || '''5/4''', 34/27, 64/51 || || 63/50
|-
|-
| -1 || 1762.2 || 4/3, 45/34 || || 168/125
| −1 || 1762.2 || 4/3, 45/34 || || 168/125
|-
|-
| 0 || 0.0 || '''1/1''' || ||
| 0 || 0.0 || '''1/1''' || ||
Line 79: Line 79:
! Srutal !! Diaschismic
! Srutal !! Diaschismic
|-
|-
| -8 || 1762.2 || 125/72 || || '''7/4'''
| −8 || 1762.2 || 125/72 || || '''7/4'''
|-
|-
| -7 || 1542.0 || 50/27 || || 28/15, 63/34
| −7 || 1542.0 || 50/27 || || 28/15, 63/34
|-
|-
| -6 || 1321.7 || 100/51, 160/81 || || 168/85
| −6 || 1321.7 || 100/51, 160/81 || || 168/85
|-  
|-  
| -5 || 1101.4 || 40/27, 25/17 || || 112/75, 126/85
| −5 || 1101.4 || 40/27, 25/17 || || 112/75, 126/85
|-
|-
| -4 || 881.1 || '''25/16''', 80/51, 128/81 || || 63/40
| −4 || 881.1 || '''25/16''', 80/51, 128/81 || || 63/40
|-
|-
| -3 || 660.9 || 5/3, 136/81 || || 42/25
| −3 || 660.9 || 5/3, 136/81 || || 42/25
|-
|-
| -2 || 440.6 || 16/9, 30/17 || || 224/125
| −2 || 440.6 || 16/9, 30/17 || || 224/125
|-
|-
| -1 || 220.3 || '''15/8''', 17/9, 32/17 || ||
| −1 || 220.3 || '''15/8''', 17/9, 32/17 || ||
|-
|-
| 0 || 0.0 || 17/12, 24/17, 45/32, 64/45 || ||
| 0 || 0.0 || 17/12, 24/17, 45/32, 64/45 || ||
Line 122: Line 122:
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Generator
! Generator
! | -17
! −17
! | -16
! −16
! | -15
! −15
! | -14
! −14
! | -13
! −13
! | -12
! −12
|-
|-
! | Cents*
! Cents*
| | 17.73
| 17.73
| | 122.57
| 122.57
| | 227.40
| 227.40
| | 332.24
| 332.24
| | 437.08
| 437.08
| | 541.92
| 541.92
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 15/14
| 15/14
| | 8/7
| 8/7
| | 17/14
| 17/14
| | 9/7
| 9/7
| | 15/11
| 15/11
|-
|-
! | Generator
! Generator
! | -11
! −11
! | -10
! −10
! | -9
! −9
! | -8
! −8
! | -7
! −7
! | -6
! −6
|-
|-
! | Cents*
! Cents*
| | 46.76
| 46.76
| | 151.60
| 151.60
| | 256.44
| 256.44
| | 361.28
| 361.28
| | 466.12
| 466.12
| | 570.96
| 570.96
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 12/11
| 12/11
| | 15/13
| 15/13
| | 16/13
| 16/13
| | 17/13
| 17/13
| | 18/13
| 18/13
|-
|-
! | Generator
! Generator
! | -5
! −5
! | -4
! −4
! | -3
! −3
! | -2
! −2
! | -1
! −1
! | 0
! 0
|-
|-
! | Cents*
! Cents*
| | 75.80
| 75.80
| | 180.64
| 180.64
| | 285.48
| 285.48
| | 390.32
| 390.32
| | 495.16
| 495.16
| | 600.00
| 600.00
|-
|-
! | Ratios
! Ratios
| | 22/21
| 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
| | 104.84
| 104.84
| | 209.68
| 209.68
| | 314.52
| 314.52
| | 419.36
| 419.36
| | 524.20
| 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*
| | 29.04
| 29.04
| | 133.88
| 133.88
| | 238.72
| 238.72
| | 343.56
| 343.56
| | 448.40
| 448.40
| | 553.24
| 553.24
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 14/13, 13/12
| 14/13, 13/12
| |  
|  
| | 11/9
| 11/9
| | 22/17, 13/10
| 22/17, 13/10
| | 11/8
| 11/8
|-
|-
! | Generator
! Generator
! | 12
! 12
! | 13
! 13
! | 14
! 14
! | 15
! 15
! | 16
! 16
! | 17
! 17
|-
|-
! | Cents*
! Cents*
| | 58.08
| 58.08
| | 162.92
| 162.92
| | 267.76
| 267.76
| | 372.60
| 372.60
| | 477.43
| 477.43
| | 582.27
| 582.27
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 11/10
| 11/10
| | 7/6
| 7/6
| | 21/17
| 21/17
| | 21/16
| 21/16
| | 7/5
| 7/5
|}
|}
<nowiki />* In 17-limit POTE tuning
<nowiki />* In 17-limit POTE tuning
Line 275: Line 275:
{| 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
| 35.19
| | 139.01
| 139.01
| | 242.82
| 242.82
| | 346.63
| 346.63
| | 450.44
| 450.44
| | 554.25
| 554.25
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 13/12
| 13/12
| |  
|  
| | 11/9
| 11/9
| | 22/17, 13/10
| 22/17, 13/10
| | 11/8
| 11/8
|-
|-
! | Generator
! Generator
! | -11
! &minus;11
! | -10
! &minus;10
! | -9
! &minus;9
! | -8
! &minus;8
! | -7
! &minus;7
! | -6
! &minus;6
|-
|-
! | Cents*
! Cents*
| | 58.07
| 58.07
| | 161.88
| 161.88
| | 265.69
| 265.69
| | 369.50
| 369.50
| | 473.32
| 473.32
| | 577.13
| 577.13
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 11/10
| 11/10
| | 7/6
| 7/6
| | 21/17, 26/21
| 21/17, 26/21
| | 21/16
| 21/16
| | 7/5
| 7/5
|-
|-
! | Generator
! Generator
! | -5
! &minus;5
! | -4
! &minus;4
! | -3
! &minus;3
! | -2
! &minus;2
! | -1
! &minus;1
! | 0
! &minus; 0
|-
|-
! | Cents*
! Cents*
| | 80.94
| 80.94
| | 184.75
| 184.75
| | 288.56
| 288.56
| | 392.38
| 392.38
| | 496.19
| 496.19
| | 600.00
| 600.00
|-
|-
! | Ratios
! Ratios
| | 22/21, 21/20
| 22/21, 21/20
| | 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
| 103.81
| | 207.62
| 207.62
| | 311.44
| 311.44
| | 415.25
| 415.25
| | 519.06
| 519.06
|-
|-
! | 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
Line 379: Line 379:
! | 11
! | 11
|-
|-
! | Cents*
! Cents*
| | 22.87
| 22.87
| | 126.68
| 126.68
| | 230.50
| 230.50
| | 334.31
| 334.31
| | 438.12
| 438.12
| | 541.93
| 541.93
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 15/14, 14/13
| 15/14, 14/13
| | 8/7
| 8/7
| | 17/14
| 17/14
| | 9/7
| 9/7
| | 15/11
| 15/11
|-
|-
! | Generator
! Generator
! | 12
! 12
! | 13
! 13
! | 14
! 14
! | 15
! 15
! | 16
! 16
! | 17
! 17
|-
|-
! | Cents*
! Cents*
| | 45.75
| 45.75
| | 149.56
| 149.56
| | 253.37
| 253.37
| | 357.18
| 357.18
| | 460.99
| 460.99
| | 564.81
| 564.81
|-
|-
! | Ratios
! Ratios
| |  
|  
| | 12/11
| 12/11
| | 15/13
| 15/13
| | 16/13
| 16/13
| | 17/13
| 17/13
| | 18/13
| 18/13
|}
|}
<nowiki />* In 17-limit POTE tuning
<nowiki />* In 17-limit POTE tuning

Revision as of 19:09, 22 October 2024

This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean.

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value).

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&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 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. 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.

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.

Interval chains (WIP)

First period
# Cents* Approximate Ratios
2.3.5.17 subgroup Intervals of 7
Srutal Diaschismic
−8 220.3 100/81 21/17, 56/45
−7 440.6 125/96 21/16, 112/85
−6 660.9 25/18 7/5
−5 881.1 25/24 21/20
−4 1101.4 10/9, 75/68 28/25
−3 1321.7 20/17, 32/27 119/100
−2 1542.0 5/4, 34/27, 64/51 63/50
−1 1762.2 4/3, 45/34 168/125
0 0.0 1/1
1 220.3 16/15, 17/16, 18/17
2 440.6 9/8, 17/15 125/112
3 660.9 6/5, 81/68 25/21
4 881.1 32/25, 51/40, 81/64 80/63
5 1101.4 27/20, 34/25 75/56, 85/63
6 1321.7 51/50, 81/80 85/84
7 1542.0 27/25 15/14, 68/63
8 1762.2 144/125 8/7

* In 3.5.7-targeted DKW tuning for BPS

Second period
# Cents* Approximate Ratios
2.3.5.17 subgroup Intervals of 7
Srutal Diaschismic
−8 1762.2 125/72 7/4
−7 1542.0 50/27 28/15, 63/34
−6 1321.7 100/51, 160/81 168/85
−5 1101.4 40/27, 25/17 112/75, 126/85
−4 881.1 25/16, 80/51, 128/81 63/40
−3 660.9 5/3, 136/81 42/25
−2 440.6 16/9, 30/17 224/125
−1 220.3 15/8, 17/9, 32/17
0 0.0 17/12, 24/17, 45/32, 64/45
1 1762.2 3/2, 68/45 125/84
2 1542.0 8/5, 27/17, 51/32 100/63
3 1321.7 17/10, 27/16 200/119
4 1101.4 9/5, 136/75 25/14
5 881.1 48/25 40/21
6 660.9 36/25 10/7
7 440.6 192/125 32/21, 85/56
8 220.3 81/50 34/21, 45/28

* In 3.5.7-targeted DKW tuning for BPS

Intervals

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


Intervals of diaschismic (46 & 58)
Generator −17 −16 −15 −14 −13 −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 −11 −10 −9 −8 −7 −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 −5 −4 −3 −2 −1 − 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,
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

* In 17-limit POTE tuning

Scales

See also

Retrieved from "https://en.xen.wiki/index.php?title=Diaschismic_extensions&oldid=161914"