Diaschismic extensions: Difference between revisions

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| 8 || 1762.2 || 144/125 || || 8/7, 135/119
| 8 || 1762.2 || 144/125 || || 8/7, 135/119
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* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
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| 8 || 220.3 || 81/50
| 8 || 220.3 || 81/50
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* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
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| | 7/5
| | 7/5
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<nowiki>*</nowiki> in 17-limit POTE tuning
<nowiki />* In 17-limit POTE tuning
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| | 18/13
| | 18/13
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<nowiki>*</nowiki> in 17-limit POTE tuning
<nowiki />* In 17-limit POTE tuning
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</div></div>


Revision as of 18:43, 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.

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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 naturally 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
-7 440.6 125/96
-6 660.9 25/18
-5 881.1 25/24
-4 1101.4 10/9, 75/68
-3 1321.7 20/17, 32/27
-2 1542.0 5/4, 34/27, 64/51
-1 1762.2 4/3, 45/34
0 0.0 1/1
1 220.3 16/15, 17/16, 18/17 200/189
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, 150/119
5 1101.4 27/20, 34/25 75/56, 85/63
6 1321.7 51/50, 81/80 85/84, 120/119, 225/224
7 1542.0 27/25 15/14, 68/63, 128/119
8 1762.2 144/125 8/7, 135/119

* 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 1542.0 50/27
-6 1321.7 100/51, 160/81
-5 1101.4 40/27, 25/17
-4 881.1 25/16, 80/51, 128/81
-3 660.9 5/3, 136/81
-2 440.6 16/9, 30/17
-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
2 1542.0 8/5, 27/17, 51/32
3 1321.7 17/10, 27/16
4 1101.4 9/5, 136/75
5 881.1 48/25
6 660.9 36/25
7 440.6 192/125
8 220.3 81/50

* 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

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