Diaschismic extensions: Difference between revisions

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m Lériendil moved page Srutal vs diaschismic to Srutal and diaschismic: remaking it into an article on the actual diaschismic temperament; similar principles apply in naming to "hanson and cata"
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{{URWTC}}
{{URWTC}}
{{Infobox Regtemp
| Title = Srutal; srutal archagall
| Subgroups = 2.3.5, 2.3.5.17
| Comma basis = [[15625/15552]] (2.3.5); <br> [[325/324]], [[625/624]] (2.3.5.13)
| 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''' and '''diaschismic''', both discussed at [[diaschismic family]], are two different temperaments in the 7-limit. This page compares and contrasts them in detail.
'''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]].


Temperaments of diaschismic family has a half-octave period and tempers out 2048/2025, [[diaschisma]]. Not only the fifth or fourth, but also the diatonic semitone (~16/15) can be used as a generator. Extending diaschismic temperament to the 7-limit, there are several mappings that are comparable in complexity and error: [[pajara]] (12&amp;22), keen (22&amp;34), srutal (34d&amp;46) and diaschismic (46&amp;58).
== 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).


In srutal, 7/4 is represented by 15 diatonic semitones minus half octave, or five 6/5s minus half octave.
In srutal, 7/4 is represented by 15 diatonic semitones minus half octave, or five 6/5s minus half octave.

Revision as of 00:35, 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).

In srutal, 7/4 is represented by 15 diatonic semitones minus half octave, or five 6/5s minus half octave.

In diaschismic, 7/4 is represented by one and half octaves minus 8 diatonic semitones, or four 5/4s minus half octave.

They 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.

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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