Diaschismic extensions: Difference between revisions
will need to rework this into an actual page on srutal/diaschismic (in 2.3.5, 2.3.5.17, and 7-lim) rather than a "compare and contrast" |
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== Scales == | == Scales == | ||
* [[Srutal12]] - [[10L 2s]] | * [[Srutal12]] - proper [[10L 2s]] | ||
* [[Srutal22]] - improper [[12L 10s]] | * [[Srutal22]] - improper [[12L 10s]] | ||
* [[Diaschismic12]] - 10L 2s | * [[Diaschismic12]] - proper [[10L 2s]] | ||
* [[Diaschismic22]] - improper 12L 10s | * [[Diaschismic22]] - improper [[12L 10s]] | ||
* [[Diaschismic34]] - improper [[12L 22s]] | * [[Diaschismic34]] - improper [[12L 22s]] | ||
Revision as of 13:08, 19 October 2024
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This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean. |
Srutal and diaschismic, both discussed at diaschismic family, are two different temperaments in the 7-limit. This page compares and contrasts them in detail.
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&22), keen (22&34), srutal (34d&46) and diaschismic (46&58).
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
| 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
| 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
- Srutal12 - proper 10L 2s
- Srutal22 - improper 12L 10s
- Diaschismic12 - proper 10L 2s
- Diaschismic22 - improper 12L 10s
- Diaschismic34 - improper 12L 22s