Diaschismic extensions
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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)
| # | Cents* | Approximate Ratios | ||
|---|---|---|---|---|
| 2.3.5.17 subgroup | Intervals of 7 | |||
| Srutal | Diaschismic | |||
| -8 | 220.3 | |||
| -7 | 440.6 | |||
| -6 | 660.9 | |||
| -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 | ||
| 2 | 440.6 | 9/8, 17/15 | ||
| 3 | 660.9 | 6/5, 81/68 | ||
| 4 | 881.1 | 32/25, 51/40, 81/64 | ||
| 5 | 1101.4 | 27/20, 34/25 | ||
| 6 | 1321.7 | 51/50, 81/80 | ||
| 7 | 1542.0 | 27/25 | ||
| 8 | 1762.2 | 144/125 | ||
| # | Cents* | Approximate Ratios | ||
|---|---|---|---|---|
| 2.3.5.17 subgroup | Intervals of 7 | |||
| Srutal | Diaschismic | |||
| 1 | 1681.7 | 45/17, 119/45, 459/175 | ||
| 2 | 1461.4 | 7/3, 81/35 | ||
| 3 | 1241.1 | 35/17, 51/25, 243/119 | ||
| 4 | 1020.8 | 9/5, 49/27 | ||
| 5 | 800.6 | 27/17, 119/75, 245/153 | ||
| 6 | 580.3 | 7/5, 243/175 | ||
| 7 | 360.0 | 21/17, 153/125 | ||
| 8 | 139.7 | 27/25, 49/45 | ||
| 0 | 0.0 | 1/1 | ||
| 1 | 1681.7 | 45/17, 119/45, 459/175 | ||
| 2 | 1461.4 | 7/3, 81/35 | ||
| 3 | 1241.1 | 35/17, 51/25, 243/119 | ||
| 4 | 1020.8 | 9/5, 49/27 | ||
| 5 | 800.6 | 27/17, 119/75, 245/153 | ||
| 6 | 580.3 | 7/5, 243/175 | ||
| 7 | 360.0 | 21/17, 153/125 | ||
| 8 | 139.7 | 27/25, 49/45 | ||
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