Srutal and diaschismic
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| Srutal; srutal archagall |
136/135, 256/255 (2.3.5.17)
((2.3.5.17) 25-odd limit) ???c
((2.3.5.17) 25-odd limit) 22 notes
Srutal, known interchangeably as diaschismic in the 5-limit, is a regular temperament defined by tempering out the comma 2048/2025 = [11 -4 -2⟩, 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.
For technical data, see Diaschismic family #Srutal aka diaschismic.
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.
For technical data on 7-limit and higher-limit diaschismic: see Diaschismic family #Diaschismic.
For technical data on 7-limit and higher-limit srutal: see Diaschismic family #Srutal.
Interval chains
Diaschismic
| # | Cents* | Approximate Ratios | |
|---|---|---|---|
| 2.3.5.17 subgroup | Intervals of 7 | ||
| −8 | 370.6 | 100/81 | 21/17, 56/45 |
| −7 | 474.2 | 125/96 | 21/16, 112/85 |
| −6 | 577.9 | 25/18 | 7/5 |
| −5 | 81.6 | 25/24 | 21/20 |
| −4 | 185.3 | 10/9, 75/68 | 28/25 |
| −3 | 289.0 | 20/17, 32/27 | 119/100 |
| −2 | 392.6 | 5/4, 34/27, 64/51 | 63/50 |
| −1 | 496.3 | 4/3, 45/34 | 168/125 |
| 0 | 0.0 | 1/1 | 126/125 |
| 1 | 103.7 | 16/15, 17/16, 18/17 | |
| 2 | 207.4 | 9/8, 17/15 | 125/112 |
| 3 | 311.0 | 6/5, 81/68 | 25/21 |
| 4 | 414.7 | 32/25, 51/40, 81/64 | 80/63 |
| 5 | 518.4 | 27/20, 34/25 | 75/56, 85/63 |
| 6 | 22.1 | 51/50, 81/80 | 85/84 |
| 7 | 125.8 | 27/25 | 15/14, 68/63 |
| 8 | 229.5 | 144/125 | 8/7 |
* In 7-limit POTE tuning
| # | Cents* | Approximate Ratios | |
|---|---|---|---|
| 2.3.5.17 subgroup | Intervals of 7 | ||
| −8 | 970.6 | 125/72 | 7/4 |
| −7 | 1074.2 | 50/27 | 28/15, 63/34 |
| −6 | 1177.9 | 100/51, 160/81 | 168/85 |
| −5 | 681.6 | 40/27, 25/17 | 112/75, 126/85 |
| −4 | 785.3 | 25/16, 80/51, 128/81 | 63/40 |
| −3 | 889.0 | 5/3, 136/81 | 42/25 |
| −2 | 992.6 | 16/9, 30/17 | 224/125 |
| −1 | 1096.3 | 15/8, 17/9, 32/17 | |
| 0 | 600.0 | 17/12, 24/17, 45/32, 64/45 | |
| 1 | 703.7 | 3/2, 68/45 | 125/84 |
| 2 | 807.4 | 8/5, 27/17, 51/32 | 100/63 |
| 3 | 911.0 | 17/10, 27/16 | 200/119 |
| 4 | 1014.7 | 9/5, 136/75 | 25/14 |
| 5 | 1118.4 | 48/25 | 40/21 |
| 6 | 622.1 | 36/25 | 10/7 |
| 7 | 725.8 | 192/125 | 32/21, 85/56 |
| 8 | 829.5 | 81/50 | 34/21, 45/28 |
* In 7-limit POTE tuning
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