Diaschismic extensions
In the 5-limit, diaschismic is a regular temperament (also known as srutal, though they refer to different extensions in higher limits) 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 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. There are multiple ways to extend diaschismic to primes 7, 11, and 13.
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
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.
For technical data on 7-limit and higher-limit srutal: see Diaschismic family #Srutal.
Diaschismic
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.
For technical data on 7-limit and higher-limit diaschismic: see Diaschismic family #Septimal diaschismic.
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. The extensions to prime 11 and 13 can be characterized by parapyth, which makes sense as the fifth is tuned slightly sharp, and prime 17 is found via srutal archagall.
Pajara
Pajara combines diaschismic with archy, tempering the fifth to about 709 cents. The interval of two stacked fifths is equated to 16/7, and the harmonic seventh 7/4 and the just major third 5/4 are separated by a perfect semioctave.
For technical data on 7-limit and higher-limit pajara, see Diaschismic family #Pajara.
Keen
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Todo: complete section |
Intervals
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Todo: cleanup, complete section |
Diaschismic (2.3.5.7.17)
| # | 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
Srutal (17-limit)
| 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
Scales
- Srutal12 – proper 10L 2s
- Srutal22 – improper 12L 10s
- Diaschismic12 – proper 10L 2s
- Diaschismic22 – improper 12L 10s
- Diaschismic34 – improper 12L 22s