Diaschismic
Diaschismic, [10 & 12], is the temperament (usually 2.3.5.17) generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.
12edo and 22edo are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are 34edo and 46edo.
Extensions
Archytas (10 & 12) (Pajara)
Main article: Pajara
The edo join [10 & 12] given above represents the extension pajara, which tempers out 64/63 such that a sharpened generator of about 110 cents is also 15/14 and stacks twice to a flatly tuned 8/7. The semioctave is interpreted as 7/5 and 10/7. This is a particularly common practical 7-limit interpretation of Diaschismic supported by most of the small diaschismic edos, and especially 22edo.
Further extensions of Pajara are discussed on its own page.
Aberschismic (46 & 58)
A more accurate but complex option is to interpret Diaschismic as a Hemifamity temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).
In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in Pele; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.
Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.
Only the mappings for the 2.3.5.7.17 group will be given here.
Generator chain
| Period 1 | Period 2 | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Up | Down | Up | Down | ||||||
| # | Cents | JI | Cents | JI | # | Cents | JI | Cents | JI |
| 0 | 0 | 1/1 | 600 | 24/17, 64/45 | 0 | 600 | 17/12, 45/32 | 1200 | 2/1 |
| 1 | 104 | 18/17, 17/16, 16/15 | 496 | 4/3 | 1 | 704 | 3/2 | 1096 | 15/8, 32/17, 17/9 |
| 2 | 208 | 9/8 | 392 | 5/4 | 2 | 808 | 8/5 | 992 | 16/9 |
| 3 | 312 | 6/5 | 288 | 32/27 | 3 | 912 | 27/16 | 888 | 5/3 |
| 4 | 416 | 81/64 | 184 | 10/9 | 4 | 1016 | 9/5 | 784 | 128/81 |
| 5 | 520 | 27/20 | 80 | 21/20, 25/24 | 5 | 1120 | 40/21, 48/25 | 680 | 40/27 |
| 6 | 24 | 81/80, 64/63 | 576 | 7/5 | 6 | 624 | 10/7 | 1176 | 63/32, 160/81 |
| 7 | 128 | 27/25, 15/14 | 472 | 21/16 | 7 | 728 | 32/21 | 1072 | 50/27, 28/15 |
| 8 | 232 | 8/7 | 368 | 21/17 | 8 | 832 | 34/21 | 968 | 7/4 |
| 9 | 336 | 17/14 | 264 | 7/6 | 9 | 936 | 12/7 | 864 | 28/17 |
| 10 | 440 | 9/7 | 160 | 35/32 | 10 | 1040 | 51/28 | 760 | 14/9 |
List of patent vals
The following patent vals support 2.3.5 Diaschismic, not including vals contorted in 2.3.5.
| Edo | Extension | Generator (3/2) |
|---|---|---|
| 2 | 600.000 | |
| 14 | 685.714 | |
| 12 | Pajara, Septimal Diaschismic | 700.000 |
| 70 | Septimal Diaschismic | 702.857 |
| 58 | Septimal Diaschismic | 703.448 |
| 46 | Septimal Diaschismic (alternate 11 mapping) | 704.348 |
| 126 | (alternate 11 mapping) | 704.762 |
| 80 | (alternate 11 mapping) | 705.000 |
| 114 | 705.263 | |
| 148 | 705.405 | |
| 34 | Septimal Diaschismic (alternate 11 mapping) | 705.882 |
| 124 | 706.452 | |
| 90 | 706.667 | |
| 56 | 707.143 | |
| 78 | 707.692 | |
| 22 | Pajara | 709.091 |
| 54 | Pajara | 711.111 |
| 32 | Pajara | 712.500 |
| 10 | Pajara | 720.000 |
Shrub temperament
Shrub temperament is a restriction of diaschismic temperament to the 2.3.25 subgroup, removing the tritone. It bridges the gap between gentle and superpyth tunings, with reasonable tunings around 707 cents, which leads to a 428 cents major third representing 32/25 and a 279 cents minor third representing 75/64. It may also set the major third to 23/18 and 41/32. Overall, however, this largely ends up just functioning as an excuse to tune the fifth to 707 cents, which should be done directly if no particular JI interpretation is desired.
Interestingly, the comma 82/81 is the next superparticular up after the meantone comma 81/80. Fittingly, shrub tunings are detuned the same amount from 3/2 as meantone tunings, but in the opposite direction.
