Cassaschismic
Cassaschismic is a rank-3 temperament that expands the chain of fifths of gary into the full 11-limit by adding an independent generator for the 5th harmonic. It is therefore a member of the garischismic family and olympic clan.
| Cassaschismic |
2080/2079, 4096/4095, 19712/19683 (13-limit);
1216/1215, 1540/1539, 1729/1728,
2080/2079 (2.3.5.7.11.13.19)
By moving the generators around, the generator for 5 can be used for 13 and 19. It can also be taken to be a 3–5 ¢ generic aberschisma, which represents the schisma, the aberschisma, the undevicesimal schisma, and many other important commas around that size. Tempering out this tiny interval results in cassandra, so cassaschismic may be viewed as a rank-3 detemperament thereof, modifying its mapping by ±1 aberschisma step to reach the rest of primes.
Other rank-2 temperaments of cassaschismic include cotoneum, gariwizmic, newt, satin, and vulture; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.
Cassaschismic is supported by notable equal temperaments such as 217, 270, 311, and 364, where the aberschisma step is well represented by one edostep. It is also trivially supported by edos of cassandra, these being 41, 53, 94, and of course, 12edo through the 12e val, where both the comma step and the aberschisma step are tempered out, so it can be used in any of those forms as well.
See Garischismic family #Cassaschismic for technical data.
Interval lattice
Here is a quick compressed cheat sheet of octave-reduced intervals. This is a simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to 8 fifths - 1 aberschisma step, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 aberschisma step. There are no octave reduced primes or prime inverses with positive fifth step and aberschisma step.
| # | Aberschisma offset -1 | Aberschisma offset 0 | ||
|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 1195.83 | 351/176 | 0.00 | 1/1 |
| 1 | 698.06 | 256/171 | 702.23 | 3/2 |
| 2 | 200.29 | 64/57 | 204.46 | 9/8 |
| 3 | 902.52 | 32/19 | 906.69 | 27/16 |
| 4 | 404.75 | 24/19 | 408.92 | 19/15 |
| 5 | 1106.98 | 36/19 | 1111.15 | 19/10 |
| 6 | 609.21 | 27/19 | 613.38 | 57/40 |
| 7 | 111.44 | 16/15 | 115.62 | 77/72 |
| 8 | 813.68 | 8/5 | 817.85 | 77/48 |
| 9 | 315.91 | 6/5 | 320.08 | 77/64 |
| 10 | 1018.14 | 9/5 | 1022.31 | 65/36 |
| 11 | 520.37 | 27/20 | 524.54 | 65/48 |
| 12 | 22.60 | 81/80 | 26.77 | 64/63 |
| 13 | 724.83 | 38/25 | 729.00 | 32/21 |
| 14 | 227.06 | 57/50 | 231.23 | 8/7 |
| 15 | 929.29 | 77/45 | 933.46 | 12/7 |
| 16 | 431.52 | 77/60 | 435.69 | 9/7 |
| 17 | 1133.75 | 52/27 | 1137.92 | 27/14 |
| 18 | 635.98 | 13/9 | 640.15 | 81/56 |
| 19 | 138.21 | 13/12 | 142.38 | 88/81 |
| 20 | 840.44 | 13/8 | 844.61 | 44/27 |
| 21 | 342.67 | 39/32 | 346.85 | 11/9 |
| 22 | 1044.91 | 64/35 | 1049.08 | 11/6 |
| 23 | 547.14 | 48/35 | 551.31 | 11/8 |
| 24 | 49.37 | 36/35 | 53.54 | 33/32 |
- In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced
Notation
Cassaschismic is easily notated with chain-of-fifths notation with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to highly accurate (217, 270, 311, …).
As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. In this scheme, 4:5:6:7:9:11:13 on a C is notated as C–^↓E–G–↓B♭–D–↑↑F–v↑↑A♭.