Cassaschismic

Cassaschismic

Subgroups	2.3.5.7.11.13.19
Comma basis	19712/19683, 41503/41472
Reduced mapping	⟨1; 0 1 0 -14 23 12 5; 0 0 1 0 0 -1 1]
ET join	217 & 270 & 311
Generators (TBA)
MOS scales	2L 3s, 5L 2s, 5L 7s, 12L 5s
Ploidacot	Haploid monocot
Pergen	(P8, P5, ^1)
Color name	Salozo & Sasaru
Minimax error	15-odd-limit: TBA ¢; 13-limit 225-odd-limit: TBA ¢
Target scale size	15-odd-limit: TBD notes; 13-limit 225-odd-limit: TBD notes

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community. Terms: Many of the information and terminology here has been ported and adapted from Eufalesio's user page "Ultimate" and his own TAMNAMS extensions.

Cassaschismic is a rank-3 temperament for the 11-, 13-limit and beyond. It is a member of garischismic clan and olympic clan. It extends the chain-of-fifths sequence of gary into the full 11-limit by adding an independent generator for 5/4, which naturally can be then used for 13/8 and 19/16.

By moving the generators around, this generator can be instead taken to be a tiny 3~5c minicomma (from this point forward refered to as "MC") that represents 385/384, 352/351, 5120/5103, 513/512, the schisma... etc; acting as a rank-3 detemper of cassandra, where cassandra mappings are modified by ±1 MC to reach the rest of primes.

The pergen is (P8, P5, ^1), where ^1 is the MC. 4:5:6:7:9:11:13 is notated as P1 ^↓M3 P5 ↓m7 M9 ↑↑11 v↑↑m13, where ↑/↓ represents alteration by a pyth-septimal comma (from this point forward refered to as "PC")

It is supported by notable edos such as 41, 53, 94, 217, 270, 311. 41 and 53 are known for their incredibly accurate fifths; 270edo provides an astonishingly accurate equal tuning for cassaschismic in the no-17 19-odd-limit; 311edo notably provides an optimized extension to the full 41-odd-limit with a slightly worse 13-limit. It is also trivially supported by 12edo through the 12e val, where the pythagorean comma and MC are tempered out. See Garischismic family#Cassaschismic for technical data.

The temperament is also known by Kite Giedraitis and Eufalesio as Ultimate, being nicknamed as such in 2026.

Interval chain

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 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.

	MC-span
Fifth-span	-1	0
0	720/361	1/1
1	256/171	3/2
2	64/57	9/8
3	32/19	27/16
4	24/19	81/64
5	36/19	243/128
6	64/45	729/512
7	16/15	77/72
8	8/5	77/48
9	6/5	77/64
10	9/5	65/36
11	27/20	65/48
12	81/80	64/63
13	243/160	32/21
14	729/640	8/7
15	416/243	12/7
16	104/81	9/7
17	52/27	27/14
18	13/9	81/56
19	13/12	88/81
20	13/8	44/27
21	39/32	11/9
22	64/35	11/6
23	48/35	11/8
24	36/35	33/32

Scales

• 5L 7s (p-chromatic)
• 12L 5s (p-enharmonic)
• 12L 17s (pythagotonic)
• 12L 29s (pythamystonic)
• 41L 12s (pythomerc)
• 41L 53s (garytonic)

Notation

Much like schismic, using cassaschismic can be a challenge because it defies the tradition of diatonic tertian harmony in chain-of-fifths notation; The just major triad on C is not C–E–G like in meantone, and it isn't C–F♭–G or C-↓E-G like in schismic either. Because it is a rank-3 temperament, it needs two extra pairs of accidentals, one for the PC (like ↑/↓), and one for the MC (like ^/v).

It can be instead see as an "addition" to the schismic/garibaldi/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 accurate (41,53,94), to incredibly accurate (217,270,311...).

Tunings

TBA