Septischismic

Septischismic

Subgroups	2.3.5.7.11, 2.3.5.7.11.13, 2.3.5.7.11.13.19
Comma basis	19712/19683, 41503/41472 (11-limit); 2080/2079, 4096/4095, 19712/19683 (13-limit); 1216/1215, 1540/1539, 1729/1728, 2080/2079 (2.3.5.7.11.13.19)
Reduced mapping	⟨1; 1 0 -14 23 12 5; 0 1 0 0 -1 1]
ET join	41 & 53 & 270
Generators (CWE)	~3/2 = 702.2307 ¢, ~5/4 = 386.3245 ¢
MOS scales	n/a
Ploidacot	n/a
Pergen	(P8, P5, ^1)
Color name	Salozo & Sasaru + Ya Salozo & Sasaru (& Sathoyo (& Sanogu))
Minimax error	11-odd-limit: 0.588 ¢
Target scale size	11-odd-limit: ? notes

Septischismic is a rank-3 temperament that expands gary's chain of fifths 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.

The generator for 5 can be used for 13 and 19. By moving the generators around, it can also be taken to be a ~4.5 ¢ generic aberschisma, which represents the schisma, the aberschisma, the undevicesimal schisma, the major minthma and many other important commas around that size. Tempering out this aberschisma results in cassandra, so septischismic is a rank-3 detemperament of it, modifying its mapping by ±1 aberschisma to reach primes 5, 13, and 19.

Other rank-2 temperaments of septischismic include cotoneum, newt, gariwizmic, satin, vulture, paramity and heptacot; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.

Generators needed to reach the aberschisma

• Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;
• Newt (41 & 270): -41 hemififths;
• Gariwizmic (94 & 270): +53 fifths (mercator comma) - 1/2 pythagorean comma;
• Satin (94 & 217): -94 1/3-fourths;
• Vulture (53 & 217): -41 1/4-fifths;
• Paramity (53 & 311): -53 1/5-elevenths;
• Heptacot (12e & 311): 12 1/7-fifths.

Septischismic 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. 12edo supports it trivially through the 12e val, where both the comma step and the aberschisma step are tempered out. It can be used in any of those forms.

See Septischismic family #Septischismic 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.

An image is provided instead showing copies of a Standard Lumatone mapping for Pythagorean-like arrangement of the chain of fifths, copied thrice to access most of the no-17 27-odd-limit and then some.

A visualization of septischismic, with 3 copies of a chain of fifths laid out in a Lumatone-like fashion. You see and download the Blender file used to make this picture here.

#	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

Septischismic 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.

Nomenclature of selected intervals

Ratio	Example on C
3/2	C–G (perfect fifth)
5/4	C–^↓E (upsubmajor third)
7/4	C–↓Bb (subminor seventh)
11/8	C–↑↑F (hyperfourth)
13/8	C–v↑↑Ab (downhyperminor sixth)
19/16	C–^Eb (upminor third)

See also

• User:Eufalesio/Ultimate – An opinion-based derivation of septischismic

External links

• Ultimate mapgraph – A Desmos graph by Eufalesio showing Septischismic edos up to 311, harmonic mode 8 (green), and 5L 7s 6|5 (red). The purple line on 12 is the patent val prime 11, which is not used in Septischismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.