Schismic

Schismic

Subgroups	2.3.5, 2.3.5.19
Reduced mapping	⟨1; 1 -8 -3]
ET join	12 & 29
Generators (CWE)	~3/2 = 701.7¢
MOS scales	5L 2s, 5L 7s, 12L 5s
Ploidacot	monocot
Comma basis	32805/32768 (2.3.5); 361/360, 513/512 (2.3.5.19)
Minimax error	5-odd-limit: 0.217¢
Target scale size	5-odd-limit: 12 notes

Schismic or schismatic[a], [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the schisma, which is about 2 cents; this means that Schismic can be tuned to perfect Pythagorean tuning (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent.

The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third (and consequently 19/15 with the diatonic major third), tempering out 513/512.

The primary MOS to use for schismic is schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting porcupine.

Extensions

Prime 7

The primary 7-limit extension of schismic is garibaldi (also 12 & 29), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just), but is still more accurate than meantone as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category.

Primes 11 and 13

A reasonable extension to the 11-limit assuming garibaldi is cassandra (41 & 53), which sets 33/32 to twice the garibaldi comma; alternatively there is andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.

Prime 17

Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).

Intervals

These interpretations assume cassandra (in Pythagorean tuning).

Up from the unison			Down from the octave
#	Cents	JI	#	Cents	JI
0	0.00	1/1	0	1,200.00	2/1
1	701.96	3/2	1	498.04	4/3
2	203.91	9/8	2	996.09	16/9
3	905.87	32/19, 27/16	3	294.13	19/16
4	407.82	19/15	4	792.18	19/12
5	1,109.78	19/10	5	90.22	19/18, 20/19, 21/20
6	611.73	10/7	6	588.27	7/5
7	113.69	16/15, 15/14	7	1,086.31	15/8
8	815.64	8/5	8	384.36	5/4
9	317.60	5/3	9	882.40	6/5
10	1,019.55	9/5	10	180.45	10/9
11	521.51	27/20	11	678.49	40/27
12	23.46	50/49	12	1,176.54	49/25
13	725.42	32/21	13	474.58	21/16
14	227.37	8/7	14	972.63	7/4
15	929.33	12/7	15	270.67	7/6
16	431.28	9/7	16	768.72	14/9
17	1,133.24	28/27	17	66.76	27/14
18	635.19	13/9	18	564.81	18/13
19	137.15	13/12	19	1,062.85	24/13
20	839.10	13/8	20	360.90	16/13
21	341.06	11/9	21	858.94	18/11
22	1,043.01	11/6	22	156.99	12/11
23	544.97	11/8	23	655.03	16/11
24	46.92	33/32	24	1,153.08	64/33
25	748.88	54/35	25	451.12	35/27
26	250.83	81/70	26	949.17	140/81
27	952.79	26/15	27	247.21	15/13
28	454.74	13/10	28	745.26	20/13
29	1,156.70	39/20	29	43.30	40/39

[a] Despite ending in -isma, schisma is a 3rd-declension comma name, therefore its temperaments are schismatic and schismic (which, as it is a 2.3.5 comma, refer to the same temperament).