Schismic

Schismic, schismatic, or helmholtz is a 5-limit temperament which takes a roughly justly tuned perfect fifth and stacks it eight times to reach 8/5, thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭) and tempering out the schisma, 32805/32768. 5/4 can be respelled as a major third flattened by one Pythagorean comma, and thus, the Pythagorean and syntonic commas are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a microtemperament, as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between 8192/6561 and 5/4, the schisma being tempered out, is approximately 2 cents, which is unnoticeable to most people). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings on both sides of the just interval work fine.

Extensions to schismic include garibaldi, which equates the generalized comma further to 64/63 and 50/49 (tempering out 225/224 and 5120/5103) to provide an efficient framework for 7-limit harmony, and unlike 5-limit schismic performs best with a fifth tuned slightly sharp of just; pontiac, which tempers out 4375/4374 to induce very little damage on schismic harmonies, at the cost of 7 being quite complex; and the 2.3.5.19 subgroup extension nestoria, which equates the minor third to 19/16, major third to 19/15 and 24/19, and the minor second to 19/18 and 20/19 (tempering out 513/512 and 361/360). This page, however, focuses on the basic 5-limit temperament.

See Schismatic family #Schismic, schismatic, a.k.a. helmholtz for technical data.

Interval chain

In the following table, odd harmonics 1–9 and their inverses are in bold.

#	Cents*	Approximate ratios
0	0.00	1/1
1	701.73	3/2
2	203.46	9/8
3	905.19	27/16
4	406.92	81/64
5	1108.65	243/128, 256/135
6	610.38	64/45
7	112.12	16/15
8	813.85	8/5
9	315.58	6/5
10	1017.31	9/5
11	519.04	27/20
12	20.77	81/80

* In 5-limit CWE tuning

Tunings

Target tunings

Target tunings

Target	Minimax		Least squares
	Generator	Eigenmonzo*	Generator	Eigenmonzo*
5-odd-limit	~3/2 = 701.7379 ¢	5/3	~3/2 = 701.728 ¢	[0 -10 17⟩