Schismic

This page is about a regular temperament sometimes known as "helmholtz". For the music theorist, see Hermann von Helmholtz.

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

Notation

Using schismic can be a challenge because it defies the tradition of tertian harmony in chain-of-fifths notation. The just major triad on C is C–F♭–G, for example. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.

Scales

Scala files

• Clipper32805 – in a 1–3–5 equal-beating tuning

Tunings

Norm-based tunings

5-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 701.7187 ¢	CWE: ~3/2 = 701.7308 ¢	POTE: ~3/2 = 701.7359 ¢

Target tunings

Delta-rational tunings

Optimized chord	Generator value	Polynomial	Further notes
3:4:5 (+1 +1)	~3/2 = 701.6910 ¢	g9 - 4g8 + 64 = 0	1–3–5 equal-beating tuning
4:5:6 (+1 +1)	~3/2 = 701.7278 ¢	g9 + g8 - 64 = 0	1–3–5 equal-beating tuning

Odd-limit-based 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⟩

Tuning spectrum

Edo generator	Eigenmonzo (Unchanged-interval)*	Generator (¢)	Comments
7\12		700.0000	Lower bound of 5-limit 9-odd-limit diamond monotone
52\89		701.1236
45\77		701.2987
38\65		701.5385
	15/8	701.6759	1/7-comma
69\118		701.6949
	5/4	701.7108	1/8-comma
	25/24	701.7252	2/17-comma
169\289		701.7301
	5/3	701.7379	1/9-comma, 5-odd-limit minimax
100\171		701.7544
	9/5	701.7596	1/10-comma
31\53		701.8868
	3/2	701.9550	Pythagorean tuning
24\41		702.4390
17\29		703.4483
10\17		705.8824	Upper bound of 5-limit 9-odd-limit diamond monotone

* Besides the octave

External links

• Schismic Temperaments by Graham Breed