Schismic

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

Schismic

Subgroups	2.3.5
Comma basis	32805/32768
Reduced mapping	⟨1; 1 -8]
ET join	12 & 53
Generators (CWE)	~3/2 = 701.731 ¢
MOS scales	2L 3s, 5L 2s, 5L 7s, 12L 5s
Ploidacot	monocot
Pergen	(P8, P5)
Color name	Layoti
Minimax error	5-odd-limit: 0.217 ¢; 5-limit 125-odd-limit: 0.837 ¢
Target scale size	5-odd-limit: 12 notes; 5-limit 125-odd-limit: 29 notes

Schismic, schismatic, or helmholtz is a 5-limit temperament which takes an almost just perfect fifth and stacks it eightfold to reach 8/5, mapping 5/4 to the diminished fourth (e.g. C–F♭) and tempering out the schisma, 32805/32768.

5/4 maps equivalently to a major third minus one Pythagorean comma, and thus, the Pythagorean and syntonic commas are equated into one tempered comma, splitting octaves into two diatonic major thirds and one downmajor third representing 5/4.

Schismic is one of the simplest microtemperaments, as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the schisma is practically unnoticeable). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings with sharper fifths (and worse 5-limit, like in 41- or 94edo) still work fine.

Extensions of schismic include garibaldi and pontiac. Garibaldi 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, though with worse 5-limit intonation since the tuning favors slightly sharp fifths; pontiac, which tempers out 4375/4374 to induce very little damage on schismic harmonies, at the cost of 7 being quite complex. Besides these, there is 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).

A notable example of a weak extension is sesquiquartififths, which tempers out 2401/2400 and splits the fifth in fourths, inducing very little damage with a less complex mapping of 7 at the cost of quadrupling the complexity of 3 and 5.

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 diatonic tertian harmony in chain-of-fifths notation; The just major triad on C is not C–E–G like in meantone, but rather C–F♭–G. 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

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: The later mos names are proposals that can be found on the page TAMNAMS Extension.

• 5L 7s (p-chromatic)
• 12L 5s (p-enharmonic)
• 12L 17s (pythagotonic)
• 12L 29s (pythamystonic)
• 12L 41s (antipythomerc)
• 53L 12s (m-chro antipythomerc)

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
	45/32	701.6294	1/6-comma
	15/8	701.6759	1/7-comma
69\118		701.6949
	5/4	701.7108	1/8-comma, lower bound of 5-odd-limit diamond tradeoff
	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
	81/80	701.7922	1/12-comma
31\53		701.8868
	3/2	701.9550	Pythagorean tuning, upper bound of 5-odd-limit diamond tradeoff
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