The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is [-15 8 1⟩, and flipping that yields ⟨⟨ 1 -8 -15 ]] for the wedgie. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth. In fact, 10 = (4/3)⁸ × 32805/32768.

Schismatic aka Helmholtz

The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz, schismic or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismatic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.

Subgroup: 2.3.5

Comma list: 32805/32768

Mapping: [⟨1 0 15], ⟨0 1 -8]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 701.736

Tuning ranges:

valid range: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
nice range: ~3/2 = [701.711, 701.955]
strict range: ~3/2 = [701.711, 701.955]

Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc

Badness: 0.004259

Seven-limit extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

Garibaldi adds [25 -14 0 -1⟩,
Grackle adds [-44 26 0 1⟩,
Schism adds [6 -2 0 -1⟩,
Pontiac adds [-59 39 0 -1⟩.

Those all have a fifth as generator.

Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period.
Guiron adds [-10 1 0 3⟩, with an 8/7 generator, three of which give the fifth.
Term adds [-94 54 0 3⟩ with a 1/3 octave period.
Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.

Temperaments not discussed here include salsa, guiron, hemischis and karadeniz.

Garibaldi

Main article: Garibaldi temperament

Subgroup: 2.3.5.7

Comma list: 225/224, 3125/3087

Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -8 -14 -15 -25 -10 ]]

POTE generator: ~3/2 = 702.085

Minimax tuning:

7-odd-limit

[[1 0 0 0⟩, [5/3 1/15 0 -1/15⟩, [5/3 -8/15 0 8/15⟩, [5/3 -14/15 0 14/15⟩]

Eigenmonzos: 2, 7/6

9-odd-limit

[[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]

Eigenmonzos: 2, 9/7

Tuning ranges:

valid range: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
nice range: ~3/2 = [701.711, 702.915]
strict range: ~3/2 = [701.711, 702.915]

Template:Val list

Badness: 0.021644

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2200/2187

Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]

Mapping generators: ~2, ~3

POTE generator: ~3/2 = 702.157

Minimax tuning:

11-odd-limit

[[1 0 0 0 0⟩, [25/16 1/8 0 -1/16 0⟩, [5/2 -1 0 1/2 0⟩, [25/8 -7/4 0 7/8 0⟩, [47/16 23/8 0 -23/16 0⟩]

Eigenmonzos: 2, 9/7

Schismatic family

Schismatic aka Helmholtz

Seven-limit extensions

Garibaldi

11-limit

13-limit

Andromeda

13-limit

Helenus

13-limit

Hemigari

13-limit

Sanjaab

13-limit

Schism

11-limit

Pontiac

Grackle

Bischismic

11-limit

13-limit

Kleischismic

11-limit

13-limit

Pogo

11-limit

13-limit

Squirrel

11-limit

13-limit

Term

Sesquiquartififths

Sesquart

13-limit

Bisesqui

Sextilififths

11-limit

13-limit

Tsaharuk

Quanharuk

11-limit

13-limit

Octant

11-limit

13-limit

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