Schismatic family: Difference between revisions

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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 {{monzo| -15 8 1 }}, and flipping that yields {{multival| 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.

[[POTE generator]]: ~3/2 = 701.736

Mapping generators: ~2, ~3

[[Mapping]]: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}]

* 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]

{{Vals|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}

Badness: 0.00426

* Garibaldi adds [[garischisma|{{monzo|25 -14 0 -1}}]],

* Grackle adds {{monzo|-44 26 0 1}},

* Schism adds [[64/63|{{monzo|6 -2 0 -1}}]],

* Pontiac adds {{monzo|-59 39 0 -1}}.

Those all have a fifth as generator.  

* Bischismic adds {{monzo|-69 40 0 2}} and has a fifth generator with a half-octave period.  

* Guiron adds [[1029/1024|{{monzo|-10 1 0 3}}]], with an 8/7 generator, three of which give the fifth.  

* Term adds {{monzo|-94 54 0 3}} with a 1/3 octave period.  

* Sesquiquartififths adds {{monzo|-35 15 0 4}} and slices the fifth in four.

Temperaments not discussed here include [[Sensamagic clan #Salsa|salsa]], [[Porwell temperaments #Hemischis|hemischis]] and [[Turkish maqam music temperaments #Karadeniz temperament|karadeniz]].

Commas: 225/224, 3125/3087

7-limit minimax tuning:

7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,

|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;<nowiki>]</nowiki>

[[Eigenmonzo]]s: 2, 7/6

9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,

|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;<nowiki>]</nowiki>

Eigenmonzos: 2, 9/7

[[POTE generator]]: ~3/2 = 702.085

Mapping generator: ~3

Map: [&lt;1 0 15 25|, &lt;0 1 -8 -14|]

Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||

* 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]