Schismatic family: Difference between revisions

Line 9:

= 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.

[[Comma list]]: 32805/32768

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

Mapping ~~generator~~: ~3

Mapping generators: ~2, ~3

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

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

[[Tuning ranges]]:

Line 26:

Line 28:

== Seven-limit extensions ==

The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. ~~Adding~~ [[garischisma|{{monzo|25 -14 0 -1}}]] ~~gives garibaldi~~, {{monzo|-44 26 0 1}} ~~grackle~~, [[64/63|{{monzo|6 -2 0 -1}}]] ~~schism and~~ {{monzo|-59 39 0 -1}} ~~pontiac; these~~ 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~~, and 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.

The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.

* 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]].

@@ Line 9: / Line 9: @@
 = 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.
+[[Comma list]]: 32805/32768
 [[POTE generator]]: ~3/2 = 701.736
-Mapping generator: ~3
+Mapping generators: ~2, ~3
-Map: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}]
+[[Mapping]]: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}]
 [[Tuning ranges]]:
@@ Line 26: / Line 28: @@
 == Seven-limit extensions ==
-The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding [[garischisma|{{monzo|25 -14 0 -1}}]] gives garibaldi, {{monzo|-44 26 0 1}} grackle, [[64/63|{{monzo|6 -2 0 -1}}]] schism and {{monzo|-59 39 0 -1}} pontiac; these 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, and 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.
+The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
+* 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]].