Schismatic family: Difference between revisions

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This defies the tradition of tertian harmony, as the just major triad on C is {{nowrap|{{dash|C, F♭, G|hair|med}}}}, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as {{nowrap|{{dash|C, vE, G|hair|med}}}}.

== Schismic ==

== Schismic, schismatic, a.k.a. helmholtz ==

{{Main|Schismic ~~and garibaldi~~}}The 5-limit version of the temperament is a [[microtemperament]], ~~sometimes~~ called '''~~schismic~~, ~~helmholtz~~''', or ~~historically~~ ''~~'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 schismic; [[118edo]] ~~is a more expansive schismic system with a particularly flat tuning~~. In exact analogy with 1/4 comma meantone there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, 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, ~~or even simply~~ leaving the fifths just ~~and~~ collapsing schismic to a simple relabeling of the 3-limit~~, although the differences would be very hard to distinguish unless using a large gamut~~.

The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz'', 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 schismic, 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 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, 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. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.

[[Subgroup]]: 2.3.5

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[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7187

* [[POTE]]: ~2 = 1\1, ~3/2 = 701.7359

* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7187

[[Tuning ranges]]:

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== Garibaldi ==

Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.

@@ Line 10: / Line 10: @@
 This defies the tradition of tertian harmony, as the just major triad on C is {{nowrap|{{dash|C, F&#x266D;, G|hair|med}}}}, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as {{nowrap|{{dash|C, vE, G|hair|med}}}}.
-== Schismic ==
+== Schismic, schismatic, a.k.a. helmholtz ==
-{{Main|Schismic and garibaldi}}The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''schismic, helmholtz''', or historically '''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 schismic; [[118edo]] is a more expansive schismic system with a particularly flat tuning. In exact analogy with 1/4 comma meantone there is also 1/8 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, 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, or even simply leaving the fifths just and collapsing schismic to a simple relabeling of the 3-limit, although the differences would be very hard to distinguish unless using a large gamut.
+{{Main| Schismic }}
+The 5-limit version of the temperament is a [[microtemperament]], called ''schismic'', ''schismatic'', or ''helmholtz'', 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 schismic, 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 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, 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. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.
 [[Subgroup]]: 2.3.5
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 [[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7187
 * [[POTE]]: ~2 = 1\1, ~3/2 = 701.7359
-* [[CTE]]: ~2 = 1\1, ~3/2 = 701.7187
 [[Tuning ranges]]:
@@ Line 55: / Line 57: @@
 == Garibaldi ==
+{{Main| Garibaldi }}
 Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.