Schisma: Difference between revisions

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== Temperaments ==

Tempering out this comma gives a [[5-limit]] microtemperament called [[schismatic family #Schismatic aka helmholtz|schismatic, schismic or helmholtz]], which if extended to larger ~~subgroups~~ leads to the [[schismatic family]] of temperaments.

Tempering out this comma gives a [[5-limit]] microtemperament called [[schismatic family #Schismatic aka helmholtz|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.

=== Nestoria ===

As the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]])~~, we can derive the 12 & 53 temperament:~~

Nestoria tempers out [[361/360]] and [[513/512]], and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]).

=== Garibaldi ===

As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]])~~, we can derive the 41&53 temperament:~~

Garibaldi tempers out [[225/224]] and [[5120/5103]], and can be described as the 41 & 53 temperament in the 7-limit. This is derived since the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]).

==== 2.3.5.7.19 subgroup ====

Adding ~~Nestoria~~ to ~~Garibaldi~~ (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that ~~Garibaldi~~ has but which includes a mapping for 19 through ~~Nestoria~~.

Adding nestoria to garibaldi (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that garibaldi has but which includes a mapping for 19 through nestoria.

=== 2.3.5.7.17 12 & 118 & 171 (unnamed) ===

As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12&41 temperament:

As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12 & 118 & 171 temperament:

[[Subgroup]]: 2.3.5.7.17

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{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}

== Trivia ==

@@ Line 16: / Line 16: @@
 == Temperaments ==
-Tempering out this comma gives a [[5-limit]] microtemperament called [[schismatic family #Schismatic aka helmholtz|schismatic, schismic or helmholtz]], which if extended to larger subgroups leads to the [[schismatic family]] of temperaments.
+Tempering out this comma gives a [[5-limit]] microtemperament called [[schismatic family #Schismatic aka helmholtz|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.
 === Nestoria ===
 {{See also| No-sevens subgroup temperaments #Nestoria }}
-As the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]), we can derive the 12 & 53 temperament:
+Nestoria tempers out [[361/360]] and [[513/512]], and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]).
 === Garibaldi ===
 {{Main| Garibaldi }}
-As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament:
+Garibaldi tempers out [[225/224]] and [[5120/5103]], and can be described as the 41 & 53 temperament in the 7-limit. This is derived since the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]).
 ==== 2.3.5.7.19 subgroup ====
 {{Main| Garibaldi }}
-Adding Nestoria to Garibaldi (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that Garibaldi has but which includes a mapping for 19 through Nestoria.
+Adding nestoria to garibaldi (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that garibaldi has but which includes a mapping for 19 through nestoria.
 === 2.3.5.7.17 12 & 118 & 171 (unnamed) ===
-As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12&41 temperament:
+As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12 & 118 & 171 temperament:
 [[Subgroup]]: 2.3.5.7.17
@@ Line 62: / Line 62: @@
 {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}
+{{Todo| improve readability }}
 == Trivia ==