Septischismic family: Difference between revisions

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The '''garischismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo|legend=1| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).

The '''garischismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo|legend=1| 25 -14 0 -1 }}).

The ~~best extension to the 11~~-~~limit identifies the~~ [[11/8]] ~~at +23 fifths~~. ~~This is also~~ the ~~mapping used in~~ [[~~cassandra~~]], ~~so we call it~~ [[~~#Cassaschismic|cassaschismic~~]]~~. An alternative~~, ~~supported by~~ [[~~andromeda~~]]~~, is~~ [[~~#Androschismic|androschismic~~]].

== Garischismic ==

The head of this family is garischismic, which is generated by a [[3/2|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.

Garischismic can be easily notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].

~~== Garischismic ==~~

[[Subgroup]]: 2.3.5.7

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[[Badness]] (Sintel): 5.79

=== Overview to extensions ===

The best extension to the 11-limit identifies the [[11/8]] at +23 fifths. This is also the mapping used in [[cassandra]], so we call it [[#Cassaschismic|cassaschismic]]. An alternative, supported by [[andromeda]], is [[#Androschismic|androschismic]].

== Cassaschismic ==

Cassaschismic is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.

[[Subgroup]]: 2.3.5.7.11

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 {{Technical data page}}
-The '''garischismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo|legend=1| 25 -14 0 -1 }}). The head of this family is garischismic, which is generated by a [[3/2|perfect fifth]] and an independent generator for [[5/4]]. Two [[apotome]]s i.e. 14 fifths octave-reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).
+The '''garischismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ([[ratio]]: 33554432/33480783, {{monzo|legend=1| 25 -14 0 -1 }}).
-The best extension to the 11-limit identifies the [[11/8]] at +23 fifths. This is also the mapping used in [[cassandra]], so we call it [[#Cassaschismic|cassaschismic]]. An alternative, supported by [[andromeda]], is [[#Androschismic|androschismic]].
+== Garischismic ==
+The head of this family is garischismic, which is generated by a [[3/2|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.
+Garischismic can be easily notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].
-== Garischismic ==
 [[Subgroup]]: 2.3.5.7
@@ Line 21: / Line 23: @@
 [[Badness]] (Sintel): 5.79
+=== Overview to extensions ===
+The best extension to the 11-limit identifies the [[11/8]] at +23 fifths. This is also the mapping used in [[cassandra]], so we call it [[#Cassaschismic|cassaschismic]]. An alternative, supported by [[andromeda]], is [[#Androschismic|androschismic]].
 == Cassaschismic ==
+{{Main| Cassaschismic }}
+Cassaschismic is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.
 [[Subgroup]]: 2.3.5.7.11