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 '''septischismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33 554 432 / 33 480 783).

== ~~Garischismic~~ ==

== Septischismic ==

~~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]].

The head of this family is septischismic (formerly ''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.

Septischismic can be easily notated with [[chain-of-fifths notation]] with two additional sets 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]].

[[Subgroup]]: 2.3.5.7

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=== 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]].

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

== ~~Cassaschismic~~ ==

== Undecimal septischismic (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.

Undecimal septischismic, a.k.a. cassaschismic, maps [[prime interval|prime]] [[11/1|11]] to +23 perfect fifths, so it is an [[expansion]] of the [[2.3.7.11 subgroup|2.3.7.11-subgroup]] version of [[gary]]. It 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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Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 43904/43875~~, 154880/154791~~

Comma list: 2080/2079, 10648/10647, 43904/43875

Mapping: {{mapping| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}

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Badness (Sintel): 0.942

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 2080/2079, 3136/3135, 10648/10647

Mapping: {{mapping| 1 0 0 25 62 82 -6 | 0 1 0 -14 -34 -43 5 | 0 0 1 0 -2 -3 1 }}

Optimal tunings:

* WE: ~2 = 1199.9282{{c}}, ~3/2 = 702.1718{{c}}, ~5/4 = 386.5228{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2181{{c}}, ~5/4 = 386.5012{{c}}

{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 851, 1133, 1403, 1984, 3117bcef, 3387bcef }}

Badness (Sintel): 0.580

[[Category:Septischismic family| ]]

[[Category:Temperament families]]

[[Category:~~Garischismic family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-3 temperaments]]

~~[[Category:Rank~~ 3]]

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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 '''septischismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33 554 432 / 33 480 783).
-== Garischismic ==
+== Septischismic ==
-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.
+{{Main| Septischismic }}
-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]].
+The head of this family is septischismic (formerly ''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.
+Septischismic can be easily notated with [[chain-of-fifths notation]] with two additional sets 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]].
 [[Subgroup]]: 2.3.5.7
@@ Line 25: / Line 27: @@
 === 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]].
+The best extension to the 11-limit identifies the [[11/8]] at +23 fifths. This is also the mapping used in [[cassandra]], so we also call it [[#Undecimal septischismic (cassaschismic)|cassaschismic]]. An alternative, supported by [[andromeda]], is [[#Androschismic|androschismic]].
-== Cassaschismic ==
+== Undecimal septischismic (cassaschismic) ==
-{{Main| Cassaschismic }}
+{{Main| Septischismic }}
-Cassaschismic is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.
+Undecimal septischismic, a.k.a. cassaschismic, maps [[prime interval|prime]] [[11/1|11]] to +23 perfect fifths, so it is an [[expansion]] of the [[2.3.7.11 subgroup|2.3.7.11-subgroup]] version of [[gary]]. It 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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 Subgroup: 2.3.5.7.11.13
-Comma list: 2080/2079, 43904/43875, 154880/154791
+Comma list: 2080/2079, 10648/10647, 43904/43875
 Mapping: {{mapping| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}
@@ Line 110: / Line 112: @@
 Badness (Sintel): 0.942
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 1216/1215, 2080/2079, 3136/3135, 10648/10647
+Mapping: {{mapping| 1 0 0 25 62 82 -6 | 0 1 0 -14 -34 -43 5 | 0 0 1 0 -2 -3 1 }}
+Optimal tunings:
+* WE: ~2 = 1199.9282{{c}}, ~3/2 = 702.1718{{c}}, ~5/4 = 386.5228{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2181{{c}}, ~5/4 = 386.5012{{c}}
+{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 851, 1133, 1403, 1984, 3117bcef, 3387bcef }}
+Badness (Sintel): 0.580
+[[Category:Septischismic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Garischismic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-3 temperaments]]
-[[Category:Rank 3]]