Garischismic family: Difference between revisions

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

Cassaschismic maps [[prime interval|prime]] [[11/1|11]] to +23 perfect fifths, so it is an expansion 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:Temperament families]]

[[Category:Garischismic family| ]]

[[Category:Rank 3]]

@@ Line 5: / Line 5: @@
 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 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 30: / Line 30: @@
 {{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.
+Cassaschismic maps [[prime interval|prime]] [[11/1|11]] to +23 perfect fifths, so it is an expansion 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
@@ Line 98: / Line 98: @@
 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 109: / Line 109: @@
 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:Temperament families]]
 [[Category:Garischismic family| ]] <!-- main article -->
 [[Category:Rank 3]]