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Septischismic family: Difference between revisions

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{{Technical data page}}
{{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 '''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).  


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]].
== Septischismic ==
{{Main| Septischismic }}
 
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]].  


== Garischismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2124{{c}}, ~5/4 = 386.4496{{c}}
: error map: {{val| 0.000 +0.257 +0.136 +0.201 }}
: error map: {{val| 0.000 +0.257 +0.136 +0.201 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2224{{c}}, ~5/4 = 386.3137{{c}} -->


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd }}
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[[Badness]] (Sintel): 5.79
[[Badness]] (Sintel): 5.79


== Cassaschismic ==
=== 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 also call it [[#Undecimal septischismic (cassaschismic)|cassaschismic]]. An alternative, supported by [[andromeda]], is [[#Androschismic|androschismic]].
 
== Undecimal septischismic (cassaschismic) ==
{{Main| Septischismic }}
 
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
[[Subgroup]]: 2.3.5.7.11


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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2290{{c}}, ~5/4 = 386.3819{{c}}
: error map: {{val| 0.000 +0.274 +0.068 -0.032 -0.051 }}
: error map: {{val| 0.000 +0.274 +0.068 -0.032 -0.051 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2280{{c}}, ~5/4 = 386.3137{{c}} -->


{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}
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* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}
* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.2869{{c}} -->


{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
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* WE: ~2 = 1199.9817{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}
* WE: ~2 = 1199.9817{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.3021{{c}} -->


{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2178{{c}}, ~5/4 = 386.5048{{c}}
: error map: {{val| 0.000 +0.263 +0.191 +0.125 +0.266 }}
: error map: {{val| 0.000 +0.263 +0.191 +0.125 +0.266 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2308{{c}}, ~5/4 = 386.3806{{c}} -->


{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}
{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}
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Subgroup: 2.3.5.7.11.13
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 }}
Mapping: {{mapping| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}
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* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}
* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2174{{c}}, ~5/4 = 386.4968{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2305{{c}}, ~5/4 = 386.3748{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}
{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}
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Badness (Sintel): 0.942
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:Temperament families]]
[[Category:Garischismic family| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Garischismic| ]] <!-- key article -->
[[Category:Rank 3]]
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