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Septischismic family: Difference between revisions

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The '''garischismic family''' of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out|tempers out]] the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 33554432/33480783. 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).
{{Technical data page}}
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


Line 9: Line 14:


{{Mapping|legend=1| 1 0 0 25 | 0 1 0 -14 | 0 0 1 0 }}
{{Mapping|legend=1| 1 0 0 25 | 0 1 0 -14 | 0 0 1 0 }}
: mapping generators: ~2, ~3, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2224, ~5/4 = 386.3137
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9155{{c}}, ~3/2 = 702.1584{{c}}, ~5/4 = 386.4827{{c}}
: [[error map]]: {{val| -0.085 +0.119 -0.000 +0.027 }}
* [[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 }}


{{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 }}


[[Badness]]: 1.31 × 10<sup>-3</sup>
[[Badness]] (Sintel): 5.79


== Cassaschismic ==
=== Overview to extensions ===
{{See also| Olympic clan #Cassaschismic }}
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
Line 25: Line 40:
{{Mapping|legend=1| 1 0 0 25 -33 | 0 1 0 -14 23 | 0 0 1 0 0 }}
{{Mapping|legend=1| 1 0 0 25 -33 | 0 1 0 -14 23 | 0 0 1 0 0 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2280, ~5/4 = 386.3137
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}, ~5/4 = 386.3874{{c}}
: [[error map]]: {{val| -0.037 +0.216 -0.000 -0.139 -0.173 }}
* [[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 }}


{{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 }}


[[Badness]]: 1.41 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 38: Line 57:
Mapping: {{mapping| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}
Mapping: {{mapping| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2289, ~5/4 = 386.2869
Optimal tunings:  
* 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}}


{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}


Badness: 0.872 × 10<sup>-3</sup>
Badness (Sintel): 0.815


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 49: Line 70:
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079


Sval mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}
Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2293, ~5/4 = 386.3021
Optimal tunings:  
* 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}}


{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851 }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}


Badness: 0.496 × 10<sup>-3</sup>
Badness (Sintel): 0.486


== Androschismic ==
== Androschismic ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Comma list: 151263/151250, [[Reef comma|200704/200475]]
[[Comma list]]: 151263/151250, 200704/200475


Mapping: {{mapping| 1 0 0 25 62 | 0 1 0 -14 -34 | 0 0 1 0 -2 }}
{{Mapping|legend=1| 1 0 0 25 62 | 0 1 0 -14 -34 | 0 0 1 0 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.231, ~5/4 = 386.381
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9118{{c}}, ~3/2 = 702.1606{{c}}, ~5/4 = 386.5301{{c}}
: [[error map]]: {{val| -0.088 +0.117 +0.040 -0.045 +0.044 }}
* [[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 }}
 
{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}
 
[[Badness]] (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, [[Punctisma|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 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.230, ~5/4 = 386.375
Optimal tunings:
* 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}}
 
{{Optimal ET sequence|legend=0| 12f, 29, 41, …, 229, 241, 270, 552, 581, 822, 851, 863ef, 1133, 1403, 2536bcdef, 3117bcef, 4250bcdeeff, 4520bcdeeff, 5653bbccddeeeff }}
 
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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