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


Line 10: 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
: mapping generators: ~2, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.2224, ~5/4 = 386.3137
* [[WE]]: ~2 = 1199.9155{{c}}, ~3/2 = 702.1584{{c}}, ~5/4 = 386.4827{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.2124, ~5/4 = 386.4496
: [[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
 
=== 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.


== Cassaschismic ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 29: Line 41:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.2280, ~5/4 = 386.3137
* [[WE]]: ~2 = 1199.9631{{c}}, ~3/2 = 702.2077{{c}}, ~5/4 = 386.3874{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 702.2290, ~5/4 = 386.3819
: [[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 44: Line 58:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.2289, ~5/4 = 386.2869
* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}
* CWE: ~2 = 1\1, ~3/2 = 702.2303, ~5/4 = 386.3031
* 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 56: 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|legend=1| 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]]s:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~3/2 = 702.2308, ~5/4 = 386.3806
* [[WE]]: ~2 = 1199.9118{{c}}, ~3/2 = 702.1606{{c}}, ~5/4 = 386.5301{{c}}
* CWE: ~2 = 1\1, ~3/2 = 702.2178, ~5/4 = 386.5048
: [[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 }}
{{Optimal ET sequence|legend=1| 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee }}


[[Badness]]: 1.64 × 10<sup>-3</sup>
[[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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.2305, ~5/4 = 386.3748
* WE: ~2 = 1199.9121{{c}}, ~3/2 = 702.1603{{c}}, ~5/4 = 386.5212{{c}}
* CWE: ~2 = 1\1, ~3/2 = 702.2174, ~5/4 = 386.4968
* 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: {{Optimal ET sequence| 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, 851, 1133, 1403, 1984, 3117bcef, 3387bcef }}


Badness: 1.01 × 10<sup>-3</sup>
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]]

Latest revision as of 14:16, 15 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The septischismic family of rank-3 temperaments tempers out the septischisma (monzo: [25 -14 0 -1⟩, ratio: 33 554 432 / 33 480 783).

Septischismic

Main article: Septischismic

The head of this family is septischismic (formerly garischismic), which is generated by a perfect fifth and an independent generator for 5/4. Two Pythagorean apotomes i.e. 14 fifths octave-reduced make a septimal major second (8/7). Equivalently stated, the 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

Comma list: 33554432/33480783

Mapping: [⟨1 0 0 25], ⟨0 1 0 -14], ⟨0 0 1 0]]

mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1199.9155 ¢, ~3/2 = 702.1584 ¢, ~5/4 = 386.4827 ¢
error map: ⟨-0.085 +0.119 -0.000 +0.027]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2124 ¢, ~5/4 = 386.4496 ¢
error map: ⟨0.000 +0.257 +0.136 +0.201]

Optimal ET sequence: 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd

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 also call it cassaschismic. An alternative, supported by andromeda, is androschismic.

Undecimal septischismic (cassaschismic)

Main article: Septischismic

Undecimal septischismic, a.k.a. cassaschismic, maps prime 11 to +23 perfect fifths, so it is an expansion of the 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

Comma list: 19712/19683, 41503/41472

Mapping: [⟨1 0 0 25 -33], ⟨0 1 0 -14 23], ⟨0 0 1 0 0]]

Optimal tunings:

  • WE: ~2 = 1199.9631 ¢, ~3/2 = 702.2077 ¢, ~5/4 = 386.3874 ¢
error map: ⟨-0.037 +0.216 -0.000 -0.139 -0.173]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2290 ¢, ~5/4 = 386.3819 ¢
error map: ⟨0.000 +0.274 +0.068 -0.032 -0.051]

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 1121

Badness (Sintel): 1.69

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 19712/19683

Mapping: [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9785 ¢, ~3/2 = 702.2180 ¢, ~5/4 = 386.2991 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2303 ¢, ~5/4 = 386.3031 ¢

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 2283b

Badness (Sintel): 0.815

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079

Subgroup-val mapping: [⟨1 0 0 25 -33 -13 -6], ⟨0 1 0 -14 23 12 5], ⟨0 0 1 0 0 -1 1]]

Optimal tunings:

  • WE: ~2 = 1199.9817 ¢, ~3/2 = 702.2203 ¢, ~5/4 = 386.3225 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2307 ¢, ~5/4 = 386.3245 ¢

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851

Badness (Sintel): 0.486

Androschismic

Subgroup: 2.3.5.7.11

Comma list: 151263/151250, 200704/200475

Mapping: [⟨1 0 0 25 62], ⟨0 1 0 -14 -34], ⟨0 0 1 0 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9118 ¢, ~3/2 = 702.1606 ¢, ~5/4 = 386.5301 ¢
error map: ⟨-0.088 +0.117 +0.040 -0.045 +0.044]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2178 ¢, ~5/4 = 386.5048 ¢
error map: ⟨0.000 +0.263 +0.191 +0.125 +0.266]

Optimal ET sequence: 12, 29, 41, …, 229, 270, 581, 822, 851, 863e, 1133, 1403, 3117bce, 3387bce, 4520bcdee, 4790bbcdee, 5923bbccddeee

Badness (Sintel): 1.97

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 25 62 82], ⟨0 1 0 -14 -34 -43], ⟨0 0 1 0 -2 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9121 ¢, ~3/2 = 702.1603 ¢, ~5/4 = 386.5212 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2174 ¢, ~5/4 = 386.4968 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.1718 ¢, ~5/4 = 386.5228 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2181 ¢, ~5/4 = 386.5012 ¢

Optimal ET sequence: 12f, 29, 41, …, 229, 241, 270, 552, 581, 851, 1133, 1403, 1984, 3117bcef, 3387bcef

Badness (Sintel): 0.580

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