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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


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 the related lower accuracy temperament [[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.
 
Septischismic has two reasonable extensions to add 17, which coalesce in the newt extension 270 & 311. Septinoellismic finds it a tad unintuitively as two aberschismas up and 4 commas down from a major second (^^4sM2) yet supported by 94edo, and septisperasmic finds it 4 aberschismas up from a minor second (4^m2), which can be deemed the more intuitive mapping, albeit it is not supported by any cassandra tuning except for 12edo, which notably has a very well tuned 17/16.
 
Septischismic has two reasonable extensions to add 29 and 31 into the system, that like before, coalesce in the newt extension 270 & 311 too. Septiplanar finds them in the chain of fifths, without need of third generators, thus they in the same plane to the rank-2 chain of fifths, mapped respectively to a major seventh – three commas (3sM7) and a perfect octave minus two commas (hP8). Because this extension is reached only through the septischismic chain of fifths, it is also available in [[gary]].


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


Line 29: Line 45:


[[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 41: Line 59:
Comma list: 2080/2079, 4096/4095, 19712/19683
Comma list: 2080/2079, 4096/4095, 19712/19683


Mapping: {{mapping| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}
{{Mapping|legend=0| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}


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
: error map: {{val| -0.022 +0.241 -0.058 -0.114 -0.089 -0.146 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2303{{c}}, ~5/4 = 386.3031{{c}}
: error map: {{val| 0.000 +0.275 -0.011 -0.050 -0.021 -0.067 }}


{{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 ====
Subgroup: 2.3.5.7.11.13.19
Subgroup: 2.3.5.7.11.13.19


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 }}
{{Mapping|legend=2| 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{{c}}, ~3/2 = 702.2203{{c}}, ~5/4 = 386.3225{{c}}
: error map: {{val| -0.018 +0.247 -0.028 -0.111 -0.069 -0.152 -0.107 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
: error map: {{val| 0.000 +0.276 +0.011 -0.056 -0.011 -0.084 -0.035 }}
 
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}
 
Badness (Sintel): 0.486
 
===== Septiplanar =====
Subgroup: 2.3.5.7.11.13.19.29
 
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 23 | 0 1 0 -14 23 12 5 -31 | 0 0 1 0 0 -1 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1199.9476{{c}}, ~3/2 = 702.2134{{c}}, ~5/4 = 386.3794{{c}}
: error map: {{val| -0.052 +0.206 -0.039 -0.389 +0.113 -0.190 -0.119 +0.603 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2440{{c}}, ~5/4 = 386.3884{{c}}
: error map: {{val| 0.000 +0.289 +0.075 -0.242 +0.293 +0.012 +0.095 +0.860 }}
 
{{Optimal ET sequence|legend=0| 41, 53j, 94, 176, 217, 270, 487, 581j, 1068j }}
 
Badness (Sintel): 0.758
 
====== 2.3.5.7.11.13.19.29.31 subgroup ======
Subgroup: 2.3.5.7.11.13.19.29.31
 
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2233/2232
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 23 19 | 0 1 0 -14 23 12 5 -31 -24 | 0 0 1 0 0 -1 1 0 0 }}
 
Optimal tunings:
* WE: ~2 = 1199.9298{{c}}, ~3/2 = 702.2082{{c}}, ~5/4 = 386.4062{{c}}
: error map: {{val| -0.070 +0.183 -0.048 -0.513 +0.173 -0.225 -0.136 +0.353 +0.633 }}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.2495{{c}}, ~5/4 = 386.4198{{c}}
: error map: {{val| 0.000 +0.295 +0.106 -0.319 +0.421 +0.047 +0.154 +0.688 +0.976 }}
 
{{Optimal ET sequence|legend=0| 41, 94, 135, 176, 217, 270, 311, 487, 581jk, 798k, 1068jk }}
 
Badness (Sintel): 0.793
 
===== Septifantasious =====
Septifantasious tempers out [[1625/1624]] – the fantasia, and thus finds 29/16 as 3 aberschismas above 9/5 – that is to say, superminor seventh + 2 aberschismas (^^Sm7), and 31/16 one apotome higher as a hypermajor seventh + 2 aberschismas (^^HM7).
 
Subgroup: 2.3.5.7.11.13.19.29
 
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 -41 | 0 1 0 -14 23 12 5 26 | 0 0 1 0 0 -1 1 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9833{{c}}, ~3/2 = 702.2132{{c}}, ~5/4 = 386.1721{{c}}
: error map: {{val| -0.017 +0.242 -0.175 +0.005 -0.246 -0.091 -0.291 +0.495 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2228{{c}}, ~5/4 = 386.1741{{c}}
: error map: {{val| 0.000 +0.268 -0.140 +0.055 -0.193 -0.028 -0.225 +0.564 }}
 
{{Optimal ET sequence|legend=0| 41, 53, 94j, 217j, 258j, 270, 581j, 904, 945, 1215j }}
 
Badness (Sintel): 0.716
 
====== 2.3.5.7.11.13.19.29.31 subgroup ======
Subgroup: 2.3.5.7.11.13.19.29.31
 
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079, 2233/2232
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 -41 -52 | 0 1 0 -14 23 12 5 26 33 | 0 0 1 0 0 -1 1 2 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.985{{c}}, ~3/2 = 702.210{{c}}, ~5/4 = 386.120{{c}}
: error map: {{val| -0.015 +0.241 -0.223 +0.066 -0.331 -0.079 -0.356 +0.296 +0.370 }}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.219{{c}}, ~5/4 = 386.122{{c}}
: error map: {{val| 0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431 }}
 
{{Optimal ET sequence|legend=0| 41, 53k, 94jk, 217jk, 229j, 270, 311, 581jk, 634, 904, 945, 1215jk }}
 
Badness (Sintel): 0.796
 
=== Septinoellismic ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 1225/1224, 2080/2079, 2431/2430, 4096/4095
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 47 | 0 1 0 -14 23 12 -30 | 0 0 1 0 0 -1 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9654{{c}}, ~3/2 = 702.2159{{c}}, ~5/4 = 386.2053{{c}}
: error map: {{val| -0.035 +0.226 -0.178 -0.229 -0.006 -0.038 +0.252 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~5/4 = 386.2080{{c}}
: error map: {{val| 0.000 +0.281 -0.106 -0.129 +0.109 +0.096 +0.383 }}
 
{{Optimal ET sequence|legend=0| 53, 94, 176g, 217, 270, 311, 487, 581 }}
 
Badness (Sintel): 0.950
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2080/2079
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 47 -6 | 0 1 0 -14 23 12 -30 5 | 0 0 1 0 0 -1 2 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9699{{c}}, ~3/2 = 702.2200{{c}}, ~5/4 = 386.2373{{c}}
: error map: {{val| -0.030 +0.235 -0.137 -0.238 +0.044 -0.034 +0.286 -0.206 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2374{{c}}, ~5/4 = 386.2384{{c}}
: error map: {{val| 0.000 +0.282 -0.075 -0.150 +0.143 +0.083 +0.399 -0.088 }}
 
{{Optimal ET sequence|legend=0| 53, 94, 176g, 217, 270, 311, 487, 581 }}
 
Badness (Sintel): 0.682
 
=== Septisperasmic ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2080/2079, 2500/2499, 4096/4095, 19712/19683
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 -48 | 0 1 0 -14 23 12 27 | 0 0 1 0 0 -1 4 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2293, ~5/4 = 386.3021
Optimal tunings:
* WE: ~2 = 1199.9797{{c}}, ~3/2 = 702.2164{{c}}, ~5/4 = 386.2286{{c}}
: error map: {{val| -0.020 +0.241 -0.126 -0.080 -0.136 -0.098 +0.068 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2281{{c}}, ~5/4 = 386.2332{{c}}
: error map: {{val| 0.000 +0.273 -0.081 -0.020 -0.071 -0.023 +0.137 }}
 
{{Optimal ET sequence|legend=0| 258, 270, 311, 581, 1485, 1796, 2066, 2647b }}
 
Badness (Sintel): 1.76
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2500/2499
 
{{Mapping|legend=2| 1 0 0 25 -33 -13 -48 -6 | 0 1 0 -14 23 12 27 5 | 0 0 1 0 0 -1 4 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9861{{c}}, ~3/2 = 702.2198{{c}}, ~5/4 = 386.2353{{c}}
: error map: {{val| -0.014 +0.251 -0.106 -0.056 -0.123 -0.084 +0.101 -0.193 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2278{{c}}, ~5/4 = 386.2383{{c}}
: error map: {{val| 0.000 +0.273 -0.075 -0.015 -0.078 -0.032 +0.148 -0.136 }}


{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851 }}
{{Optimal ET sequence|legend=0| 258, 270, 311, 581, 1485, 1796, 2066 }}


Badness: 0.496 × 10<sup>-3</sup>
Badness (Sintel): 1.10


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 702.2305, ~5/4 = 386.3748
* WE: ~2 = 1199.9282{{c}}, ~3/2 = 702.1718{{c}}, ~5/4 = 386.5228{{c}}
* CWE: ~2 = 1\1, ~3/2 = 702.2174, ~5/4 = 386.4968
* 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]]
Retrieved from "https://en.xen.wiki/w/Septischismic_family"