Septischismic family: Difference between revisions

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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]].

[[Subgroup]]: 2.3.5.7.11

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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:

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Line 71:

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

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Badness (Sintel): 0.486

==== Septiplanar ====

===== Septiplanar =====

Septischismic has two reasonable extensions to add 29 and 31 into the system, both of which coalesce in the structurally very similar temperament [[newt]]. 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]].

Subgroup: 2.3.5.7.11.13.19.29

Line 94:

Line 96:

Optimal tunings:

* WE: ~2 = 1199.~~948~~{{c}}, ~3/2 = 702.~~213~~{{c}}, ~5/4 = 386.~~379~~{{c}}

* 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.~~000~~{{c}}, ~3/2 = 702.~~244~~{{c}}, ~5/4 = 386.~~388~~{{c}}

* 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~~: TBA~~

{{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 =====

====== 2.3.5.7.11.13.19.29.31 subgroup ======

Subgroup: 2.3.5.7.11.13.19.29.31

Line 111:

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Optimal tunings:

* WE: ~2 = 1199.~~930~~{{c}}, ~3/2 = 702.~~208~~{{c}}, ~5/4 = 386.~~406~~{{c}}

* 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.~~250~~{{c}}, ~5/4 = 386.~~420~~{{c}}

* 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~~: TBA~~

{{Optimal ET sequence|legend=0| 41, 94, 135, 176, 217, 270, 311, 487, 581jk, 798k, 1068jk }}

Badness (Sintel): 0.793

==== Septifantasious ====

===== 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).

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Optimal tunings:

* WE: ~2 = 1199.~~983~~{{c}}, ~3/2 = 702.~~213~~{{c}}, ~5/4 = 386.~~172~~{{c}}

* 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.~~000~~{{c}}, ~3/2 = 702.~~233~~{{c}}, ~5/4 = 386.~~174~~{{c}}

* 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~~: TBA~~

{{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 =====

====== 2.3.5.7.11.13.19.29.31 subgroup ======

Subgroup: 2.3.5.7.11.13.19.29.31

Line 152:

Line 154:

: error map: {{val| 0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431 }}

Optimal ET sequence~~: TBA~~

{{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

Subgroup-val mapping: {{mapping| 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

Subgroup-val mapping: {{mapping| 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

Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -48 | 0 1 0 -14 23 12 27 | 0 0 1 0 0 -1 4 }}

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

Subgroup-val mapping: {{mapping| 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=0| 258, 270, 311, 581, 1485, 1796, 2066 }}

Badness (Sintel): 1.10

== Androschismic ==

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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|legend=0| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}

Optimal tunings:

Line 193:

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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 }}

{{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:

@@ Line 33: / Line 33: @@
 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]].
 [[Subgroup]]: 2.3.5.7.11
@@ Line 55: / Line 59: @@
 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:
@@ Line 67: / Line 71: @@
 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
@@ Line 84: / Line 88: @@
 Badness (Sintel): 0.486
-==== Septiplanar ====
+===== Septiplanar =====
-Septischismic has two reasonable extensions to add 29 and 31 into the system, both of which coalesce in the structurally very similar temperament [[newt]]. 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]].
 Subgroup: 2.3.5.7.11.13.19.29
@@ Line 94: / Line 96: @@
 Optimal tunings:
-* WE: ~2 = 1199.948{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.379{{c}}
+* 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 = 1200.000{{c}}, ~3/2 = 702.244{{c}}, ~5/4 = 386.388{{c}}
+* 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: TBA
+{{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 =====
+====== 2.3.5.7.11.13.19.29.31 subgroup ======
 Subgroup: 2.3.5.7.11.13.19.29.31
@@ Line 111: / Line 113: @@
 Optimal tunings:
-* WE: ~2 = 1199.930{{c}}, ~3/2 = 702.208{{c}}, ~5/4 = 386.406{{c}}
+* 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 = 702.250{{c}}, ~5/4 = 386.420{{c}}
+* 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: TBA
+{{Optimal ET sequence|legend=0| 41, 94, 135, 176, 217, 270, 311, 487, 581jk, 798k, 1068jk }}
 Badness (Sintel): 0.793
-==== Septifantasious ====
+===== 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).
@@ Line 130: / Line 132: @@
 Optimal tunings:
-* WE: ~2 = 1199.983{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.172{{c}}
+* 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 = 1200.000{{c}}, ~3/2 = 702.233{{c}}, ~5/4 = 386.174{{c}}
+* 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: TBA
+{{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 =====
+====== 2.3.5.7.11.13.19.29.31 subgroup ======
 Subgroup: 2.3.5.7.11.13.19.29.31
@@ Line 152: / Line 154: @@
 : error map: {{val| 0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431 }}
-Optimal ET sequence: TBA
+{{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
+Subgroup-val mapping: {{mapping| 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
+Subgroup-val mapping: {{mapping| 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
+Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -48 | 0 1 0 -14 23 12 27 | 0 0 1 0 0 -1 4 }}
+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
+Subgroup-val mapping: {{mapping| 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=0| 258, 270, 311, 581, 1485, 1796, 2066 }}
+Badness (Sintel): 1.10
 == Androschismic ==
@@ Line 178: / Line 248: @@
 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|legend=0| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}
 Optimal tunings:
@@ Line 193: / Line 263: @@
 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 }}
+{{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: