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Septischismic family: Difference between revisions

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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]].  
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 (etc.) septischismic (cassaschismic) ==
== Undecimal septischismic (cassaschismic) ==
{{Main| Septischismic }}
{{Main| Septischismic }}


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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9785{{c}}, ~3/2 = 702.2180{{c}}, ~5/4 = 386.2991{{c}}error map: {{Val|-0.022, 0.241, -0.058, -0.114, -0.089, -0.146}}
* 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}}error map: {{Val|0.000, 0.275, -0.011, -0.050, -0.021, -0.067}}
: 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=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
Line 73: Line 75:


Optimal tunings:  
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}}
* 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}}error map: {{Val|0.000, 0.276, 0.011, -0.056, -0.011, -0.084, -0.035}}
: 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 }}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851 }}
Line 80: Line 84:
Badness (Sintel): 0.486
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]].
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
Subgroup: 2.3.5.7.11.13.19.29
Line 87: Line 91:
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079


Mapping: {{mapping| 1 0 0 25 -33 -13 -6 23| 0 1 0 -14 23 12 5 -31| 0 0 1 0 0 -1 1 0}}
Subgroup-val mapping: {{mapping| 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:
Optimal tunings:
* WE: ~2 = 1199.948{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.379{{c}} error map: {{Val|-0.052, 0.206, -0.039, -0.389, 0.113, -0.190, -0.119, 0.603}}
* WE: ~2 = 1199.948{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.379{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.244{{c}}, ~5/4 = 386.388{{c}} error map: {{Val|0.000, 0.289, 0.075, -0.242, 0.293, 0.012, 0.095, 0.860}}
: 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}}
: 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: TBA


Badness (Sintel): 0.758
Badness (Sintel): 0.758


==== Subgroup: 2.3.5.7.11.13.19.29.31 ====
===== 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
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2233/2232


Mapping: {{mapping| 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}}
Subgroup-val mapping: {{mapping| 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:
Optimal tunings:
* WE: ~2 = 1199.930{{c}}, ~3/2 = 702.208{{c}}, ~5/4 = 386.406{{c}}
* WE: ~2 = 1199.930{{c}}, ~3/2 = 702.208{{c}}, ~5/4 = 386.406{{c}}
{{Val|-0.070, 0.183, -0.048, -0.513, 0.173, -0.225, -0.136, 0.353, 0.633}}
: 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 = 702.250{{c}}, ~5/4 = 386.420{{c}}
{{Val|0.000, 0.295, 0.106, -0.319, 0.421, 0.047, 0.154, 0.688, 0.976}}
: error map: {{val| 0.000 +0.295 +0.106 -0.319 +0.421 +0.047 +0.154 +0.688 +0.976 }}
Error map:
 
* WE: ~2 = -0.070, 0.183, -0.048, -0.513, 0.173, -0.225, -0.136, 0.353, 0.633
* CWE: ~2 = 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: TBA


Badness (Sintel): 0.793
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).
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
Subgroup: 2.3.5.7.11.13.19.29
Line 118: Line 127:
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079


Mapping: {{mapping| 1 0 0 25 -33 -13 -6 -41| 0 1 0 -14 23 12 5 26| 0 0 1 0 0 -1 1 2}}
Subgroup-val mapping: {{mapping| 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:
Optimal tunings:
* WE: ~2 = 1199.983{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.172{{c}} error map: {{Val|-0.017, 0.242, -0.175, 0.005, -0.246, -0.091, -0.291, 0.495}}
* WE: ~2 = 1199.983{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.172{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.233{{c}}, ~5/4 = 386.174{{c}} error map: {{Val|0.000, 0.268, -0.140, 0.055, -0.193, -0.028, -0.225, 0.564}}
: 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}}
: 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: TBA


Badness (Sintel): 0.716
Badness (Sintel): 0.716


==== Subgroup: 2.3.5.7.11.13.19.29.31 ====
===== 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
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079, 2233/2232


Mapping: {{mapping| 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}}
Subgroup-val mapping: {{mapping| 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:
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}}
* WE: ~2 = 1199.985{{c}}, ~3/2 = 702.210{{c}}, ~5/4 = 386.120{{c}}
* 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}}
: error map: {{val| -0.015 +0.241 -0.223 +0.066 -0.331 -0.079 -0.356 +0.296 +0.370 }}
Error map:
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.219{{c}}, ~5/4 = 386.122{{c}}
* WE: ~2 = -0.015, 0.241, -0.223, 0.066, -0.331, -0.079, -0.356, 0.296, 0.370
: error map: {{val| 0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431 }}
* CWE: ~2 = 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: TBA


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