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Semicathartismic clan: Difference between revisions

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The '''semicanousmic clan''' of rank-3 temperaments tempers out the [[semicanousma]], 14641/14580 = {{monzo| -2 -6 -1 0 4 }}.
{{Technical data page}}
The '''semicathartismic clan''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[semicathartisma]] ({{monzo|legend=1| -2 -6 -1 0 4 }}, [[ratio]]: 14641/14580).


Temperaments discussed elsewhere are:
For the rank-4 semicathartismic temperament, see [[Catalog of rank-4 temperaments #Semicathartismic (14641/14580)]].
* ''[[Urania]]'', {81/80, 121/120} → [[Didymus rank three family #Urania|Didymus rank three family]]
* ''[[Triglav]]'', {3025/3024, 3136/3125} → [[Hemimean family #Triglav|Hemimean family]]
* ''[[Semicanou]]'', {9801/9800, 14641/14580} → [[Canou family #Semicanou|Canou family]]


Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.
== Deuteromere ==
[[File:Lattice Semicanousmic.png|thumb|Lattice for deutermere.]]
[[File:Lattice Semicanousmic Rearranged.png|thumb|Ditto, but re-arranged into neutral thirds]]


For the rank-4 semicanousmic temperament, see [[Rank-4 temperament #Semicanousmic (14641/14580)]].
== Deuteromere ==
[[Subgroup]]: 2.3.5.11
[[Subgroup]]: 2.3.5.11


[[Comma list]]: 14641/14580
[[Comma list]]: 14641/14580


[[Sval]] [[mapping]]: [{{val| 1 0 2 1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 -4 -1 }}]
{{Mapping|legend=2| 1 0 -2 0 | 0 1 2 2 | 0 0 4 1 }}
: mapping generators: ~2, ~3, ~11/9


Sval mapping generators: ~2, ~3, ~18/11
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0495{{c}}, ~3/2 = 702.2793{{c}}, ~11/9 = 345.4721{{c}}
: [[error map]]: {{val| +0.049 +0.374 +0.133 -1.188 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2918{{c}}, ~11/9 = 345.4516{{c}}
: error map: {{val| 0.000 +0.337 +0.077 -1.283 }}


Optimal tuning ([[CTE]]): ~3/2 = 702.3342, ~18/11 = 854.5548
{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}


{{Val list|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}
[[Badness]] (Sintel): 1.18


[[Badness]]: 0.142 × 10<sup>-3</sup>
=== Overview to extensions ===
 
Temperaments discussed elsewhere are:
=== 2.3.5.11.17 subgroup ===
* ''[[Urania]]'' (+81/80 or 121/120) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
Subgroup: 2.3.5.11.17
* ''[[Triglav]]'' (+3025/3024 or 3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
 
* ''[[Semicathart]]'' (+9801/9800) → [[Cathartic family #Semicathart|Cathartic family]]
Comma list: 1089/1088, 14641/14580
 
Sval mapping: [{{val| 1 0 2 1 -4 }}, {{val| 0 1 2 2 6 }}, {{val| 0 0 -4 -1 -2 }}]
 
Optimal tuning (CTE): ~3/2 = 702.3427, ~18/11 = 854.5587
 
Optimal GPV sequence: {{val list| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}
 
Badness: 0.481 × 10<sup>-3</sup>
 
=== 2.3.5.11.17.19 subgroup ===
Subgroup: 2.3.5.11.17.19
 
Comma list: 1089/1088, 1216/1215, 1445/1444
 
Sval mapping: [{{val| 1 0 2 1 -4 -4 }}, {{val| 0 1 2 2 6 7 }}, {{val| 0 0 -4 -1 -2 -4 }}]
 
Optimal tuning (CTE): ~3/2 = 702.2987, ~18/11 = 854.5520
 
Optimal GPV sequence: {{val list| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}


Badness: 0.403 × 10<sup>-3</sup>
Considered below are syndeute and hemireiwa, in addition to the add-17 extension of deuteromere.  


== Syndeute ==
== Syndeute ==
Line 56: Line 38:
[[Comma list]]: 14641/14580, 19712/19683
[[Comma list]]: 14641/14580, 19712/19683


[[Mapping]]: [{{val| 1 0 2 -9 1 }}, {{val| 0 1 2 7 2 }}, {{val| 0 0 -4 1 -1 }}]
{{Mapping|legend=1| 1 0 -2 -8 0 | 0 1 2 7 2 | 0 0 4 -1 1 }}


Mapping generators: ~2, ~3, ~18/11
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1265{{c}}, ~3/2 = 702.0849{{c}}, ~11/9 = 345.5759{{c}}
: [[error map]]: {{val| +0.126 +0.256 +0.159 +0.066 -1.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0357{{c}}, ~11/9 = 345.5568{{c}}
: error map: {{val| 0.000 +0.081 -0.015 -0.133 -1.690 }}


Optimal tuning ([[CTE]]): ~3/2 = 702.0812, ~18/11 = 854.4115
{{Optimal ET sequence|legend=1| 94, 111, 118, 212, 323, 441e }}


{{Val list|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}
[[Badness]] (Sintel): 3.91
 
[[Badness]]: 3.26 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 71: Line 55:
Comma list: 1573/1568, 2080/2079, 14641/14580
Comma list: 1573/1568, 2080/2079, 14641/14580


Mapping: [{{val| 1 0 2 -9 1 -15 }}, {{val| 0 1 2 7 2 10 }}, {{val| 0 0 -4 1 -1 4 }}]
Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 1 2 7 2 10 | 0 0 4 -1 1 -4 }}


Optimal tuning (CTE): ~3/2 = 702.1682, ~18/11 = 854.5074
Optimal tunings:
* WE: ~2 = 1200.1164{{c}}, ~3/2 = 702.1730{{c}}, ~11/9 = 854.4793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1272{{c}}, ~11/9 = 345.4623{{c}}


Optimal GPV sequence: {{Val list| 94, 111, 205, 212, 323, 535ef }}
{{Optimal ET sequence|legend=0| 94, 111, 205, 212, 323, 535ef }}


Badness: 3.01 × 10<sup>-3</sup>
Badness (Sintel): 2.82


=== 17-limit ===
=== 17-limit ===
Line 84: Line 70:
Comma list: 442/441, 561/560, 715/714, 14641/14580
Comma list: 442/441, 561/560, 715/714, 14641/14580


Mapping: [{{val| 1 0 2 -9 1 -15 -4 }}, {{val| 0 1 2 7 2 10 6 }}, {{val| 0 0 -4 1 -1 4 -2 }}]
Mapping: {{mapping| 1 0 -2 -8 0 -11 -6 | 0 1 2 7 2 10 6 | 0 0 4 -1 1 -4 2 }}


Optimal tuning (CTE): ~3/2 = 702.1910, ~18/11 = 854.4751
Optimal tunings:
* WE: ~2 = 1200.1171{{c}}, ~3/2 = 702.1958{{c}}, ~11/9 = 345.5113{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1497{{c}}, ~11/9 = 345.4943{{c}}


Optimal GPV sequence: {{Val list| 94, 111, 205, 212g, 323 }}
{{Optimal ET sequence|legend=0| 94, 111, 205, 212g, 323 }}


Badness: 1.69 × 10<sup>-3</sup>
Badness (Sintel): 1.61


=== 19-limit ===
=== 19-limit ===
Line 97: Line 85:
Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444
Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444


Mapping: [{{val| 1 0 2 -9 1 -15 -4 -4 }}, {{val| 0 1 2 7 2 10 6 7 }}, {{val| 0 0 -4 1 -1 4 -2 -4 }}]
Mapping: {{mapping| 1 0 -2 -8 0 -11 -6 -8 | 0 1 2 7 2 10 6 7 | 0 0 4 -1 1 -4 2 4 }}


Optimal tuning (CTE): ~3/2 = 702.1925, ~18/11 = 854.4721
Optimal tunings:
* WE: ~2 = 1200.1120{{c}}, ~3/2 = 702.1996{{c}}, ~11/9 = 345.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1543{{c}}, ~11/9 = 345.5018{{c}}


Optimal GPV sequence: {{Val list| 94, 111, 205, 212gh, 323 }}
{{Optimal ET sequence|legend=0| 94, 111, 205, 212gh, 323 }}


Badness: 1.27 × 10<sup>-3</sup>
Badness (Sintel): 1.31


== Hemireiwa ==
== Hemireiwa ==
Hemireiwa is in a certain sense dual to semicanou. It splits the perfect twelfth into two, whereas semicanou splits the octave.  
Hemireiwa is in a certain sense dual to [[semicathart]]. It splits the perfect twelfth into two, whereas semicathart splits the octave.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 112: Line 102:
[[Comma list]]: 14641/14580, 160083/160000
[[Comma list]]: 14641/14580, 160083/160000


[[Mapping]]: [{{val| 1 0 2 7 1 }}, {{val| 0 2 0 -6 3 }}, {{val| 0 0 4 7 1 }}]
{{Mapping|legend=1| 1 0 2 7 1 | 0 2 0 -6 3 | 0 0 4 7 1 }}
: mapping generators: ~2, ~400/231, ~200/189


Mapping generators: ~2, ~400/231, ~200/189
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0073{{c}}, ~400/231 = 951.1989{{c}}, ~200/189 = 96.5797{{c}}
: [[error map]]: {{val| +0.007 +0.443 +0.020 +0.090 -1.134 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.1950{{c}}, ~200/189 = 96.5816{{c}}
: error map: {{val| 0.000 +0.435 +0.013 +0.076 -1.151 }}


Optimal tuning ([[CTE]]): ~400/231 = 951.1975, ~200/189 = 96.5851
{{Optimal ET sequence|legend=1| 87, 111, 125, 198, 212, 323, 410 }}


{{Val list|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
[[Badness]] (Sintel): 4.74
 
[[Badness]]: 3.95 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 127: Line 120:
Comma list: 676/675, 1001/1000, 14641/14580
Comma list: 676/675, 1001/1000, 14641/14580


Mapping: [{{val| 1 0 2 7 1 1 }}, {{val| 0 2 0 -6 3 3 }}, {{val| 0 0 4 7 1 4 }}]
Mapping: {{mapping| 1 0 2 7 1 1 | 0 2 0 -6 3 3 | 0 0 4 7 1 4 }}


Optimal tuning (CTE): ~26/15 = 951.2316, ~55/52 = 96.6155
Optimal tuning:  
* WE: ~2 = 1199.9814{{c}}, ~26/15 = 951.2264{{c}}, ~55/52 = 96.6277{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.2372{{c}}, ~55/52 = 96.6235{{c}}


Optimal GPV sequence: {{val list| 87, 111, 198, 323, 410 }}
{{Optimal ET sequence|legend=0| 87, 111, 198, 323, 410 }}


Badness: 1.89 × 10<sup>-3</sup>
Badness (Sintel): 1.77


=== 17-limit ===
=== 17-limit ===
Line 140: Line 135:
Comma list: 676/675, 715/714, 1001/1000, 14641/14580
Comma list: 676/675, 715/714, 1001/1000, 14641/14580


Mapping: [{{val| 1 0 2 7 1 1 -4 }}, {{val| 0 2 0 -6 3 3 10 }}, {{val| 0 0 4 7 1 4 2 }}]
Mapping: {{mapping| 1 0 2 7 1 1 -4 | 0 2 0 -6 3 3 10 | 0 0 4 7 1 4 2 }}


Optimal tuning (CTE): ~26/15 = 951.2022, ~55/52 = 96.6006
Optimal tuning:  
* WE: ~2 = 1200.0093{{c}}, ~26/15 = 951.2058{{c}}, ~55/52 = 96.5949{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.1999{{c}}, ~55/52 = 96.5966{{c}}


Optimal GPV sequence: {{val list| 87, 111, 198g, 212g, 299, 323, 410 }}
{{Optimal ET sequence|legend=0| 87, 111, 198g, 212g, 299, 323, 410 }}


Badness: 1.90 × 10<sup>-3</sup>
Badness (Sintel): 1.81


=== 19-limit ===
=== 19-limit ===
Line 153: Line 150:
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444


Mapping: [{{val| 1 0 2 7 1 1 -4 -4 }}, {{val| 0 2 0 -6 3 3 10 10 }}, {{val| 0 0 4 7 1 4 2 4 }}]
Mapping: {{mapping| 1 0 2 7 1 1 -4 -4 | 0 2 0 -6 3 3 10 10 | 0 0 4 7 1 4 2 4 }}
 
Optimal tuning:
* WE: ~2 = 1200.0492{{c}}, ~26/15 = 951.1982{{c}}, ~55/52 = 96.5541{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.1648{{c}}, ~55/52 = 96.5612{{c}}
 
{{Optimal ET sequence|legend=0| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}
 
Badness (Sintel): 1.78
 
== Subgroup extensions ==
=== Deuteromere (2.3.5.11.17) ===
Subgroup: 2.3.5.11.17
 
Comma list: 1089/1088, 14641/14580
 
Subgroup-val mapping: {{mapping| 1 0 -2 0 -6 | 0 1 2 2 6 | 0 0 4 1 2 }}
 
Optimal tunings:
* WE: ~2 = 1200.0291{{c}}, ~3/2 = 702.3333{{c}}, ~11/9 = 345.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3266{{c}}, ~11/9 = 345.4394{{c}}
 
{{Optimal ET sequence|legend=0| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}
 
Badness (Sintel): 1.01
 
==== 2.3.5.11.17.19 subgroup ====
Subgroup: 2.3.5.11.17.19
 
Comma list: 1089/1088, 1216/1215, 1445/1444
 
Subgroup-val mapping: {{mapping| 1 0 -2 0 -6 -8 | 0 1 2 2 6 7 | 0 0 4 1 2 4 }}


Optimal tuning (CTE): ~26/15 = 951.1749, ~55/52 = 96.5813
Optimal tunings:  
* WE: ~2 = 1200.0556{{c}}, ~3/2 = 702.2904{{c}}, ~11/9 = 345.4565{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2728{{c}}, ~11/9 = 345.4432{{c}}


Optimal GPV sequence: {{val list| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}
{{Optimal ET sequence|legend=0| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}


Badness: 1.73 × 10<sup>-3</sup>
Badness (Sintel): 0.630


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Semicanousmic clan| ]] <!-- main article -->
[[Category:Semicathartismic clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Semicathartismic_clan"