Semicanousmic clan: Difference between revisions

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-These are rank-3 temperaments tempering out the [[semicanousma]], 14641/14580 = {{monzo|-2 -6 -1 0 4}}, but we can begin by looking at the rank-4 temperament.
+{{Technical data page}}
+The '''semicanousmic clan''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[semicanousma]], 14641/14580 = {{monzo| -2 -6 -1 0 4 }}.
-Temperaments not discussed here include [[Hemimean family #Triglav|triglav]] and [[Canou family #Semicanou|semicanou]].
+Temperaments discussed elsewhere are:
+* ''[[Urania]]'' (+81/80 or 121/120) → [[Didymus rank three family #Urania|Didymus rank-3 family]]
+* ''[[Triglav]]'' (+3025/3024 or 3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
+* ''[[Semicanou]]'' (+9801/9800) → [[Canou family #Semicanou|Canou family]]
-Comma: 14641/14580
+Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.
-Map: [<1 0 2 0 1|, <0 1 2 0 2|, <0 0 4 0 1|, <0 0 0 1 0|]
+For the rank-4 semicanousmic temperament, see [[Rank-4 temperament #Semicanousmic (14641/14580)]].
-POTE generators: ~3/2 = 702.2503, ~11/9 = 345.4579, ~7/4 = 968.6866
+== Deuteromere ==
+[[File:Lattice Semicanousmic.png|thumb|Lattice]]
+[[File:Lattice Semicanousmic Rearranged.png|thumb|Lattice rearranged into neutral thirds]]
-{{EDOs|legend=1| 7, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410}}
+[[Subgroup]]: 2.3.5.11
-Badness: 0.3506 × 10<sup>-6</sup>
+[[Comma list]]: 14641/14580
-== No-seven semicanousmic ==
+{{Mapping|legend=2| 1 0 2 1 | 0 1 2 2 | 0 0 -4 -1 }}
-Comma: 14641/14580
-Map: [<1 0 2 0 1|, <0 1 2 0 2|, <0 0 4 0 1|]
+: sval mapping generators: ~2, ~3, ~18/11
-POTE generators: ~3/2 = 702.2503, ~11/9 = 345.4579
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.3342, ~18/11 = 854.5548
-{{EDOs|legend=1| 7, 24, 31, 56, 73, 80, 87, 118, 205, 323 }}
+{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}
-Badness: 0.1425 × 10<sup>-3</sup>
+[[Badness]]: 0.142 × 10<sup>-3</sup>
-[[Category:Theory]]
+=== 2.3.5.11.17 subgroup ===
-[[Category:Temperament]]
+Subgroup: 2.3.5.11.17
-[[Category:Semicanousmic]]
+Comma list: 1089/1088, 14641/14580
+Sval mapping: {{mapping| 1 0 2 1 -4 | 0 1 2 2 6 | 0 0 -4 -1 -2 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3427, ~18/11 = 854.5587
+{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 2 1 -4 -4 | 0 1 2 2 6 7 | 0 0 -4 -1 -2 -4 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2987, ~18/11 = 854.5520
+{{Optimal ET sequence|legend=1| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}
+Badness: 0.403 × 10<sup>-3</sup>
+== Syndeute ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 14641/14580, 19712/19683
+{{Mapping|legend=1| 1 0 2 -9 1 | 0 1 2 7 2 | 0 0 -4 1 -1 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.0812, ~18/11 = 854.4115
+{{Optimal ET sequence|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}
+[[Badness]]: 3.26 × 10<sup>-3</sup>
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 1573/1568, 2080/2079, 14641/14580
+Mapping: {{mapping| 1 0 2 -9 1 -15 | 0 1 2 7 2 10 | 0 0 -4 1 -1 4 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1682, ~18/11 = 854.5074
+{{Optimal ET sequence|legend=1| 94, 111, 205, 212, 323, 535ef }}
+Badness: 3.01 × 10<sup>-3</sup>
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 442/441, 561/560, 715/714, 14641/14580
+Mapping: {{mapping| 1 0 2 -9 1 -15 -4 | 0 1 2 7 2 10 6 | 0 0 -4 1 -1 4 -2 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1910, ~18/11 = 854.4751
+{{Optimal ET sequence|legend=1| 94, 111, 205, 212g, 323 }}
+Badness: 1.69 × 10<sup>-3</sup>
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444
+Mapping: {{mapping| 1 0 2 -9 1 -15 -4 -4 | 0 1 2 7 2 10 6 7 | 0 0 -4 1 -1 4 -2 -4 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1925, ~18/11 = 854.4721
+{{Optimal ET sequence|legend=1| 94, 111, 205, 212gh, 323 }}
+Badness: 1.27 × 10<sup>-3</sup>
+== Hemireiwa ==
+Hemireiwa is in a certain sense dual to semicanou. It splits the perfect twelfth into two, whereas semicanou splits the octave.
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 14641/14580, 160083/160000
+{{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
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~400/231 = 951.1975, ~200/189 = 96.5851
+{{Optimal ET sequence|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
+[[Badness]]: 3.95 × 10<sup>-3</sup>
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 676/675, 1001/1000, 14641/14580
+Mapping: {{mapping| 1 0 2 7 1 1 | 0 2 0 -6 3 3 | 0 0 4 7 1 4 }}
+Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.2316, ~55/52 = 96.6155
+{{Optimal ET sequence|legend=1| 87, 111, 198, 323, 410 }}
+Badness: 1.89 × 10<sup>-3</sup>
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 676/675, 715/714, 1001/1000, 14641/14580
+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): ~2 = 1\1, ~26/15 = 951.2022, ~55/52 = 96.6006
+{{Optimal ET sequence|legend=1| 87, 111, 198g, 212g, 299, 323, 410 }}
+Badness: 1.90 × 10<sup>-3</sup>
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444
+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 (CTE): ~2 = 1\1, ~26/15 = 951.1749, ~55/52 = 96.5813
+{{Optimal ET sequence|legend=1| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}
+Badness: 1.73 × 10<sup>-3</sup>
+[[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
+[[Category:Semicanousmic clan| ]] <!-- main article -->
+[[Category:Semicanousmic| ]] <!-- key article -->
 [[Category:Rank 3]]