Semicanousmic clan: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 24: Line 24:
Optimal tuning ([[CTE]]): ~3/2 = 702.3342, ~18/11 = 854.5548
Optimal tuning ([[CTE]]): ~3/2 = 702.3342, ~18/11 = 854.5548


{{Val list|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}
{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}


[[Badness]]: 0.142 × 10<sup>-3</sup>
[[Badness]]: 0.142 × 10<sup>-3</sup>
Line 37: Line 37:
Optimal tuning (CTE): ~3/2 = 702.3427, ~18/11 = 854.5587
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 }}
{{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>
Badness: 0.481 × 10<sup>-3</sup>
Line 50: Line 50:
Optimal tuning (CTE): ~3/2 = 702.2987, ~18/11 = 854.5520
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 }}
{{Optimal ET sequence|legend=1| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}


Badness: 0.403 × 10<sup>-3</sup>
Badness: 0.403 × 10<sup>-3</sup>
Line 65: Line 65:
Optimal tuning ([[CTE]]): ~3/2 = 702.0812, ~18/11 = 854.4115
Optimal tuning ([[CTE]]): ~3/2 = 702.0812, ~18/11 = 854.4115


{{Val list|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}


[[Badness]]: 3.26 × 10<sup>-3</sup>
[[Badness]]: 3.26 × 10<sup>-3</sup>
Line 78: Line 78:
Optimal tuning (CTE): ~3/2 = 702.1682, ~18/11 = 854.5074
Optimal tuning (CTE): ~3/2 = 702.1682, ~18/11 = 854.5074


Optimal GPV sequence: {{Val list| 94, 111, 205, 212, 323, 535ef }}
{{Optimal ET sequence|legend=1| 94, 111, 205, 212, 323, 535ef }}


Badness: 3.01 × 10<sup>-3</sup>
Badness: 3.01 × 10<sup>-3</sup>
Line 91: Line 91:
Optimal tuning (CTE): ~3/2 = 702.1910, ~18/11 = 854.4751
Optimal tuning (CTE): ~3/2 = 702.1910, ~18/11 = 854.4751


Optimal GPV sequence: {{Val list| 94, 111, 205, 212g, 323 }}
{{Optimal ET sequence|legend=1| 94, 111, 205, 212g, 323 }}


Badness: 1.69 × 10<sup>-3</sup>
Badness: 1.69 × 10<sup>-3</sup>
Line 104: Line 104:
Optimal tuning (CTE): ~3/2 = 702.1925, ~18/11 = 854.4721
Optimal tuning (CTE): ~3/2 = 702.1925, ~18/11 = 854.4721


Optimal GPV sequence: {{Val list| 94, 111, 205, 212gh, 323 }}
{{Optimal ET sequence|legend=1| 94, 111, 205, 212gh, 323 }}


Badness: 1.27 × 10<sup>-3</sup>
Badness: 1.27 × 10<sup>-3</sup>
Line 121: Line 121:
Optimal tuning ([[CTE]]): ~400/231 = 951.1975, ~200/189 = 96.5851
Optimal tuning ([[CTE]]): ~400/231 = 951.1975, ~200/189 = 96.5851


{{Val list|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 111, 125, 198, 212, 323, 410 }}


[[Badness]]: 3.95 × 10<sup>-3</sup>
[[Badness]]: 3.95 × 10<sup>-3</sup>
Line 134: Line 134:
Optimal tuning (CTE): ~26/15 = 951.2316, ~55/52 = 96.6155
Optimal tuning (CTE): ~26/15 = 951.2316, ~55/52 = 96.6155


Optimal GPV sequence: {{val list| 87, 111, 198, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 111, 198, 323, 410 }}


Badness: 1.89 × 10<sup>-3</sup>
Badness: 1.89 × 10<sup>-3</sup>
Line 147: Line 147:
Optimal tuning (CTE): ~26/15 = 951.2022, ~55/52 = 96.6006
Optimal tuning (CTE): ~26/15 = 951.2022, ~55/52 = 96.6006


Optimal GPV sequence: {{val list| 87, 111, 198g, 212g, 299, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 111, 198g, 212g, 299, 323, 410 }}


Badness: 1.90 × 10<sup>-3</sup>
Badness: 1.90 × 10<sup>-3</sup>
Line 160: Line 160:
Optimal tuning (CTE): ~26/15 = 951.1749, ~55/52 = 96.5813
Optimal tuning (CTE): ~26/15 = 951.1749, ~55/52 = 96.5813


Optimal GPV sequence: {{val list| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}
{{Optimal ET sequence|legend=1| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}


Badness: 1.73 × 10<sup>-3</sup>
Badness: 1.73 × 10<sup>-3</sup>

Revision as of 16:15, 7 May 2023

The semicanousmic clan of rank-3 temperaments tempers out the semicanousma, 14641/14580 = [-2 -6 -1 0 4⟩.

Temperaments discussed elsewhere are:

Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.

For the rank-4 semicanousmic temperament, see Rank-4 temperament #Semicanousmic (14641/14580).

Deuteromere

Lattice
Lattice rearranged into neutral thirds

Subgroup: 2.3.5.11

Comma list: 14641/14580

Sval mapping: [⟨1 0 2 1], ⟨0 1 2 2], ⟨0 0 -4 -1]]

Sval mapping generators: ~2, ~3, ~18/11

Optimal tuning (CTE): ~3/2 = 702.3342, ~18/11 = 854.5548

Optimal ET sequence: 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e

Badness: 0.142 × 10-3

2.3.5.11.17 subgroup

Subgroup: 2.3.5.11.17

Comma list: 1089/1088, 14641/14580

Sval mapping: [⟨1 0 2 1 -4], ⟨0 1 2 2 6], ⟨0 0 -4 -1 -2]]

Optimal tuning (CTE): ~3/2 = 702.3427, ~18/11 = 854.5587

Optimal ET sequence: 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e

Badness: 0.481 × 10-3

2.3.5.11.17.19 subgroup

Subgroup: 2.3.5.11.17.19

Comma list: 1089/1088, 1216/1215, 1445/1444

Sval mapping: [⟨1 0 2 1 -4 -4], ⟨0 1 2 2 6 7], ⟨0 0 -4 -1 -2 -4]]

Optimal tuning (CTE): ~3/2 = 702.2987, ~18/11 = 854.5520

Optimal ET sequence: 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e

Badness: 0.403 × 10-3

Syndeute

Subgroup: 2.3.5.7.11

Comma list: 14641/14580, 19712/19683

Mapping: [⟨1 0 2 -9 1], ⟨0 1 2 7 2], ⟨0 0 -4 1 -1]]

Mapping generators: ~2, ~3, ~18/11

Optimal tuning (CTE): ~3/2 = 702.0812, ~18/11 = 854.4115

Optimal ET sequence: 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e

Badness: 3.26 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1573/1568, 2080/2079, 14641/14580

Mapping: [⟨1 0 2 -9 1 -15], ⟨0 1 2 7 2 10], ⟨0 0 -4 1 -1 4]]

Optimal tuning (CTE): ~3/2 = 702.1682, ~18/11 = 854.5074

Optimal ET sequence: 94, 111, 205, 212, 323, 535ef

Badness: 3.01 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 561/560, 715/714, 14641/14580

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): ~3/2 = 702.1910, ~18/11 = 854.4751

Optimal ET sequence: 94, 111, 205, 212g, 323

Badness: 1.69 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444

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): ~3/2 = 702.1925, ~18/11 = 854.4721

Optimal ET sequence: 94, 111, 205, 212gh, 323

Badness: 1.27 × 10-3

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: [⟨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): ~400/231 = 951.1975, ~200/189 = 96.5851

Optimal ET sequence: 87, 111, 125, 198, 212, 323, 410

Badness: 3.95 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 14641/14580

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 ET sequence: 87, 111, 198, 323, 410

Badness: 1.89 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 1001/1000, 14641/14580

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 ET sequence: 87, 111, 198g, 212g, 299, 323, 410

Badness: 1.90 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444

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): ~26/15 = 951.1749, ~55/52 = 96.5813

Optimal ET sequence: 87, 111, 198gh, 212gh, 299, 323, 410, 622ef

Badness: 1.73 × 10-3

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