Semicathartismic clan: Difference between revisions

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-semicanousmic to save some space
POTE > CTE
Line 19: Line 19:
Sval mapping generators: ~2, ~3, ~18/11
Sval mapping generators: ~2, ~3, ~18/11


Optimal tuning ([[POTE]]): ~3/2 = 702.2503, ~18/11 = 854.5421
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 }}
{{Val list|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}
Line 32: Line 32:
Sval mapping: [{{val| 1 0 2 1 -4 }}, {{val| 0 1 2 2 6 }}, {{val| 0 0 -4 -1 -2 }}]
Sval mapping: [{{val| 1 0 2 1 -4 }}, {{val| 0 1 2 2 6 }}, {{val| 0 0 -4 -1 -2 }}]


Optimal tuning (POTE): ~3/2 = 702.3162, ~18/11 = 854.5619
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 GPV sequence: {{val list| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}
Line 45: Line 45:
Sval mapping: [{{val| 1 0 2 1 -4 -4 }}, {{val| 0 1 2 2 6 7 }}, {{val| 0 0 -4 -1 -2 -4 }}]
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 (POTE): ~3/2 = 702.2578, ~18/11 = 854.5595
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 GPV sequence: {{val list| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}
Line 60: Line 60:
Mapping generators: ~2, ~3, ~18/11
Mapping generators: ~2, ~3, ~18/11


Optimal tuning ([[POTE]]): ~3/2 = 702.0109, ~18/11 = 854.4606
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 }}
{{Val list|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}
Line 73: Line 73:
Mapping: [{{val| 1 0 2 -9 1 -15 }}, {{val| 0 1 2 7 2 10 }}, {{val| 0 0 -4 1 -1 4 }}]
Mapping: [{{val| 1 0 2 -9 1 -15 }}, {{val| 0 1 2 7 2 10 }}, {{val| 0 0 -4 1 -1 4 }}]


Optimal tuning (POTE): ~3/2 = 702.1049, ~18/11 = 854.5542
Optimal tuning (CTE): ~3/2 = 702.1682, ~18/11 = 854.5074


Optimal GPV sequence: {{Val list| 94, 111, 205, 212, 323, 535ef }}
Optimal GPV sequence: {{Val list| 94, 111, 205, 212, 323, 535ef }}
Line 86: Line 86:
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: [{{val| 1 0 2 -9 1 -15 -4 }}, {{val| 0 1 2 7 2 10 6 }}, {{val| 0 0 -4 1 -1 4 -2 }}]


Optimal tuning (POTE): ~3/2 = 702.1272, ~18/11 = 854.5224
Optimal tuning (CTE): ~3/2 = 702.1910, ~18/11 = 854.4751


Optimal GPV sequence: {{Val list| 94, 111, 205, 212g, 323 }}
Optimal GPV sequence: {{Val list| 94, 111, 205, 212g, 323 }}
Line 99: Line 99:
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: [{{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 }}]


Optimal tuning (POTE): ~3/2 = 702.1340, ~18/11 = 854.5120
Optimal tuning (CTE): ~3/2 = 702.1925, ~18/11 = 854.4721


Optimal GPV sequence: {{Val list| 94, 111, 205, 212gh, 323 }}
Optimal GPV sequence: {{Val list| 94, 111, 205, 212gh, 323 }}
Line 116: Line 116:
Mapping generators: ~2, ~400/231, ~200/189
Mapping generators: ~2, ~400/231, ~200/189


Optimal tuning ([[POTE]]): ~400/231 = 951.1931, ~200/189 = 96.5791
Optimal tuning ([[CTE]]): ~400/231 = 951.1975, ~200/189 = 96.5851


{{Val list|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
{{Val list|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
Line 129: Line 129:
Mapping: [{{val| 1 0 2 7 1 1 }}, {{val| 0 2 0 -6 3 3 }}, {{val| 0 0 4 7 1 4 }}]
Mapping: [{{val| 1 0 2 7 1 1 }}, {{val| 0 2 0 -6 3 3 }}, {{val| 0 0 4 7 1 4 }}]


Optimal tuning (POTE): ~26/15 = 951.2412, ~55/52 = 96.6292
Optimal tuning (CTE): ~26/15 = 951.2316, ~55/52 = 96.6155


Optimal GPV sequence: {{val list| 87, 111, 198, 323, 410 }}
Optimal GPV sequence: {{val list| 87, 111, 198, 323, 410 }}
Line 142: Line 142:
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: [{{val| 1 0 2 7 1 1 -4 }}, {{val| 0 2 0 -6 3 3 10 }}, {{val| 0 0 4 7 1 4 2 }}]


Optimal tuning (POTE): ~26/15 = 951.1985, ~55/52 = 96.5941
Optimal tuning (CTE): ~26/15 = 951.2022, ~55/52 = 96.6006


Optimal GPV sequence: {{val list| 87, 111, 198g, 212g, 299, 323, 410 }}
Optimal GPV sequence: {{val list| 87, 111, 198g, 212g, 299, 323, 410 }}
Line 155: Line 155:
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: [{{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 }}]


Optimal tuning (POTE): ~26/15 = 951.1592, ~55/52 = 96.5502
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 GPV sequence: {{val list| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}

Revision as of 17:53, 14 August 2022

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 synsem and vonacimes, in addition to deuteromere, the no-7 subgroup temperament.

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

Deuteromere

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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.403 × 10-3

Synsem

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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 1.27 × 10-3

Vonacimes

Vonacimes (semicanou spelt backwards) 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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 1.73 × 10-3

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