Semicanousmic clan

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

Temperaments discussed elsewhere are:

Urania, {81/80, 121/120} → Didymus rank three family
Triglav, {3025/3024, 3136/3125} → Hemimean family
Semicanou, {9801/9800, 14641/14580} → Canou family

Considered below are synsem and vonacimes, in addition to deuteromere, the no-7 subgroup temperament, but we can begin by looking at the rank-4 temperament.

Semicanousmic

For the full 19-limit extension, see Rank-4 temperament #Semicanousmic (14641/14580).

Subgroup: 2.3.5.7.11

Comma list: 14641/14580

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

Mapping generators: ~2, ~3, ~18/11, ~7

Optimal tuning (POTE): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866

Template:Val list

Badness: 0.351 × 10^-6

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 (POTE): ~3/2 = 702.2503, ~18/11 = 854.5421

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 (POTE): ~3/2 = 702.3162, ~18/11 = 854.5619

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 (POTE): ~3/2 = 702.2578, ~18/11 = 854.5595

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 (POTE): ~3/2 = 702.0109, ~18/11 = 854.4606

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 (POTE): ~3/2 = 702.1049, ~18/11 = 854.5542

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 (POTE): ~3/2 = 702.1272, ~18/11 = 854.5224

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 (POTE): ~3/2 = 702.1340, ~18/11 = 854.5120

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 (POTE): ~400/231 = 951.1931, ~200/189 = 96.5791

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 (POTE): ~26/15 = 951.2412, ~55/52 = 96.6292

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 (POTE): ~26/15 = 951.1985, ~55/52 = 96.5941

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 (POTE): ~26/15 = 951.1592, ~55/52 = 96.5502

Optimal GPV sequence: Template:Val list

Badness: 1.73 × 10^-3