Sensamagic family

The sensamagic family of rank-3 temperaments tempers out 245/243. For a list of rank-2 temperaments, see Sensamagic clan.

Sensamagic

Main article: Sensamagic

Subgroup: 2.3.5.7

Comma list: 245/243

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

Mapping generators: ~2, ~3, ~9/7

Mapping to lattice: [⟨0 1 1 2], ⟨0 0 2 -1]]

Lattice basis:

3/2 length = 0.9644, 9/7 length = 1.0807

Angle (3/2, 9/7) = 86.5288°

Optimal tuning (POTE): ~3/2 = 703.7424, ~9/7 = 440.9020

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [0 0 1/5 2/5⟩, [0 0 1 0⟩, [0 0 0 1⟩]

Eigenmonzos (unchanged intervals): 2, 8/7, 5/4

• 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]

Eigenmonzos (unchanged intervals): 2, 4/3, 7/5

Badness: 0.129 × 10^-3

Projection pair: 5 243/49 to 2.3.7

Minkowski blocks

{2, 3, 7} subgroup

• 12: 729/686, 64/63
• 17: 64/63, 19683/19208
• 19: 49/48, 177147/175616
• 22: 64/63, 537824/531441
• 24: 64/63, 15059072/14348907

Overview to extensions

Temperaments discussed elsewhere include supernatural (→ Keemic family). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.

Undecimal sensamagic

Main article: Sensamagic

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384

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

Optimal tuning (POTE): ~3/2 = 703.8004, ~9/7 = 440.9178

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [21/13 6/13 -1/13 1/13 -3/13⟩, [35/13 10/13 7/13 -7/13 -5/13⟩, [35/13 10/13 -6/13 6/13 -5/13⟩, [42/13 -14/13 -2/13 2/13 7/13⟩]

Eigenmonzos (unchanged intervals): 2, 11/9, 7/5

Badness: 0.722 × 10^-3

Projection pairs: 5 243/49 11 896/81 to 2.3.7

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 1 2 -2 -5], ⟨0 0 2 -1 -1 -1]]

Optimal GPV sequence: Template:Val list

Badness: 1.20 × 10^-3

Sensawer

Subgroup: 2.3.5.7.11

Comma list: 245/243, 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 1 2 5], ⟨0 0 2 -1 -4]]

Optimal tuning (POTE): ~3/2 = 703.1900, ~9/7 = 441.1359

Badness: 0.796 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 1 2 5 2], ⟨0 0 2 -1 -4 -4]]

Optimal GPV sequence: Template:Val list

Badness: 0.928 × 10^-3

Octarod

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243

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

Optimal tuning (POTE): ~3/2 = 705.0464, ~9/7 = 439.5050

Badness: 0.581 × 10^-3

Scales: octarod1, octarod2, octarod3, octarod4, octarod5

Shrusus

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243

Mapping: [⟨1 0 0 0 -4], ⟨0 1 1 2 4], ⟨0 0 2 -1 3]]

Optimal tuning (POTE): ~3/2 = 706.3702, ~9/7 = 442.1147

Badness: 0.877 × 10^-3

Shrusic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243

Mapping: [⟨1 0 0 0 -4 1], ⟨0 1 1 2 4 1], ⟨0 0 2 -1 3 3]]

Optimal GPV sequence: Template:Val list

Badness: 1.125 × 10^-3

Bisector

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243

Mapping: [⟨2 0 0 0 3], ⟨0 1 1 2 1], ⟨0 0 2 -1 1]]

Mapping generators: ~77/54, ~3, ~9/7

Optimal tuning (POTE): ~3/2 = 703.0884, ~9/7 = 441.1060

Badness: 1.089 × 10^-3

Sensigh

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168

Mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10], ⟨0 0 0 0 1 0]]

Mapping generators: ~2, ~9/7, ~11

Optimal GPV sequence: Template:Val list

Badness: 0.939 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168

Mapping: [⟨1 6 8 11 0 10 0], ⟨0 -7 -9 -13 0 -10 1], ⟨0 0 0 0 1 0 1]]

Optimal GPV sequence: Template:Val list