Keemic family

The keemic family of rank-3 temperaments tempers out the keema, 875/864 = [-5 -3 3 1⟩.

Supermagic

Subgroup: 2.3.5.7

Comma list: 875/864

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

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 0 -1 3], ⟨0 1 1 0]]

Lattice basis:

6/5 length = 0.8879, 3/2 length = 1.3391

Angle (6/5, 3/2) = 77.834

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8733, ~5/4 = 380.4684

Minimax tuning:

• 7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [5/4 3/4 1/4 -1/4⟩, [5/4 3/4 -3/4 3/4⟩]

eigenmonzo (unchanged-interval) basis: 2.3.7/5

Optimal ET sequence: 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd

Badness: 0.212 × 10^-3

Projection pair: 7 864/125

Scales: supermagic15

Undecimal supermagic

See also: Ptolemismic clan § Supermagic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1985, ~5/4 = 381.5394

Optimal ET sequence: 7, 15, 19, 22, 37, 41, 104

Badness: 0.641 × 10^-3

Scales: supermagic15

Supernatural

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/243

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

mapping generators: ~2, ~5, ~11

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.3520, ~11/8 = 547.5758

Optimal ET sequence: 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde *

* optimal patent val: 104

Badness: 0.888 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/243

Mapping: [⟨1 0 2 -1 0 -2], ⟨0 5 1 12 0 18], ⟨0 0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 380.0401, ~11/8 = 546.6672

Optimal ET sequence: 19, 22f, 38df, 41, 60e, 79d, 101cd

Badness: 1.14 × 10^-3