Orwellismic family

The orwellismic family of rank-3 temperaments tempers out the orwellisma, 1728/1715.

Orwellismic

Subgroup: 2.3.5.7

Comma list: 1728/1715

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

mapping generators: ~2, ~3, ~12/7

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

Lattice basis:

7/6 length = 0.7526, 3/2 length = 1.3931

Angle (7/6, 3/2) = 97.5081

Minimax tuning:

• 7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [3/2 3/4 3/4 -3/4⟩, [3/2 3/4 -1/4 1/4⟩]

Eigenmonzo (unchanged-interval) basis: 2.3.7/5

Optimal ET sequence: 4, 5, 9, 17, 18, 22, 27, 31, 53, 80, 84, 111, 164d, 195d, 275d, 306d

Badness: 0.187 × 10^-3

Projection pair: ~5 = 1728/343 to 2.3.7

Scales: orwellismic9

Guanyin

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539

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

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

Lattice basis:

7/6 length = 0.5167, 3/2 length = 1.3055

Angle (7/6, 3/2) = 79.0940

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [-2/9 0 -5/9 5/9 4/9⟩, [4/3 0 1/3 -1/3 1/3⟩, [4/3 0 -2/3 2/3 1/3⟩, [0 0 0 0 1⟩]

Eigenmonzo (unchanged-interval) basis: 2.11.7/5

Optimal ET sequence: 22, 27e, 31, 53, 58, 80, 111, 280cd

Badness: 0.411 × 10^-3

Projection pairs: ~5 = 1728/343, ~11 = 186624/16807 to 2.3.7

Scales: guanyin22

Music

• Trio for SoftSaturn, NebulaSing and TromBonehead (archived 2010) by Gene Ward Smith – SoundCloud | details | play – guanyin[22] in 111edo tuning

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539

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

Optimal ET sequence: 22, 27e, 31, 53, 58, 80, 111, 280cdf

Badness: 0.758 × 10^-3

Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 221184/16897 to 2.3.7

Georgian

This temperament is also known as avalokita.

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195

Mapping: [⟨1 0 0 2 -2 6], ⟨0 1 0 1 1 1], ⟨0 0 3 -1 5 -5]]

Optimal ET sequence: 22f, 27e, 31, 49f, 58, 138cdeff, 196cdefff, 254ccdeeffff

Badness: 0.904 × 10^-3

Projection pairs: ~5 = 1728/343, ~11 = 186624/16807, ~13 = 16807/1296 to 2.3.7

Orwellian

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384

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

Optimal ET sequence: 4, 5, 9, 17, 22, 31, 53, 84e

Badness: 0.715 × 10^-3