Octagar family

The octagar family of temperaments are planar temperaments tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.

Octagar

Comma: 4000/3969

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

Mapping generators: 2, ~3, ~63/40

Minimax tuning:

• 7-limit minimax:

[[1 0 0 0⟩, [5/6 1/3 1/2 -1/3⟩, [0 0 1 0⟩, [5/6 -2/3 1/2 2/3⟩]

Eigenmonzos (unchanged intervals): 2, 7/6, 5/4

• 9-limit minimax:

[[1 0 0 0⟩, [5/8 1/2 3/8 -1/4⟩, [0 0 1 0⟩, [5/4 -1 3/4 1/2⟩]

Eigenmonzos (unchanged intervals): 2, 5/4, 9/7

Lattice basis: 63/50 length = 0.8966, 21/20 length = 1.0605

Angle(63/50, 21/20) = 97.743 cents

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

Badness: 0.000216

Projection pairs: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5

Hobbit bases

{2, 3, 7/5} subgroup

12: 50/49, 256000/250047

15: 256000/250047, 1029/1000

23: 12800000/12252303, 107163/102400

Nakika

Comma list: 100/99, 245/242

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

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

Associated linear temperament: octacot

Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906

Angle(11/7, 22/21) = 97.747 degrees

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

Vals: Template:Val list

Badness: 0.000539

Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7

Scales: nakika12

13-limit

Comma list: 100/99, 105/104, 245/242

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

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

Vals: Template:Val list