Kleismic rank three family

The kleismic rank-3 family of temperaments tempers out the kleisma, 15625/15552. If nothing else is tempered out in the 7-limit, we have a planar temperament. With an 11-limit comma we get an 11-limit temperament, and so forth.

Rank-3 kleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552

Mapping: [⟨1 0 1 0], ⟨0 6 5 0], ⟨0 0 0 1]]

Mapping generators: ~2, ~6/5, ~7

Minimax tuning:

• 7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [1 5/6 0 0⟩, [0 0 0 1⟩]

Eigenmonzos (unchanged intervals): 2, 7/6, 4/3

Optimal GPV sequence: 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c

Enlil

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237

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

Mapping generators: ~2, ~6/5, ~7

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [1 5/6 0 0 0⟩, [3 1/12 0 1/2 -1/2⟩, [3 1/12 0 -1/2 1/2⟩]

Eigenmonzos (unchanged intervals): 2, 4/3, 14/11

Optimal GPV sequence: 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee

Badness: 0.798 × 10^-3

13-limit (aka sumatra)

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624

Mapping: [⟨1 0 1 0 6 0], ⟨0 6 5 0 1 14], ⟨0 0 0 1 -1 0]]

Mapping generators: ~2, ~6/5, ~7

Minimax tuning:

• 13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [4/3 0 0 -2/9 -2/9 4/9⟩, [19/9 0 0 -5/27 -5/27 10/27⟩, [28/9 0 0 13/27 -14/27 1/27⟩, [28/9 0 0 -14/27 13/27 1/27⟩, [28/9 0 0 -14/27 -14/27 28/27⟩]

Eigenmonzos (unchanged intervals): 2, 13/11, 14/11

Vals: 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef

Badness: 0.680 × 10^-3

Scales: enlil19_13

Enki

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4356

Mapping: [⟨1 0 1 0 1], ⟨0 6 5 0 4], ⟨0 0 0 2 1]]

Mapping generators: 2, 6/5, 66/25

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [-18/13 0 6/13 -6/13 12/13⟩, [-2/13 0 5/13 -5/13 10/13⟩, [-2/13 0 -8/13 8/13 10/13⟩, [0 0 0 0 1⟩]

Eigenmonzos (unchanged intervals): 2, 11/8, 7/5

Optimal GPV sequence: 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424

Badness: 1.09 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624

Mapping: [⟨1 0 1 0 1 0], ⟨0 6 5 0 4 14], ⟨0 0 0 2 1 0]]

Mapping generators: 2, 6/5, 66/25

Minimax tuning:

• 13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [2/3 0 -2/3 0 0 2/3⟩, [14/9 0 -5/9 0 0 5/9⟩, [0 0 0 1 0 0⟩, [13/9 0 -4/9 1/2 0 4/9⟩, [14/9 0 -14/9 0 0 14/9⟩]

Eigenmonzos (unchanged intervals): 2, 8/7 13/10

Vals: 34d, 49f, 72, 87, 159, 193, 280

Badness: 0.874 × 10^-3

Isimud

Subgroup: 2.3.5.7.11

Comma list: 4125/4096, 8019/8000

Mapping: [⟨1 0 1 0 9], ⟨0 6 5 0 -21], ⟨0 0 0 1 0]]

Optimal GPV sequence: 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee

Badness: 3.690 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1287/1280

Mapping: [⟨1 0 1 0 9 0], ⟨0 6 5 0 -21 14], ⟨0 0 0 1 0 0]]

Vals: 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff

Badness: 2.273 × 10^-3