Diaschismic rank three family

The diaschismic rank-3 family of temperaments tempers out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Rank-3 diaschismic

Subgroup: 2.3.5.7

Comma list: 2048/2025

Mapping: [⟨2 0 11 0], ⟨0 1 -2 0], ⟨0 0 0 1]]

mapping generators: ~45/32, ~3, ~7

Optimal tuning (POTE): ~3/2 = 704.8982, ~7/4 = 972.1330

Optimal ET sequence: 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd

Badness: 0.442 × 10^-3

Julius aka varda

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891

Mapping: [⟨2 0 11 0 14], ⟨0 1 -2 0 -4], ⟨0 0 0 1 1]]

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

Lattice basis:

3/2, 9/7

Angle (3/2, 9/7) = 82.289 degrees

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [-1/8 3/2 -5/4 0 1⟩, [11/8 -1/2 -1/4 0 1⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.11/3

Optimal ET sequence: 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde

Badness: 0.728 × 10^-3

Julius[22] hobbit transversal:

28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2

Julius[24] hobbit transversal

28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2

Varda

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363

Mapping: [⟨2 0 11 0 14 24], ⟨0 1 -2 0 -4 -7], ⟨0 0 0 1 1 1]]

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde

Badness: 0.970 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 351/350

Mapping: [⟨2 0 11 0 14 24 5], ⟨0 1 -2 0 -4 -7 1], ⟨0 0 0 1 1 1 0]]

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 126, 138cde

Julius

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 325/324, 896/891

Mapping: [⟨2 0 11 0 14 -18], ⟨0 1 -2 0 -4 8], ⟨0 0 0 1 1 0]]

Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd

Badness: 1.78 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 325/324

Mapping: [⟨2 0 11 0 14 -18 5], ⟨0 1 -2 0 -4 8 1], ⟨0 0 0 1 1 0 0]]

Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 176/175, 190/189, 256/255, 286/285

Mapping: [⟨2 0 11 0 14 -18 5 -13], ⟨0 1 -2 0 -4 8 1 5], ⟨0 0 0 1 1 0 0 1]]

Optimal ET sequence: 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd

Augustus

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1232/1215

Mapping: [⟨2 0 11 0 3], ⟨0 1 -2 0 3], ⟨0 0 0 1 -1]]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [45/16 3/4 -3/8 1/2 -1/2⟩, [45/16 3/4 -3/8 -1/2 1/2⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.11/7

Optimal ET sequence: 10, 12e, 22, 46, 68, 114, 170b

Badness: 1.58 × 10^-3

Tiberius

Subgroup: 2.3.5.7.11

Comma list: 441/440, 2048/2025

Mapping: [⟨2 0 11 0 -17], ⟨0 1 -2 0 4], ⟨0 0 0 1 2]]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [0 0 0 1 0⟩, [-3 2 -1 2 0⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.7

Optimal ET sequence: 46, 58, 104c, 160, 162ce, 172ce, 218ce

Badness: 1.71 × 10^-3

Claudius

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2048/2025

Mapping: [⟨2 0 11 0 15], ⟨0 1 -2 0 1], ⟨0 0 0 1 -2]]

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [9/7 0 -3/7 2/7 1/7⟩, [41/14 0 6/7 -4/7 -2/7⟩, [41/14 0 -1/7 3/7 -2/7⟩, [41/14 0 -1/7 -4/7 5/7⟩]

eigenmonzo (unchanged-interval) basis: 2.7/5.11/5

Optimal ET sequence: 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee

Badness: 1.97 × 10^-3