Archytas family
The archytas family of rank-3 temperaments tempers out 64/63. The head of this family, archytas, tempers out 64/63 alone in the full 7-limit, so it has the same 2.3.7-subgroup structure as archy but giving prime 5 an independent generator.
See Archytas clan for the rank-2 temperament without the last generator of archytas, and its various extensions.
Archytas
Subgroup: 2.3.5.7
Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]
- mapping generators: ~2, ~3, ~5
Mapping to lattice: [⟨0 1 0 -2], ⟨0 0 1 0]]
Lattice basis:
- 3/2 length = 1.0508, 5/4 length = 2.3219
- Angle (3/2, 5/4) = 90 degrees
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 709.3213, ~5/4 = 393.3747
- [[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
- eigenmonzo (unchanged-interval) basis: 2.5/3.7/3
- [[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]
- eigenmonzo (unchanged-interval) basis: 2.5/3.9/7
Optimal ET sequence: 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd
Scales: archytas12, archytas12synch
- Music
- Chris Vaisvil – 12 archytas hobbits with ale
- Joel Taylor – Portrait of a Tragedy (Death of a Son) - Archytas12 sync Study 3
Ares
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99
Mapping: [⟨1 0 0 6 2], ⟨0 1 0 -2 -2], ⟨0 0 1 0 2]]
Mapping to lattice: [⟨0 1 1 -2 0], ⟨0 0 -1 0 -2]]
Lattice basis:
- 3/2 length = 0.9878, 6/5 length = 1.4312
- Angle (3/2, 6/5) = 75.0475 degrees
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 709.6869, ~5/4 = 391.4507
- [[1 0 0 0 0⟩, [3/2 1/2 0 -1/4 0⟩, [2 0 0 -1/2 1/2⟩, [3 -1 0 1/2 0⟩, [3 -1 0 -1/2 1⟩]
- eigenmonzo (unchanged-interval) basis: 2.11/9.9/7
Optimal ET sequence: 12, 15, 22, 27e, 34d, 49, 181bbcddeeee
Badness: 0.469 × 10-3
Scales: ares12
Vulcan
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63
Mapping: [⟨1 0 0 6 9], ⟨0 1 0 -2 -2], ⟨0 0 1 0 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.9720, ~5/4 = 401.2699
Optimal ET sequence: 5, 10, 12, 15, 27e
Badness: 0.577 × 10-3