Archytas family

The archytas family of temperaments tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord, which is familiar from 12edo and also a feature of 22edo and 27edo.

Archytas

Subgroup: 2.3.5.7

Comma list: 64/63

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

Minimax tuning:

• 7-odd-limit

[[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

• 9-odd-limit

[[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

Main article: Ares

See also: Ptolemismic clan #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

Minimax tuning:

• 11-odd-limit

[[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