Archytas family

From Xenharmonic Wiki
Jump to navigation Jump to search

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): ~3/2 = 709.3213, ~5/4 = 393.3747

Minimax tuning:

[[1 0 0 0, [2 1/3 0 -1/3, [2 -2/3 1 -1/3, [2 -2/3 0 2/3]
Eigenmonzo basis (unchanged intervals): 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 basis (unchanged intervals): 2.5/3.9/7

Optimal GPV sequence5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd

Scales: archytas12, archytas12synch

Music

Ares

Main article: 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 generators: ~2, ~3, ~5

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): ~3/2 = 709.6869, ~5/4 = 391.4507

Minimax tuning:

[[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 basis (unchanged intervals): 2.11/9.9/7

Optimal GPV sequence12, 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]]

Mapping generators: ~2, ~3, ~5

Optimal tuning (POTE): ~3/2 = 710.9720, ~5/4 = 401.2699

Optimal GPV sequence5, 10, 12, 15, 27e

Badness: 0.577 × 10-3