Archytas family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

Main article: Archytas and ares

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 tunings:

• WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢, ~5/4 = 392.3765 ¢

error map: ⟨-3.045 +2.522 -0.027 +3.952]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢, ~5/4 = 391.5995 ¢

error map: ⟨0.000 +7.435 +5.286 +12.394]

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⟩]

unchanged-interval (eigenmonzo) 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⟩]

unchanged-interval (eigenmonzo) basis: 2.5/3.9/7

Optimal ET sequence: 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd

Badness (Sintel): 0.435

Scales: archytas12, archytas12synch

Ares

Main article: Archytas and 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 tunings:

• WE: ~2 = 1197.0279 ¢, ~3/2 = 707.9291 ¢, ~5/4 = 390.4812 ¢

error map: ⟨-2.972 +3.002 -1.777 +3.427 +1.898]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.6947 ¢, ~5/4 = 390.0248 ¢

error map: ⟨0.000 +7.740 +3.711 +11.785 +9.342]

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⟩]

unchanged-interval (eigenmonzo) basis: 2.11/9.9/7

Optimal ET sequence: 12, 15, 22, 27e, 34d, 49, 181bbcddeeee

Badness (Sintel): 0.563

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 tunings:

• CWE: ~2 = 1195.4265 ¢, ~3/2 = 708.2623 ¢, ~5/4 = 399.7405 ¢

error map: ⟨-4.573 +1.734 +4.280 -3.644 +9.550]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5280 ¢, ~5/4 = 400.4798 ¢

error map: ⟨0.000 +9.573 +14.166 +8.118 +25.146]

Optimal ET sequence: 5, 10, 12, 15, 27e

Badness (Sintel): 0.693