Semaphoresmic family

The semaphoresmic family of rank-3 temperaments tempers out 49/48 in the full 7-limit, and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-limit 49/48 temperament semaphore, and the 2.3.5.7 49/48 and 81/80 temperament godzilla.

Semiphore

This temperament is also known as semaphoresmic.

Subgroup: 2.3.5.7

Comma list: 49/48

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

mapping generators: ~2, ~7/4, ~5

Lattice basis:

7/6 length = 0.7627, 5/4 length = 2.322

Angle (7/6, 5/4) = 90 degrees

Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 949.6154, ~5/4 = 379.7035

Optimal ET sequence: 4, 5, 9, 10, 14c, 15, 19

Badness: 0.116 × 10^-3

Selenium

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55

Mapping: [⟨1 0 2 2 5], ⟨0 2 0 1 1], ⟨0 0 1 0 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 951.7290, ~5/4 = 390.6487

Optimal ET sequence: 4, 5, 9, 10, 15, 19, 24, 34

Badness: 0.665 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90

Mapping: [⟨1 0 2 2 5 -1], ⟨0 2 0 1 1 3], ⟨0 0 1 0 -1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 951.2441, ~5/4 = 389.5957

Optimal ET sequence: 5, 9, 10, 15, 19, 24, 34

Badness: 0.787 × 10^-3

Negric

Subgroup: 2.3.5.7.11

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3 0], ⟨0 -4 3 -2 0], ⟨0 0 0 0 1]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 125.6080, ~11/8 = 539.2342

Optimal ET sequence: 9, 10, 19, 29, 38d, 67cde

Badness: 1.087 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90

Mapping: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3], ⟨0 0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 125.5675, ~11/8 = 538.4845

Optimal ET sequence: 9, 10, 19, 29, 38df, 67cdef

Badness: 0.8076 × 10^-3