10th-octave temperaments

10edo is notable for having close approximations of 15/14 to one step, and 13/8 to 7 steps. 10th-octave temperaments naturally occur between any equal divisions of the octave whose greatest common divisor is 10.

Temperaments discussed elsewhere include: decoid, deca, decal, decimetra, decistearn, and decavish.

Linus (rank-3)

Tempering out the linus comma, 578509309952 / 576650390625 = [11 -10 -10 10⟩ leads a number of regular temperaments, some of which are listed above. Linus rank three temperament can be described as the 130 & 190 & 270 temperament, which tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.

Subgroup: 2.3.5.7

Comma list: 578509309952/576650390625

Mapping: [⟨10 0 0 -11], ⟨0 1 0 1], ⟨0 0 1 1]]

mapping generators: ~15/14 = 1\10, ~3/2 = 702.095, ~5/4 = 386.574

Optimal ET sequence: 10, 40, 50, 80, 130, 140, 190, 270, 1270, 1400, 1540, 1670, 1810, 1940

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 391314/390625

Mapping: [⟨10 0 0 -11 4], ⟨0 1 0 1 -1], ⟨0 0 1 1 2]]

mapping generators: ~15/14 = 1\10, ~3/2 = 702.034, ~5/4 = 386.648

Optimal ET sequence: 10e, 50, 80, 130, 190, 270, 860, 940, 1130, 1400, 1670

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4225/4224, 4459/4455

Mapping: [⟨10 0 0 -11 4 37], ⟨0 1 0 1 -1 0], ⟨0 0 1 1 2 0]]

mapping generators: ~15/14 = 1\10, ~3/2 = 701.931, ~5/4 = 386.473

Optimal ET sequence: 10e, 50, 80, 130, 190, 270, 590, 730, 860, 1130