10th-octave temperaments
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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
Neon
Neon tempers out the [21 60 -50⟩ comma in the 5-limit. Neon extensions discussed elsewhere include deca, calcium, and zinc.
Subgroup: 2.3.5
Comma list: [21 60 -50⟩
Mapping: [⟨10 4 9], ⟨0 5 6]]
- mapping generators: ~3125/2916 = 1\10, [10 29 -24⟩ = 284.389
Optimal tuning (CTE): ~[10 29 -24⟩ = 284.389
Supporting ETs: 80, 190, 270, 460, 730, 1000, 1190, 1270, 1460, 2460