Landscape microtemperaments

From Xenharmonic Wiki
Jump to: navigation, search


Landscape microtemperaments are rank two temperaments tempering out the landscape comma, |-4 6 -6 3> = 250047/250000. This is the amount by which three 63/50 intervals exceed an octave. Tempering 63/50 interval into one degree of 3edo (exact 400 cents) is highly characteristic of landscape tempering; any tuning system (41edo, for example) which the number of divisions of the octave is not multiple of 3 cannot be tempering out the landscape comma.

Landscape rank-two temperaments include sextile, septichrome, pnict, and domain. Considered elsewhere are ennealimmal, misty, nessafof, and chromat.

Sextile

Commas: 250047/250000, 33554432/33480783

POTE generator: ~3/2 = 702.212

Map: [<6 0 71 150|, <0 1 -6 -14|]

Wedgie: <<6 -36 -84 -71 -150 -94||

EDOs: 12, 258, 270, 1362c, 1632c, 1902c, 2172c

Badness: 0.070

11-limit

Commas: 5632/5625, 9801/9800, 151263/151250

POTE generator: ~3/2 = 702.202

Map: [<6 0 71 150 230|, <0 1 -6 -14 -22|]

EDOs: 12, 270, 822, 1092

Badness: 0.0297

Septichrome

Commas: 250047/250000, 2460375/2458624

POTE generator: ~243/224 = 140.367

Map: [<3 3 1 0|, <0 5 17 24|]

Wedgie: <<15 51 72 46 72 24||

EDOs: 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b

Badness: 0.0168

Pnict

Commas: 250047/250000, 2100875/2097152

POTE generator: ~192/175 = 161.399

Map: [<3 10 11 6|, <0 -13 -10 6|]

Wedgie: <<39 30 -18 -43 -138 -126||

EDOs: 15, 141, 156, 171

Badness: 0.0457

Domain

Commas: 250047/250000, 645700815/645657712

POTE generator: ~10/9 = 182.467

Map: [<3 -3 -9 -8|, <0 17 35 36|]

Wedgie: <<-51 -105 -108 -48 -28 44||

EDOs: 171, 1335, 1506, 1677, 1848, 2019, 2190