Landscape 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 landscape family of rank-3 temperaments tempers out the landscape comma, 250047/250000.

Discussed elsewhere are:

• Odin (+9801/9800 or +151263/151250) → Kalismic temperaments
• Tyr (+3025/3024 or +102487/102400) → Lehmerismic temperaments

Considered below are landy and house, besides the 7-limit temperament.

Landscape

Subgroup: 2.3.5.7

Comma list: 250047/250000

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

mapping generators: ~63/50, ~3, ~5

Optimal tunings:

• WE: ~63/50 = 400.0012 ¢, ~3/2 = 701.9383 ¢, ~5/4 = 386.3350 ¢

error map: ⟨+0.004 -0.013 +0.028 -0.021]

• CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.9387 ¢, ~5/4 = 386.3385 ¢

error map: ⟨0.000 -0.016 +0.025 -0.026]

Minimax tuning:

• 7-odd-limit

[[9 0 0 0⟩, [4 3 6 -3⟩, [0 0 9 0⟩, [4 -6 6 6⟩]/9

unchanged-interval (eigenmonzo) basis: 2.5.7/3

• 9-odd-limit

[[12 0 0 0⟩, [4 6 6 -3⟩, [0 0 12 0⟩, [8 -12 12 6⟩]/12

unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 12, 27, 39d, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555, 22920d

Badness (Sintel): 0.0940

Landy

Landy tempers out the werckisma and fantaresma, alongside the landscape comma.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356

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

Optimal tunings:

• WE: ~63/50 = 400.078 ¢, ~3/2 = 701.696 ¢, ~5/4 = 385.162 ¢

error map: ⟨+0.234 -0.025 -0.684 -1.116 +1.632]

• CWE: ~63/50 = 400.000 ¢, ~3/2 = 701.709 ¢, ~5/4 = 385.341 ¢

error map: ⟨0.000 -0.246 -0.972 -1.562 +1.287]

Optimal ET sequence: 12, 15, 27e, 45e, 57, 60e, 72, 159, 231, 330, 402d

Badness (Sintel): 0.792

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624

Mapping: [⟨3 0 0 4 -1 -12], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 2 3 4]]

Optimal tunings:

• WE: ~63/50 = 400.0548 ¢, ~3/2 = 701.8141 ¢, ~5/4 = 385.3981 ¢
• CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.7720 ¢, ~5/4 = 385.4420 ¢

Optimal ET sequence: 12f, 15, 27eff, 45efff, 57f, 60eff, 72, 87, 159, 243, 330

Badness (Sintel): 0.923

House

House tempers out the quartisma alongside the landscape comma.

Subgroup: 2.3.5.7.11

Comma list: 250047/250000, 117440512/117406179

Mapping: [⟨3 0 -13 -22 10], ⟨0 1 4 6 0], ⟨0 0 5 10 -2]]

mapping generators: ~63/50, ~3, ~10240/9801

Optimal tunings:

• WE: ~63/50 = 399.9958 ¢, ~3/2 = 701.9604 ¢, ~10240/9801 = 75.7013 ¢

error map: ⟨-0.012 -0.007 +0.039 -0.034 +0.043]

• CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.9652 ¢, ~10240/9801 = 75.7024 ¢

error map: ⟨0.000 +0.010 +0.059 -0.011 +0.087]

Optimal ET sequence: 111, 159, 270, 696, 966, 1236, 1506, 2631, 2901, 4137, 4407, 8544e, 12951ce, 14187cee

Badness (Sintel): 2.31