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 = 400.001 ¢, ~3/2 = 701.938 ¢, ~5/4 = 386.335 ¢
• CWE: ~63/50 = 400.000 ¢, ~3/2 = 701.939 ¢, ~5/4 = 386.339 ¢

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: 3, 12, 27, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555

Badness (Sintel): 0.094

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]]

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

Optimal tunings:

• WE: ~63/50 = 400.078 ¢, ~3/2 = 701.696 ¢, ~5/4 = 385.162 ¢
• CWE: ~63/50 = 400.000 ¢, ~3/2 = 701.709 ¢, ~5/4 = 385.341 ¢

Optimal ET sequence: 12, 15, 27e, 45e, 57, 60e, 72, 159

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 = 400.055 ¢, ~3/2 = 701.814 ¢, ~5/4 = 385.398 ¢
• CWE: ~63/50 = 400.000 ¢, ~3/2 = 701.772 ¢, ~5/4 = 385.442 ¢

Optimal ET sequence: 12f, 15, 27eff, 57f, 60eff, 72, 84, 87, 99ef, 159, 171

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 2 8 16], ⟨0 1 4 6 0], ⟨0 0 -5 -10 2]]

mapping generators: ~63/50, ~3, ~9801/5120

Optimal tunings:

• WE: ~63/50 = 399.996 ¢, ~3/2 = 701.960 ¢, ~9801/5120 = 1124.286 ¢
• CWE: ~63/50 = 400.000 ¢, ~3/2 = 701.965 ¢, ~9801/5120 = 1124.298 ¢

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

Badness (Sintel): 2.305