Landscape family: Difference between revisions

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The '''landscape family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[landscape comma]], 250047/250000.
The '''landscape family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[landscape comma]], 250047/250000.
Discussed elsewhere are:
* ''[[Odin]]'' (+9801/9800 or +151263/151250) → [[Kalismic temperaments #Odin|Kalismic temperaments]]
* ''[[Tyr]]'' (+3025/3024 or +102487/102400) → [[Lehmerismic temperaments #Tyr|Lehmerismic temperaments]]


Considered below are landy and house, besides the 7-limit temperament.
Considered below are landy and house, besides the 7-limit temperament.
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Badness (Sintel): 0.923
Badness (Sintel): 0.923
== Odin ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 9801/9800, 151263/151250
{{Mapping|legend=1| 6 0 0 8 17 | 0 1 0 -2 -4 | 0 0 1 2 3 }}
: mapping generators: ~55/49, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~55/49 = 200.0008{{c}}, ~3/2 = 701.9318{{c}}, ~5/4 = 386.3342{{c}}
: [[error map]]: {{val| +0.005 -0.018 +0.031 -0.004 -0.018 }}
* [[CWE]]: ~55/49 = 200.0000{{c}}, ~3/2 = 701.9320{{c}}, ~5/4 = 386.3393{{c}}
: error map: {{val| 0.000 -0.023 +0.026 -0.011 -0.028 }}
{{Optimal ET sequence|legend=1| 12, 42, 48dee, 54c, 60e, 72, 198, 270, 342, 612, 954, 1236, 1506, 1578, 1848, 3426, 4038, 5616, 7464, 17118e, 18966e }}
[[Badness]] (Sintel): 0.140
== Tyr ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 3025/3024, 102487/102400
{{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }}
: mapping generators: ~63/50, ~400/231, ~5
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0162{{c}}, ~400/231 = 950.9828{{c}}, ~5/4 = 386.2400{{c}}
: [[error map]]: {{val| +0.049 +0.011 +0.024 -0.017 -0.205 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~400/231 = 950.9591{{c}}, ~5/4 = 386.2764{{c}}
: error map: {{val| 0.000 -0.037 -0.037 -0.109 -0.359 }}
{{Optimal ET sequence|legend=1| 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}
[[Badness]] (Sintel): 0.311
=== Borneo ===
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024
Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }}
Optimal tunings:
* WE: ~63/50 = 400.0002{{c}}, ~26/15 = 951.0850{{c}}, ~5/4 = 386.5817{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.0846{{c}}, ~5/4 = 386.5821{{c}}
{{Optimal ET sequence|legend=0| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}
Badness (Sintel): 0.514
Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7


== House ==
== House ==

Revision as of 13:25, 20 April 2026

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.

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:

[[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
[[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

Named by Xenllium in 2026, 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

Odin

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250

Mapping: [⟨6 0 0 8 17], ⟨0 1 0 -2 -4], ⟨0 0 1 2 3]]

mapping generators: ~55/49, ~3, ~5

Optimal tunings:

  • WE: ~55/49 = 200.0008 ¢, ~3/2 = 701.9318 ¢, ~5/4 = 386.3342 ¢
error map: ⟨+0.005 -0.018 +0.031 -0.004 -0.018]
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.9320 ¢, ~5/4 = 386.3393 ¢
error map: ⟨0.000 -0.023 +0.026 -0.011 -0.028]

Optimal ET sequence: 12, 42, 48dee, 54c, 60e, 72, 198, 270, 342, 612, 954, 1236, 1506, 1578, 1848, 3426, 4038, 5616, 7464, 17118e, 18966e

Badness (Sintel): 0.140

Tyr

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 102487/102400

Mapping: [⟨3 0 0 4 8], ⟨0 2 0 -4 1], ⟨0 0 1 2 0]]

mapping generators: ~63/50, ~400/231, ~5

Optimal tunings:

  • WE: ~63/50 = 400.0162 ¢, ~400/231 = 950.9828 ¢, ~5/4 = 386.2400 ¢
error map: ⟨+0.049 +0.011 +0.024 -0.017 -0.205]
  • CWE: ~63/50 = 400.0000 ¢, ~400/231 = 950.9591 ¢, ~5/4 = 386.2764 ¢
error map: ⟨0.000 -0.037 -0.037 -0.109 -0.359]

Optimal ET sequence: 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee

Badness (Sintel): 0.311

Borneo

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024

Mapping: [⟨3 0 0 4 8 -3], ⟨0 2 0 -4 1 3], ⟨0 0 1 2 0 1]]

Optimal tunings:

  • WE: ~63/50 = 400.0002 ¢, ~26/15 = 951.0850 ¢, ~5/4 = 386.5817 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~26/15 = 951.0846 ¢, ~5/4 = 386.5821 ¢

Optimal ET sequence: 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270

Badness (Sintel): 0.514

Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7

House

Named by Eliora in 2024, 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

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