Landscape family: Difference between revisions

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→House: + very obvious 13-limit extension
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-03 19:40:41 UTC</tt>.<br>
: The original revision id was <tt>506252638</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]


The landscape family of rank three temperaments tempers out the landscape comma, 250047/250000.
The '''landscape family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[landscape comma]], 250047/250000.


=Landscape=
== Landscape ==
Comma: 250047/250000
[[Subgroup]]: 2.3.5.7


7-limit minimax tuning
[[Comma list]]: 250047/250000
[|9 0 0 0&gt;, |4 3 6 -3&gt;, |0 0 9 0&gt;, |4 -6 6 6&gt;]/9
Eigenmonzos: 7/6, 5/4


9-limit minimax tuning
{{Mapping|legend=1| 3 0 0 4 | 0 1 0 -2 | 0 0 1 2 }}
[|12 0 0 0&gt;, |4 6 6 -3&gt;, |0 0 12 0&gt;, |8 -12 12 6&gt;]/12
: mapping generators: ~63/50, ~3, ~5
Eigenmonzos: 5/4, 9/7


Map: [&lt;3 0 0 4|, &lt;0 1 0 -2|, &lt;0 0 1 2|]
[[Optimal tuning]]s:  
Generators: 63/50, 3, 5
* [[WE]]: ~63/50 = 400.0012{{c}}, ~3/2 = 701.9383{{c}}, ~5/4 = 386.3350{{c}}
EDOs: 3, 12, 27, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555
: [[error map]]: {{val| +0.004 -0.013 +0.028 -0.021 }}
10^5*Badness: 2.130</pre></div>
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.9387{{c}}, ~5/4 = 386.3385{{c}}
<h4>Original HTML content:</h4>
: error map: {{val| 0.000 -0.016 +0.025 -0.026 }}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Landscape family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:2:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:2 --&gt;&lt;!-- ws:start:WikiTextTocRule:3: --&gt;&lt;a href="#Landscape"&gt;Landscape&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:3 --&gt;&lt;!-- ws:start:WikiTextTocRule:4: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:4 --&gt;&lt;br /&gt;
[[Minimax tuning]]:
The landscape family of rank three temperaments tempers out the landscape comma, 250047/250000.&lt;br /&gt;
* [[7-odd-limit]]
&lt;br /&gt;
: {{monzo list| 9 0 0 0 | 4 3 6 -3 | 0 0 9 0 | 4 -6 6 6 }}/9
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Landscape"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Landscape&lt;/h1&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
Comma: 250047/250000&lt;br /&gt;
* [[9-odd-limit]]
&lt;br /&gt;
: {{monzo list| 12 0 0 0 | 4 6 6 -3 | 0 0 12 0 | 8 -12 12 6 }}/12
7-limit minimax tuning&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.5.9/7
[|9 0 0 0&amp;gt;, |4 3 6 -3&amp;gt;, |0 0 9 0&amp;gt;, |4 -6 6 6&amp;gt;]/9&lt;br /&gt;
 
Eigenmonzos: 7/6, 5/4&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 27, 39d, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555, 22920d }}
&lt;br /&gt;
 
9-limit minimax tuning&lt;br /&gt;
[[Badness]] (Sintel): 0.0940
[|12 0 0 0&amp;gt;, |4 6 6 -3&amp;gt;, |0 0 12 0&amp;gt;, |8 -12 12 6&amp;gt;]/12&lt;br /&gt;
 
Eigenmonzos: 5/4, 9/7&lt;br /&gt;
=== Overview to extensions ===
&lt;br /&gt;
Extensions of landscape include landy, odin, tyr and house, all considered below.  
Map: [&amp;lt;3 0 0 4|, &amp;lt;0 1 0 -2|, &amp;lt;0 0 1 2|]&lt;br /&gt;
 
Generators: 63/50, 3, 5&lt;br /&gt;
== Landy ==
EDOs: 3, 12, 27, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555&lt;br /&gt;
Named by [[Xenllium]] in 2026, landy tempers out the [[441/440|werckisma]] and [[4375/4356|fantaresma]], alongside the landscape comma.
10^5*Badness: 2.130&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 4375/4356
 
{{Mapping|legend=1| 3 0 0 4 -1 | 0 1 0 -2 -2 | 0 0 1 2 3 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.078{{c}}, ~3/2 = 701.696{{c}}, ~5/4 = 385.162{{c}}
: [[error map]]: {{val| +0.234 -0.025 -0.684 -1.116 +1.632 }}
* [[CWE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.709{{c}}, ~5/4 = 385.341{{c}}
: error map: {{val| 0.000 -0.246 -0.972 -1.562 +1.287 }}
 
{{Optimal ET sequence|legend=1| 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: {{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{{c}}, ~3/2 = 701.8141{{c}}, ~5/4 = 385.3981{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7720{{c}}, ~5/4 = 385.4420{{c}}
 
{{Optimal ET sequence|legend=0| 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|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 ==
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|legend=1| 3 0 -13 -22 10 | 0 1 4 6 0 | 0 0 5 10 -2 }}
: mapping generators: ~63/50, ~3, ~10240/9801
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9958{{c}}, ~3/2 = 701.9604{{c}}, ~10240/9801 = 75.7013{{c}}
: [[error map]]: {{val| -0.012 -0.007 +0.039 -0.034 +0.043 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.9652{{c}}, ~10240/9801 = 75.7024{{c}}
: error map: {{val| 0.000 +0.010 +0.059 -0.011 +0.087 }}
 
{{Optimal ET sequence|legend=1| 111, 159, 270, 696, 966, 1236, 1506, 2631, 2901, 4137, 4407, 8544e, 12951ce, 14187cee }}
 
[[Badness]] (Sintel): 2.31
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 6656/6655, 105644/105625, 123201/123200
 
Mapping: {{mapping| 3 0 -13 -22 10 -10 | 0 1 4 6 0 4 | 0 0 5 10 -2 11 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9957{{c}}, ~3/2 = 701.9629{{c}}, ~224/117 = 75.6993{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.9689{{c}}, ~224/117 = 75.6997{{c}}
 
{{Optimal ET sequence|legend=0| 111, 159, 270, 855, 966, 1125, 1236, 1506, 2901, 4137, 4407, 12951cef }}
 
Badness (Sintel): 0.851
 
[[Category:Temperament families]]
[[Category:Landscape family| ]] <!-- main article -->
[[Category:Rank 3]]

Latest revision as of 14:13, 21 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.

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

Overview to extensions

Extensions of landscape include landy, odin, tyr and house, all considered below.

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

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 6656/6655, 105644/105625, 123201/123200

Mapping: [⟨3 0 -13 -22 10 -10], ⟨0 1 4 6 0 4], ⟨0 0 5 10 -2 11]]

Optimal tunings:

  • WE: ~63/50 = 399.9957 ¢, ~3/2 = 701.9629 ¢, ~224/117 = 75.6993 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.9689 ¢, ~224/117 = 75.6997 ¢

Optimal ET sequence: 111, 159, 270, 855, 966, 1125, 1236, 1506, 2901, 4137, 4407, 12951cef

Badness (Sintel): 0.851

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