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Landscape family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
__FORCETOC__
The '''landscape family''' of [[rank-3 temperament]]s tempers out the [[landscape comma]], 250047/250000.


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


== Landscape ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 3 0 0 4 | 0 1 0 -2 | 0 0 1 2 }}
{{Mapping|legend=1| 3 0 0 4 | 0 1 0 -2 | 0 0 1 2 }}
: mapping generators: ~63/50, ~3, ~5


: mapping generators: ~63/50, ~3, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.0012{{c}}, ~3/2 = 701.9383{{c}}, ~5/4 = 386.3350{{c}}
: [[error map]]: {{val| +0.004 -0.013 +0.028 -0.021 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.9387{{c}}, ~5/4 = 386.3385{{c}}
: error map: {{val| 0.000 -0.016 +0.025 -0.026 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: {{monzo list| 9 0 0 0 | 4 3 6 -3 | 0 0 9 0 | 4 -6 6 6 }}/9
: {{monzo list| 9 0 0 0 | 4 3 6 -3 | 0 0 9 0 | 4 -6 6 6 }}/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]
* [[9-odd-limit]]
: {{monzo list| 12 0 0 0 | 4 6 6 -3 | 0 0 12 0 | 8 -12 12 6 }}/12
: {{monzo list| 12 0 0 0 | 4 6 6 -3 | 0 0 12 0 | 8 -12 12 6 }}/12
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
: unchanged-interval (eigenmonzo) basis: 2.5.9/7
 
{{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 }}
 
[[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 [[441/440|werckisma]] and [[4375/4356|fantaresma]], alongside the landscape comma.
 
[[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=1| 3, 12, 27, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555 }}
{{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


[[Badness]]: 0.02130 × 10<sup>-3</sup>
== House ==
== House ==
House tempers out the [[quartisma]] alongside the landscape comma.
Named by [[Eliora]] in 2024, house tempers out the [[quartisma]] alongside the landscape comma.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 31: Line 126:
[[Comma list]]: 250047/250000, 117440512/117406179
[[Comma list]]: 250047/250000, 117440512/117406179


{{Mapping|legend=1| 3 0 2 8 16 | 0 1 4 6 0 | 0 0 -5 -10 2 }}  
{{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


: mapping generators: ~63/50, ~3, ~9801/5120
[[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 tuning]] ([[CTE]]) ~63/50 = 1\3, ~3/2 = 701.9592, ~9801/5120 = 1124.2969
{{Optimal ET sequence|legend=1| 111, 159, 270, 696, 966, 1236, 1506, 2631, 2901, 4137, 4407, 8544e, 12951ce, 14187cee }}


{{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 }}  


[[Badness]]: 1.92 × 10<sup>-3</sup>
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}}


== Odin  ==
{{Optimal ET sequence|legend=0| 111, 159, 270, 855, 966, 1125, 1236, 1506, 2901, 4137, 4407, 12951cef }}
{{See also| Kalismic temperaments #Odin }}


== Tyr  ==
Badness (Sintel): 0.851
{{See also| Lehmerismic temperaments #Tyr }}


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Landscape family| ]] <!-- main article -->
[[Category:Landscape family| ]] <!-- main article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Landscape_family"