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'''Landscape microtemperaments''' are [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[landscape comma]] ({{monzo| -4 6 -6 3 }} = 250047/250000).
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''landscape microtemperaments''', which [[tempering out|temper out]] the [[landscape comma]] ({{monzo|legend=1| -4 6 -6 3 }}, [[ratio]]: 250047/250000). For the [[rank-3 temperament]], see [[Landscape family #Landscape]].  


Landscape rank-2 temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
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* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Tricot family #Tritricot|Tricot family]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]


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== Sextile ==
== Sextile ==
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 & 270 temperament.  
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 & 270 temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~4096/3645, ~3
: mapping generators: ~4096/3645, ~3
{{Multival|legend=1| 6 -36 -84 -71 -150 -94 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Multival|legend=1| 15 51 72 46 72 24 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
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{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
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{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
{{Multival|legend=1| 51 105 108 48 28 -44 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
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{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}


{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~63/50 = 1\3, ~1024/945 = 137.772
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~1024/945 = 137.768
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768


{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
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Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~693/640 = 137.771
Optimal tunings:
* CTE: ~63/50 = 1\3, ~693/640 = 137.773
* POTE: ~63/50 = 1\3, ~693/640 = 137.771


{{Optimal ET sequence|legend=1| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 1263, 1533, 1803c }}


Badness: 0.023085
Badness: 0.023085
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Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.777
* POTE: ~63/50 = 1\3, ~13/12 = 137.777


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}


Badness: 0.015557
Badness: 0.015557
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Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.758
* POTE: ~63/50 = 1\3, ~13/12 = 137.761


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 453 }}


Badness: 0.015557
Badness: 0.0171


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 3025/3024, 4096/4095
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024


Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.763
* POTE: ~63/50 = 1\3, ~13/12 = 137.767


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}


Badness: 0.015557
Badness: 0.0153


== Terture ==
== Terture ==
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Badness: 0.014915
Badness: 0.014915
== Slendscape ==
Slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 12884901888/12867859375
{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
: Mapping generators: ~8575/8192, ~1152/875
[[Optimal tuning]] ([[CTE]]): ~8575/8192 = 80.0000, ~1152/875 = 475.4847 (~6/5 = 315.4847 or ~21/20 = 84.5153)
{{Optimal ET sequence|legend=1| 255, 270, 525, 795, 1065 }}
[[Badness]]: 0.058002
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 102487/102400, 180224/180075
Mapping: {{Mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
Optimal tuning (CTE): ~22/21 = 80.0000, ~968/735 = 475.4912 (~6/5 = 315.4912 or ~21/20 = 84.5088)
Optimal ET sequence: {{EDOs| 255, 270, 525, 795, 1065 }}
Badness: 0.026246
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
Mapping: {{Mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Optimal tuning (CTE): ~22/21 = 80.0000, ~154/117 = 475.4935 (~6/5 = 315.4935 or ~21/20 = 84.5065)
Optimal ET sequence: {{EDOs| 255, 270, 795, 1065 }}
Badness: 0.021230


== Akjayland ==
== Akjayland ==
{{See also| 21st-octave temperaments }}
{{See also| 21st-octave temperaments }}


Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1＼21.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~1323/1280, ~131072/91875
: mapping generators: ~1323/1280, ~131072/91875


[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 1\21, ~131072/91875 = 614.9354
[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 1＼21, ~131072/91875 = 614.9354


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
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=== Vasca ===
=== Vasca ===
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of twenty-one [[23/16]]'s equal with eleven octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of twenty-one [[23/16]]'s equal with eleven octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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: mapping generators: ~1323/1280, ~6615/5632
: mapping generators: ~1323/1280, ~6615/5632


Optimal tuning (CTE): ~1323/1280 = 1\21, ~6615/5632 = 278.8998
Optimal tuning (CTE): ~1323/1280 = 1＼21, ~6615/5632 = 278.8998


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
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Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9058
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9058


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
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Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
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Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
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Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.8971
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.8971


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
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: ''For the 5-limit version, see [[24th-octave temperaments#Chromium (5-limit)]].
: ''For the 5-limit version, see [[24th-octave temperaments#Chromium (5-limit)]].


[[Chromium]] is defined by associating the porcupine comma [[256/243]] to the 24th of an octave. Named after the 24th element for being period 24.
[[Chromium]] is defined by associating the porcupine comma [[250/243]] to the 24th of an octave. Named after the 24th element for being period 24.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Zinc ==
== Zinc ==
{{See also|30th-octave temperaments#Zinc}}
{{See also| 10th-octave temperaments #Neon }}
{{See also| 60th-octave temperaments #Neodymium }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, {{monzo| -45 -24 14 18 }}
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}


{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}


[[Optimal tuning]] ([[CTE]]): ~53747712/52521875 = 1\30, ~216/175 = 364.389
[[Optimal tuning]]s:
* [[CTE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3896 (~{{monzo| 21 3 1 -10 }} = 4.3896)
* [[CWE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3879 (~{{monzo| 21 3 1 -10 }} = 4.3879)


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}


[[Badness]]: 0.0742
[[Badness]]: 0.0742
=== 11-limit===
=== 11-limit===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 250047/250000, 97869261875/97844723712
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}


Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}


Optimal tuning (CTE): ~18865/18432 = 1\30, ~216/175 = 364.389
Optimal tunings:
* CTE: ~18865/18432 = 1\30, ~216/175 = 364.3886 (~385/384 = 4.3886)
* CWE: ~18865/18432 = 1\30, ~216/175 = 364.3849 (~385/384 = 4.3849)


{{Optimal ET sequence|legend=1| 270, 1110, 1380, 1650, 1920, 2190 }}
Optimal ET sequence: {{Optimal ET sequence| 270, 1110, 1380, 1650, 1920, 2190 }}


Badness: 0.0220
Badness: 0.0220
===13-limit===
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608
 
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}
 
Optimal tunings:
* CTE: ~351/343 = 1\30, ~216/175 = 364.3879 (~385/384 = 4.3879)
* CWE: ~351/343 = 1\30, ~216/175 = 364.3867 (~385/384 = 4.3867)
 
Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
 
Badness: 0.0155
 
=== 2.3.5.7.11.13.19 subgroup (neozinc) ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375
 
Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}
 
Optimal tunings:
* CTE: ~175/171 = 1\30, ~216/175 = 364.3876 (~400/399 = 4.3876)
* CWE: ~175/171 = 1\30, ~216/175 = 364.3864 (~400/399 = 4.3864)
 
Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}
 
Badness: 0.00921
 
== Poe ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}
 
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2240/2187 = 1\30, ~3/2 = 702.141 (~81/80 = 22.141)
* [[CWE]]: ~2240/2187 = 1\30, ~3/2 = 702.166 (~81/80 = 22.166)
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 123201/123200, 250047/250000, 1990656/1990625
Comma list: 9801/9800, 19712/19683, 250047/250000


Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}
Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
 
Optimal tunings:
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CWE: ~45/44 = 1\30, ~3/2 = 702.213 (~81/80 = 22.213)
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 19712/19683, 1001/1000, 4459/4455, 4225/4224


Optimal tuning (CTE): ~351/343 = 1\30, ~216/175 = 364.389
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
Optimal tunings:
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CWE: ~45/44 = 1\30, ~3/2 = 702.170 (~81/80 = 22.170)


Badness: 0.0150
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Microtemperaments]]
Retrieved from "https://en.xen.wiki/w/Landscape_microtemperaments"