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Landscape microtemperaments: Difference between revisions

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'''Landscape microtemperaments''' are rank-2 temperaments tempering out the [[landscape comma]] ({{Monzo| -4 6 -6 3 }} = 250047/250000).
'''Landscape microtemperaments''' are rank-2 temperaments tempering out the [[landscape comma]] ({{Monzo| -4 6 -6 3 }} = 250047/250000).


Landscape rank-2 temperaments include sextile, septichrome, pnict, and domain. Considered elsewhere are:  
Landscape rank-2 temperaments considered elsewhere are:  
* [[Augene]], {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
* [[Augene]], {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
* ''[[Waage]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* ''[[Waage]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
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* ''[[Stearnscape]]'', {118098/117649, 250047/250000} → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Stearnscape]]'', {118098/117649, 250047/250000} → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Tritricot]]'', {250047/250000, 11785390260224/11767897353375} → [[Tricot family #Tritricot|Tricot family]]
* ''[[Tritricot]]'', {250047/250000, 11785390260224/11767897353375} → [[Tricot family #Tritricot|Tricot family]]
Discussed below are sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.


== Sextile ==
== Sextile ==
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== Septichrome ==
== Septichrome ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 2460375/2458624
[[Comma list]]: 250047/250000, 2460375/2458624
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{{Multival|legend=1| 15 51 72 46 72 24 }}
{{Multival|legend=1| 15 51 72 46 72 24 }}


[[POTE generator]]: ~243/224 = 140.367
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367


{{Val list|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
{{Val list|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
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{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}
{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}


[[POTE generator]]: ~192/175 = 161.399
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399


{{Val list|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
{{Val list|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
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{{see also| Minortonic family }}
{{see also| Minortonic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 645700815/645657712
[[Comma list]]: 250047/250000, 645700815/645657712
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{{Multival|legend=1| 51 105 108 48 28 -44 }}
{{Multival|legend=1| 51 105 108 48 28 -44 }}


[[POTE generator]]: ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467


{{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
{{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
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== Avicenna ==
== Avicenna ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 29360128/29296875
[[Comma list]]: 250047/250000, 29360128/29296875
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{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}


[[POTE generator]]: ~1024/945 = 137.768
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~1024/945 = 137.768


{{Val list|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Val list|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
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Mapping: [{{val|3 2 8 16 9}}, {{val|0 8 -3 -22 4}}]
Mapping: [{{val|3 2 8 16 9}}, {{val|0 8 -3 -22 4}}]


POTE generator: ~693/640 = 137.771
Optimal tuning (POTE): ~63/50 = 1\3, ~693/640 = 137.771


Optimal GPV sequence: {{Val list| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
Optimal GPV sequence: {{Val list| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
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Mapping: [{{val|3 2 8 16 9 8}}, {{val|0 8 -3 -22 4 9}}]
Mapping: [{{val|3 2 8 16 9 8}}, {{val|0 8 -3 -22 4 9}}]


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


Optimal GPV sequence: {{Val list| 87, 183, 270 }}
Optimal GPV sequence: {{Val list| 87, 183, 270 }}
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== Terture ==
== Terture ==
{{see also| Vulture family }}
{{See also| Vulture family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 359661568/358722675
[[Comma list]]: 250047/250000, 359661568/358722675
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[[Mapping]]: [{{val| 3 4 3 2 }}, {{val| 0 4 21 34 }}]
[[Mapping]]: [{{val| 3 4 3 2 }}, {{val| 0 4 21 34 }}]


[[POTE generator]]: ~392/375 = 75.555
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555


{{Val list|legend=1| 111, 159, 270 }}
{{Val list|legend=1| 111, 159, 270 }}
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Mapping: [{{val| 3 4 3 2 10 }}, {{val| 0 4 21 34 2 }}]
Mapping: [{{val| 3 4 3 2 10 }}, {{val| 0 4 21 34 2 }}]


POTE generator: ~392/375 = 75.550
Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550


Optimal GPV sequence: {{Val list| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
Optimal GPV sequence: {{Val list| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
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Mapping: [{{val| 3 4 3 2 10 6 }}, {{val| 0 4 21 34 2 27 }}]
Mapping: [{{val| 3 4 3 2 10 6 }}, {{val| 0 4 21 34 2 27 }}]


POTE generator: ~117/112 = 75.553
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
Optimal GPV sequence: {{Val list| 111, 159, 270 }}
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Mapping: [{{val| 3 4 3 2 10 6 10 }}, {{val| 0 4 21 34 2 27 12 }}]
Mapping: [{{val| 3 4 3 2 10 6 10 }}, {{val| 0 4 21 34 2 27 12 }}]


POTE generator: ~117/112 = 75.560
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
Optimal GPV sequence: {{Val list| 111, 159, 270 }}
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Mapping: [{{val| 3 4 3 2 10 6 10 5 }}, {{val| 0 4 21 34 2 27 12 41 }}]
Mapping: [{{val| 3 4 3 2 10 6 10 5 }}, {{val| 0 4 21 34 2 27 12 41 }}]


POTE generator: ~95/91 = 75.560
Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
Optimal GPV sequence: {{Val list| 111, 159, 270 }}
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Mapping: [{{val| 3 4 3 2 10 6 10 5 13 }}, {{val| 0 4 21 34 2 27 12 41 3 }}]
Mapping: [{{val| 3 4 3 2 10 6 10 5 13 }}, {{val| 0 4 21 34 2 27 12 41 3 }}]


POTE generator: ~24/23 = 75.548
Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
Optimal GPV sequence: {{Val list| 111, 159, 270 }}
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Mapping generators: ~1323/1280, ~131072/91875
Mapping generators: ~1323/1280, ~131072/91875


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


[[Optimal GPV sequence]]: {{val list| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
[[Optimal GPV sequence]]: {{val list| 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 21 [[23/16]]'s equal with 11 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): ~6615/5632 = 278.8998
Optimal tuning (CTE): ~1323/1280 = 1\21, ~6615/5632 = 278.8998


Optimal GPV sequence: {{val list| 168, 357, 525, 882, 1407, 2289e }}
Optimal GPV sequence: {{val list| 168, 357, 525, 882, 1407, 2289e }}
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Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]
Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]


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


Optimal GPV sequence: {{val list| 168, 357, 525, 882 }}
Optimal GPV sequence: {{val list| 168, 357, 525, 882 }}
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Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]
Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]


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


Optimal GPV sequence: {{val list| 168, 357, 525, 882 }}
Optimal GPV sequence: {{val list| 168, 357, 525, 882 }}
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Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]
Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]


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


Optimal GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
Optimal GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
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Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211


Mapping: [{{val| 21 34 49 58 73 77 85 91 95 }}, {{val| 0 -6 -2 8 -3 6 7 -15 0 }}]
Mapping: [{{val| 21 4 39 98 58 107 120 16 95 }}, {{val| 0 -6 -2 8 -3 6 7 -15 0 }}]


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


Optimal GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
Optimal GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
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Mapping generators: ~250/243, ~10/7
Mapping generators: ~250/243, ~10/7


[[Optimal tuning]] ([[CTE]]): ~10/7 = 617.2710
[[Optimal tuning]] ([[CTE]]): ~250/243 = 1\24, ~10/7 = 617.2710


{{Val list|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
{{Val list|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
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Mapping: [{{val| 24 1 -6 18 46 }}, {{val| 0 3 5 4 3 }}]
Mapping: [{{val| 24 1 -6 18 46 }}, {{val| 0 3 5 4 3 }}]


Optimal tuning (CTE): ~10/7 = 617.2597
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2597


Optimal GPV sequence: {{Val list| 72, …, 480, 552, 624 }}
Optimal GPV sequence: {{Val list| 72, …, 480, 552, 624 }}
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Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 }}]
Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 }}]


Optimal tuning (CTE): ~10/7 = 617.2869
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2869


Optimal GPV sequence: {{Val list| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
Optimal GPV sequence: {{Val list| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
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Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 9 }}]
Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 9 }}]


Optimal tuning (CTE): ~10/7 = 617.2732
Optimal tuning (CTE): ~35/34 = 1\24, ~10/7 = 617.2732


Optimal GPV sequence: {{Val list| 72, …, 480fgg, 552g, 624 }}
Optimal GPV sequence: {{Val list| 72, …, 480fgg, 552g, 624 }}
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