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Landscape microtemperaments: Difference between revisions

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== Sextile ==
== Sextile ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Sextile (5-limit)]].''  
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Sextile (5-limit)]].''  


Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2347{{c}}
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
* [[POTE]]: ~4096/3645 = 200.000{{c}}, ~3/2 = 702.212{{c}}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}


{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc }}


[[Badness]] (Smith): 0.070097
[[Badness]] (Sintel): 1.77


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2225{{c}}
* CTE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* POTE: ~55/49 = 200.000{{c}}, ~3/2 = 702.202{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}


{{Optimal ET sequence|legend=0| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=0| 12, …, 258e, 270, 822, 1092, 1362c }}


Badness (Smith): 0.029677
Badness (Sintel): 0.981


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2141{{c}}
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}


{{Optimal ET sequence|legend=0| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}
{{Optimal ET sequence|legend=0| 12f, …, 258ef, 270, 552, 822, 1092, 1914cde }}


Badness (Smith): 0.0191
Badness (Sintel): 0.788


==== 17-limit ====
==== 17-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2117{{c}}
* WE: ~55/49 = 199.9669{{c}}, ~3/2 = 702.0643{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}


{{Optimal ET sequence|legend=0| 12f, 270, 552g }}
{{Optimal ET sequence|legend=0| 12f, 270, 552g }}


Badness (Smith): 0.0209
Badness (Sintel): 1.06


==== 19-limit ====
==== 19-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2118{{c}}
* CTE: ~55/49 = 199.9711{{c}}, ~3/2 = 702.0829{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}


{{Optimal ET sequence|legend=0| 12f, 270, 552g, 822gg }}
{{Optimal ET sequence|legend=0| 12f, 270, 552g, 822gg }}


Badness (Smith): 0.0156
Badness (Sintel): 0.948


=== Sextilia ===
=== Sextilia ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2296{{c}}
* WE: ~55/49 = 199.9975{{c}}, ~3/2 = 702.2196{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}


{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness (Smith): 0.0391
Badness (Sintel): 1.62


==== 17-limit ====
==== 17-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2264{{c}}
* WE: ~55/49 = 199.9862{{c}}, ~3/2 = 702.1714{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}


{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness (Smith): 0.0384
Badness (Sintel): 1.95


==== 19-limit ====
==== 19-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2266{{c}}
* WE: ~55/49 = 199.9866{{c}}, ~3/2 = 702.1731{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}


{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness (Smith): 0.0252
Badness (Sintel): 1.53


== Septichrome ==
== Septichrome ==
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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 }}
: mapping generators: ~2, ~243/224


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~243/224 = 140.367{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
: [[error map]]: {{val| +0.030 -0.074 -0.010 +0.059 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~243/224 = 140.3685{{c}}
: error map: {{val| 0.000 -0.113 -0.050 +0.017 }}


{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}


[[Badness]] (Smith): 0.016814
[[Badness]] (Sintel): 0.426


=== Semiseptichrome ===
=== Semiseptichrome ===
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: mapping generators: ~55/49, ~375/308
: mapping generators: ~55/49, ~375/308


Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3687{{c}} (~1760/1701 = 59.6313{{c}})
Optimal tunings:
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})


{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}
{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}


Badness (Smith): 0.0194
Badness (Sintel): 0.642


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}


Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3778{{c}} (~121/117 = 59.6222{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Smith): 0.0397
Badness (Sintel): 1.64


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}


Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3763{{c}} (~88/85 = 59.6237{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Smith): 0.0272
Badness (Sintel): 1.39


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
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Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}


Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~162/133 = 340.3767{{c}} (~88/85 = 59.6233{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Smith): 0.0222
Badness (Sintel): 1.35


==== 23-limit ====
==== 23-limit ====
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Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}


Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~162/133 = 340.3757{{c}} (~88/85 = 59.6243{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Smith): 0.0160
Badness (Sintel): 1.14


== Pnict ==
== Pnict ==
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[[Comma list]]: 250047/250000, 2100875/2097152
[[Comma list]]: 250047/250000, 2100875/2097152


{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Mapping|legend=1| 3 -3 1 12 | 0 13 10 -6 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~192/175 = 161.399{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0312{{c}}, ~147/128 = 238.6196{{c}} (~192/175 = 161.4116{{c}})
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~147/128 = 238.6038{{c}} (~192/175 = 161.3962{{c}})


{{Optimal ET sequence|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
{{Optimal ET sequence|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}


[[Badness]] (Smith): 0.045660
[[Badness]] (Sintel): 1.16


== Domain ==
== Domain ==
Retrieved from "https://en.xen.wiki/w/Landscape_microtemperaments"