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159th-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|159}}
{{Infobox fractional-octave|159}}
[[159edo]] is a notable [[17-limit]] system. It first contains [[53edo]], tempering out Mercator's comma.  
[[159edo]] is a notable [[17-limit]] system. It first contains [[53edo]], tempering out Mercator's comma.  


159edo has attracted a lot of attention from prominent microtonalists like [[Ozan Yarman]], [[Dawson Berry]] (also known as [[Aura]]), [[Gene Ward Smith]]. In addition, some of its multiples have large consistency limits as well, and this naturally commends itself to 159th-octave temperaments.
159edo has attracted a lot of attention from prominent microtonalists like [[Ozan Yarman]], [[Dawson Berry]] (also known as [[Aura]]), [[Gene Ward Smith]]. In addition, some of its multiples have large consistency limits as well, and this naturally commends itself to 159th-octave temperaments.


This page collects temperaments that are of theoretical interest specific to 159edo, beyond just [[53edo]].
This page collects temperaments that are of theoretical interest specific to 159edo, beyond just [[53edo]].  
== Auric ==
''Auric'', named after [[Aura]], is defined by setting the 2.3.5.11 mapping to that of 159edo and the generator for the prime 7 being independent.


Subgroup: 2.3.5.7.11
Temperaments discussed elsewhere include [[mercator family #Auric|auric]] and [[mercator family #Aemillic|aemillic]].  
 
[[Comma list]]: 4000/3993, 15625/15552, 32805/32768
 
{{Mapping|legend=1|159 252 369 0 550|0 0 0 1 0}}
 
: mappping generators: ~243/242 = 1\159, ~7
 
[[Optimal tuning]] ([[CTE]]): ~7/4 = 968.726
 
{{Optimal ET sequence|legend=1|159, 318, 477c, 636c, 795cd}}


== Aemic (rank-3) ==
== Aemic (rank-3) ==
''Aemic'', a shortening of aemilic (see below) tempers out the [[frameshift comma]] and the [[nexus comma]]. Aemic is alternately defined by setting the 2.3.11 mapping to that of 159edo and the rest being independent generators. On the patent val it is supported by first 10 multiples of 159edo.
''Aemic'', a shortening of [[mercator family #Aemillic|aemilic]], tempers out the [[frameshift comma]] and the [[nexus comma]]. It sets the [[2.3.11 subgroup]] to that of 159edo and the rest are independent generators. On the [[patent val]] it is [[support]]ed by first 10 multiples of 159edo.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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[[Comma list]]: 1771561/1769472, 22876792454961/22866405883904
[[Comma list]]: 1771561/1769472, 22876792454961/22866405883904


{{Mapping|legend=1|159 252 0 0 550|0 0 1 0 0|0 0 0 1 0}}
{{Mapping|legend=1| 159 252 0 0 550 | 0 0 1 0 0 | 0 0 0 1 0 }}
 
: mapping generators: ~243/242, ~5, ~7
: mapping generators: ~243/242 = 1\159, ~5, ~7
 
[[Support]]ing [[ET]]s: {{EDOs|159, 318, 477, 636, 795, 954, 1113, 1272, 1431, 1590}}
 
== Aemilic ==
''Aemilic'' is described as the 159 & 954 temperament in the 17-limit and is named after the minor planet [[wikipedia:159 Aemilia|159 Aemilia]].
 
In the 7-limit, aemilic tempers out the [[landscape comma]] alongisde the [[Mercator comma]], and also {{monzo|-76 41 12 -6}}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo|-84 53}}
 
{{Mapping|legend=1|159 252 0 -292|0 0 1 2}}
 
: mapping generators: ~225/224 = 1\159, ~5
 
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.303
 
[[Support]]ing [[ET]]s: {{EDOs|159, 795, 954, 1749, 2544, 2703}}
 
=== 11-limit ===
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 160083/160000, {{monzo|34 19 8 4 -1}}
 
{{Mapping|legend=1|159 252 0 -292 550|0 0 1 2 0}}
 
: mapping generators: ~225/224 = 1\159, ~5
 
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.303
 
[[Support]]ing [[ET]]s: {{EDOs|159, 795, 954}}
 
=== 13-limit ===
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 123201/123200, 160083/160000, 4100625/4100096
 
{{Mapping|legend=1|159 252 0 -292 550 -150|0 0 1 2 0 2}}
 
: mapping generators: ~225/224 = 1\159, ~5
 
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.303
 
[[Support]]ing [[ET]]s: {{EDOs|159, 795, 954}}
 
=== 17-limit ===
 
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Comma list]]: 2431/2430, 3025/3024, 57375/57344, 123201/123200, 160083/160000


{{Mapping|legend=1|159 252 0 -292 550 -150 1019|0 0 1 2 0 2 -1}}
[[Optimal tuning]]s:
* [[WE]]: ~243/242 = 7.5475{{c}}, ~5/4 = 386.2128{{c}}, ~7/4 = 968.7250{{c}}
: [[error map]]: {{val| +0.050 +0.012 -0.000 -0.000 -0.200 }}
* [[CWE]]: ~243/242 = 7.5472{{c}}, ~5/4 = 386.2259{{c}}, ~7/4 = 968.7197{{c}}
: error map: {{val| 0.000 -0.068 -0.088 -0.106 -0.375 }}


: mapping generators: ~225/224 = 1\159, ~5
{{Optimal ET sequence|legend=1| 159, 318, 477(c), 636, 795, 954, 1749e, 2703e, 3657ee }}


[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.263
[[Badness]] (Sintel): 71.2


[[Support]]ing [[ET]]s: {{EDOs|159, 795g, 954}}
{{Navbox fractional-octave}}
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