159th-octave temperaments: Difference between revisions
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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]]. | ||
Temperaments discussed elsewhere include [[mercator family #Auric|auric]] and [[mercator family #Aemillic|aemillic]]. | |||
[[ | |||
== Aemic (rank-3) == | == Aemic (rank-3) == | ||
''Aemic'', a shortening of aemilic | ''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 | |||
: | [[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 }} | |||
{{Optimal ET sequence|legend=1| 159, 318, 477(c), 636, 795, 954, 1749e, 2703e, 3657ee }} | |||
[[ | [[Badness]] (Sintel): 71.2 | ||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
Latest revision as of 06:47, 10 September 2026
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.
This page collects temperaments that are of theoretical interest specific to 159edo, beyond just 53edo.
Temperaments discussed elsewhere include auric and aemillic.
Aemic (rank-3)
Aemic, a shortening of 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 supported by first 10 multiples of 159edo.
Subgroup: 2.3.5.7.11
Comma list: 1771561/1769472, 22876792454961/22866405883904
Mapping: [⟨159 252 0 0 550], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]
- mapping generators: ~243/242, ~5, ~7
- WE: ~243/242 = 7.5475 ¢, ~5/4 = 386.2128 ¢, ~7/4 = 968.7250 ¢
- error map: ⟨+0.050 +0.012 -0.000 -0.000 -0.200]
- CWE: ~243/242 = 7.5472 ¢, ~5/4 = 386.2259 ¢, ~7/4 = 968.7197 ¢
- error map: ⟨0.000 -0.068 -0.088 -0.106 -0.375]
Optimal ET sequence: 159, 318, 477(c), 636, 795, 954, 1749e, 2703e, 3657ee
Badness (Sintel): 71.2