159th-octave temperaments
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.
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
Comma list: 4000/3993, 15625/15552, 32805/32768
Mapping: [⟨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: 159, 318, 477c, 636c, 795cd
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.
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 = 1\159, ~5, ~7
Supporting ETs: 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 159 Aemilia.
In the 7-limit, aemilic tempers out the landscape comma alongisde the Mercator comma, and also [-76 41 12 -6⟩.
Subgroup: 2.3.5.7
Comma list: 250047/250000, [-84 53⟩
Mapping: [⟨159 252 0 -292], ⟨0 0 1 2]]
- mapping generators: ~225/224 = 1\159, ~5
Optimal tuning (CTE): ~5/4 = 386.303
Supporting ETs: 159, 795, 954, 1749, 2544, 2703
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 160083/160000, [34 19 8 4 -1⟩
Mapping: [⟨159 252 0 -292 550], ⟨0 0 1 2 0]]
- mapping generators: ~225/224 = 1\159, ~5
Optimal tuning (CTE): ~5/4 = 386.303
Supporting ETs: 159, 795, 954
13-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 123201/123200, 160083/160000, 4100625/4100096
Mapping: [⟨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
Supporting ETs: 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: [⟨159 252 0 -292 550 -150 1019], ⟨0 0 1 2 0 2 -1]]
- mapping generators: ~225/224 = 1\159, ~5
Optimal tuning (CTE): ~5/4 = 386.263
Supporting ETs: 159, 795g, 954