97th-octave temperaments
Berkelium is a remarkable high-limit subgroup temperament with equally remarkable full 31-limit branchings. It is named after the 97th element, as it divides the octave into 97 equal parts.
Berkelium comes in two variants, berkelium-247, named after the most stable isotope, is described as the 388 & 3395 temperament, of which 388edo is consistent in the 37-odd-limit and 3395edo is a zeta edo and a strong 19-limit tuning. Another 31-limit variety, named berkelium-247 is described as a 388 & 2619 temperament, and while 2619edo is not remarkably strong in harmonic approximation, it is consistent in the 33-odd-limit, meaning it is natural for it to be temperament-merged with 388edo, and the end result is a 97th-octave temperament.
Different branchings of berkelium also map 1 step of 97edo to drastically different intervals, each of which could be used in a comma pump. Berkelium-247 maps the period in the higher limits to 144/143, the grossma.
Temperament data
Subgroup: 2.3.5.13.17.23.29.31
Comma list: 10881/10880, 13312/13311, 86411/86400, 96876/96875, 4784000/4782969, 223171875/223135744
Sval mapping: [⟨97 97 55 -95 283 609 301 821], ⟨0 1 3 8 2 -3 3 -6]]
Sval mapping generators: ~6075/6032, ~3/2
Optimal tuning (CTE): ~3/2 = 701.9...
Vals: 388, 2619, 3395...
Berkelium-248
The temperament with higher TE error of the two branchings, therefore named after the second most stable berkelium isotope.
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-266 81 23 30⟩
Mapping: [⟨97 97 55 556], ⟨0 1 3 -5]]
Mapping generators: ~[82 -27 -6 -9⟩ = 1\97, ~3/2 = 701.929
Optimal tuning (CTE): ~3/2 = 701.929
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 8595365625/8589934592, 68641485507/68594841920
Mapping: [⟨97 97 55 556 676], ⟨0 1 3 -5 -6]]
Mapping generators: ~1617165/1605632 = 1\97, ~3/2 = 701.928
Optimal tuning (CTE): ~3/2 = 701.928
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4375/4374, 405769/405504, 1063348/1063125, 25694955/25690112
Mapping: [⟨97 97 55 556 676 -95], ⟨0 1 3 -5 -6 8]]
Mapping generators: ~144/143, ~3/2
Optimal tuning (CTE): ~3/2 = 701.945
Berkelium-247
The temperament with lower TE error of the two branchings, therefore named after the most stable berkelium isotope.
Subgroup: 2.3.5.7
Comma list: 12824703626379264/12822723388671875, [56 -57 16 -1⟩
Mapping: [⟨97 97 55 783], ⟨0 1 3 -9]]
Mapping generators: ~13839047287569/13743895347200 = 1\97, ~3/2 = 701.973
Optimal tuning (CTE):~ 3/2 = 701.973
11-limit
Subgroup: 2.3.5.7.11
Comma list: 21437500/21434787, 44660948992/44659644435, 1573159698432/1572763671875
Mapping: [⟨97 97 55 783 903], ⟨0 1 3 -9 -10]]
Mapping generators: ~4125/4096 = 1\97, ~3/2 = 701.976
Optimal tuning (CTE):~ 3/2 = 701.976
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1990656/1990625, 1146880/1146717, 492128/492075, 2662250409/2662000000
Mapping: [⟨97 97 55 783 903 -95], ⟨0 1 3 -9 -10 8]]
Mapping generators: ~16038/15925, ~3/2
Optimal tuning (CTE): ~3/2 = 701.976
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 12376/12375, 37180/37179, 1990656/1990625, 1146880/1146717, 263299491/263296000
Mapping: [⟨97 97 55 783 903 -95 283], ⟨0 1 3 -9 -10 8 2]]
Mapping generators: ~1547/1536, ~3/2
Optimal tuning (CTE): ~3/2 = 701.976
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 12376/12375, 13377/13376, 14080/14079, 27456/27455, 37180/37179, 165376/165375, 722007/722000
Mapping: [⟨97 97 55 783 903 -95 283 89 1642], ⟨0 1 3 -9 -10 8 2]]
Mapping generators: ~? = 1\97, ~3/2 = 701.976
Optimal tuning (CTE): ~3/2 = 701.976