97th-octave temperaments
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97th-octave temperaments occur between any equal divisions whose greatest common divisor is 97.
Although harmonic quality of 97edo is not visible at first glance, some of its multiples are highly notable large edos. 388edo is the first edo to be consistent in the 37-odd-limit, 3395edo is a zeta edo and a strong 19-limit tuning, having lowest 19-limit relative error than any division before it, and while 2619edo is not remarkably strong in harmonic approximation, it is consistent in the 33-odd-limit.
Berkelium
Berkelium is a remarkable high-limit subgroup temperament with equally remarkable full 31-limit branchings. It is named after the 97th element,
Berkelium comes in two variants, berkelium-247, named after the most stable isotope, is described as the 388 & 3395 temperament. Another 31-limit variety, named berkelium-248 is described as a 388 & 2619 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.
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...
Optimal ET sequence: 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
Optimal ET sequence: 388, 2619, ...
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