60th-octave temperaments
← 62nd-octave • 63rd-octave • 64th-octave →
← 59th-octave • 60th-octave • 61st-octave →
← 56th-octave • 57th-octave • 58th-octave →
60edo is a highly composite EDO, and some its multiples are notable for their consistency limits, such as 2460edo, which is a zeta edo.
Minutes
Defined as the 2460 & 4320 temperament, starting with the 13-limit. Named "minutes" for period-60, since there's 60 minutes in an hour. In light of 12 being a divisor of 60, minutes tempers out the Kirnberger's atom, and in the limits below 13, it's a contorted atomic temperament.
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 250047/250000, 371293/371250, 184549376/184528125
Mapping: [⟨60 60 385 730 1085 573], ⟨0 1 7 -16 -25 -10]]
Mapping generators: ~2704/2673, ~3/2
Optimal tuning (CTE): ~3/2 = 701.948
Optimal ET sequence: 1860, 2460, 6780, 7380
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 12376/12375, 28561/28560, 250047/250000, 253755392/253746675
Mapping: [⟨60 60 385 730 1085 573 877], ⟨0 1 7 -16 -25 -10 -18]]
Mapping generators: ~3520/3213, ~3/2
Optimal tuning (CTE): ~3/2 = 701.948
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 9801/9800, 12376/12375, 27456/27455, 250047/250000, 401408/401375, 1549184/1549125
Mapping: [⟨60 60 385 730 1085 573 877 -61], ⟨0 1 7 -16 -25 -10 -18 9]]
Mapping generators: ~3520/3213, ~3/2
Optimal tuning (CTE): ~3/2 = 701.948
Neodymium
Starts with the 17-limit since it is contorted in 13-limit and below, can be expressed as as 1920 & 4380 or 1920 & 2460.
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 123201/123200, 250047/250000, 31213/31212, 1990656/1990625
Mapping: [⟨60 4 30 132 244 386 391], ⟨0 5 6 2 -2 -9 -8]
Mapping generators: ~612/605, ~216/175
Optimal tuning (CTE): ~216/175 = 364.387
Optimal ET sequence: 540, 1380, 1920, 2460, 3000, 3840, 4380, 4920, 6300, 6840e
19-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 10241/10240, 13377/13376, 5929/5928, 89376/89375, 23409/23408
Mapping: [⟨60 4 30 132 244 386 391 182], ⟨0 5 6 2 -2 -9 -8 4]
Mapping generators: ~612/605, ~216/175
Optimal tuning (CTE): ~216/175 = 364.387
Magnetismic microtemperaments
All these temperaments temper out the magnetisma, a 2.3.29.43 subgroup comma which when tempered sets the 87/86 interval to 1/60th of the octave, and thus all these temperaments have a period that maps to 87/86.
Neodymium magnet
An extension of neodymium. Defined just as neodymium in 1920 & 4380, except adds a mapping for 29 and 43 via the fact that 87/86 is very close to 1/60th of the octave. and thus the extension is called "neodymium magnet". Defined starting with 2.3.5.29.43 all the way into the 2.3.5.7.11.13.17.19.23.29.43 subgroup, and unlike plain neodymium, addition of .29.43 harmonics saves it from contorsion.
Subgroup: 2.3.5.29.43
Comma list: 46235367/46225000, [-3 13 -5 1 -2⟩, [29 20 -35 2 2⟩
Sval mapping: ⟨60 4 30 -164 -221], ⟨0 5 6 25 30]
Sval mapping generators: ~87/86, ~7533637632/6103515625
Optimal tuning (CTE): ~7533637632/6103515625 = 364.385
Optimal ET sequence: 540jn, 1380jn, 1920, 2460, 3000jjnn, 3840jn, 4920jn 4380, 6300, 6840
2.3.5.7.29.43 subgroup
Subgroup: 2.3.5.7.29.43
Comma list: 250047/250000, 6890625/6889472, 634230/634207, 104487018125/104485552128
Sval mapping: ⟨60 4 30 132 -164 -221], ⟨0 5 6 2 25 30]
Sval mapping generators: ~87/86, ~216/175
Optimal tuning (CTE): ~216/175 = 364.385
2.3.5.7.11.29.43 subgroup
Subgroup: 2.3.5.7.11.29.43
Comma list: 7425/7424, 9801/9800, 250047/250000, 2278125/2277968, 10320758525/10319560704
Sval mapping: ⟨60 4 30 132 244 -164 -221], ⟨0 5 6 2 -2 25 30]
Sval mapping generators: ~87/86, ~216/175
Optimal tuning (CTE): ~216/175 = 364.385