60th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|60}} | |||
[[60edo]] is a highly composite EDO, and some its multiples are notable for their consistency limits, such as 2460edo, which is a zeta edo. | [[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== | |||
== 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. | 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 | [[Subgroup]]: 2.3.5.7.11.13 | ||
Comma list: 9801/9800, 250047/250000, 371293/371250, 184549376/184528125 | [[Comma list]]: 9801/9800, 250047/250000, 371293/371250, 184549376/184528125 | ||
Mapping: [{{val|60 60 385 730 1085 573}}, {{val|0 1 7 -16 -25 -10}}] | [[Mapping]]: [{{val|60 60 385 730 1085 573}}, {{val|0 1 7 -16 -25 -10}}] | ||
Mapping generators: ~2704/2673, ~3/2 | Mapping generators: ~2704/2673, ~3/2 | ||
Optimal tuning (CTE): ~3/2 = 701.948 | Optimal tuning (CTE): ~3/2 = 701.948 | ||
{{Optimal ET sequence|legend=1|1860, 2460, 6780, 7380}} | |||
=== 17-limit === | === 17-limit === | ||
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Optimal tuning (CTE): ~3/2 = 701.948 | Optimal tuning (CTE): ~3/2 = 701.948 | ||
==Neodymium== | ==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. | Starts with the 17-limit since it is contorted in 13-limit and below, can be expressed as as 1920 & 4380 or 1920 & 2460. | ||
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Optimal tuning (CTE): ~216/175 = 364.387 | Optimal tuning (CTE): ~216/175 = 364.387 | ||
{{Optimal ET sequence|legend=1|540, 1380, 1920, 2460, 3000, 3840, 4380, 4920, 6300, 6840e}} | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 9801/9800, 10241/10240, 13377/13376, 5929/5928, 89376/89375, 23409/23408 | |||
Mapping: [{{val|60 4 30 132 244 386 391 182}}, {{val|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=== | ===Neodymium magnet=== | ||
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. | 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 | Subgroup: 2.3.5.29.43 | ||
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Optimal tuning (CTE): ~7533637632/6103515625 = 364.385 | Optimal tuning (CTE): ~7533637632/6103515625 = 364.385 | ||
{{Optimal ET sequence|legend=1|540jn, 1380jn, 1920, 2460, 3000jjnn, 3840jn, 4920jn 4380, 6300, 6840}} | |||
==== 2.3.5.7.29.43 subgroup ==== | ==== 2.3.5.7.29.43 subgroup ==== | ||
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Optimal tuning (CTE): ~216/175 = 364.385 | Optimal tuning (CTE): ~216/175 = 364.385 | ||
{{Navbox fractional-octave}} | |||
[[Category:60edo]] | |||