60th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|60}} | {{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 [[ | [[60edo]] is a [[highly composite edo]], and some its multiples are notable for their consistency limits, such as [[2460edo]], which is a [[Riemann zeta function #Zeta edo lists|zeta record edo]]. | ||
See [[minutes]] and [[neodymium]]. | See [[minutes]] and [[neodymium]]. | ||
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== Magnetismic microtemperaments == | == 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. | 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. | ||
=== Magnetismic (rank-3) === | |||
Subgroup: 2.3.29.43 | |||
Comma list: 2.3.29.43 {{monzo| -61 60 60 -60 }} | |||
{{mapping|legend=2| 60 60 240 239 | 0 1 0 1 | 0 0 1 1 }} | |||
: mapping generators: ~87/86, ~3/2 ~29/16 | |||
Optimal tunings: | |||
* WE: ~87/86 = 20.000{{c}}, ~3/2 = 701.954{{c}}, ~29/16 = 1029.570{{c}} | |||
* CWE: ~87/86 = 20.000{{c}}, ~3/2 = 709.954{{c}}, ~29/16 = 1029.570{{c}} | |||
[[Support]]ing [[ET]]s: {{EDOs|600, 660, 720, 780, 1260, 1140, 1920, 2460, 4380}} | |||
Badness (Sintel): 797.107 | |||
=== Neodymium magnet === | === Neodymium magnet === | ||
{{Main|Landscape microtemperaments#Neodymium}} | |||
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. | 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. | ||
Latest revision as of 16:35, 8 August 2026
60edo is a highly composite edo, and some its multiples are notable for their consistency limits, such as 2460edo, which is a zeta record edo.
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.
Magnetismic (rank-3)
Subgroup: 2.3.29.43
Comma list: 2.3.29.43 [-61 60 60 -60⟩
Subgroup-val mapping: [⟨60 60 240 239], ⟨0 1 0 1], ⟨0 0 1 1]]
- mapping generators: ~87/86, ~3/2 ~29/16
Optimal tunings:
- WE: ~87/86 = 20.000 ¢, ~3/2 = 701.954 ¢, ~29/16 = 1029.570 ¢
- CWE: ~87/86 = 20.000 ¢, ~3/2 = 709.954 ¢, ~29/16 = 1029.570 ¢
Supporting ETs: 600, 660, 720, 780, 1260, 1140, 1920, 2460, 4380
Badness (Sintel): 797.107
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