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60th-octave temperaments: Difference between revisions

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[[60edo]] is a highly composite EDO, and some its multiples are notable for their consistency limits, such as 2460edo, which is a zeta edo.
{{Infobox fractional-octave|60}}
==Minutes==
[[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]].  
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
See [[minutes]] and [[neodymium]].  


Comma list: 9801/9800, 250047/250000, 371293/371250, 184549376/184528125
== 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.


Mapping: [{{val|60 60 385 730 1085 573}}, {{val|0 1 7 -16 -25 -10}}]
=== Magnetismic (rank-3) ===
Subgroup: 2.3.29.43


Mapping generators: ~2704/2673, ~3/2
Comma list: 2.3.29.43 {{monzo| -61 60 60 -60 }}


Optimal tuning (CTE): ~3/2 = 701.948
{{mapping|legend=2| 60 60 240 239 | 0 1 0 1 | 0 0 1 1 }}
: mapping generators: ~87/86, ~3/2 ~29/16


=== 17-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17
* 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}}


Comma list: 9801/9800, 12376/12375, 28561/28560, 250047/250000, 253755392/253746675
[[Support]]ing [[ET]]s: {{EDOs|600, 660, 720, 780, 1260, 1140, 1920, 2460, 4380}}


Mapping: [{{val|60 60 385 730 1085 573 877}}, {{val|0 1 7 -16 -25 -10 -18}}]
Badness (Sintel): 797.107


Mapping generators: ~3520/3213, ~3/2
=== Neodymium magnet ===
 
{{Main|Landscape microtemperaments#Neodymium}}
Optimal tuning (CTE): ~3/2 = 701.948
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.
 
=== 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: [{{val|60 60 385 730 1085 573 877 -61}}, {{val|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: [{{val|60 4 30 132 244 386 391}}, {{val|0 5 6 2 -2 -9 -8}}
 
Mapping generators: ~612/605, ~216/175
 
Optimal tuning (CTE): ~216/175 = 364.387
 
===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. Comma which closes a stack of 60 87/86's at the octave, is named [[magnetisma]] by [[Eliora]], 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


Vals: {{EDOs|540jn, 1380jn, 1920, 2460, 3000jjnn, 3840jn, 4920jn 4380, 6300, 6840}}
{{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]]
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