2460edo: Difference between revisions

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 edo is [[consistency|distinctly consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see [[#Approximation to JI]]). It is also a [[zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.
-As a micro- (or nano-) temperament, it is a [[landscape]] system in the [[7-limit]], [[tempering out]] [[250047/250000]], and in the [[11-limit]] it tempers out [[9801/9800]]. Beyond that, it tempers out [[10648/10647]] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].
+In higher limits, it is ''almost'' consistent in the [[29-odd-limit]] missing [[29/22]], [[29/17]], [[34/29]], [[44/29]]. It is also fully consistent in the no-29 [[39-odd-limit]].
+As a micro- (or nano-) temperament, it tempers [[Kirnberger's atom]] in the [[5-limit]], [[250047/250000]] (landscape comma) in the [[7-limit]], [[9801/9800]] [kalisma] in the [[11-limit]], [[10648/10647]] [harmonisma] in the [[13-limit]], [[12376/12375]] in the [[17-limit]], 5929/5928 and 6860/6859 in the [[19-limit]]; and 8281/8280 in the [[23-limit]].
 === Prime harmonics ===
-{{Harmonics in equal|2460|columns=9}}
+{{Harmonics in equal|2460|columns=11}}
-{{Harmonics in equal|2460|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}
+{{Harmonics in equal|2460|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}
 === Subsets and supersets ===
@@ Line 17: / Line 19: @@
 edo is also notable for being the smallest edo that is a multiple of 12 to be [[purely consistent]] in the 15-odd-limit (i.e. it is the smallest edo that is a multiple of 12 which maintains [[relative interval error|relative error]]s of less than 25% on all of the first 16 harmonics of the harmonic series). [[72edo]] comes close, but its approximations to [[13/8]] and [[15/8]] are somewhat inaccurate.
+== Notation ==
+edo is special in the [[Sagittal notation]], as it has been the model for the Olympian set, which offers "extreme" precision. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor 145\2460 onwards).
+<div style="overflow-x:auto;">
+{| class="wikitable" style="text-align:center"
+|-
+!'''Steps'''
+! 0 !! 7 !! 8
+!18
+!20
+!25
+!30
+!34
+!36
+!41
+!44
+!51
+!56
+!63
+!65
+!68
+!73
+!78
+!80
+!83
+!88
+!95
+!100
+!102
+!109
+!112
+!116
+|'''121'''
+|'''124'''
+|'''131'''
+|'''133'''
+|'''138'''
+|'''145'''
+|'''150'''
+|'''153'''
+!'''155'''
+|'''160'''
+|'''165'''
+|'''168'''
+|'''170'''
+|'''177'''
+|'''184'''
+|'''189'''
+|'''192'''
+|'''197'''
+|'''199'''
+|'''203'''
+|'''208'''
+|'''213'''
+|'''215'''
+|'''225'''
+|'''226'''
+|'''233'''
+|-
+|Symbol
+|<big>{{sagittal|h}}</big>
+|<big>{{sagittal|)|}}</big>
+|<big>{{sagittal||(}}</big>
+|<big>{{sagittal|~|}}</big>
+|<big>{{sagittal|)|(}}</big>
+|<big>{{sagittal|)~|}}</big>
+|<big>{{sagittal|~|(}}</big>
+|<big>{{sagittal||~}}</big>
+|<big>{{sagittal|~~|}}</big>
+|<big>{{sagittal|)|~}}</big>
+|<big>{{sagittal|/|}}</big>
+|<big>{{sagittal|)/|}}</big>
+|<big>{{sagittal||)}}</big>
+|<big>{{sagittal|)|)}}</big>
+|<big>{{sagittal||\}}</big>
+|<big>{{sagittal|(|}}</big>
+|<big>{{sagittal|~|)}}</big>
+|<big>{{sagittal|/|~}}</big>
+|<big>{{sagittal|(|(}}</big>
+|<big>{{sagittal|~|\}}</big>
+|<big>{{sagittal|//|}}</big>
+|<big>{{sagittal|)//|}}</big>
+|<big>{{sagittal|/|)}}</big>
+|<big>{{sagittal|(|~}}</big>
+|<big>{{sagittal|/|\}}</big>
+|<big>{{sagittal|(/|}}</big>
+|<big>{{sagittal|)/|\}}</big>
+|<big>{{sagittal||\)}}</big>
+|<big>{{sagittal|(|)}}</big>
+|<big>{{sagittal||\\}}</big>
+|<big>{{sagittal|(|\}}</big>
+|<big>{{sagittal|)|\\}}</big>
+|<big>{{sagittal|)||(}}</big>
+|<big>{{sagittal|)~||}}</big>
+|<big>{{sagittal|~||(}}</big>
+|<big>{{sagittal|||~}}</big>
+|<big>{{sagittal|~~||}}</big>
+|<big>{{sagittal|)||~}}</big>
+|<big>{{sagittal|/||}}</big>
+|<big>{{sagittal|)/||}}</big>
+|<big>{{sagittal|||)}}</big>
+|<big>{{sagittal|)||)}}</big>
+|<big>{{sagittal|||\}}</big>
+|<big>{{sagittal|(||}}</big>
+|<big>{{sagittal|~||)}}</big>
+|<big>{{sagittal|/||~}}</big>
+|<big>{{sagittal|(||(}}</big>
+|<big>{{sagittal|~||\}}</big>
+|<big>{{sagittal|//||}}</big>
+|<big>{{sagittal|)//||}}</big>
+|<big>{{sagittal|/||)}}</big>
+|<big>{{sagittal|(||~}}</big>
+|<big>{{sagittal|/||\}}</big>
+|}
+</div>
+{| class="wikitable data-darkreader-inline-color="
+|+Olympian diacritics
+!'''Steps'''
+|1
+|2
+|3
+|4
+!5
+|6
+|-
+|Symbol
+|<big>{{sagittal|`}}</big>
+|<big>{{sagittal|``}}</big>
+|<big>{{sagittal|'}}{{sagittal|,}}</big>
+|<big>{{sagittal|'}}</big>
+|<big>{{sagittal|'}}{{sagittal|`}}</big>
+|<big>{{sagittal|'}}{{sagittal|``}}</big>
+|}
 == Approximation to JI ==
@@ Line 127: / Line 262: @@
 | 1021\2460<br>(1\2460)
 | 498.049<br>(0.488)
-| 4/3<br />({{monzo| 215 -121 -10 }})
+| 4/3<br>({{monzo| 215 -121 -10 }})
 | [[Niobium]]
 |-
@@ Line 142: / Line 277: @@
 | [[Minutes]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 [[Category:Mina]]