@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2460}}
+{{ED intro}}
 == Theory ==
-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 below). It is also a [[zeta edo|zeta peak and zeta peak integer edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.
+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, [[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=11}}
+{{Harmonics in equal|2460|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}
-=== JI approximation in the 27-odd-limit ===
-{{15-odd-limit|2460|27}}
 === Subsets and supersets ===
-is divisible by {{EDOs| 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230 }}, and its [[abundancy index]] is 1.868. Of the divisors, [[12edo]] is too well-known to need any introduction, [[41edo]] is an important system, and [[205edo]] has proponents such as [[Aaron Andrew Hunt]], who uses it as the default tuning for [http://www.h-pi.com/theory/measurement3.html Hi-pi Instruments] (and as a unit: [[mem]]). Aside from these, [[15edo]] is notable for use by [[Easley Blackwood Jr.]], [[60edo]] is a [[Highly composite equal division|highly composite edo]]. In addition, 2460edo maps the [[schisma]] to an exact fraction of the octave, 4 steps. However, such mapping does not hold in [[615edo]].
+is divisible by {{EDOs| 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230 }}, and its [[abundancy index]] is 1.868. Of the divisors, [[12edo]] is too well-known to need any introduction, [[41edo]] is an important system, and [[205edo]] has proponents such as [[Aaron Andrew Hunt]], who uses it as the default tuning for [http://www.h-pi.com/theory/measurement3.html Hi-pi Instruments] (and as a unit: [[mem]]). Aside from these, [[15edo]] is notable for use by [[Easley Blackwood Jr.]], [[60edo]] is a [[highly composite edo]]. In addition, 2460edo maps the [[schisma]] to an exact fraction of the octave, 4 steps. However, such mapping does not hold in [[615edo]].
 In light of having a large amount of divisors and precise approximation of just intonation, 2460edo has been proposed as the basis for a unit, the [[mina]], which could be used in place of the cent. Moreover, a cent is exactly 2.05 [[mina]]s, and a mem, 1\205, is exactly 12 minas.
+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 ==
+{{15-odd-limit|2460|27}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 30: / Line 169: @@
 |-
 | 2.3
-| {{monzo| -3899 4320 }}
+| {{Monzo| -3899 4320 }}
-| {{mapping| 2460 3899 }}
+| {{Mapping| 2460 3899 }}
-| 0.001
+| +0.001
 | 0.001
 | 0.24
 |-
 | 2.3.5
-| {{monzo| 91 -12 -31 }}, {{monzo| -70  72 -19 }}
+| {{Monzo| 91 -12 -31 }}, {{monzo| -70  72 -19 }}
-| {{mapping| 2460 3899 5712 }}
+| {{Mapping| 2460 3899 5712 }}
-| -0.003
+| −0.003
 | 0.006
 | 1.29
@@ Line 45: / Line 184: @@
 | 2.3.5.7
 | 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }}
-| {{mapping| 2460 3899 5712 6096 }}
+| {{Mapping| 2460 3899 5712 6096 }}
-| 0.002
+| +0.002
 | 0.010
 | 2.05
 |-
 | 2.3.5.7.11
-| 9801/9800, 250047/250000, {{monzo| 24 -10 -5  0 1 }}, {{monzo| -3 -16 -1 6 4 }}
+| 9801/9800, 151263/151250, {{monzo| 24 -10 -5  0 1 }}, {{monzo| -3 -16 -1 6 4 }}
-| {{mapping| 2460 3899 5712 6096 8510 }}
+| {{Mapping| 2460 3899 5712 6096 8510 }}
-| 0.007
+| +0.007
 | 0.014
 | 2.86
@@ Line 59: / Line 198: @@
 | 2.3.5.7.11.13
 | 9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125
-| {{mapping| 2460 3899 5712 6096 8510 9103 }}
+| {{Mapping| 2460 3899 5712 6096 8510 9103 }}
-| 0.008
+| +0.008
 | 0.013
 | 2.63
@@ Line 66: / Line 205: @@
 | 2.3.5.7.11.13.17
 | 9801/9800, 10648/10647, 12376/12375, 31213/31212, 37180/37179, 221221/221184
-| {{mapping| 2460 3899 5712 6096 8510 9103 10055 }}
+| {{Mapping| 2460 3899 5712 6096 8510 9103 10055 }}
-| 0.009
+| +0.009
 | 0.013
 | 2.56
@@ Line 76: / Line 215: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 94: / Line 235: @@
 | [[Gaster]]
 |-
-|10
+| 10
-|583\2460<br>(91\2460)
+| 583\2460<br>(91\2460)
-|284.390<br>(44.390)
+| 284.390<br>(44.390)
-|{{Monzo|10 29 -24}}<br>(?)
+| {{Monzo| 10 29 -24 }}<br>(?)
-|[[Neon]]
+| [[Neon]]
 |-
 | 12
@@ Line 107: / Line 248: @@
 |-
 | 20
-| 353\2460<br>(107\2460)
+| 353\2460<br>(16\2460)
-| 172.195<br>(52.195)
+| 172.195<br>(7.805)
 | 169/153<br>(?)
 | [[Calcium]]
@@ Line 115: / Line 256: @@
 | 747\2460<br>(9\2460)
 | 364.390<br>(4.390)
-| 216/175<br>(?)
+| 216/175<br>(385/384)
 | [[Zinc]]
 |-
@@ Line 127: / Line 268: @@
 | 747\2460<br>(9\2460)
 | 364.390<br>(4.390)
-| 216/175<br>(?)
+| 216/175<br>(385/384)
 | [[Neodymium]] / [[neodymium magnet]]
 |-
@@ Line 136: / Line 277: @@
 | [[Minutes]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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]]

2460edo: Difference between revisions