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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-08-20 12:03:10 UTC</tt>.<br>
: The original revision id was <tt>557042207</tt>.<br>
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The **2460** equal division divides the [[octave]] into 2460 equal parts of 0.4878 [[cent]]s each. It has been used in [[Sagittal notation]] to define the "olympian level" of JI notation, and has been proposed as the basis for a unit, the [[mina]], which could be used in place of the [[cent]]. It is uniquely [[consistent]] through to the [[27-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-limit intervals. It has a lower 19-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any edo until [[3395edo|3395]], and a lower 23-limit relative error than any until [[8269edo|8269]]. It also has a lower 23-limit [[Tenney-Euclidean metrics#Logflat TE badness| TE loglfat badness]] than any smaller edo and less than any until [[16808edo|16808]].


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.
== Theory ==
2460edo 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.


Since its prime factorization is 2^2*3*5*41, 2460 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230. Of these, [[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]], [[20edo]], [[30edo]], [[60edo]], and [[164edo]] all have drawn some attention. Moreover a cent is exactly 2.05 [[mina]]s, and a mem, 1\205 octaves, is exactly 12 minas.</pre></div>
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]].
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;2460edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;2460&lt;/strong&gt; equal division divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 2460 equal parts of 0.4878 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It has been used in &lt;a class="wiki_link" href="/Sagittal%20notation"&gt;Sagittal notation&lt;/a&gt; to define the &amp;quot;olympian level&amp;quot; of JI notation, and has been proposed as the basis for a unit, the &lt;a class="wiki_link" href="/mina"&gt;mina&lt;/a&gt;, which could be used in place of the &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;. It is uniquely &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through to the &lt;a class="wiki_link" href="/27-limit"&gt;27-limit&lt;/a&gt;, which is not very remarkable in itself (&lt;a class="wiki_link" href="/388edo"&gt;388edo&lt;/a&gt; is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It has a lower 19-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; than any edo until &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, and a lower 23-limit relative error than any until &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;. It also has a lower 23-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness"&gt; TE loglfat badness&lt;/a&gt; than any smaller edo and less than any until &lt;a class="wiki_link" href="/16808edo"&gt;16808&lt;/a&gt;.&lt;br /&gt;
=== Prime harmonics ===
&lt;br /&gt;
{{Harmonics in equal|2460|columns=9}}
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.&lt;br /&gt;
{{Harmonics in equal|2460|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}
&lt;br /&gt;
 
Since its prime factorization is 2^2*3*5*41, 2460 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230. Of these, &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; is too well-known to need any introduction, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; is an important system, and &lt;a class="wiki_link" href="/205edo"&gt;205edo&lt;/a&gt; has proponents such as &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;, who uses it as the default tuning for &lt;a class="wiki_link_ext" href="http://www.h-pi.com/theory/measurement3.html" rel="nofollow"&gt;Hi-pi Instruments&lt;/a&gt; (and as a unit: &lt;a class="wiki_link" href="/Mem"&gt;Mem&lt;/a&gt;). Aside from these, &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/20edo"&gt;20edo&lt;/a&gt;, &lt;a class="wiki_link" href="/30edo"&gt;30edo&lt;/a&gt;, &lt;a class="wiki_link" href="/60edo"&gt;60edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/164edo"&gt;164edo&lt;/a&gt; all have drawn some attention. Moreover a cent is exactly 2.05 &lt;a class="wiki_link" href="/mina"&gt;mina&lt;/a&gt;s, and a mem, 1\205 octaves, is exactly 12 minas.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
2460 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.
 
2460edo 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.
 
== 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]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -3899 4320 }}
| {{Mapping| 2460 3899 }}
| +0.001
| 0.001
| 0.24
|-
| 2.3.5
| {{Monzo| 91 -12 -31 }}, {{monzo| -70  72 -19 }}
| {{Mapping| 2460 3899 5712 }}
| −0.003
| 0.006
| 1.29
|-
| 2.3.5.7
| 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }}
| {{Mapping| 2460 3899 5712 6096 }}
| +0.002
| 0.010
| 2.05
|-
| 2.3.5.7.11
| 9801/9800, 151263/151250, {{monzo| 24 -10 -5  0 1 }}, {{monzo| -3 -16 -1 6 4 }}
| {{Mapping| 2460 3899 5712 6096 8510 }}
| +0.007
| 0.014
| 2.86
|-
| 2.3.5.7.11.13
| 9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125
| {{Mapping| 2460 3899 5712 6096 8510 9103 }}
| +0.008
| 0.013
| 2.63
|-
| 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 }}
| +0.009
| 0.013
| 2.56
|}
* 2460edo has lower 23-limit relative error than any edo until [[8269edo|8269]]. Also it has a lower 23-limit [[TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]].
* In addition, it has the lowest relative error in the 19-limit, being only bettered by [[3395edo]].
 
=== 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*
! Temperaments
|-
| 1
| 271\2460
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 1219\2460
| 594.634
| {{Monzo| -70 72 -19 }}
| [[Gaster]]
|-
| 10
| 583\2460<br>(91\2460)
| 284.390<br>(44.390)
| {{Monzo| 10 29 -24 }}<br>(?)
| [[Neon]]
|-
| 12
| 1021\2460<br>(4\2460)
| 498.049<br>(1.951)
| 4/3<br>(32805/32768)
| [[Atomic]]
|-
| 20
| 353\2460<br>(16\2460)
| 172.195<br>(7.805)
| 169/153<br>(?)
| [[Calcium]]
|-
| 30
| 747\2460<br>(9\2460)
| 364.390<br>(4.390)
| 216/175<br>(385/384)
| [[Zinc]]
|-
| 41
| 1021\2460<br>(1\2460)
| 498.049<br>(0.488)
| 4/3<br />({{monzo| 215 -121 -10 }})
| [[Niobium]]
|-
| 60
| 747\2460<br>(9\2460)
| 364.390<br>(4.390)
| 216/175<br>(385/384)
| [[Neodymium]] / [[neodymium magnet]]
|-
| 60
| 1021\2460<br>(4\2460)
| 498.049<br>(1.951)
| 4/3<br>(32805/32768)
| [[Minutes]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 
[[Category:Mina]]
Retrieved from "https://en.xen.wiki/w/2460edo"