Xenharmonic Wiki

2460edo: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 236681372 - Original comment: **
Overthink (talk | contribs)
autopatrolled
2,773 edits
→Theory: Those all involve 29
 
(59 intermediate revisions by 14 users not shown)
Line 1: Line 1:
<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>2011-06-14 18:10:21 UTC</tt>.<br>
: The original revision id was <tt>236681372</tt>.<br>
: The revision comment was: <tt></tt><br>
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 cents 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.


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.


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]]. Aside from these, [[15edo]], [[20edo]], [[30edo]], [[60edo]], and [[164edo]] all have drawn some attention. Moreover a cent is exactly 2.05 minas, and a mem, 1/205 octaves, is exactly 12 minas.
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]].
</pre></div>
 
<h4>Original HTML content:</h4>
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]].
<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 2460 equal division divides the octave into 2460 equal parts of 0.4878 cents 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 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.&lt;br /&gt;
 
&lt;br /&gt;
=== Prime harmonics ===
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=11}}
&lt;br /&gt;
{{Harmonics in equal|2460|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}
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;. 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 minas, 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.
 
== Notation ==
2460edo 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]]
! 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 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]]
Retrieved from "https://en.xen.wiki/w/2460edo"