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8539edo: Difference between revisions

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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:xenwolf|xenwolf]] and made on <tt>2011-06-15 09:56:13 UTC</tt>.<br>
 
: The original revision id was <tt>236820600</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists|strict zeta]] tuning, and is also the first [[trivial temperament|non-trivial]] edo to be consistent in the [[odd prime sum limit|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo|20203]].
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>
Some of the simpler commas it [[tempering out|tempers out]] include the [[senior|senior comma]] in the [[5-limit]]; [[very high accuracy temperaments #Sepbizo-asegu (171 & 3566 & 4973)|{{monzo| -3 2 -17 14 }}]] in the [[7-limit]]; [[3294225/3294172]] in the [[11-limit]]; [[123201/123200]] in the [[13-limit]]; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the [[17-limit]]; 27456/27455, 43681/43680, 89376/89375 in the [[19-limit]]; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others 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;white-space: pre-wrap ! important" class="old-revision-html">Dividing the [[octave]] into 8539 equal parts gives a [[tina]], see http://www.tonalsoft.com/enc/t/tina.aspx</pre></div>
 
<h4>Original HTML content:</h4>
Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-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;8539edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Dividing the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 8539 equal parts gives a &lt;a class="wiki_link" href="/tina"&gt;tina&lt;/a&gt;, see &lt;!-- ws:start:WikiTextUrlRule:2:http://www.tonalsoft.com/enc/t/tina.aspx --&gt;&lt;a class="wiki_link_ext" href="http://www.tonalsoft.com/enc/t/tina.aspx" rel="nofollow"&gt;http://www.tonalsoft.com/enc/t/tina.aspx&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:2 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Prime harmonics ===
{{Harmonics in equal|8539|columns=11}}
{{Harmonics in equal|8539|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 8539edo (continued)}}
 
=== Subsets and supersets ===
8539edo is the 1065th [[prime edo]]. On that basis, the tina as a unit of measure could be criticized; however, some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.
 
== Notation ==
8539edo is special in the [[Sagittal notation]], as it has been the model for the Magrathean set, which offers "insane" precision. This is, and will be, the highest precision available in Sagittal notation. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor (503\8539 onwards).
<div style="overflow-x:auto;">
{| class="wikitable" style="text-align:center"
|-
!'''Steps'''
! 0 !! 24 !! 41
!62
!69
!86
!105
!118
!124
!143
!153
!177
!194
!218
!226
!236
!254
!271
!277
!285
!306
!330
!347
!357
!379
!389
!402
!420
!430
!452
!462
!479
!503
!524
!532
!538
!555
!573
!583
!591
!615
!632
!656
!666
!671
!691
!704
!723
!740
!747
!768
!785
!809
|-
|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="
|+Magrathean diacritics
!'''Steps'''
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
!21
!22
!23
|-
|Symbol
|<big>{{sagittal|@1}}</big>
|<big>{{sagittal|@2}}</big>
|<big>{{sagittal|@3}}</big>
|<big>{{sagittal|@4}}</big>
|<big>{{sagittal|@5}}</big>
|<big>{{sagittal|@6}}</big>
|<big>{{sagittal|@7}}</big>
|<big>{{sagittal|@8}}</big>
|<big>{{sagittal|@9}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|l4}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|l3}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|l2}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|l1}}</big>
|<big>{{sagittal|'}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@1}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@3}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@3}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@4}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@5}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@6}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@7}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@8}}</big>
|<big>{{sagittal|'}}</big><big>{{sagittal|@9}}</big>
|}
 
== External links ==
* [http://www.tonalsoft.com/enc/t/tina.aspx Tina] on [[Tonalsoft Encyclopedia]]
 
[[Category:Tina]]
Retrieved from "https://en.xen.wiki/w/8539edo"