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:genewardsmith|genewardsmith]] and made on <tt>2014-05-25 17:55:53 UTC</tt>.<br>
 
: The original revision id was <tt>511123830</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">The 8539 equal temperament divides the octave into 8539 equal parts of 0.1405 cents each. While it may strike many people as too large to be practical, it's seen actual use as a bookeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the [[tina]] (see http://www.tonalsoft.com/enc/t/tina.aspx.) This is because it is a very strong higher limit system, distinctly consistent through the 27 limit, and is both a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak]] and zeta integral tuning. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203</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;The 8539 equal temperament divides the octave into 8539 equal parts of 0.1405 cents each. While it may strike many people as too large to be practical, it's seen actual use as a bookeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the &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;.) This is because it is a very strong higher limit system, distinctly consistent through the 27 limit, and is both a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak&lt;/a&gt; and zeta integral tuning. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203&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]]

Latest revision as of 20:27, 20 January 2026

← 8538edo 8539edo 8540edo →
Prime factorization 8539 (prime)
Step size 0.140532 ¢ 
Fifth 4995\8539 (701.956 ¢)
Semitones (A1:m2) 809:642 (113.7 ¢ : 90.22 ¢)
Consistency limit 27
Distinct consistency limit 27

8539 equal divisions of the octave (abbreviated 8539edo or 8539ed2), also called 8539-tone equal temperament (8539tet) or 8539 equal temperament (8539et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8539 equal parts of about 0.141 ¢ each. Each step represents a frequency ratio of 21/8539, or the 8539th root of 2.

Theory

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 unit of interval measure, the tina. This is because it is a very strong higher-limit system, distinctly consistent through the 27-odd-limit. It is a strict zeta tuning, and is also the first non-trivial edo to be consistent in the 27-odd-prime-sum-limit. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203.

Some of the simpler commas it tempers out include the senior comma in the 5-limit; [-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.

Since it tempers out 12168/12167, it allows vicetertismic chords in the 23-odd-limit.

Prime harmonics

Approximation of prime harmonics in 8539edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0000 +0.0007 +0.0079 -0.0005 -0.0122 -0.0077 +0.0218 -0.0075 +0.0428 -0.0421 +0.0165
Relative (%) +0.0 +0.5 +5.6 -0.4 -8.7 -5.5 +15.5 -5.3 +30.4 -30.0 +11.8
Steps
(reduced) 		8539
(0) 		13534
(4995) 		19827
(2749) 		23972
(6894) 		29540
(3923) 		31598
(5981) 		34903
(747) 		36273
(2117) 		38627
(4471) 		41482
(7326) 		42304
(8148)
Approximation of prime harmonics in 8539edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.0671 -0.0192 +0.0176 +0.0514 +0.0404 +0.0156 +0.0610 -0.0472 +0.0435 +0.0124 +0.0023
Relative (%) +47.8 -13.7 +12.5 +36.6 +28.7 +11.1 +43.4 -33.6 +30.9 +8.8 +1.6
Steps
(reduced) 		44484
(1789) 		45748
(3053) 		46335
(3640) 		47431
(4736) 		48911
(6216) 		50232
(7537) 		50643
(7948) 		51798
(564) 		52513
(1279) 		52855
(1621) 		53828
(2594)

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).

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
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

External links

Retrieved from "https://en.xen.wiki/index.php?title=8539edo&oldid=222047"