8269edo: Difference between revisions

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**Imported revision 557168695 - Original comment: **
 
m →Theory: extend the harmonics table to the 61-limit
 
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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>2015-08-22 12:46:09 UTC</tt>.<br>
 
: The original revision id was <tt>557168695</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo|311]], [[581edo|581]], [[1578edo|1578]] and [[2460edo|2460]]. While [[8539edo|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''.
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 [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 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 8269 division divides the octave into 8269 equal parts of 0.14512 cents each. It is both a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong 19- and 23-limit division. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean metrics#Logflat TE badness| TE loglfat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 2460. While [[8539edo|8539]] has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes.</pre></div>
 
<h4>Original HTML content:</h4>
=== Prime harmonics ===
<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;8269edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 8269 division divides the octave into 8269 equal parts of 0.14512 cents each. It is both a  &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak and zeta integral edo&lt;/a&gt;, which has to do with the fact that it is a very strong 19- and 23-limit division. It has a lower 19-limit and a lower 23-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;relative error&lt;/a&gt; than any smaller division, a lower 19-limit &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness"&gt; TE loglfat badness&lt;/a&gt; than any smaller division, and a lower 23-limit  logflat badness than any excepting 311, 581, 1578 and 2460. While &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt; has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|8269|columns=9}}
{{Harmonics in equal|8269|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 8269edo (continued)}}
 
=== Subsets and supersets ===
8269edo is the 1037th [[prime edo]].
 
== Music ==
; [[Francium]]
* "Details To Follow" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/12kSjGHgwE4J6AMA4Pxd5d Spotify] | [https://francium223.bandcamp.com/track/details-to-follow Bandcamp] | [https://www.youtube.com/watch?v=a9HqvhEXvBk YouTube] – totziensmic in 8269edo

Latest revision as of 14:21, 11 March 2025

← 8268edo 8269edo 8270edo →
Prime factorization 8269 (prime)
Step size 0.14512 ¢ 
Fifth 4837\8269 (701.947 ¢)
Semitones (A1:m2) 783:622 (113.6 ¢ : 90.26 ¢)
Consistency limit 27
Distinct consistency limit 27

8269 equal divisions of the octave (abbreviated 8269edo or 8269ed2), also called 8269-tone equal temperament (8269tet) or 8269 equal temperament (8269et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8269 equal parts of about 0.145 ¢ each. Each step represents a frequency ratio of 21/8269, or the 8269th root of 2.

Theory

8269edo is both a zeta peak and zeta integral edo, which has to do with the fact that it is a very strong 19- and 23-limit system. It has a lower 19-limit and a lower 23-limit relative error than any smaller division, a lower 19-limit TE logflat badness than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 2460. While 8539 has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an interval size measure, the major tina.

Some of the simpler commas it tempers out include 123201/123200 in the 13-limit; 194481/194480, 336141/336140 in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

Prime harmonics

Approximation of prime harmonics in 8269edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.0000 -0.0080 -0.0034 -0.0026 -0.0058 +0.0093 -0.0334 -0.0163 -0.0484
Relative (%) +0.0 -5.5 -2.3 -1.8 -4.0 +6.4 -23.0 -11.3 -33.4
Steps
(reduced) 		8269
(0) 		13106
(4837) 		19200
(2662) 		23214
(6676) 		28606
(3799) 		30599
(5792) 		33799
(723) 		35126
(2050) 		37405
(4329)
Approximation of prime harmonics in 8269edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) +0.0515 -0.0362 +0.0044 +0.0584 +0.0315 +0.0152 -0.0253 +0.0616 -0.0388
Relative (%) +35.5 -24.9 +3.0 +40.2 +21.7 +10.5 -17.4 +42.5 -26.7
Steps
(reduced) 		40171
(7095) 		40966
(7890) 		43077
(1732) 		44302
(2957) 		44870
(3525) 		45931
(4586) 		47364
(6019) 		48644
(7299) 		49041
(7696)

Subsets and supersets

8269edo is the 1037th prime edo.

Music

Francium
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