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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>
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| : The original revision id was <tt>557168695</tt>.<br>
| | == Theory == |
| : The revision comment was: <tt></tt><br>
| | 8269edo is a very strong [[19-limit|19-]] and [[23-limit]] system, with a lower 19-limit and a lower 23-limit [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] than any smaller edo. It is both a [[Riemann zeta function #Zeta edo lists|zeta peak and zeta integral edo]]. |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
| | 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. The sum of 8269 and 8539 is [[16808edo|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo|270]], which does well in the 13-limit, and also beyond. A step of 8269edo has also been similarly proposed as an [[interval size measure]], the '''major tina'''. |
| <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>
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| <h4>Original HTML 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, 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>8269edo</title></head><body>The 8269 division divides the octave into 8269 equal parts of 0.14512 cents each. It is both a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta peak and zeta integral edo</a>, 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 <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">relative error</a> than any smaller division, a lower 19-limit <a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness"> TE loglfat badness</a> than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 2460. While <a class="wiki_link" href="/8539edo">8539</a> 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.</body></html></pre></div>
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| | === Prime harmonics === |
| | {{Harmonics in equal|8269|columns=11}} |
| | {{Harmonics in equal|8269|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 8269edo (continued)}} |
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| | === Subsets and supersets === |
| | 8269edo is the 1037th [[prime edo]]. |
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| | == 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 |