265edo: 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>2011-07-08 00:22:23 UTC</tt>.<br>
 
: The original revision id was <tt>240470073</tt>.<br>
265 = 5 × 53, and 265edo is [[enfactoring|enfactored]] in the 5-limit, [[tempering out]] the same [[comma]]s as [[53edo]], including [[15625/15552]] and [[32805/32768]]. In the 7-limit it tempers out [[16875/16807]] and [[420175/419904]], so that it [[support]]s [[sqrtphi]], for which it provides the [[optimal patent val]]. In the 11-limit it tempers out [[540/539]], 1375/1372 and 4375/4356, and gives the optimal patent val for 11-limit sqrtphi temperament.
: 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>
=== Prime harmonics ===
<h4>Original Wikitext content:</h4>
{{Harmonics in equal|265}}
<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 //265 equal division// divides the octave into 265 equal parts of 4.528 cents each. It is contorted in the 5-limit, tempering out the same commas as [[53edo]], including 15625/15552 and 32805/32768. In the 7-limit it tempers out 16875/16807 and 420175/419904, so that it supports [[Kleismic family#Sqrtphi|sqrtphi temperament]], for which it provides the [[optimal patent val]]. In the 11-limit it tempers out 540/539, 1375/1372 and 4375/4356, and gives the optimal patent val for 11-limit sqrtphi temperament.</pre></div>
 
<h4>Original HTML content:</h4>
=== Subsets and supersets ===
<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;265edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;265 equal division&lt;/em&gt; divides the octave into 265 equal parts of 4.528 cents each. It is contorted in the 5-limit, tempering out the same commas as &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, including 15625/15552 and 32805/32768. In the 7-limit it tempers out 16875/16807 and 420175/419904, so that it supports &lt;a class="wiki_link" href="/Kleismic%20family#Sqrtphi"&gt;sqrtphi temperament&lt;/a&gt;, for which it provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. In the 11-limit it tempers out 540/539, 1375/1372 and 4375/4356, and gives the optimal patent val for 11-limit sqrtphi temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
265edo contains [[5edo]] and [[53edo]] as subsets. [[795edo]], which triples it, corrects its harmonic 5 to near-just quality.
 
A step of 265edo is exactly 40 [[türk sent]]s.  
 
[[Category:Sqrtphi]]

Latest revision as of 14:26, 20 February 2025

← 264edo 265edo 266edo →
Prime factorization 5 × 53
Step size 4.5283 ¢ 
Fifth 155\265 (701.887 ¢) (→ 31\53)
Semitones (A1:m2) 25:20 (113.2 ¢ : 90.57 ¢)
Consistency limit 9
Distinct consistency limit 9

265 equal divisions of the octave (abbreviated 265edo or 265ed2), also called 265-tone equal temperament (265tet) or 265 equal temperament (265et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 265 equal parts of about 4.53 ¢ each. Each step represents a frequency ratio of 21/265, or the 265th root of 2.

265 = 5 × 53, and 265edo is enfactored in the 5-limit, tempering out the same commas as 53edo, including 15625/15552 and 32805/32768. In the 7-limit it tempers out 16875/16807 and 420175/419904, so that it supports sqrtphi, for which it provides the optimal patent val. In the 11-limit it tempers out 540/539, 1375/1372 and 4375/4356, and gives the optimal patent val for 11-limit sqrtphi temperament.

Prime harmonics

Approximation of prime harmonics in 265edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.07 -1.41 +0.23 +1.13 +1.74 -0.80 +1.35 +1.16 -1.65 +0.62
Relative (%) +0.0 -1.5 -31.1 +5.1 +25.1 +38.3 -17.8 +29.9 +25.6 -36.5 +13.8
Steps
(reduced) 		265
(0) 		420
(155) 		615
(85) 		744
(214) 		917
(122) 		981
(186) 		1083
(23) 		1126
(66) 		1199
(139) 		1287
(227) 		1313
(253)

Subsets and supersets

265edo contains 5edo and 53edo as subsets. 795edo, which triples it, corrects its harmonic 5 to near-just quality.

A step of 265edo is exactly 40 türk sents.

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