378edo: 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>2011-07-13 01:12:32 UTC</tt>.<br>
 
: The original revision id was <tt>241117695</tt>.<br>
The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit and [[3136/3125]] in the 7-limit, so that it [[support]]s [[bischismic]], and in fact provides the [[optimal patent val]]. It tempers out [[441/440]] and [[8019/8000]] in the 11-limit and [[729/728]] and [[1001/1000]] in the 13-limit so that it supports 11- and 13-limit bischismatic, and it also gives the optimal patent val for 13-limit bischismic.
: 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|378}}
<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 //378 equal division// divides the octave into 378 equal parts of 3.175 cents each. It tempers out 32805/32768 in the 5-limit and 31363125 in the 7-limit, so that it supports [[Schismatic family#Bischismatic|bischismatic temperament]] and in fact provides the [[optimal patent val]]. It tempers out 441/440 and 8019/8000 in the 11-limit and 729/728 and 1001/1000 in the 13-limit so that it supports 11 and 13 limit bischismatic, and it also gives the optimal patent val for 13-limit bischismatic.</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;378edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;378 equal division&lt;/em&gt; divides the octave into 378 equal parts of 3.175 cents each. It tempers out 32805/32768 in the 5-limit and 31363125 in the 7-limit, so that it supports &lt;a class="wiki_link" href="/Schismatic%20family#Bischismatic"&gt;bischismatic temperament&lt;/a&gt; and in fact provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;. It tempers out 441/440 and 8019/8000 in the 11-limit and 729/728 and 1001/1000 in the 13-limit so that it supports 11 and 13 limit bischismatic, and it also gives the optimal patent val for 13-limit bischismatic.&lt;/body&gt;&lt;/html&gt;</pre></div>
Since 378 factors into {{factorization|378}}, 378edo has subset edos {{EDOs| 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, and 189 }}.
 
[[Category:Bischismic]]

Latest revision as of 14:48, 20 February 2025

← 377edo 378edo 379edo →
Prime factorization 2 × 33 × 7
Step size 3.1746 ¢ 
Fifth 221\378 (701.587 ¢)
Semitones (A1:m2) 35:29 (111.1 ¢ : 92.06 ¢)
Consistency limit 7
Distinct consistency limit 7

378 equal divisions of the octave (abbreviated 378edo or 378ed2), also called 378-tone equal temperament (378tet) or 378 equal temperament (378et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 378 equal parts of about 3.17 ¢ each. Each step represents a frequency ratio of 21/378, or the 378th root of 2.

The equal temperament tempers out 32805/32768 (schisma) in the 5-limit and 3136/3125 in the 7-limit, so that it supports bischismic, and in fact provides the optimal patent val. It tempers out 441/440 and 8019/8000 in the 11-limit and 729/728 and 1001/1000 in the 13-limit so that it supports 11- and 13-limit bischismatic, and it also gives the optimal patent val for 13-limit bischismic.

Prime harmonics

Approximation of prime harmonics in 378edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.37 +0.99 -0.57 +1.06 +0.74 -0.19 +0.90 +0.30 -1.01 +1.00
Relative (%) +0.0 -11.6 +31.1 -18.0 +33.5 +23.4 -6.1 +28.3 +9.4 -31.7 +31.4
Steps
(reduced) 		378
(0) 		599
(221) 		878
(122) 		1061
(305) 		1308
(174) 		1399
(265) 		1545
(33) 		1606
(94) 		1710
(198) 		1836
(324) 		1873
(361)

Subsets and supersets

Since 378 factors into 2 × 33 × 7, 378edo has subset edos 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, and 189.

Retrieved from "https://en.xen.wiki/index.php?title=378edo&oldid=181760"