162edo: 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>2010-12-10 17:49:11 UTC</tt>.<br>
: The original revision id was <tt>187245061</tt>.<br>
: 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>
<h4>Original Wikitext content:</h4>
<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">162edo divides the octave into 162 parts of 7.407 cents each. In the 7-limit it tempers out 4000/3969, 10976/10935 and 65625/65536.


The non-patent val &lt;162 257 377| and its extensions are of considerable interest, as this tempers out 2048/2025. In the 7-limit, &lt;162 257 377 455| tempers out 126/125 and 2048/2025 both, giving a tuning for 7-limit diaschismic. In the 11-limit &lt;162 257 377 455 561| tempers out 126/125, 176/175 and 896/891, and so supports 11-limit diaschismic, and in fact has a fifth only 0.01 cents flatter than the [[POTE tuning]]. The 13-limit is even closer: the 13-limit val &lt;162 257 377 455 561 600| tempers out 126/125, 196/195, 364/363, 2048/2025 giving 13-limit diaschismic, and the 162 fifth of 95/162 octave is a mere 0.0000383 cents sharp of the 13-limit POTE tuning.</pre></div>
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[4000/3969]], [[10976/10935]] and [[65625/65536]].
<h4>Original HTML content:</h4>
 
<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;162edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;162edo divides the octave into 162 parts of 7.407 cents each. In the 7-limit it tempers out 4000/3969, 10976/10935 and 65625/65536. &lt;br /&gt;
The non-patent val {{val| 162 257 '''377''' }} (162c) and its [[extension]]s are of considerable interest, as this tempers out [[2048/2025]]. In the 7-limit, {{val| 162 257 '''377''' 455 }} tempers out [[126/125]] and 2048/2025 both, giving a tuning for 7-limit [[diaschismic]]. In the 11-limit {{val| 162 257 '''377''' 455 '''561''' }} (162ce) tempers out 126/125, [[176/175]] and [[896/891]], and so [[support]]s 11-limit diaschismic, and in fact has a fifth only 0.01 cents flatter than the [[POTE tuning]]. The 13-limit is even closer: {{val| 162 257 '''377''' 455 '''561''' '''600''' }} (162cef) tempers out 126/125, 176/175, [[196/195]], [[364/363]] giving 13-limit diaschismic, and the fifth of 95\162 is a mere 0.0000383 cents sharp of the 13-limit POTE tuning.
&lt;br /&gt;
 
The non-patent val &amp;lt;162 257 377| and its extensions are of considerable interest, as this tempers out 2048/2025. In the 7-limit, &amp;lt;162 257 377 455| tempers out 126/125 and 2048/2025 both, giving a tuning for 7-limit diaschismic. In the 11-limit &amp;lt;162 257 377 455 561| tempers out 126/125, 176/175 and 896/891, and so supports 11-limit diaschismic, and in fact has a fifth only 0.01 cents flatter than the &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt;. The 13-limit is even closer: the 13-limit val &amp;lt;162 257 377 455 561 600| tempers out 126/125, 196/195, 364/363, 2048/2025 giving 13-limit diaschismic, and the 162 fifth of 95/162 octave is a mere 0.0000383 cents sharp of the 13-limit POTE tuning.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Prime harmonics ===
{{Harmonics in equal|162}}
 
=== Subsets and supersets ===
Since 162 factors into {{factorization|162}}, 162edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 27, 54, and 81 }}.

Latest revision as of 18:03, 19 February 2025

← 161edo 162edo 163edo →
Prime factorization 2 × 34
Step size 7.40741 ¢ 
Fifth 95\162 (703.704 ¢)
Semitones (A1:m2) 17:11 (125.9 ¢ : 81.48 ¢)
Consistency limit 7
Distinct consistency limit 7

162 equal divisions of the octave (abbreviated 162edo or 162ed2), also called 162-tone equal temperament (162tet) or 162 equal temperament (162et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 162 equal parts of about 7.41 ¢ each. Each step represents a frequency ratio of 21/162, or the 162nd root of 2.

Using the patent val, the equal temperament tempers out 4000/3969, 10976/10935 and 65625/65536.

The non-patent val ⟨162 257 377] (162c) and its extensions are of considerable interest, as this tempers out 2048/2025. In the 7-limit, ⟨162 257 377 455] tempers out 126/125 and 2048/2025 both, giving a tuning for 7-limit diaschismic. In the 11-limit ⟨162 257 377 455 561] (162ce) tempers out 126/125, 176/175 and 896/891, and so supports 11-limit diaschismic, and in fact has a fifth only 0.01 cents flatter than the POTE tuning. The 13-limit is even closer: ⟨162 257 377 455 561 600] (162cef) tempers out 126/125, 176/175, 196/195, 364/363 giving 13-limit diaschismic, and the fifth of 95\162 is a mere 0.0000383 cents sharp of the 13-limit POTE tuning.

Prime harmonics

Approximation of prime harmonics in 162edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.75 -1.13 +1.54 -3.17 -3.49 -1.25 -1.22 +1.36 +0.05 +3.11
Relative (%) +0.0 +23.6 -15.2 +20.9 -42.8 -47.1 -16.9 -16.4 +18.3 +0.7 +42.0
Steps
(reduced) 		162
(0) 		257
(95) 		376
(52) 		455
(131) 		560
(74) 		599
(113) 		662
(14) 		688
(40) 		733
(85) 		787
(139) 		803
(155)

Subsets and supersets

Since 162 factors into 2 × 34, 162edo has subset edos 2, 3, 6, 9, 18, 27, 54, and 81.

Retrieved from "https://en.xen.wiki/index.php?title=162edo&oldid=181460"