18edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 22:44:54 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 22:47:01 UTC</tt>.<br>

: The original revision id was <tt>~~232646824~~</tt>.<br>

: The original revision id was <tt>232647018</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>

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==Basic Properties==

The //18 equal division// divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a >30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach centering on the chords in the 17-limit [[k*N subgroups|4*18 subgroup]] [[just intonation subgroup]] 2.9.75.21.55.39.51 . On this subgroup it tempers out exactly the same commas as 72 does on the full [[17-limit]], and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.

The //18 equal division// divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a >30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach centering on the chords in the 17-limit [[k*N subgroups|4*18 subgroup]] [[Just intonation subgroups|just intonation subgroup]] 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full [[17-limit]], and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.

===Relationship to Other EDOs===

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<br />

<h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a>Basic Properties</h2>

The <em>18 equal division</em> divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably &quot;non-common-practice&quot; approach centering on the chords in the 17-limit <a class="wiki_link" href="/k%2AN%20subgroups">4*18 subgroup</a> <a class="wiki_link" href="/~~just~~%20intonation%~~20subgroup~~">just intonation subgroup</a> 2.9.75.21.55.39.51 . On this subgroup it tempers out exactly the same commas as 72 does on the full <a class="wiki_link" href="/17-limit">17-limit</a>, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.<br />

The <em>18 equal division</em> divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably &quot;non-common-practice&quot; approach centering on the chords in the 17-limit <a class="wiki_link" href="/k%2AN%20subgroups">4*18 subgroup</a> <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full <a class="wiki_link" href="/17-limit">17-limit</a>, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.<br />

<br />

<h3 id="toc2"><a name="x18 Equal Divisions of the Octave-Basic Properties-Relationship to Other EDOs"></a>Relationship to Other EDOs</h3>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 22:44:54 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 22:47:01 UTC</tt>.<br>
-: The original revision id was <tt>232646824</tt>.<br>
+: The original revision id was <tt>232647018</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>
@@ Line 10: / Line 10: @@
 ==Basic Properties==
-The //18 equal division// divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach centering on the chords in the 17-limit [[k*N subgroups|4*18 subgroup]] [[just intonation subgroup]] 2.9.75.21.55.39.51 . On this subgroup it tempers out exactly the same commas as 72 does on the full [[17-limit]], and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.
+The //18 equal division// divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach centering on the chords in the 17-limit [[k*N subgroups|4*18 subgroup]] [[Just intonation subgroups|just intonation subgroup]] 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full [[17-limit]], and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.
 ===Relationship to Other EDOs===
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x18 Equal Divisions of the Octave-Basic Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Basic Properties&lt;/h2&gt;
-  The &lt;em&gt;18 equal division&lt;/em&gt; divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &amp;gt;30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably &amp;quot;non-common-practice&amp;quot; approach centering on the chords in the 17-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;4*18 subgroup&lt;/a&gt; &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt; 2.9.75.21.55.39.51 . On this subgroup it tempers out exactly the same commas as 72 does on the full &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt;, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.&lt;br /&gt;
+  The &lt;em&gt;18 equal division&lt;/em&gt; divides the octave into 18 equal parts of 66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &amp;gt;30¢-error is considered acceptable. In order to access the excellent consonances actually available, one must take a considerably &amp;quot;non-common-practice&amp;quot; approach centering on the chords in the 17-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;4*18 subgroup&lt;/a&gt; &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intonation subgroup&lt;/a&gt; 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt;, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x18 Equal Divisions of the Octave-Basic Properties-Relationship to Other EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Relationship to Other EDOs&lt;/h3&gt;

18edo: Difference between revisions

Wikispaces>genewardsmith