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:igliashon|igliashon]] and made on <tt>2010-05-03 22:08:56 UTC</tt>.<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2010-05-03 22:13:17 UTC</tt>.<br>

: The original revision id was <tt>~~139203501~~</tt>.<br>

: The original revision id was <tt>139204613</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">=18 Equal Divisions of the Octave=

AKA The Third-Tone System

==~~Basis~~==

== ==

=== ===

===**Basic Properties**===

===**Representations of Just Intervals**~~===~~

<span style="font-size: 14px; line-height: 21px;">**Representations of Just Intervals**</span>

|| Degree || Cents || Nearest Ratio || Error (cents) ||

|| 0 || 0 || 1/1 || 0 ||

Line 31:

|| 17 || 1133.333 || 52/27 || -1.329 ||

|| 18 || 1200 || 2/1 || 0 ||

18-EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does

18-EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably "non-common-practice" approach.

offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably "non-common-practice" approach.

<span style="font-size: 14px; line-height: 21px;">**Relationship to Other EDOs** </span>

18-EDO, aka the "third-tone" system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET "whole tone" is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are "Father" temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all "Amity" temperaments ("Amity" is derived from the acronym of "Acute Minor Thirds", meaning a minor third sharper than 6/5 but still flatter than a neutral third).</pre></div>

18-EDO, aka the "third-tone" system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET "whole tone" is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are "Father" temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all "Amity" temperaments ("Amity" is derived from the acronym of "Acute Minor Thirds", meaning a minor third sharper than 6/5 but still flatter than a neutral third).

==Useful Moment-of-Symmetry Scales==

===Pentatonics:===

===Hexatonics:===

===Heptatonics:===

===Octatonics:===

===Enneatonics:===

===Decatonics:=== </pre></div>

<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"><html><head><title>18edo</title></head><body><h1 id="toc0"><a name="x18 Equal Divisions of the Octave"></a>18 Equal Divisions of the Octave</h1>

<br />

AKA The Third-Tone System<br />

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

<h2 id="toc1"> </h2>

<h3 id="toc2"><~~!-- ws:end:WikiTextHeadingRule:4~~ --> </h3>

<h3 id="toc2"><a name="x18 Equal Divisions of the Octave--Basic Properties"></a><strong>Basic Properties</strong></h3>

<~~h3 id="toc3"~~><~~a name="x18 Equal Divisions of the Octave-Basis-Representations of Just Intervals"~~></a><!-- ws:~~end:WikiTextHeadingRule:6 --~~><strong>Representations of Just Intervals</strong></h3>

<span style="font-size: 14px; line-height: 21px;"><strong>Representations of Just Intervals</strong></span><br />

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</table>

18-EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does ~~<br />~~

18-EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably &quot;non-common-practice&quot; approach.<br />

offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably &quot;non-common-practice&quot; approach.<br />

<br />

<span style="font-size: 14px; line-height: 21px;"><strong>Relationship to Other EDOs</strong> </span><br />

18-EDO, aka the &quot;third-tone&quot; system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET &quot;whole tone&quot; is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are &quot;Father&quot; temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all &quot;Amity&quot; temperaments (&quot;Amity&quot; is derived from the acronym of &quot;Acute Minor Thirds&quot;, meaning a minor third sharper than 6/5 but still flatter than a neutral third).</body></html></pre></div>

18-EDO, aka the &quot;third-tone&quot; system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET &quot;whole tone&quot; is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are &quot;Father&quot; temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all &quot;Amity&quot; temperaments (&quot;Amity&quot; is derived from the acronym of &quot;Acute Minor Thirds&quot;, meaning a minor third sharper than 6/5 but still flatter than a neutral third).<br />

<br />

<h2 id="toc3"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales"></a>Useful Moment-of-Symmetry Scales</h2>

<h3 id="toc4"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonics:"></a>Pentatonics:</h3>

<h3 id="toc5"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonics:"></a>Hexatonics:</h3>

<h3 id="toc6"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonics:"></a>Heptatonics:</h3>

<h3 id="toc7"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonics:"></a>Octatonics:</h3>

<h3 id="toc8"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Enneatonics:"></a>Enneatonics:</h3>

<h3 id="toc9"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonics:"></a>Decatonics:</h3>

</body></html></pre></div>

@@ 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:igliashon|igliashon]] and made on <tt>2010-05-03 22:08:56 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2010-05-03 22:13:17 UTC</tt>.<br>
-: The original revision id was <tt>139203501</tt>.<br>
+: The original revision id was <tt>139204613</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">=18 Equal Divisions of the Octave=
+AKA The Third-Tone System
-==Basis==
+== ==
-=== ===
+===**Basic Properties**===
-===**Representations of Just Intervals**===
+&lt;span style="font-size: 14px; line-height: 21px;"&gt;**Representations of Just Intervals**&lt;/span&gt;
 || Degree || Cents || Nearest Ratio || Error (cents) ||
 || 0 || 0 || 1/1 || 0 ||
@@ Line 31: / Line 31: @@
 || 17 || 1133.333 || 52/27 || -1.329 ||
 || 18 || 1200 || 2/1 || 0 ||
--EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does
+-EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably "non-common-practice" approach.
-offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably "non-common-practice" approach.
 &lt;span style="font-size: 14px; line-height: 21px;"&gt;**Relationship to Other EDOs** &lt;/span&gt;
--EDO, aka the "third-tone" system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET "whole tone" is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are "Father" temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all "Amity" temperaments ("Amity" is derived from the acronym of "Acute Minor Thirds", meaning a minor third sharper than 6/5 but still flatter than a neutral third).</pre></div>
+-EDO, aka the "third-tone" system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET "whole tone" is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are "Father" temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all "Amity" temperaments ("Amity" is derived from the acronym of "Acute Minor Thirds", meaning a minor third sharper than 6/5 but still flatter than a neutral third).
+==Useful Moment-of-Symmetry Scales==
+===Pentatonics:===
+===Hexatonics:===
+===Heptatonics:===
+===Octatonics:===
+===Enneatonics:===
+===Decatonics:=== </pre></div>
 <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;18edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x18 Equal Divisions of the Octave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;18 Equal Divisions of the Octave&lt;/h1&gt;
-  &lt;br /&gt;
+  AKA The Third-Tone System&lt;br /&gt;
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-  &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt; &lt;/h3&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"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;strong&gt;Basic Properties&lt;/strong&gt;&lt;/h3&gt;
- &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x18 Equal Divisions of the Octave-Basis-Representations of Just Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;strong&gt;Representations of Just Intervals&lt;/strong&gt;&lt;/h3&gt;
+ &lt;span style="font-size: 14px; line-height: 21px;"&gt;&lt;strong&gt;Representations of Just Intervals&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;table class="wiki_table"&gt;
@@ Line 247: / Line 254: @@
 &lt;/table&gt;
--EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does &lt;br /&gt;
+-EDO does not approximate the 3rd Harmonic at all, unless a 33.333¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably &amp;quot;non-common-practice&amp;quot; approach.&lt;br /&gt;
-offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably &amp;quot;non-common-practice&amp;quot; approach.&lt;br /&gt;
 &lt;br /&gt;
 &lt;span style="font-size: 14px; line-height: 21px;"&gt;&lt;strong&gt;Relationship to Other EDOs&lt;/strong&gt; &lt;/span&gt;&lt;br /&gt;
--EDO, aka the &amp;quot;third-tone&amp;quot; system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET &amp;quot;whole tone&amp;quot; is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are &amp;quot;Father&amp;quot; temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all &amp;quot;Amity&amp;quot; temperaments (&amp;quot;Amity&amp;quot; is derived from the acronym of &amp;quot;Acute Minor Thirds&amp;quot;, meaning a minor third sharper than 6/5 but still flatter than a neutral third).&lt;/body&gt;&lt;/html&gt;</pre></div>
+-EDO, aka the &amp;quot;third-tone&amp;quot; system, is related to 12-tET by the whole-tone scale (which is 6-EDO), since 18=6*3 and 12=6*2; hence a 12-tET &amp;quot;whole tone&amp;quot; is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of 9-EDO, offset from each other by a third-tone. 18-EDO is related to 13-EDO, 21-EDO, 23-EDO, and 28-EDO in that all are &amp;quot;Father&amp;quot; temperaments (they temper out 16/15--the difference between a major third and perfect fourth). It is related to 11-EDO, 15-EDO, 25-EDO, and 29-EDO in that they are all &amp;quot;Amity&amp;quot; temperaments (&amp;quot;Amity&amp;quot; is derived from the acronym of &amp;quot;Acute Minor Thirds&amp;quot;, meaning a minor third sharper than 6/5 but still flatter than a neutral third).&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Useful Moment-of-Symmetry Scales&lt;/h2&gt;
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+&lt;/body&gt;&lt;/html&gt;</pre></div>