18edo: Difference between revisions

Revision as of 22:08, 3 May 2010

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This revision was by author igliashon and made on 2010-05-03 22:08:56 UTC.

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Original Wikitext content:

=18 Equal Divisions of the Octave= 

==Basis== 
=== === 
===**Representations of Just Intervals**=== 
|| Degree || Cents || Nearest Ratio || Error (cents) ||
|| 0 || 0 || 1/1 || 0 ||
|| 1 || 66.667 || 27/26 || +1.329 ||
|| 2 || 133.333 || 27/25 || +0.096 ||
|| 3 || 200 || 9/8 || -3.910 ||
|| 4 || 266.667 || 7/6 || -0.204 ||
|| 5 || 333.333 || 17/14 or 40/33 || -2.796 +0.293 ||
|| 6 || 400 || 5/4 or 44/35 || +13.686 +3.822 ||
|| 7 || 466.667 || 21/16 || -4.114 ||
|| 8 || 533.333 || 15/11 || -3.617 ||
|| 9 || 600 || 17/12 or 24/17 || -3.000 +3.000 ||
|| 10 || 666.667 || 22/15 || +3.617 ||
|| 11 || 733.333 || 32/21 || +4.114 ||
|| 12 || 800 || 8/5 or 35/22 || -13.686 -3.8222 ||
|| 13 || 866.667 || 28/17 or 33/20 || +2.796 -0.293 ||
|| 14 || 933.333 || 12/7 || +0.204 ||
|| 15 || 1000 || 16/9 || +3.910 ||
|| 16 || 1066.667 || 50/27 || -0.096 ||
|| 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 
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).

Original HTML content:

<html><head><title>18edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x18 Equal Divisions of the Octave"></a><!-- ws:end:WikiTextHeadingRule:0 -->18 Equal Divisions of the Octave</h1>
 <br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basis"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basis</h2>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><!-- ws:end:WikiTextHeadingRule:4 --> </h3>
 <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><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>
 

<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents<br />
</td>
        <td>Nearest Ratio<br />
</td>
        <td>Error (cents)<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>1/1<br />
</td>
        <td>0<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>66.667<br />
</td>
        <td>27/26<br />
</td>
        <td>+1.329<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>133.333<br />
</td>
        <td>27/25<br />
</td>
        <td>+0.096<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>200<br />
</td>
        <td>9/8<br />
</td>
        <td>-3.910<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>266.667<br />
</td>
        <td>7/6<br />
</td>
        <td>-0.204<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>333.333<br />
</td>
        <td>17/14 or 40/33<br />
</td>
        <td>-2.796 +0.293<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>400<br />
</td>
        <td>5/4 or 44/35<br />
</td>
        <td>+13.686 +3.822<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>466.667<br />
</td>
        <td>21/16<br />
</td>
        <td>-4.114<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>533.333<br />
</td>
        <td>15/11<br />
</td>
        <td>-3.617<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>600<br />
</td>
        <td>17/12 or 24/17<br />
</td>
        <td>-3.000 +3.000<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>666.667<br />
</td>
        <td>22/15<br />
</td>
        <td>+3.617<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>733.333<br />
</td>
        <td>32/21<br />
</td>
        <td>+4.114<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>800<br />
</td>
        <td>8/5 or 35/22<br />
</td>
        <td>-13.686 -3.8222<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>866.667<br />
</td>
        <td>28/17 or 33/20<br />
</td>
        <td>+2.796 -0.293<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>933.333<br />
</td>
        <td>12/7<br />
</td>
        <td>+0.204<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>1000<br />
</td>
        <td>16/9<br />
</td>
        <td>+3.910<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>1066.667<br />
</td>
        <td>50/27<br />
</td>
        <td>-0.096<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>1133.333<br />
</td>
        <td>52/27<br />
</td>
        <td>-1.329<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>1200<br />
</td>
        <td>2/1<br />
</td>
        <td>0<br />
</td>
    </tr>
</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 />
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>

18edo: Difference between revisions

Revision as of 22:08, 3 May 2010

IMPORTED REVISION FROM WIKISPACES

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