18edo: Difference between revisions

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=18 Equal Divisions of the Octave= 
AKA The Third-Tone System

==Basic Properties== 
18-EDO 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, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6). 

In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach, meaning to avoid the usual closed-voice "root-3rd-5th" type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of 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.

18-EDO contains sub-EDOs 2, 3, 6, and 9, and itself is half of 36-EDO and one-fourth of 72-EDO. It bears some similarities to 13-EDO (with its very flat 4ths and nice subminor 3rds), 11-EDO (with its very sharp minor 3rds, two of which span a very flat 5th), 16-EDO (with its sharp 4ths and flat 5ths), and 17-EDO and 19-EDO (with its narrow semitone, three of which comprise a whole-tone). It is an excellent tuning for those seeking a forceful deviation from the common practice.

===Representations of Just Intervals=== 
|| Degree || Cents || Nearest Ratio || Error (cents) || 17-Limit Ratios* ||
|| 0 || 0 || 1/1 || 0 || **1/1** ||
|| 1 || 66.667 || 27/26 || +1.329 ||> 78/75, 75/72 ||
|| 2 || 133.333 || 27/25 || +0.096 ||> 51/55, 42/39 ||
|| 3 || 200 || 9/8 || -3.910 || **9/8** ||
|| 4 || 266.667 || 7/6 || -0.204 || **75/64** ||
|| 5 || 333.333 || 17/14 or 40/33 || -2.796 +0.293 || **39/32** ||
|| 6 || 400 || 5/4 or 44/35 || +13.686 +3.822 ||> 64/55 ||
|| 7 || 466.667 || 21/16 || -4.114 || **21/16** ||
|| 8 || 533.333 || 15/11 || -3.617 ||> 102/75 ||
|| 9 || 600 || 17/12 or 24/17 || -3.000 +3.000 ||> 17/12 ||
|| 10 || 666.667 || 22/15 || +3.617 ||> 75/51 ||
|| 11 || 733.333 || 32/21 || +4.114 ||> 32/21 ||
|| 12 || 800 || 8/5 or 35/22 || -13.686 -3.8222 || **51/32** ||
|| 13 || 866.667 || 28/17 or 33/20 || +2.796 -0.293 ||> 64/39 ||
|| 14 || 933.333 || 12/7 || +0.204 || **55/32** ||
|| 15 || 1000 || 16/9 || +3.910 ||> 16/9 ||
|| 16 || 1066.667 || 50/27 || -0.096 ||> 39/21 ||
|| 17 || 1133.333 || 52/27 || -1.329 ||> 75/39 ||
|| 18 || 1200 || 2/1 || 0 || **2/1** ||
*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament

==<span style="font-size: 1.3em;">Useful Moment-of-Symmetry Scales</span>== 
Note: This list excludes scales found in 9-EDO.
===<span style="font-size: 1.1em;">Pentatonic:</span>=== 
3L2s Father Pentatonic: 4 4 3 4 3
===<span style="font-size: 1.1em;">Hexatonic:</span>=== 
6-Equal Whole-Tone Scale: 3 3 3 3 3 3
4L2s Bicycle: 4 4 1 4 4 1
2L4s Rice Hexatonic: 2 5 2 2 5 2
===<span style="font-size: 1.1em;">Heptatonic:</span>=== 
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
===<span style="font-size: 1.1em;">Octatonic:</span>=== 
5L3s Father Octatonic: 3 1 3 3 1 3 3 1
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
===<span style="font-size: 1.1em;">Decatonic:</span>=== 
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2

==<span style="font-size: 1.3em;">Application to Guitar</span>== 
18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!

The "Father Octatonic" scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).

==Commas== 
18 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] < 18 29 42 51 62 67 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
||= 128/125 || | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||
||= 1212717/1210381 || | 23 6 -14 > ||> 3.34 ||= Vishnuzma ||= Semisuper ||
||= 50/49 || | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||
||= 686/675 || | 1 -3 -2 3 > ||> 27.99 ||= Senga ||=   ||
||= 875/864 || | -5 -3 3 1 > ||> 21.90 ||= Keema ||=   ||
||= 1728/1715 || | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||
||= 16875/16807 || | 0 3 4 -5 > ||> 6.99 ||= Mirkwai ||=   ||
||= 3136/3125 || | 6 0 -5 2 > ||> 6.08 ||= Hemimean ||=   ||
||= 99/98 || | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||=   ||
||= 100/99 || | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||=   ||
||= 385/384 || | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0.18 ||= Kalisma ||= Gauss' Comma ||
||= 91/90 || | -1 -2 -1 1 1 > ||> 19.13 ||= Superleap ||=   ||

==Listen== 
* [[http://www.h-pi.com/mp3/18ETPrelude.mp3|18ETPrelude]] by [[Aaron Andrew Hunt]]
* [[http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3|Prelude in 18et]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] => [[@http://chrisvaisvil.com/?p=3|composer notes]]
* [[http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3|Flippertronics]] by Chris Vaisvil

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>
 AKA The Third-Tone System<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basic Properties</h2>
 18-EDO 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, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6). <br />
<br />
In order to access the excellent consonances actually available, one must take a considerably &quot;non-common-practice&quot; approach, meaning to avoid the usual closed-voice &quot;root-3rd-5th&quot; type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of 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 />
18-EDO contains sub-EDOs 2, 3, 6, and 9, and itself is half of 36-EDO and one-fourth of 72-EDO. It bears some similarities to 13-EDO (with its very flat 4ths and nice subminor 3rds), 11-EDO (with its very sharp minor 3rds, two of which span a very flat 5th), 16-EDO (with its sharp 4ths and flat 5ths), and 17-EDO and 19-EDO (with its narrow semitone, three of which comprise a whole-tone). It is an excellent tuning for those seeking a forceful deviation from the common practice.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x18 Equal Divisions of the Octave-Basic Properties-Representations of Just Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Representations of Just Intervals</h3>
 

<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents<br />
</td>
        <td>Nearest Ratio<br />
</td>
        <td>Error (cents)<br />
</td>
        <td>17-Limit Ratios*<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>1/1<br />
</td>
        <td>0<br />
</td>
        <td><strong>1/1</strong><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>66.667<br />
</td>
        <td>27/26<br />
</td>
        <td>+1.329<br />
</td>
        <td style="text-align: right;">78/75, 75/72<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>133.333<br />
</td>
        <td>27/25<br />
</td>
        <td>+0.096<br />
</td>
        <td style="text-align: right;">51/55, 42/39<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>200<br />
</td>
        <td>9/8<br />
</td>
        <td>-3.910<br />
</td>
        <td><strong>9/8</strong><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>266.667<br />
</td>
        <td>7/6<br />
</td>
        <td>-0.204<br />
</td>
        <td><strong>75/64</strong><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>
        <td><strong>39/32</strong><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>
        <td style="text-align: right;">64/55<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>466.667<br />
</td>
        <td>21/16<br />
</td>
        <td>-4.114<br />
</td>
        <td><strong>21/16</strong><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>533.333<br />
</td>
        <td>15/11<br />
</td>
        <td>-3.617<br />
</td>
        <td style="text-align: right;">102/75<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>
        <td style="text-align: right;">17/12<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>666.667<br />
</td>
        <td>22/15<br />
</td>
        <td>+3.617<br />
</td>
        <td style="text-align: right;">75/51<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>733.333<br />
</td>
        <td>32/21<br />
</td>
        <td>+4.114<br />
</td>
        <td style="text-align: right;">32/21<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>
        <td><strong>51/32</strong><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>
        <td style="text-align: right;">64/39<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>933.333<br />
</td>
        <td>12/7<br />
</td>
        <td>+0.204<br />
</td>
        <td><strong>55/32</strong><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>1000<br />
</td>
        <td>16/9<br />
</td>
        <td>+3.910<br />
</td>
        <td style="text-align: right;">16/9<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>1066.667<br />
</td>
        <td>50/27<br />
</td>
        <td>-0.096<br />
</td>
        <td style="text-align: right;">39/21<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>1133.333<br />
</td>
        <td>52/27<br />
</td>
        <td>-1.329<br />
</td>
        <td style="text-align: right;">75/39<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>1200<br />
</td>
        <td>2/1<br />
</td>
        <td>0<br />
</td>
        <td><strong>2/1</strong><br />
</td>
    </tr>
</table>

*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="font-size: 1.3em;">Useful Moment-of-Symmetry Scales</span></h2>
 Note: This list excludes scales found in 9-EDO.<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonic:"></a><!-- ws:end:WikiTextHeadingRule:8 --><span style="font-size: 1.1em;">Pentatonic:</span></h3>
 3L2s Father Pentatonic: 4 4 3 4 3<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="font-size: 1.1em;">Hexatonic:</span></h3>
 6-Equal Whole-Tone Scale: 3 3 3 3 3 3<br />
4L2s Bicycle: 4 4 1 4 4 1<br />
2L4s Rice Hexatonic: 2 5 2 2 5 2<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonic:"></a><!-- ws:end:WikiTextHeadingRule:12 --><span style="font-size: 1.1em;">Heptatonic:</span></h3>
 4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonic:"></a><!-- ws:end:WikiTextHeadingRule:14 --><span style="font-size: 1.1em;">Octatonic:</span></h3>
 5L3s Father Octatonic: 3 1 3 3 1 3 3 1<br />
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonic:"></a><!-- ws:end:WikiTextHeadingRule:16 --><span style="font-size: 1.1em;">Decatonic:</span></h3>
 8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="x18 Equal Divisions of the Octave-Application to Guitar"></a><!-- ws:end:WikiTextHeadingRule:18 --><span style="font-size: 1.3em;">Application to Guitar</span></h2>
 18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!<br />
<br />
The &quot;Father Octatonic&quot; scale maps very simply to a 6-string guitar tuned in &quot;reverse-standard&quot; tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x18 Equal Divisions of the Octave-Commas"></a><!-- ws:end:WikiTextHeadingRule:20 -->Commas</h2>
 18 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the <a class="wiki_link" href="/val">val</a> &lt; 18 29 42 51 62 67 |.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td>| 7 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">41.06<br />
</td>
        <td style="text-align: center;">Diesis<br />
</td>
        <td style="text-align: center;">Augmented Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1212717/1210381<br />
</td>
        <td>| 23 6 -14 &gt;<br />
</td>
        <td style="text-align: right;">3.34<br />
</td>
        <td style="text-align: center;">Vishnuzma<br />
</td>
        <td style="text-align: center;">Semisuper<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50/49<br />
</td>
        <td>| 1 0 2 -2 &gt;<br />
</td>
        <td style="text-align: right;">34.98<br />
</td>
        <td style="text-align: center;">Tritonic Diesis<br />
</td>
        <td style="text-align: center;">Jubilisma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">686/675<br />
</td>
        <td>| 1 -3 -2 3 &gt;<br />
</td>
        <td style="text-align: right;">27.99<br />
</td>
        <td style="text-align: center;">Senga<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">875/864<br />
</td>
        <td>| -5 -3 3 1 &gt;<br />
</td>
        <td style="text-align: right;">21.90<br />
</td>
        <td style="text-align: center;">Keema<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1728/1715<br />
</td>
        <td>| 6 3 -1 -3 &gt;<br />
</td>
        <td style="text-align: right;">13.07<br />
</td>
        <td style="text-align: center;">Orwellisma<br />
</td>
        <td style="text-align: center;">Orwell Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16875/16807<br />
</td>
        <td>| 0 3 4 -5 &gt;<br />
</td>
        <td style="text-align: right;">6.99<br />
</td>
        <td style="text-align: center;">Mirkwai<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3136/3125<br />
</td>
        <td>| 6 0 -5 2 &gt;<br />
</td>
        <td style="text-align: right;">6.08<br />
</td>
        <td style="text-align: center;">Hemimean<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">99/98<br />
</td>
        <td>| -1 2 0 -2 1 &gt;<br />
</td>
        <td style="text-align: right;">17.58<br />
</td>
        <td style="text-align: center;">Mothwellsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">100/99<br />
</td>
        <td>| 2 -2 2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">17.40<br />
</td>
        <td style="text-align: center;">Ptolemisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65536/65219<br />
</td>
        <td>| 16 0 0 -2 -3 &gt;<br />
</td>
        <td style="text-align: right;">8.39<br />
</td>
        <td style="text-align: center;">Orgonisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td>| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">4.50<br />
</td>
        <td style="text-align: center;">Keenanisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9801/9800<br />
</td>
        <td>| -3 4 -2 -2 2 &gt;<br />
</td>
        <td style="text-align: right;">0.18<br />
</td>
        <td style="text-align: center;">Kalisma<br />
</td>
        <td style="text-align: center;">Gauss' Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td>| -1 -2 -1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">19.13<br />
</td>
        <td style="text-align: center;">Superleap<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x18 Equal Divisions of the Octave-Listen"></a><!-- ws:end:WikiTextHeadingRule:22 -->Listen</h2>
 <ul><li><a class="wiki_link_ext" href="http://www.h-pi.com/mp3/18ETPrelude.mp3" rel="nofollow">18ETPrelude</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3" rel="nofollow">Prelude in 18et</a> by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank">Chris Vaisvil</a> =&gt; <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=3" rel="nofollow" target="_blank">composer notes</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3" rel="nofollow">Flippertronics</a> by Chris Vaisvil</li></ul></body></html>

@@ 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:xenwolf|xenwolf]] and made on <tt>2011-07-01 11:07:02 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-19 01:23:21 UTC</tt>.<br>
-: The original revision id was <tt>239656913</tt>.<br>
+: The original revision id was <tt>241890068</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 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.
+-EDO 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, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6).
-===Relationship to Other EDOs===
+In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach, meaning to avoid the usual closed-voice "root-3rd-5th" type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of 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.
--EDO, aka the "third-tone" system, is related to [[12edo|12-tET]] by the whole-tone scale (which is [[6edo|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 [[9edo|9-EDO]], offset from each other by a third-tone. 18-EDO is related to [[13edo|13-EDO]], [[21edo|21-EDO]], [[23edo|23-EDO]], and [[28edo|28-EDO]] in that all are [[Father Temperament|"Father" temperaments]] (they temper out 16/15--the difference between a major third and perfect fourth). It is related to [[11edo|11-EDO]], [[15edo|15-EDO]], [[25edo|25-EDO]], and 29-EDO in that they are all [[Amity Temperament|"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==
+-EDO contains sub-EDOs 2, 3, 6, and 9, and itself is half of 36-EDO and one-fourth of 72-EDO. It bears some similarities to 13-EDO (with its very flat 4ths and nice subminor 3rds), 11-EDO (with its very sharp minor 3rds, two of which span a very flat 5th), 16-EDO (with its sharp 4ths and flat 5ths), and 17-EDO and 19-EDO (with its narrow semitone, three of which comprise a whole-tone). It is an excellent tuning for those seeking a forceful deviation from the common practice.
+===Representations of Just Intervals===
+|| Degree || Cents || Nearest Ratio || Error (cents) || 17-Limit Ratios* ||
+|| 0 || 0 || 1/1 || 0 || **1/1** ||
+|| 1 || 66.667 || 27/26 || +1.329 ||&gt; 78/75, 75/72 ||
+|| 2 || 133.333 || 27/25 || +0.096 ||&gt; 51/55, 42/39 ||
+|| 3 || 200 || 9/8 || -3.910 || **9/8** ||
+|| 4 || 266.667 || 7/6 || -0.204 || **75/64** ||
+|| 5 || 333.333 || 17/14 or 40/33 || -2.796 +0.293 || **39/32** ||
+|| 6 || 400 || 5/4 or 44/35 || +13.686 +3.822 ||&gt; 64/55 ||
+|| 7 || 466.667 || 21/16 || -4.114 || **21/16** ||
+|| 8 || 533.333 || 15/11 || -3.617 ||&gt; 102/75 ||
+|| 9 || 600 || 17/12 or 24/17 || -3.000 +3.000 ||&gt; 17/12 ||
+|| 10 || 666.667 || 22/15 || +3.617 ||&gt; 75/51 ||
+|| 11 || 733.333 || 32/21 || +4.114 ||&gt; 32/21 ||
+|| 12 || 800 || 8/5 or 35/22 || -13.686 -3.8222 || **51/32** ||
+|| 13 || 866.667 || 28/17 or 33/20 || +2.796 -0.293 ||&gt; 64/39 ||
+|| 14 || 933.333 || 12/7 || +0.204 || **55/32** ||
+|| 15 || 1000 || 16/9 || +3.910 ||&gt; 16/9 ||
+|| 16 || 1066.667 || 50/27 || -0.096 ||&gt; 39/21 ||
+|| 17 || 1133.333 || 52/27 || -1.329 ||&gt; 75/39 ||
+|| 18 || 1200 || 2/1 || 0 || **2/1** ||
+*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament
+==&lt;span style="font-size: 1.3em;"&gt;Useful Moment-of-Symmetry Scales&lt;/span&gt;==
 Note: This list excludes scales found in 9-EDO.
-===Pentatonic:===
+===&lt;span style="font-size: 1.1em;"&gt;Pentatonic:&lt;/span&gt;===
-Father Pentatonic: 4 4 3 4 3
+L2s Father Pentatonic: 4 4 3 4 3
-===Hexatonic:===
+===&lt;span style="font-size: 1.1em;"&gt;Hexatonic:&lt;/span&gt;===
-Whole-Tone Scale: 3 3 3 3 3 3
+-Equal Whole-Tone Scale: 3 3 3 3 3 3
-Bicycle: 4 4 1 4 4 1
+L2s Bicycle: 4 4 1 4 4 1
-Rice Hexatonic: 2 5 2 2 5 2
+L4s Rice Hexatonic: 2 5 2 2 5 2
-===Heptatonic:===
+===&lt;span style="font-size: 1.1em;"&gt;Heptatonic:&lt;/span&gt;===
-Amity/Mish Heptatonic: 3 2 3 2 3 3 2
+L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
-===Octatonic:===
+===&lt;span style="font-size: 1.1em;"&gt;Octatonic:&lt;/span&gt;===
-Father Octatonic: 3 1 3 3 1 3 3 1
+L3s Father Octatonic: 3 1 3 3 1 3 3 1
-Rice Octatonic: 2 2 3 2 2 2 3 2
+L6s Rice Octatonic: 2 2 3 2 2 2 3 2
-===Decatonic:===
+===&lt;span style="font-size: 1.1em;"&gt;Decatonic:&lt;/span&gt;===
-Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
+L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
-==Application to Guitar==
+==&lt;span style="font-size: 1.3em;"&gt;Application to Guitar&lt;/span&gt;==
 -EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!
-The "Father Octatonic" scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21.
+The "Father Octatonic" scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).
-===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 ||
 ==Commas==
@@ Line 85: / Line 87: @@
 &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%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;
+-EDO 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, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6). &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;
+In order to access the excellent consonances actually available, one must take a considerably &amp;quot;non-common-practice&amp;quot; approach, meaning to avoid the usual closed-voice &amp;quot;root-3rd-5th&amp;quot; type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of 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;
--EDO, aka the &amp;quot;third-tone&amp;quot; system, is related to &lt;a class="wiki_link" href="/12edo"&gt;12-tET&lt;/a&gt; by the whole-tone scale (which is &lt;a class="wiki_link" href="/6edo"&gt;6-EDO&lt;/a&gt;), 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 &lt;a class="wiki_link" href="/9edo"&gt;9-EDO&lt;/a&gt;, offset from each other by a third-tone. 18-EDO is related to &lt;a class="wiki_link" href="/13edo"&gt;13-EDO&lt;/a&gt;, &lt;a class="wiki_link" href="/21edo"&gt;21-EDO&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23-EDO&lt;/a&gt;, and &lt;a class="wiki_link" href="/28edo"&gt;28-EDO&lt;/a&gt; in that all are &lt;a class="wiki_link" href="/Father%20Temperament"&gt;&amp;quot;Father&amp;quot; temperaments&lt;/a&gt; (they temper out 16/15--the difference between a major third and perfect fourth). It is related to &lt;a class="wiki_link" href="/11edo"&gt;11-EDO&lt;/a&gt;, &lt;a class="wiki_link" href="/15edo"&gt;15-EDO&lt;/a&gt;, &lt;a class="wiki_link" href="/25edo"&gt;25-EDO&lt;/a&gt;, and 29-EDO in that they are all &lt;a class="wiki_link" href="/Amity%20Temperament"&gt;&amp;quot;Amity&amp;quot; temperaments&lt;/a&gt; (&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;
+-EDO contains sub-EDOs 2, 3, 6, and 9, and itself is half of 36-EDO and one-fourth of 72-EDO. It bears some similarities to 13-EDO (with its very flat 4ths and nice subminor 3rds), 11-EDO (with its very sharp minor 3rds, two of which span a very flat 5th), 16-EDO (with its sharp 4ths and flat 5ths), and 17-EDO and 19-EDO (with its narrow semitone, three of which comprise a whole-tone). It is an excellent tuning for those seeking a forceful deviation from the common practice.&lt;br /&gt;
- Note: This list excludes scales found in 9-EDO.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Pentatonic:&lt;/h3&gt;
- Father Pentatonic: 4 4 3 4 3&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Hexatonic:&lt;/h3&gt;
- Whole-Tone Scale: 3 3 3 3 3 3&lt;br /&gt;
-Bicycle: 4 4 1 4 4 1&lt;br /&gt;
-Rice Hexatonic: 2 5 2 2 5 2&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Heptatonic:&lt;/h3&gt;
- Amity/Mish Heptatonic: 3 2 3 2 3 3 2&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Octatonic:&lt;/h3&gt;
- Father Octatonic: 3 1 3 3 1 3 3 1&lt;br /&gt;
-Rice Octatonic: 2 2 3 2 2 2 3 2&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Decatonic:&lt;/h3&gt;
- Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="x18 Equal Divisions of the Octave-Application to Guitar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Application to Guitar&lt;/h2&gt;
--EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!&lt;br /&gt;
 &lt;br /&gt;
-The &amp;quot;Father Octatonic&amp;quot; scale maps very simply to a 6-string guitar tuned in &amp;quot;reverse-standard&amp;quot; tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21.&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-Representations of Just Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Representations of Just Intervals&lt;/h3&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x18 Equal Divisions of the Octave-Application to Guitar-Representations of Just Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Representations of Just Intervals&lt;/h3&gt;
@@ Line 123: / Line 105: @@
 &lt;/td&gt;
          &lt;td&gt;Error (cents)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;17-Limit Ratios*&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 133: / Line 117: @@
 &lt;/td&gt;
          &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;1/1&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 143: / Line 129: @@
 &lt;/td&gt;
          &lt;td&gt;+1.329&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;78/75, 75/72&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 153: / Line 141: @@
 &lt;/td&gt;
          &lt;td&gt;+0.096&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;51/55, 42/39&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 163: / Line 153: @@
 &lt;/td&gt;
          &lt;td&gt;-3.910&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;9/8&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 173: / Line 165: @@
 &lt;/td&gt;
          &lt;td&gt;-0.204&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;75/64&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 183: / Line 177: @@
 &lt;/td&gt;
          &lt;td&gt;-2.796 +0.293&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;39/32&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 193: / Line 189: @@
 &lt;/td&gt;
          &lt;td&gt;+13.686 +3.822&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;64/55&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 203: / Line 201: @@
 &lt;/td&gt;
          &lt;td&gt;-4.114&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;21/16&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 213: / Line 213: @@
 &lt;/td&gt;
          &lt;td&gt;-3.617&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;102/75&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 223: / Line 225: @@
 &lt;/td&gt;
          &lt;td&gt;-3.000 +3.000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;17/12&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 233: / Line 237: @@
 &lt;/td&gt;
          &lt;td&gt;+3.617&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;75/51&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 243: / Line 249: @@
 &lt;/td&gt;
          &lt;td&gt;+4.114&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;32/21&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 253: / Line 261: @@
 &lt;/td&gt;
          &lt;td&gt;-13.686 -3.8222&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;51/32&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 263: / Line 273: @@
 &lt;/td&gt;
          &lt;td&gt;+2.796 -0.293&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;64/39&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 273: / Line 285: @@
 &lt;/td&gt;
          &lt;td&gt;+0.204&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;55/32&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 283: / Line 297: @@
 &lt;/td&gt;
          &lt;td&gt;+3.910&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;16/9&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 293: / Line 309: @@
 &lt;/td&gt;
          &lt;td&gt;-0.096&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;39/21&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 303: / Line 321: @@
 &lt;/td&gt;
          &lt;td&gt;-1.329&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: right;"&gt;75/39&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 313: / Line 333: @@
 &lt;/td&gt;
          &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;2/1&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
 &lt;/table&gt;
+*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament&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;&lt;span style="font-size: 1.3em;"&gt;Useful Moment-of-Symmetry Scales&lt;/span&gt;&lt;/h2&gt;
+ Note: This list excludes scales found in 9-EDO.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;span style="font-size: 1.1em;"&gt;Pentatonic:&lt;/span&gt;&lt;/h3&gt;
+L2s Father Pentatonic: 4 4 3 4 3&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;span style="font-size: 1.1em;"&gt;Hexatonic:&lt;/span&gt;&lt;/h3&gt;
+-Equal Whole-Tone Scale: 3 3 3 3 3 3&lt;br /&gt;
+L2s Bicycle: 4 4 1 4 4 1&lt;br /&gt;
+L4s Rice Hexatonic: 2 5 2 2 5 2&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;span style="font-size: 1.1em;"&gt;Heptatonic:&lt;/span&gt;&lt;/h3&gt;
+L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;span style="font-size: 1.1em;"&gt;Octatonic:&lt;/span&gt;&lt;/h3&gt;
+L3s Father Octatonic: 3 1 3 3 1 3 3 1&lt;br /&gt;
+L6s Rice Octatonic: 2 2 3 2 2 2 3 2&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonic:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;span style="font-size: 1.1em;"&gt;Decatonic:&lt;/span&gt;&lt;/h3&gt;
+L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="x18 Equal Divisions of the Octave-Application to Guitar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;span style="font-size: 1.3em;"&gt;Application to Guitar&lt;/span&gt;&lt;/h2&gt;
+-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!&lt;br /&gt;
+&lt;br /&gt;
+The &amp;quot;Father Octatonic&amp;quot; scale maps very simply to a 6-string guitar tuned in &amp;quot;reverse-standard&amp;quot; tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x18 Equal Divisions of the Octave-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x18 Equal Divisions of the Octave-Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Commas&lt;/h2&gt;
 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 18 29 42 51 62 67 |.)&lt;br /&gt;
@@ Line 506: / Line 550: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="x18 Equal Divisions of the Octave-Listen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Listen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x18 Equal Divisions of the Octave-Listen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Listen&lt;/h2&gt;
   &lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/18ETPrelude.mp3" rel="nofollow"&gt;18ETPrelude&lt;/a&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3" rel="nofollow"&gt;Prelude in 18et&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&gt;Chris Vaisvil&lt;/a&gt; =&amp;gt; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=3" rel="nofollow" target="_blank"&gt;composer notes&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3" rel="nofollow"&gt;Flippertronics&lt;/a&gt; by Chris Vaisvil&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

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