18edo
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=18 Equal Divisions of the Octave=
AKA The Third-Tone System
==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 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===
18-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==
Note: This list excludes scales found in 9-EDO.
===Pentatonic:===
Father Pentatonic: 4 4 3 4 3
===Hexatonic:===
Whole-Tone Scale: 3 3 3 3 3 3
Bicycle: 4 4 1 4 4 1
Rice Hexatonic: 2 5 2 2 5 2
===Heptatonic:===
Amity/Mish Heptatonic: 3 2 3 2 3 3 2
===Octatonic:===
Father Octatonic: 3 1 3 3 1 3 3 1
Rice Octatonic: 2 2 3 2 2 2 3 2
===Decatonic:===
Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
==Application to Guitar==
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.
===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 ||
==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 VaisvilOriginal HTML content:
<html><head><title>18edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><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:<h2> --><h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->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 >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 <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 />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x18 Equal Divisions of the Octave-Basic Properties-Relationship to Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:4 -->Relationship to Other EDOs</h3>
18-EDO, aka the "third-tone" system, is related to <a class="wiki_link" href="/12edo">12-tET</a> by the whole-tone scale (which is <a class="wiki_link" href="/6edo">6-EDO</a>), 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 <a class="wiki_link" href="/9edo">9-EDO</a>, offset from each other by a third-tone. 18-EDO is related to <a class="wiki_link" href="/13edo">13-EDO</a>, <a class="wiki_link" href="/21edo">21-EDO</a>, <a class="wiki_link" href="/23edo">23-EDO</a>, and <a class="wiki_link" href="/28edo">28-EDO</a> in that all are <a class="wiki_link" href="/Father%20Temperament">"Father" temperaments</a> (they temper out 16/15--the difference between a major third and perfect fourth). It is related to <a class="wiki_link" href="/11edo">11-EDO</a>, <a class="wiki_link" href="/15edo">15-EDO</a>, <a class="wiki_link" href="/25edo">25-EDO</a>, and 29-EDO in that they are all <a class="wiki_link" href="/Amity%20Temperament">"Amity" temperaments</a> ("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).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales"></a><!-- ws:end:WikiTextHeadingRule:6 -->Useful Moment-of-Symmetry Scales</h2>
Note: This list excludes scales found in 9-EDO.<br />
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonic:"></a><!-- ws:end:WikiTextHeadingRule:8 -->Pentatonic:</h3>
Father Pentatonic: 4 4 3 4 3<br />
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 -->Hexatonic:</h3>
Whole-Tone Scale: 3 3 3 3 3 3<br />
Bicycle: 4 4 1 4 4 1<br />
Rice Hexatonic: 2 5 2 2 5 2<br />
<!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonic:"></a><!-- ws:end:WikiTextHeadingRule:12 -->Heptatonic:</h3>
Amity/Mish Heptatonic: 3 2 3 2 3 3 2<br />
<!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonic:"></a><!-- ws:end:WikiTextHeadingRule:14 -->Octatonic:</h3>
Father Octatonic: 3 1 3 3 1 3 3 1<br />
Rice Octatonic: 2 2 3 2 2 2 3 2<br />
<!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonic:"></a><!-- ws:end:WikiTextHeadingRule:16 -->Decatonic:</h3>
Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="x18 Equal Divisions of the Octave-Application to Guitar"></a><!-- ws:end:WikiTextHeadingRule:18 -->Application to Guitar</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 "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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="x18 Equal Divisions of the Octave-Application to Guitar-Representations of Just Intervals"></a><!-- ws:end:WikiTextHeadingRule:20 -->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>
</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>
<!-- ws:start:WikiTextHeadingRule:22:<h2> --><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> => <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>