18edo

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[[toc|flat]]
----
<span style="display: block; text-align: right;">[[18平均律|日本語]]
</span>
**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.

However, less accurate approximations can be used. 18 equal does temper out 28/27, which makes three "fifths" (ie. 3/2) up, a 7/4. Thus 9/8 = a near just 7/6 (and what the relatively accurate 200 cents as 9/8, is in fact 8/7 - what do you make of that? Music.) This treatment applies to the scale generated by the large fifth, known as Father. One, if one really gets into it, can generate scales from the 3/2 and the half octave: with all the sharpness, what's 18e going to hurt? Call 600 cents 11/8 and 866 cents 13/8. Hey it's possible, lots of people like mavila.

18-EDO contains sub-EDOs [[2edo|2]], [[3edo|3]], [[6edo|6]], and [[9edo|9]], and itself is half of [[36edo|36-EDO]] and one-fourth of [[72edo|72-EDO]]. It bears some similarities to [[13edo|13-EDO]] (with its very flat 4ths and nice subminor 3rds), [[11edo|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 
coarse/fine
DMS ||= 5L3s Notation ||||= [[Ups and Downs Notation|up/down ]]
[[Ups and Downs Notation|notation]] ** || Nearest Ratio || Error (cents 
coarse/fine)
(DMS) ||< 17-Limit Ratios* ||
|| 0 || 0 ||= **C** ||= C ||= P1 || 1/1 || 0 ||< **1/1** ||
|| 1 || 66.667
80
20° ||= Db ||= C^,
Dv ||= ^P1,
vM2 || 27/26 || +1.329
+1.595
+23'56" ||< 78/75, 75/72 ||
|| 2 || 133.333
160
40° ||= C# ||= D ||= M2 || 27/25 || +0.096
+0.115
+1'43" ||< 51/55, 42/39 ||
|| 3 || 200
240
60° ||= **D** ||= D^,
Ev ||= ~2 || 9/8 || -3.910
-4.692
-1°10'23" ||< **9/8** ||
|| 4 || 266.667
320
80° ||= Eb ||= E ||= m2,
M3 || 7/6 || -0.204
-0.245
-1"50" ||< **75/64** ||
|| 5 || 333.333
400
100° ||= D# ||= E^ ||= ~3 || 17/14 or 40/33 || -2.796 +0.293
-3.355 +0.351
-50'20" +5'16" ||< **39/32** ||
|| 6 || 400
480
120° ||= **E** ||= Eb,
F# ||= m3,
A4 || 5/4 or 44/35 || +13.686 +3.822
+16.4235 +4.586
+4°6'21" +1°8'47" ||< 64/55 ||
|| 7 || 466.667
560
140° ||= **F** ||= Fv ||= ^m3,
vP4 || 21/16 || -4.114
-4.937
-1°14'3" ||< **21/16** ||
|| 8 || 533.333
640
160° ||= Gb ||= F ||= P4 || 15/11 || -3.617
-4.341
-1°5'7" ||< 102/75 ||
|| 9 || 600
720
180° ||= F# ||= F^,
Gv ||= ^P4,
vP5 || 17/12 or 24/17 || -3.000 +3.000
-3.6005 +3.6005
-54' +54' ||< 17/12 ||
|| 10 || 666.667
800
200° ||= **G** ||= G ||= P5 || 22/15 || +3.617
+4.341
+1°5'7" ||< 75/51 ||
|| 11 || 733.333
880
220° ||= Hb ||= G^ ||= ^P5,
vM6 || 32/21 || +4.114
+4.937
+1°14'3" ||< 32/21 ||
|| 12 || 800
960
240° ||= G# ||= A ||= d5,
M6 || 8/5 or 35/22 || -13.686 -3.822
-16.4235 -4.586
-4°6'21" -1°8'47" ||< **51/32** ||
|| 13 || 866.667
1040
260° ||= **H** ||= A^,
Bv ||= ~6 || 28/17 or 33/20 || +2.796 -0.293
+3.355 -0.351
<span style="line-height: 15.6000003814697px;">+</span>50'20" <span style="line-height: 15.6000003814697px;">-</span><span style="line-height: 1.5;">5'16"</span> ||< 64/39 ||
|| 14 || 933.333
1120
280° ||= **A** ||= B ||= m6,
M7 || 12/7 || +0.204
+0.245
+1"50" ||< **55/32** ||
|| 15 || 1000
1200
300° ||= Bb ||= B^ ||= ~7 || 16/9 || +3.910
+4.692
+1°10'23" ||< 16/9 ||
|| 16 || 1066.667
1280
320° ||= A# ||= Bb,
C# ||= m7 || 50/27 || -0.096
-0.115
-1'43" ||< 39/21 ||
|| 17 || 1133.333
1360
340° ||= **B** ||= Cv ||= ^m7,
vP8 || 52/27 || -1.329
-1.595
-23'56" ||< 75/39 ||
|| 18 || 1200
1440
360° ||= **C** ||= C ||= P8 || 2/1 || 0 ||< **2/1** ||
*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament

** based on 18b-edo with sharp lower than flat and major narrower than minor

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

==<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>=== 
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 ||=   ||

=Music= 
* [[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
* [[http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3|Gerbils at the Wheel of Government]] by [[@http://chrisvaisvil.com/?p=1402|Chris Vaisvil (in 9 and 18 edo simultaneously)]]
* [[http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3|Do Androids Dream Of 18ED2?]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html|blog entry]])
* [[https://soundcloud.com/tomprice719/composition-of-june-2015|Composition of June 2015 by TomPrice719]]

Original HTML content:

<html><head><title>18edo</title></head><body><!-- ws:start:WikiTextTocRule:22:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#Basic Properties">Basic Properties</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: -->
<!-- ws:end:WikiTextTocRule:34 --><hr />
<span style="display: block; text-align: right;"><a class="wiki_link" href="/18%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
</span><br />
<strong>18 Equal Divisions of the Octave</strong><br />
<strong>AKA The Third-Tone System</strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basic Properties</h1>
 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 />
However, less accurate approximations can be used. 18 equal does temper out 28/27, which makes three &quot;fifths&quot; (ie. 3/2) up, a 7/4. Thus 9/8 = a near just 7/6 (and what the relatively accurate 200 cents as 9/8, is in fact 8/7 - what do you make of that? Music.) This treatment applies to the scale generated by the large fifth, known as Father. One, if one really gets into it, can generate scales from the 3/2 and the half octave: with all the sharpness, what's 18e going to hurt? Call 600 cents 11/8 and 866 cents 13/8. Hey it's possible, lots of people like mavila.<br />
<br />
18-EDO contains sub-EDOs <a class="wiki_link" href="/2edo">2</a>, <a class="wiki_link" href="/3edo">3</a>, <a class="wiki_link" href="/6edo">6</a>, and <a class="wiki_link" href="/9edo">9</a>, and itself is half of <a class="wiki_link" href="/36edo">36-EDO</a> and one-fourth of <a class="wiki_link" href="/72edo">72-EDO</a>. It bears some similarities to <a class="wiki_link" href="/13edo">13-EDO</a> (with its very flat 4ths and nice subminor 3rds), <a class="wiki_link" href="/11edo">11-EDO</a> (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:2:&lt;h2&gt; --><h2 id="toc1"><a name="Basic Properties-Representations of Just Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Representations of Just Intervals</h2>
 

<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents <br />
coarse/fine<br />
DMS<br />
</td>
        <td style="text-align: center;">5L3s Notation<br />
</td>
        <td colspan="2" style="text-align: center;"><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">up/down </a><br />
<a class="wiki_link" href="/Ups%20and%20Downs%20Notation">notation</a> <strong><br />
</td>
        <td>Nearest Ratio<br />
</td>
        <td>Error (cents <br />
coarse/fine)<br />
(DMS)<br />
</td>
        <td style="text-align: left;">17-Limit Ratios*<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;"></strong>C<strong><br />
</td>
        <td style="text-align: center;">C<br />
</td>
        <td style="text-align: center;">P1<br />
</td>
        <td>1/1<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: left;"></strong>1/1<strong><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>66.667<br />
80<br />
20°<br />
</td>
        <td style="text-align: center;">Db<br />
</td>
        <td style="text-align: center;">C^,<br />
Dv<br />
</td>
        <td style="text-align: center;">^P1,<br />
vM2<br />
</td>
        <td>27/26<br />
</td>
        <td>+1.329<br />
+1.595<br />
+23'56&quot;<br />
</td>
        <td style="text-align: left;">78/75, 75/72<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>133.333<br />
160<br />
40°<br />
</td>
        <td style="text-align: center;">C#<br />
</td>
        <td style="text-align: center;">D<br />
</td>
        <td style="text-align: center;">M2<br />
</td>
        <td>27/25<br />
</td>
        <td>+0.096<br />
+0.115<br />
+1'43&quot;<br />
</td>
        <td style="text-align: left;">51/55, 42/39<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>200<br />
240<br />
60°<br />
</td>
        <td style="text-align: center;"></strong>D<strong><br />
</td>
        <td style="text-align: center;">D^,<br />
Ev<br />
</td>
        <td style="text-align: center;">~2<br />
</td>
        <td>9/8<br />
</td>
        <td>-3.910<br />
-4.692<br />
-1°10'23&quot;<br />
</td>
        <td style="text-align: left;"></strong>9/8<strong><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>266.667<br />
320<br />
80°<br />
</td>
        <td style="text-align: center;">Eb<br />
</td>
        <td style="text-align: center;">E<br />
</td>
        <td style="text-align: center;">m2,<br />
M3<br />
</td>
        <td>7/6<br />
</td>
        <td>-0.204<br />
-0.245<br />
-1&quot;50&quot;<br />
</td>
        <td style="text-align: left;"></strong>75/64<strong><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>333.333<br />
400<br />
100°<br />
</td>
        <td style="text-align: center;">D#<br />
</td>
        <td style="text-align: center;">E^<br />
</td>
        <td style="text-align: center;">~3<br />
</td>
        <td>17/14 or 40/33<br />
</td>
        <td>-2.796 +0.293<br />
-3.355 +0.351<br />
-50'20&quot; +5'16&quot;<br />
</td>
        <td style="text-align: left;"></strong>39/32<strong><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>400<br />
480<br />
120°<br />
</td>
        <td style="text-align: center;"></strong>E<strong><br />
</td>
        <td style="text-align: center;">Eb,<br />
F#<br />
</td>
        <td style="text-align: center;">m3,<br />
A4<br />
</td>
        <td>5/4 or 44/35<br />
</td>
        <td>+13.686 +3.822<br />
+16.4235 +4.586<br />
+4°6'21&quot; +1°8'47&quot;<br />
</td>
        <td style="text-align: left;">64/55<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>466.667<br />
560<br />
140°<br />
</td>
        <td style="text-align: center;"></strong>F<strong><br />
</td>
        <td style="text-align: center;">Fv<br />
</td>
        <td style="text-align: center;">^m3,<br />
vP4<br />
</td>
        <td>21/16<br />
</td>
        <td>-4.114<br />
-4.937<br />
-1°14'3&quot;<br />
</td>
        <td style="text-align: left;"></strong>21/16<strong><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>533.333<br />
640<br />
160°<br />
</td>
        <td style="text-align: center;">Gb<br />
</td>
        <td style="text-align: center;">F<br />
</td>
        <td style="text-align: center;">P4<br />
</td>
        <td>15/11<br />
</td>
        <td>-3.617<br />
-4.341<br />
-1°5'7&quot;<br />
</td>
        <td style="text-align: left;">102/75<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>600<br />
720<br />
180°<br />
</td>
        <td style="text-align: center;">F#<br />
</td>
        <td style="text-align: center;">F^,<br />
Gv<br />
</td>
        <td style="text-align: center;">^P4,<br />
vP5<br />
</td>
        <td>17/12 or 24/17<br />
</td>
        <td>-3.000 +3.000<br />
-3.6005 +3.6005<br />
-54' +54'<br />
</td>
        <td style="text-align: left;">17/12<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>666.667<br />
800<br />
200°<br />
</td>
        <td style="text-align: center;"></strong>G<strong><br />
</td>
        <td style="text-align: center;">G<br />
</td>
        <td style="text-align: center;">P5<br />
</td>
        <td>22/15<br />
</td>
        <td>+3.617<br />
+4.341<br />
+1°5'7&quot;<br />
</td>
        <td style="text-align: left;">75/51<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>733.333<br />
880<br />
220°<br />
</td>
        <td style="text-align: center;">Hb<br />
</td>
        <td style="text-align: center;">G^<br />
</td>
        <td style="text-align: center;">^P5,<br />
vM6<br />
</td>
        <td>32/21<br />
</td>
        <td>+4.114<br />
+4.937<br />
+1°14'3&quot;<br />
</td>
        <td style="text-align: left;">32/21<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>800<br />
960<br />
240°<br />
</td>
        <td style="text-align: center;">G#<br />
</td>
        <td style="text-align: center;">A<br />
</td>
        <td style="text-align: center;">d5,<br />
M6<br />
</td>
        <td>8/5 or 35/22<br />
</td>
        <td>-13.686 -3.822<br />
-16.4235 -4.586<br />
-4°6'21&quot; -1°8'47&quot;<br />
</td>
        <td style="text-align: left;"></strong>51/32<strong><br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>866.667<br />
1040<br />
260°<br />
</td>
        <td style="text-align: center;"></strong>H<strong><br />
</td>
        <td style="text-align: center;">A^,<br />
Bv<br />
</td>
        <td style="text-align: center;">~6<br />
</td>
        <td>28/17 or 33/20<br />
</td>
        <td>+2.796 -0.293<br />
+3.355 -0.351<br />
<span style="line-height: 15.6000003814697px;">+</span>50'20&quot; <span style="line-height: 15.6000003814697px;">-</span><span style="line-height: 1.5;">5'16&quot;</span><br />
</td>
        <td style="text-align: left;">64/39<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>933.333<br />
1120<br />
280°<br />
</td>
        <td style="text-align: center;"></strong>A<strong><br />
</td>
        <td style="text-align: center;">B<br />
</td>
        <td style="text-align: center;">m6,<br />
M7<br />
</td>
        <td>12/7<br />
</td>
        <td>+0.204<br />
+0.245<br />
+1&quot;50&quot;<br />
</td>
        <td style="text-align: left;"></strong>55/32<strong><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>1000<br />
1200<br />
300°<br />
</td>
        <td style="text-align: center;">Bb<br />
</td>
        <td style="text-align: center;">B^<br />
</td>
        <td style="text-align: center;">~7<br />
</td>
        <td>16/9<br />
</td>
        <td>+3.910<br />
+4.692<br />
+1°10'23&quot;<br />
</td>
        <td style="text-align: left;">16/9<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>1066.667<br />
1280<br />
320°<br />
</td>
        <td style="text-align: center;">A#<br />
</td>
        <td style="text-align: center;">Bb,<br />
C#<br />
</td>
        <td style="text-align: center;">m7<br />
</td>
        <td>50/27<br />
</td>
        <td>-0.096<br />
-0.115<br />
-1'43&quot;<br />
</td>
        <td style="text-align: left;">39/21<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>1133.333<br />
1360<br />
340°<br />
</td>
        <td style="text-align: center;"></strong>B<strong><br />
</td>
        <td style="text-align: center;">Cv<br />
</td>
        <td style="text-align: center;">^m7,<br />
vP8<br />
</td>
        <td>52/27<br />
</td>
        <td>-1.329<br />
-1.595<br />
-23'56&quot;<br />
</td>
        <td style="text-align: left;">75/39<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>1200<br />
1440<br />
360°<br />
</td>
        <td style="text-align: center;"></strong>C<strong><br />
</td>
        <td style="text-align: center;">C<br />
</td>
        <td style="text-align: center;">P8<br />
</td>
        <td>2/1<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: left;"></strong>2/1**<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 />
<ul><ul><li>based on 18b-edo with sharp lower than flat and major narrower than minor</li></ul></ul><br />
For alternative notations, see <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs">Ups and Downs Notation -&quot;Superflat&quot; EDOs</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs">Ups and Downs Notation -&quot;Supersharp&quot; EDOs</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators">Ups and Downs Notation - Natural Generators</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Basic Properties-Useful Moment-of-Symmetry Scales"></a><!-- ws:end:WikiTextHeadingRule:4 --><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:6:&lt;h3&gt; --><h3 id="toc3"><a name="Basic Properties-Useful Moment-of-Symmetry Scales-Pentatonic:"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="font-size: 1.1em;">Pentatonic:</span></h3>
 3L2s Father Pentatonic: 4 4 3 4 3<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Basic Properties-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:8 --><span style="font-size: 1.1em;">Hexatonic:</span></h3>
 4L2s Bicycle: 4 4 1 4 4 1<br />
2L4s Rice Hexatonic: 2 5 2 2 5 2<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Basic Properties-Useful Moment-of-Symmetry Scales-Heptatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="font-size: 1.1em;">Heptatonic:</span></h3>
 4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Basic Properties-Useful Moment-of-Symmetry Scales-Octatonic:"></a><!-- ws:end:WikiTextHeadingRule:12 --><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:14:&lt;h3&gt; --><h3 id="toc7"><a name="Basic Properties-Useful Moment-of-Symmetry Scales-Decatonic:"></a><!-- ws:end:WikiTextHeadingRule:14 --><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:16:&lt;h2&gt; --><h2 id="toc8"><a name="Basic Properties-Application to Guitar"></a><!-- ws:end:WikiTextHeadingRule:16 --><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:18:&lt;h1&gt; --><h1 id="toc9"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:18 -->Commas</h1>
 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:20:&lt;h1&gt; --><h1 id="toc10"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:20 -->Music</h1>
 <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><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3" rel="nofollow">Gerbils at the Wheel of Government</a> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1402" rel="nofollow" target="_blank">Chris Vaisvil (in 9 and 18 edo simultaneously)</a></li><li><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3" rel="nofollow">Do Androids Dream Of 18ED2?</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html" rel="nofollow">blog entry</a>)</li><li><a class="wiki_link_ext" href="https://soundcloud.com/tomprice719/composition-of-june-2015" rel="nofollow">Composition of June 2015 by TomPrice719</a></li></ul></body></html>