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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 18-EDO
: This revision was by author [[User:vaisvil|vaisvil]] and made on <tt>2011-10-16 10:50:33 UTC</tt>.<br>
| en = 18edo
: The original revision id was <tt>265196840</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja = 18平均律
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=18 Equal Divisions of the Octave=
{{ED intro}}
AKA The Third-Tone System


==Basic Properties==
18edo is also known as the '''third-tone''' system.
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).


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.
== Theory ==
18edo does not include the 3rd or 7th harmonics, and contains the same controversial tuning of [[5/4]] as 12edo does. It does, however, render more accurate tunings of [[7/6]], [[21/16]], [[15/11]], [[12/7]], 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).


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.
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. 18edo may be treated as a temperament of the 17-limit [[k*N_subgroups|4*18 subgroup]] [[just intonation subgroup]] 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as [[72edo]] 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. 18edo also approximates 12:13:14:17:23:27:29 quite well, with the least maximum relative error out of any edos ≤ 100 (the worst-approximated interval is [[23/13]], with relative error 18.36%). Hence it can be viewed as an "/3 temperament" (/3 used in the [[primodality]] sense), specifically in the 2.9.13/12.7/6.17/12.23/12.29/24 subgroup. As for more simple subgroups, 18edo can be treated as a 2.9.5.11 subgroup temperament.


===Representations of Just Intervals===
However, less accurate approximations can be used, and 18edo can be treated as a 7-limit (with 3s) exotemperament with the mapping {{val| 18 29 42 51 }}. This maps 3/2 to 733.33{{c}}, 5/4 to 400{{c}} and 7/4 to 1000{{c}}; as a result, 28/27 is tempered out, and unintuitive things happen: 9/8 and 7/6 are both mapped to 266.67{{c}}, while 8/7 gets mapped below both of them to 200{{c}}, making for a rather disordered [[9-odd-limit]] [[tonality diamond]], although this may be serviceable for the more exotemperamental music. This 7-limit mapping [[support]]s 7-limit [[sixix]], and thus is strongly associated with 18edo's [[4L 3s]] [[mos]].  
|| Degree || Cents ||= 5L3s Notation || Nearest Ratio || Error (cents) || 17-Limit Ratios* ||
|| 0 || 0 ||= **C** || 1/1 || 0 || **1/1** ||
|| 1 || 66.667 ||= Db || 27/26 || +1.329 ||&gt; 78/75, 75/72 ||
|| 2 || 133.333 ||= C# || 27/25 || +0.096 ||&gt; 51/55, 42/39 ||
|| 3 || 200 ||= **D** || 9/8 || -3.910 || **9/8** ||
|| 4 || 266.667 ||= Eb || 7/6 || -0.204 || **75/64** ||
|| 5 || 333.333 ||= D# || 17/14 or 40/33 || -2.796 +0.293 || **39/32** ||
|| 6 || 400 ||= **E** || 5/4 or 44/35 || +13.686 +3.822 ||&gt; 64/55 ||
|| 7 || 466.667 ||= **F** || 21/16 || -4.114 || **21/16** ||
|| 8 || 533.333 ||= Gb || 15/11 || -3.617 ||&gt; 102/75 ||
|| 9 || 600 ||= F# || 17/12 or 24/17 || -3.000 +3.000 ||&gt; 17/12 ||
|| 10 || 666.667 ||= **G** || 22/15 || +3.617 ||&gt; 75/51 ||
|| 11 || 733.333 ||= Hb || 32/21 || +4.114 ||&gt; 32/21 ||
|| 12 || 800 ||= G# || 8/5 or 35/22 || -13.686 -3.8222 || **51/32** ||
|| 13 || 866.667 ||= **H** || 28/17 or 33/20 || +2.796 -0.293 ||&gt; 64/39 ||
|| 14 || 933.333 ||= **A** || 12/7 || +0.204 || **55/32** ||
|| 15 || 1000 ||= Bb || 16/9 || +3.910 ||&gt; 16/9 ||
|| 16 || 1066.667 ||= A# || 50/27 || -0.096 ||&gt; 39/21 ||
|| 17 || 1133.333 ||= **B** || 52/27 || -1.329 ||&gt; 75/39 ||
|| 18 || 1200 ||= **C** || 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;==
18edo contains sub-edos [[2edo|2]], [[3edo|3]], [[6edo|6]], and [[9edo|9]], and itself is half of [[36edo]] and one-fourth of 72edo. It bears some similarities to [[13edo]] (with its very flat 4ths and nice subminor 3rds), [[11edo]] (with its very sharp minor 3rds, two of which span a very flat 5th), [[16edo]] (with its sharp 4ths and flat 5ths), and [[17edo]] and [[19edo]] (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.
Note: This list excludes scales found in 9-EDO.
===&lt;span style="font-size: 1.1em;"&gt;Pentatonic:&lt;/span&gt;===
3L2s Father Pentatonic: 4 4 3 4 3
===&lt;span style="font-size: 1.1em;"&gt;Hexatonic:&lt;/span&gt;===
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
===&lt;span style="font-size: 1.1em;"&gt;Heptatonic:&lt;/span&gt;===
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
===&lt;span style="font-size: 1.1em;"&gt;Octatonic:&lt;/span&gt;===
5L3s Father Octatonic: 3 1 3 3 1 3 3 1
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
===&lt;span style="font-size: 1.1em;"&gt;Decatonic:&lt;/span&gt;===
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2


==&lt;span style="font-size: 1.3em;"&gt;Application to Guitar&lt;/span&gt;==
18edo is the basic example of a dual-fifth system (beyond perhaps 11 or 13edo), as the sharp and flat fifths multiply to a good approximation of 9/4. By alternating these fifths, a diatonic scale (5L 1m 1s) is generated which is similar to 19edo's diatonic, but cut short by one step.
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).
=== Odd harmonics ===
{{Harmonics in equal|18}}


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


==Listen==
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png|thumb]]
* [[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]] =&gt; [[@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)]]</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;18edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x18 Equal Divisions of the Octave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;18 Equal Divisions of the Octave&lt;/h1&gt;
AKA The Third-Tone System&lt;br /&gt;
&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;
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 &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;
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;
&lt;br /&gt;
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.&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-Representations of Just Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Representations of Just Intervals&lt;/h3&gt;


&lt;table class="wiki_table"&gt;
{| class="wikitable center-all right-2"
    &lt;tr&gt;
! Degree
        &lt;td&gt;Degree&lt;br /&gt;
! Cents
&lt;/td&gt;
! Nearest Ratio
        &lt;td&gt;Cents&lt;br /&gt;
! Error
&lt;/td&gt;
! 17-Limit Ratios <ref>based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament</ref>
        &lt;td style="text-align: center;"&gt;5L3s Notation&lt;br /&gt;
|-
&lt;/td&gt;
| 0
        &lt;td&gt;Nearest Ratio&lt;br /&gt;
| 0.000
&lt;/td&gt;
| 1/1
        &lt;td&gt;Error (cents)&lt;br /&gt;
| 0
&lt;/td&gt;
| 1/1
        &lt;td&gt;17-Limit Ratios*&lt;br /&gt;
|-
&lt;/td&gt;
| 1
    &lt;/tr&gt;
| 66.667
    &lt;tr&gt;
| 27/26
        &lt;td&gt;0&lt;br /&gt;
| +1.329
&lt;/td&gt;
| 26/25, 25/24
        &lt;td&gt;0&lt;br /&gt;
|-
&lt;/td&gt;
| 2
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;br /&gt;
| 133.333
&lt;/td&gt;
| 27/25
        &lt;td&gt;1/1&lt;br /&gt;
| +0.096
&lt;/td&gt;
| 55/51, 14/13
        &lt;td&gt;0&lt;br /&gt;
|-
&lt;/td&gt;
| 3
        &lt;td&gt;&lt;strong&gt;1/1&lt;/strong&gt;&lt;br /&gt;
| 200.000
&lt;/td&gt;
| 9/8
    &lt;/tr&gt;
| -3.910
    &lt;tr&gt;
| 9/8
        &lt;td&gt;1&lt;br /&gt;
|-
&lt;/td&gt;
| 4
        &lt;td&gt;66.667&lt;br /&gt;
| 266.667
&lt;/td&gt;
| 7/6
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
| -0.204
&lt;/td&gt;
| 75/64
        &lt;td&gt;27/26&lt;br /&gt;
|-
&lt;/td&gt;
| 5
        &lt;td&gt;+1.329&lt;br /&gt;
| 333.333
&lt;/td&gt;
| 17/14 or 40/33
        &lt;td style="text-align: right;"&gt;78/75, 75/72&lt;br /&gt;
| -2.796 +0.293
&lt;/td&gt;
| 39/32
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 6
        &lt;td&gt;2&lt;br /&gt;
| 400.000
&lt;/td&gt;
| 5/4 or 44/35
        &lt;td&gt;133.333&lt;br /&gt;
| +13.686 +3.822
&lt;/td&gt;
| 64/55
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
|-
&lt;/td&gt;
| 7
        &lt;td&gt;27/25&lt;br /&gt;
| 466.667
&lt;/td&gt;
| 21/16
        &lt;td&gt;+0.096&lt;br /&gt;
| -4.114
&lt;/td&gt;
| 21/16
        &lt;td style="text-align: right;"&gt;51/55, 42/39&lt;br /&gt;
|-
&lt;/td&gt;
| 8
    &lt;/tr&gt;
| 533.333
    &lt;tr&gt;
| 15/11
        &lt;td&gt;3&lt;br /&gt;
| -3.617
&lt;/td&gt;
| 34/25
        &lt;td&gt;200&lt;br /&gt;
|-
&lt;/td&gt;
| 9
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
| 600.000
&lt;/td&gt;
| 17/12 or 24/17
        &lt;td&gt;9/8&lt;br /&gt;
| -3.000 +3.000
&lt;/td&gt;
| 17/12
        &lt;td&gt;-3.910&lt;br /&gt;
|-
&lt;/td&gt;
| 10
        &lt;td&gt;&lt;strong&gt;9/8&lt;/strong&gt;&lt;br /&gt;
| 666.667
&lt;/td&gt;
| 22/15
    &lt;/tr&gt;
| +3.617
    &lt;tr&gt;
| 25/17
        &lt;td&gt;4&lt;br /&gt;
|-
&lt;/td&gt;
| 11
        &lt;td&gt;266.667&lt;br /&gt;
| 733.333
&lt;/td&gt;
| 32/21
        &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
| +4.114
&lt;/td&gt;
| 32/21
        &lt;td&gt;7/6&lt;br /&gt;
|-
&lt;/td&gt;
| 12
        &lt;td&gt;-0.204&lt;br /&gt;
| 800.000
&lt;/td&gt;
| 8/5 or 35/22
        &lt;td&gt;&lt;strong&gt;75/64&lt;/strong&gt;&lt;br /&gt;
| -13.686 -3.822
&lt;/td&gt;
| 51/32
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 13
        &lt;td&gt;5&lt;br /&gt;
| 866.667
&lt;/td&gt;
| 28/17 or 33/20
        &lt;td&gt;333.333&lt;br /&gt;
| +2.796 -0.293
&lt;/td&gt;
| 64/39
        &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
|-
&lt;/td&gt;
| 14
        &lt;td&gt;17/14 or 40/33&lt;br /&gt;
| 933.333
&lt;/td&gt;
| 12/7
        &lt;td&gt;-2.796 +0.293&lt;br /&gt;
| +0.204
&lt;/td&gt;
| 55/32
        &lt;td&gt;&lt;strong&gt;39/32&lt;/strong&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 15
    &lt;/tr&gt;
| 1000.000
    &lt;tr&gt;
| 16/9
        &lt;td&gt;6&lt;br /&gt;
| +3.910
&lt;/td&gt;
| 16/9
        &lt;td&gt;400&lt;br /&gt;
|-
&lt;/td&gt;
| 16
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;br /&gt;
| 1066.667
&lt;/td&gt;
| 50/27
        &lt;td&gt;5/4 or 44/35&lt;br /&gt;
| -0.096
&lt;/td&gt;
| 13/7
        &lt;td&gt;+13.686 +3.822&lt;br /&gt;
|-
&lt;/td&gt;
| 17
        &lt;td style="text-align: right;"&gt;64/55&lt;br /&gt;
| 1133.333
&lt;/td&gt;
| 52/27
    &lt;/tr&gt;
| -1.329
    &lt;tr&gt;
| 25/13
        &lt;td&gt;7&lt;br /&gt;
|-
&lt;/td&gt;
| 18
        &lt;td&gt;466.667&lt;br /&gt;
| 1200.000
&lt;/td&gt;
| 2/1
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;br /&gt;
| 0
&lt;/td&gt;
| 2/1**
        &lt;td&gt;21/16&lt;br /&gt;
|}
&lt;/td&gt;
<references />
        &lt;td&gt;-4.114&lt;br /&gt;
{{Clear}}
&lt;/td&gt;
== Notation ==
        &lt;td&gt;&lt;strong&gt;21/16&lt;/strong&gt;&lt;br /&gt;
=== Ups and downs notation ===
&lt;/td&gt;
18edo can be notated with [[ups and downs]]. The notational 5th is the 2nd-best approximation of 3/2, 10\18. This is only 4{{c}} worse that the best approximation, which becomes the up-fifth.
    &lt;/tr&gt;
{{Mavila}}
    &lt;tr&gt;
{| class="wikitable center-all right-2"
        &lt;td&gt;8&lt;br /&gt;
! Degree
&lt;/td&gt;
! Cents
        &lt;td&gt;533.333&lt;br /&gt;
! colspan="3" | [[Ups and downs notation|Up/down notation]] using the narrow 5th of 10\18, <br> with major wider than minor
&lt;/td&gt;
! colspan="3" | Up/down notation using the narrow 5th of 10\18, <br> with major narrower than minor
        &lt;td style="text-align: center;"&gt;Gb&lt;br /&gt;
! 5L3s Notation
&lt;/td&gt;
|-
        &lt;td&gt;15/11&lt;br /&gt;
| 0
&lt;/td&gt;
| 0
        &lt;td&gt;-3.617&lt;br /&gt;
| perfect unison
&lt;/td&gt;
| P1
        &lt;td style="text-align: right;"&gt;102/75&lt;br /&gt;
| D
&lt;/td&gt;
| perfect unison
    &lt;/tr&gt;
| P1
    &lt;tr&gt;
| D
        &lt;td&gt;9&lt;br /&gt;
| C
&lt;/td&gt;
|-
        &lt;td&gt;600&lt;br /&gt;
| 1
&lt;/td&gt;
| 67
        &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
| up unison, downminor 2nd
&lt;/td&gt;
| ^1, vm2
        &lt;td&gt;17/12 or 24/17&lt;br /&gt;
| ^D, vE
&lt;/td&gt;
| up unison, downmajor 2nd
        &lt;td&gt;-3.000 +3.000&lt;br /&gt;
| ^1, vM2
&lt;/td&gt;
| ^D, vE
        &lt;td style="text-align: right;"&gt;17/12&lt;br /&gt;
| Db
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 2
    &lt;tr&gt;
| 133
        &lt;td&gt;10&lt;br /&gt;
| minor 2nd
&lt;/td&gt;
| m2
        &lt;td&gt;666.667&lt;br /&gt;
| E
&lt;/td&gt;
| major 2nd
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;br /&gt;
| M2
&lt;/td&gt;
| E
        &lt;td&gt;22/15&lt;br /&gt;
| C#
&lt;/td&gt;
|-
        &lt;td&gt;+3.617&lt;br /&gt;
| 3
&lt;/td&gt;
| 200
        &lt;td style="text-align: right;"&gt;75/51&lt;br /&gt;
| mid 2nd
&lt;/td&gt;
| ~2
    &lt;/tr&gt;
| ^E
    &lt;tr&gt;
| mid 2nd
        &lt;td&gt;11&lt;br /&gt;
| ~2
&lt;/td&gt;
| ^E
        &lt;td&gt;733.333&lt;br /&gt;
| D
&lt;/td&gt;
|-
        &lt;td style="text-align: center;"&gt;Hb&lt;br /&gt;
| 4
&lt;/td&gt;
| 267
        &lt;td&gt;32/21&lt;br /&gt;
| major 2nd, minor 3rd
&lt;/td&gt;
| M2, m3
        &lt;td&gt;+4.114&lt;br /&gt;
| E#, Fb
&lt;/td&gt;
| minor 2nd, major 3rd
        &lt;td style="text-align: right;"&gt;32/21&lt;br /&gt;
| m2, M3
&lt;/td&gt;
| Eb, F#
    &lt;/tr&gt;
| Eb
    &lt;tr&gt;
|-
        &lt;td&gt;12&lt;br /&gt;
| 5
&lt;/td&gt;
| 333
        &lt;td&gt;800&lt;br /&gt;
| mid 3rd
&lt;/td&gt;
| ~3
        &lt;td style="text-align: center;"&gt;G#&lt;br /&gt;
| vF
&lt;/td&gt;
| mid 3rd
        &lt;td&gt;8/5 or 35/22&lt;br /&gt;
| ~3
&lt;/td&gt;
| vF
        &lt;td&gt;-13.686 -3.8222&lt;br /&gt;
| D#
&lt;/td&gt;
|-
        &lt;td&gt;&lt;strong&gt;51/32&lt;/strong&gt;&lt;br /&gt;
| 6
&lt;/td&gt;
| 400
    &lt;/tr&gt;
| major 3rd
    &lt;tr&gt;
| M3
        &lt;td&gt;13&lt;br /&gt;
| F
&lt;/td&gt;
| minor 3rd
        &lt;td&gt;866.667&lt;br /&gt;
| m3
&lt;/td&gt;
| F
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;H&lt;/strong&gt;&lt;br /&gt;
| E
&lt;/td&gt;
|-
        &lt;td&gt;28/17 or 33/20&lt;br /&gt;
| 7
&lt;/td&gt;
| 467
        &lt;td&gt;+2.796 -0.293&lt;br /&gt;
| upmajor 3rd, down 4th
&lt;/td&gt;
| ^M3, v4
        &lt;td style="text-align: right;"&gt;64/39&lt;br /&gt;
| ^F, vG
&lt;/td&gt;
| upminor 3rd, down 4th
    &lt;/tr&gt;
| ^m3, v4
    &lt;tr&gt;
| ^F, vG
        &lt;td&gt;14&lt;br /&gt;
| F
&lt;/td&gt;
|-
        &lt;td&gt;933.333&lt;br /&gt;
| 8
&lt;/td&gt;
| 533
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
| perfect 4th
&lt;/td&gt;
| P4
        &lt;td&gt;12/7&lt;br /&gt;
| G
&lt;/td&gt;
| perfect 4th
        &lt;td&gt;+0.204&lt;br /&gt;
| P4
&lt;/td&gt;
| G
        &lt;td&gt;&lt;strong&gt;55/32&lt;/strong&gt;&lt;br /&gt;
| Gb
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 9
    &lt;tr&gt;
| 600
        &lt;td&gt;15&lt;br /&gt;
| up 4th, down 5th
&lt;/td&gt;
| ^4, v5
        &lt;td&gt;1000&lt;br /&gt;
| ^G, vA
&lt;/td&gt;
| up 4th, down 5th
        &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
| ^4, v5
&lt;/td&gt;
| ^G, vA
        &lt;td&gt;16/9&lt;br /&gt;
| F#
&lt;/td&gt;
|-
        &lt;td&gt;+3.910&lt;br /&gt;
| 10
&lt;/td&gt;
| 667
        &lt;td style="text-align: right;"&gt;16/9&lt;br /&gt;
| perfect 5th
&lt;/td&gt;
| P5
    &lt;/tr&gt;
| A
    &lt;tr&gt;
| perfect 5th
        &lt;td&gt;16&lt;br /&gt;
| P5
&lt;/td&gt;
| A
        &lt;td&gt;1066.667&lt;br /&gt;
| G
&lt;/td&gt;
|-
        &lt;td style="text-align: center;"&gt;A#&lt;br /&gt;
| 11
&lt;/td&gt;
| 733
        &lt;td&gt;50/27&lt;br /&gt;
| up 5th, downminor 6th
&lt;/td&gt;
| ^5, vm6
        &lt;td&gt;-0.096&lt;br /&gt;
| ^A, vB
&lt;/td&gt;
| up fifth, downmajor 6th
        &lt;td style="text-align: right;"&gt;39/21&lt;br /&gt;
| ^5, vM6
&lt;/td&gt;
| ^A, vB
    &lt;/tr&gt;
| Hb
    &lt;tr&gt;
|-
        &lt;td&gt;17&lt;br /&gt;
| 12
&lt;/td&gt;
| 800
        &lt;td&gt;1133.333&lt;br /&gt;
| minor 6th
&lt;/td&gt;
| m6
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;B&lt;/strong&gt;&lt;br /&gt;
| B
&lt;/td&gt;
| major 6th
        &lt;td&gt;52/27&lt;br /&gt;
| M6
&lt;/td&gt;
| B
        &lt;td&gt;-1.329&lt;br /&gt;
| G#
&lt;/td&gt;
|-
        &lt;td style="text-align: right;"&gt;75/39&lt;br /&gt;
| 13
&lt;/td&gt;
| 867
    &lt;/tr&gt;
| mid 6th
    &lt;tr&gt;
| ~6
        &lt;td&gt;18&lt;br /&gt;
| ^B
&lt;/td&gt;
| mid 6th
        &lt;td&gt;1200&lt;br /&gt;
| ~6
&lt;/td&gt;
| ^B
        &lt;td style="text-align: center;"&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;br /&gt;
| H
&lt;/td&gt;
|-
        &lt;td&gt;2/1&lt;br /&gt;
| 14
&lt;/td&gt;
| 933
        &lt;td&gt;0&lt;br /&gt;
| major 6th, minor 7th
&lt;/td&gt;
| M6, m7
        &lt;td&gt;&lt;strong&gt;2/1&lt;/strong&gt;&lt;br /&gt;
| B#, Cb
&lt;/td&gt;
| minor 6th, major 7th
    &lt;/tr&gt;
| m6, M7
&lt;/table&gt;
| Bb, C#
| A
|-
| 15
| 1000
| mid 7th
| ~7
| vC
| mid 7th
| ~7
| vC
| Bb
|-
| 16
| 1067
| major 7th
| M7
| C
| minor 7th
| m7
| C
| A#
|-
| 17
| 1133
| upmajor 7th, down 8ve
| ^M7, v8
| ^C, vD
| upminor 7th, down 8ve
| ^m7, v8
| ^C, vD
| B
|-
| 18
| 1200
| perfect 8ve
| P8
| D
| perfect 8ve
| P8
| D
| C
|}


*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament&lt;br /&gt;
This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, nonotonic 5th-generated, and heptatonic 3rd-generated.  
&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;
3L2s 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;
6-Equal Whole-Tone Scale: 3 3 3 3 3 3&lt;br /&gt;
4L2s Bicycle: 4 4 1 4 4 1&lt;br /&gt;
2L4s 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;
4L3s 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;
5L3s Father Octatonic: 3 1 3 3 1 3 3 1&lt;br /&gt;
2L6s 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;
8L2s 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;
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!&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: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;
18 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;


'''<u>Pentatonic 5th-generated:</u> D * * * E * * G * * * A * * C * * * D''' (generator = wide 3/2 = 11\18 = perfect 5thoid)


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


&lt;br /&gt;
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d (s = sub-, d = -oid)
&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;li&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3" rel="nofollow"&gt;Gerbils at the Wheel of Government&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1402" rel="nofollow" target="_blank"&gt;Chris Vaisvil (in 9 and 18 edo simultaneously)&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...
 
pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)
 
'''<u>Nonatonic 5th-generated:</u> A * B * C * D * E * F * G * H * J * A''' (every other note is a generator, all notes are perfect)
 
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4- 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8 - ^8/v9 - 9 - ^9/v10 - 10
 
'''<u>heptatonic 3rd-generated:</u>  D * * E * F * * G * A * * B * C * * D''' (generator = 5\18 = perfect 3rd)
 
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
 
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
 
genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb... ("Every good boy deserves fudge and candy")
 
genchain of thirds: ...A4 - A6 - A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 - d3 - d5...
 
===Sagittal notation===
This notation is a subset of the notations for EDOs [[36edo#Sagittal notation|36]] and [[72edo#Sagittal notation|72]] and a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====
 
{{Sagittal chart|Evo}}
 
====Revo flavor====
 
{{Sagittal chart}}
 
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=18}}
 
=== Commas ===
18et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 18 29 42 51 62 67 }}.)
 
{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name(s)
|-
| 3
| [[536870912/387420489|(18 digits)]]
| {{monzo| 29 -18 }}
| 564.81
| Wa-18
| 18-comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, diesis
|-
| 5
| [[6115295232/6103515625|(20 digits)]]
| {{monzo| 23 6 -14 }}
| 3.34
| Sasa-sepbigu
| [[Vishnuzma]], Semisuper comma
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[16875/16807]]
| {{monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|}
<references/>
 
== Octave stretch or compression ==
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Suitable shrunk versions of 18edo include [[zpi|61zpi]], [[ed12|65ed12]] and [[ed6|47ed6]].
 
== Scales ==
Note: This list excludes scales found in [[9edo]].
 
=== Pentatonic ===
 
[[3L 2s]]: 4 4 3 4 3
 
=== Hexatonic ===
 
[[4L 2s]]: 4 4 1 4 4 1
 
[[2L 4s]]: 2 5 2 2 5 2
 
=== Heptatonic ===
 
[[4L 3s]]: 3 2 3 2 3 3 2
 
=== Octatonic ===
 
[[5L 3s]]: 3 1 3 3 1 3 3 1
 
[[2L 6s]]: 2 2 3 2 2 2 3 2
 
=== Enneatonic ===
 
[[3L 6s]]: 4 1 1 4 1 1 4 1 1
 
=== Decatonic ===
 
[[8L 2s]]: 2 2 1 2 2 2 2 1 2 2
 
=== Hendecatonic ===
[[7L 4s]]: 2 1 2 2 1 2 2 1 2 1 2
 
=== Dodecatonic ===
 
[[3L 9s]]: 3 1 1 1 3 1 1 1 3 1 1 1
 
[[6L 6s]]: 2 1 2 1 2 1 2 1 2 1 2 1
 
[[Werntz Nocturne scale]]: 2 1 1 2 2 1 1 2 2 1 1 2
 
=== Tridecatonic ===
 
[[5L 8s]]: 2 1 2 1 1 2 1 2 1 1 2 1 1
 
=== Pentadecatonic ===
 
[[3L 12s]]: 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1
 
== Instruments ==
=== Guitar ===
18edo 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 8-note oneirotonic scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667{{c}} intervals, with one 533.333{{c}} 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).
 
=== Keyboards ===
[[Julián Carrillo]] built at least one third-tone piano in 18edo.
 
[[Lumatone mapping for 18edo|Lumatone mappings for 18edo]] are available.
 
== Music ==
=== Modern renderings ===
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=mAcBBL2lkHo ''Bluin' The Black Keys''] (1926) – rendered by Francium (2025)
 
=== 20th century ===
; [[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=gbPPYOygNJc ''Prélude et Etude'', Op. 48], for third-tone piano of [[Julián Carrillo]] (1966)
 
=== 21st century ===
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=Cp_lTUNmtd8 ''XVIII-TET Tribute to Full Moon in Virgo''] (2024)
 
; [[Beheld]]
* [https://www.youtube.com/watch?v=Nog2LROg8Ss ''Overstrung vibe''] (2022)
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/-oi5eJA65Zc ''Waltz in 18edo''] (2025)
* [https://www.youtube.com/watch?v=r3FypUx_iIk ''Lament in 18edo''] (2025)
* [https://www.youtube.com/shorts/hNmse4IUWL0 ''18edo improv''] (2025)
 
; [[Francium]]
* "excucumber", from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/2uSQv7MbMOKMLue2FMVU9y Spotify] | [https://francium223.bandcamp.com/track/excucumber Bandcamp] | [https://www.youtube.com/watch?v=dcOsIrQEsg4 YouTube]
* [https://www.youtube.com/watch?v=KgIWvxMKrlo ''Stummy Beige''] (2026)
 
; [[groundfault]]
* "Life and Limb", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/life-and-limb-18edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1751 YouTube (29:11–33:47)]
 
; [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et ''Fuga a3 in 18ET'']{{dead link}}
 
; [[Noah Jordan]]
* ''The Moon'' (2016) – [https://noahdeanjordan.bandcamp.com/album/the-moon BandCamp] | [https://www.youtube.com/watch?v=TunyA3gwEJw YouTube] – 7-piece album recorded on the 1/3-tone piano of Sonido 13 / Julian Carrillo
* ''There and Back Again'' (2025) – [https://noahdeanjordan.bandcamp.com/album/there-and-back-again Bandcamp] | [https://www.youtube.com/watch?v=O36ZQyq6oR8 YouTube] – 3-piece album recorded on the 1/3-tone piano of Sonido 13 / Julian Carrillo
 
; [[Mandrake]]
* [https://www.youtube.com/watch?v=R1uz0ok4-Zs ''Such And Flowers''] (2022)
* [https://www.youtube.com/watch?v=2AquW_cqUQc ''That Kinda Lo-Fi Feel''] (2022)
 
; [[Leo Matarazzo]]
* [https://www.youtube.com/watch?v=G2vrqyE0ZX4 ''Eris''] (2026)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vUTHZNzBwUo ''Air Triste''] (2018)
 
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/revealing-the-path ''Revealing the Path''] (2018)
 
; [[Mundoworld]]
* [https://www.youtube.com/watch?v=iIaROmh7wD0 ''I Am the Monster I Fear''] (2023)
 
; [[No Clue Music]]
* [https://www.youtube.com/watch?v=UHFU9-eBXBo ''WORLD PORTAL''] (2024)
 
; [[norokusi]]
* [https://www.youtube.com/watch?v=pEvT2oyWEuo ''3 Bagatelles''] (2021)
 
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=hNgI6oRYNHA ''Three Worlds Order''] (2020)
* [https://www.youtube.com/watch?v=FnYxYuukgrM "Confusion"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
* [https://www.youtube.com/watch?v=1k4rNjyaZsE ''Purgatory''] (2021)
* [https://www.youtube.com/watch?v=a7AtOuX1NAE ''The Hydrogen Atom''] (2023)
 
; [[Carlo Serafini]]
* ''Do Androids Dream Of 18ED2?'' (2015) – [http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog] | [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 play]{{dead link}}
 
; [[TomPrice719]]
* [https://soundcloud.com/tomprice719/composition-of-june-2015 ''Composition of June 2015''] (2015)
 
; [[Chris Vaisvil]]
* ''Prelude in 18et'' (2009) – [https://www.chrisvaisvil.com/prelude-in-18et/ blog] | [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 play]
* [http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 ''Flippertronics'']
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 ''Gerbils at the Wheel of Government''] (in 9 and 18 edo simultaneously)
 
; [[Julia Werntz]], [[Eric Moe]] & the [[Pandelis Karayorgis Trio]]
* [https://driffrecords.bandcamp.com/album/climbing-to-sleep ''Climbing to Sleep''] (2025) – jazz album
 
; [[Xeno*n*]]
* [https://www.youtube.com/watch?v=fj_AISfnFnY ''Deranged Anger''] (2021)
 
; [[David Zaydullin]]
* [https://www.youtube.com/watch?v=QzKNP-NwHu0 ''Phaserun''] (2024)
 
== See also ==
* [[Fendo family]] - temperaments closely related to 18edo
 
[[Category:18-tone scales]]
[[Category:Listen]]
[[Category:Teentuning]]
[[Category:Oneirotonic]]
Retrieved from "https://en.xen.wiki/w/18edo"