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, and 18edo can be treated as a 7-limit exotemperament with the mapping <18 29 42 51|. This maps 3/2 to 733.33¢ and 7/4 to 1000¢; as a result, 28/27 is tempered out, and weird things happen: 9/8 and 7/6 are both mapped to 266.67¢, while 8/7 gets mapped below both of them to 200¢, making for a rather disordered 7-limit tonality diamond, but hey, whatever floats your boat! 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 || 5L3s Notation || Nearest Ratio || Error || 17-Limit Ratios* || || 0 || 0 ||= C || 1/1 || 0 ||< 1/1 || || 1 || 66.67 ||= Db || 27/26 || +1.329 ||< 78/75, 75/72 || || 2 || 133.33 ||= C# || 27/25 || +0.096 ||< 51/55, 42/39 || || 3 || 200 ||= D || 9/8 || -3.910 ||< 9/8 || || 4 || 266.67 ||= Eb || 7/6 || -0.204 ||< 75/64 || || 5 || 333.33 ||= D# || 17/14 or 40/33 || -2.796 +0.293 ||< 39/32 || || 6 || 400 ||= E || 5/4 or 44/35 || +13.686 +3.822 ||< 64/55 || || 7 || 466.67 ||= F || 21/16 || -4.114 ||< 21/16 || || 8 || 533.33 ||= Gb || 15/11 || -3.617 ||< 102/75 || || 9 || 600 ||= F# || 17/12 or 24/17 || -3.000 +3.000 ||< 17/12 || || 10 || 666.67 ||= G || 22/15 || +3.617 ||< 75/51 || || 11 || 733.33 ||= Hb || 32/21 || +4.114 ||< 32/21 || || 12 || 800 ||= G# || 8/5 or 35/22 || -13.686 -3.822 ||< 51/32 || || 13 || 866.67 ||= H || 28/17 or 33/20 || +2.796 -0.293 ||< 64/39 || || 14 || 933.33 ||= A || 12/7 || +0.204 ||< 55/32 || || 15 || 1000 ||= Bb || 16/9 || +3.910 ||< 16/9 || || 16 || 1066.67 ||= A# || 50/27 || -0.096 ||< 39/21 || || 17 || 1133.33 ||= B || 52/27 || -1.329 ||< 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 [[image:18-ED2-JI-approximations-2.png]] =[[#Notation]]Notation= 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¢ worse that the best approximation, which becomes the up-fifth. Using this 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. The second way preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo "on the fly". ||~ Degree ||~ Cents ||||||~ [[xenharmonic/Ups and Downs Notation|Up/down notation]] using the narrow 5th of 10\18, with major wider than minor ||||||~ Up/down notation using the narrow 5th of 10\18, with major narrower than minor || ||= 0 ||= 0 ||= perfect unison ||= P1 ||= D ||= perfect unison ||= P1 ||= D || ||= 1 ||= 67 ||= up unison, downminor 2nd ||= ^1, vm2 ||= D^, Ev ||= up unison, downmajor 2nd ||= ^1, vM2 ||= D^, Ev || ||= 2 ||= 133 ||= minor 2nd ||= m2 ||= E ||= major 2nd ||= M2 ||= E || ||= 3 ||= 200 ||= mid 2nd ||= ~2 ||= E^ ||= mid 2nd ||= ~2 ||= E^ || ||= 4 ||= 267 ||= major 2nd, minor 3rd ||= M2, m3 ||= E#, Fb ||= minor 2nd, major 3rd ||= m2, M3 ||= Eb, F# || ||= 5 ||= 333 ||= mid 3rd ||= ~3 ||= Fv ||= mid 3rd ||= ~3 ||= Fv || ||= 6 ||= 400 ||= major 3rd ||= M3 ||= F ||= minor 3rd ||= m3 ||= F || ||= 7 ||= 467 ||= upmajor 3rd, down 4th ||= ^M3, v4 ||= F^, Gv ||= upminor 3rd, down 4th ||= ^m3, v4 ||= F^, Gv || ||= 8 ||= 533 ||= perfect 4th ||= P4 ||= G ||= perfect 4th ||= P4 ||= G || ||= 9 ||= 600 ||= up 4th, down 5th ||= ^4, v5 ||= G^, Av ||= up 4th, down 5th ||= ^4, v5 ||= G^, Av || ||= 10 ||= 667 ||= perfect 5th ||= P5 ||= A ||= perfect 5th ||= P5 ||= A || ||= 11 ||= 733 ||= up 5th, downminor 6th ||= ^5, vm6 ||= A^, Bv ||= up fifth, downmajor 6th ||= ^5, vM6 ||= A^, Bv || ||= 12 ||= 800 ||= minor 6th ||= m6 ||= B ||= major 6th ||= M6 ||= B || ||= 13 ||= 867 ||= mid 6th ||= ~6 ||= B^ ||= mid 6th ||= ~6 ||= B^ || ||= 14 ||= 933 ||= major 6th, minor 7th ||= M6, m7 ||= B#, Cb ||= minor 6th, major 7th ||= m6, M7 ||= Bb, C# || ||= 15 ||= 1000 ||= mid 7th ||= ~7 ||= Cv ||= mid 7th ||= ~7 ||= Cv || ||= 16 ||= 1067 ||= major 7th ||= M7 ||= C ||= minor 7th ||= m7 ||= C || ||= 17 ||= 1133 ||= upmajor 7th, down 8ve ||= ^M7, v8 ||= C^, Dv ||= upminor 7th, down 8ve ||= ^m7, v8 ||= C^, Dv || ||= 18 ||= 1200 ||= perfect 8ve ||= P8 ||= D ||= perfect 8ve ||= P8 ||= D || For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated). ==<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: 1.1em;">Dodecatonic:</span>=== 6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1 ==<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 || ||= || | 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:26:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#Basic Properties">Basic Properties</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Notation">Notation</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: -->
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<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:<h1> --><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 >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 "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 <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, and 18edo can be treated as a 7-limit exotemperament with the mapping <18 29 42 51|. This maps 3/2 to 733.33¢ and 7/4 to 1000¢; as a result, 28/27 is tempered out, and weird things happen: 9/8 and 7/6 are both mapped to 266.67¢, while 8/7 gets mapped below both of them to 200¢, making for a rather disordered 7-limit tonality diamond, but hey, whatever floats your boat!<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:<h2> --><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 />
</td>
<td>5L3s Notation<br />
</td>
<td>Nearest Ratio<br />
</td>
<td>Error<br />
</td>
<td>17-Limit Ratios*<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td>1/1<br />
</td>
<td>0<br />
</td>
<td style="text-align: left;">1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>66.67<br />
</td>
<td style="text-align: center;">Db<br />
</td>
<td>27/26<br />
</td>
<td>+1.329<br />
</td>
<td style="text-align: left;">78/75, 75/72<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>133.33<br />
</td>
<td style="text-align: center;">C#<br />
</td>
<td>27/25<br />
</td>
<td>+0.096<br />
</td>
<td style="text-align: left;">51/55, 42/39<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>200<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td>9/8<br />
</td>
<td>-3.910<br />
</td>
<td style="text-align: left;">9/8<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>266.67<br />
</td>
<td style="text-align: center;">Eb<br />
</td>
<td>7/6<br />
</td>
<td>-0.204<br />
</td>
<td style="text-align: left;">75/64<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>333.33<br />
</td>
<td style="text-align: center;">D#<br />
</td>
<td>17/14 or 40/33<br />
</td>
<td>-2.796 +0.293<br />
</td>
<td style="text-align: left;">39/32<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>400<br />
</td>
<td style="text-align: center;">E<br />
</td>
<td>5/4 or 44/35<br />
</td>
<td>+13.686 +3.822<br />
</td>
<td style="text-align: left;">64/55<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>466.67<br />
</td>
<td style="text-align: center;">F<br />
</td>
<td>21/16<br />
</td>
<td>-4.114<br />
</td>
<td style="text-align: left;">21/16<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>533.33<br />
</td>
<td style="text-align: center;">Gb<br />
</td>
<td>15/11<br />
</td>
<td>-3.617<br />
</td>
<td style="text-align: left;">102/75<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>600<br />
</td>
<td style="text-align: center;">F#<br />
</td>
<td>17/12 or 24/17<br />
</td>
<td>-3.000 +3.000<br />
</td>
<td style="text-align: left;">17/12<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>666.67<br />
</td>
<td style="text-align: center;">G<br />
</td>
<td>22/15<br />
</td>
<td>+3.617<br />
</td>
<td style="text-align: left;">75/51<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>733.33<br />
</td>
<td style="text-align: center;">Hb<br />
</td>
<td>32/21<br />
</td>
<td>+4.114<br />
</td>
<td style="text-align: left;">32/21<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>800<br />
</td>
<td style="text-align: center;">G#<br />
</td>
<td>8/5 or 35/22<br />
</td>
<td>-13.686 -3.822<br />
</td>
<td style="text-align: left;">51/32<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>866.67<br />
</td>
<td style="text-align: center;">H<br />
</td>
<td>28/17 or 33/20<br />
</td>
<td>+2.796 -0.293<br />
</td>
<td style="text-align: left;">64/39<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>933.33<br />
</td>
<td style="text-align: center;">A<br />
</td>
<td>12/7<br />
</td>
<td>+0.204<br />
</td>
<td style="text-align: left;">55/32<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1000<br />
</td>
<td style="text-align: center;">Bb<br />
</td>
<td>16/9<br />
</td>
<td>+3.910<br />
</td>
<td style="text-align: left;">16/9<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>1066.67<br />
</td>
<td style="text-align: center;">A#<br />
</td>
<td>50/27<br />
</td>
<td>-0.096<br />
</td>
<td style="text-align: left;">39/21<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>1133.33<br />
</td>
<td style="text-align: center;">B<br />
</td>
<td>52/27<br />
</td>
<td>-1.329<br />
</td>
<td style="text-align: left;">75/39<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>1200<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td>2/1<br />
</td>
<td>0<br />
</td>
<td style="text-align: left;">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 />
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<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:41:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Notation" title="Anchor: Notation"/> --><a name="Notation"></a><!-- ws:end:WikiTextAnchorRule:41 -->Notation</h1>
<br />
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¢ worse that the best approximation, which becomes the up-fifth. Using this 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this.<br />
<br />
The first way preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.<br />
<br />
The second way preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo "on the fly".<br />
<br />
<table class="wiki_table">
<tr>
<th>Degree<br />
</th>
<th>Cents<br />
</th>
<th colspan="3"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">Up/down notation</a> using the narrow 5th of 10\18,<br />
with major wider than minor<br />
</th>
<th colspan="3">Up/down notation using the narrow 5th of 10\18,<br />
with major narrower than minor<br />
</th>
</tr>
<tr>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">perfect unison<br />
</td>
<td style="text-align: center;">P1<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">perfect unison<br />
</td>
<td style="text-align: center;">P1<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
<tr>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">67<br />
</td>
<td style="text-align: center;">up unison, downminor 2nd<br />
</td>
<td style="text-align: center;">^1, vm2<br />
</td>
<td style="text-align: center;">D^, Ev<br />
</td>
<td style="text-align: center;">up unison, downmajor 2nd<br />
</td>
<td style="text-align: center;">^1, vM2<br />
</td>
<td style="text-align: center;">D^, Ev<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">133<br />
</td>
<td style="text-align: center;">minor 2nd<br />
</td>
<td style="text-align: center;">m2<br />
</td>
<td style="text-align: center;">E<br />
</td>
<td style="text-align: center;">major 2nd<br />
</td>
<td style="text-align: center;">M2<br />
</td>
<td style="text-align: center;">E<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3<br />
</td>
<td style="text-align: center;">200<br />
</td>
<td style="text-align: center;">mid 2nd<br />
</td>
<td style="text-align: center;">~2<br />
</td>
<td style="text-align: center;">E^<br />
</td>
<td style="text-align: center;">mid 2nd<br />
</td>
<td style="text-align: center;">~2<br />
</td>
<td style="text-align: center;">E^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">267<br />
</td>
<td style="text-align: center;">major 2nd, minor 3rd<br />
</td>
<td style="text-align: center;">M2, m3<br />
</td>
<td style="text-align: center;">E#, Fb<br />
</td>
<td style="text-align: center;">minor 2nd, major 3rd<br />
</td>
<td style="text-align: center;">m2, M3<br />
</td>
<td style="text-align: center;">Eb, F#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5<br />
</td>
<td style="text-align: center;">333<br />
</td>
<td style="text-align: center;">mid 3rd<br />
</td>
<td style="text-align: center;">~3<br />
</td>
<td style="text-align: center;">Fv<br />
</td>
<td style="text-align: center;">mid 3rd<br />
</td>
<td style="text-align: center;">~3<br />
</td>
<td style="text-align: center;">Fv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6<br />
</td>
<td style="text-align: center;">400<br />
</td>
<td style="text-align: center;">major 3rd<br />
</td>
<td style="text-align: center;">M3<br />
</td>
<td style="text-align: center;">F<br />
</td>
<td style="text-align: center;">minor 3rd<br />
</td>
<td style="text-align: center;">m3<br />
</td>
<td style="text-align: center;">F<br />
</td>
</tr>
<tr>
<td style="text-align: center;">7<br />
</td>
<td style="text-align: center;">467<br />
</td>
<td style="text-align: center;">upmajor 3rd, down 4th<br />
</td>
<td style="text-align: center;">^M3, v4<br />
</td>
<td style="text-align: center;">F^, Gv<br />
</td>
<td style="text-align: center;">upminor 3rd, down 4th<br />
</td>
<td style="text-align: center;">^m3, v4<br />
</td>
<td style="text-align: center;">F^, Gv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8<br />
</td>
<td style="text-align: center;">533<br />
</td>
<td style="text-align: center;">perfect 4th<br />
</td>
<td style="text-align: center;">P4<br />
</td>
<td style="text-align: center;">G<br />
</td>
<td style="text-align: center;">perfect 4th<br />
</td>
<td style="text-align: center;">P4<br />
</td>
<td style="text-align: center;">G<br />
</td>
</tr>
<tr>
<td style="text-align: center;">9<br />
</td>
<td style="text-align: center;">600<br />
</td>
<td style="text-align: center;">up 4th, down 5th<br />
</td>
<td style="text-align: center;">^4, v5<br />
</td>
<td style="text-align: center;">G^, Av<br />
</td>
<td style="text-align: center;">up 4th, down 5th<br />
</td>
<td style="text-align: center;">^4, v5<br />
</td>
<td style="text-align: center;">G^, Av<br />
</td>
</tr>
<tr>
<td style="text-align: center;">10<br />
</td>
<td style="text-align: center;">667<br />
</td>
<td style="text-align: center;">perfect 5th<br />
</td>
<td style="text-align: center;">P5<br />
</td>
<td style="text-align: center;">A<br />
</td>
<td style="text-align: center;">perfect 5th<br />
</td>
<td style="text-align: center;">P5<br />
</td>
<td style="text-align: center;">A<br />
</td>
</tr>
<tr>
<td style="text-align: center;">11<br />
</td>
<td style="text-align: center;">733<br />
</td>
<td style="text-align: center;">up 5th, downminor 6th<br />
</td>
<td style="text-align: center;">^5, vm6<br />
</td>
<td style="text-align: center;">A^, Bv<br />
</td>
<td style="text-align: center;">up fifth, downmajor 6th<br />
</td>
<td style="text-align: center;">^5, vM6<br />
</td>
<td style="text-align: center;">A^, Bv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">12<br />
</td>
<td style="text-align: center;">800<br />
</td>
<td style="text-align: center;">minor 6th<br />
</td>
<td style="text-align: center;">m6<br />
</td>
<td style="text-align: center;">B<br />
</td>
<td style="text-align: center;">major 6th<br />
</td>
<td style="text-align: center;">M6<br />
</td>
<td style="text-align: center;">B<br />
</td>
</tr>
<tr>
<td style="text-align: center;">13<br />
</td>
<td style="text-align: center;">867<br />
</td>
<td style="text-align: center;">mid 6th<br />
</td>
<td style="text-align: center;">~6<br />
</td>
<td style="text-align: center;">B^<br />
</td>
<td style="text-align: center;">mid 6th<br />
</td>
<td style="text-align: center;">~6<br />
</td>
<td style="text-align: center;">B^<br />
</td>
</tr>
<tr>
<td style="text-align: center;">14<br />
</td>
<td style="text-align: center;">933<br />
</td>
<td style="text-align: center;">major 6th, minor 7th<br />
</td>
<td style="text-align: center;">M6, m7<br />
</td>
<td style="text-align: center;">B#, Cb<br />
</td>
<td style="text-align: center;">minor 6th, major 7th<br />
</td>
<td style="text-align: center;">m6, M7<br />
</td>
<td style="text-align: center;">Bb, C#<br />
</td>
</tr>
<tr>
<td style="text-align: center;">15<br />
</td>
<td style="text-align: center;">1000<br />
</td>
<td style="text-align: center;">mid 7th<br />
</td>
<td style="text-align: center;">~7<br />
</td>
<td style="text-align: center;">Cv<br />
</td>
<td style="text-align: center;">mid 7th<br />
</td>
<td style="text-align: center;">~7<br />
</td>
<td style="text-align: center;">Cv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">16<br />
</td>
<td style="text-align: center;">1067<br />
</td>
<td style="text-align: center;">major 7th<br />
</td>
<td style="text-align: center;">M7<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">minor 7th<br />
</td>
<td style="text-align: center;">m7<br />
</td>
<td style="text-align: center;">C<br />
</td>
</tr>
<tr>
<td style="text-align: center;">17<br />
</td>
<td style="text-align: center;">1133<br />
</td>
<td style="text-align: center;">upmajor 7th, down 8ve<br />
</td>
<td style="text-align: center;">^M7, v8<br />
</td>
<td style="text-align: center;">C^, Dv<br />
</td>
<td style="text-align: center;">upminor 7th, down 8ve<br />
</td>
<td style="text-align: center;">^m7, v8<br />
</td>
<td style="text-align: center;">C^, Dv<br />
</td>
</tr>
<tr>
<td style="text-align: center;">18<br />
</td>
<td style="text-align: center;">1200<br />
</td>
<td style="text-align: center;">perfect 8ve<br />
</td>
<td style="text-align: center;">P8<br />
</td>
<td style="text-align: center;">D<br />
</td>
<td style="text-align: center;">perfect 8ve<br />
</td>
<td style="text-align: center;">P8<br />
</td>
<td style="text-align: center;">D<br />
</td>
</tr>
</table>
<br />
For alternative notations, see <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs">Ups and Downs Notation -"Supersharp" EDOs</a> (pentatonic and nonatonic fifth-generated) and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators">Ups and Downs Notation - Natural Generators</a> (heptatonic third-generated).<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Notation-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:<h3> --><h3 id="toc4"><a name="Notation-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:<h3> --><h3 id="toc5"><a name="Notation-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 --><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:12:<h3> --><h3 id="toc6"><a name="Notation-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:<h3> --><h3 id="toc7"><a name="Notation-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:<h3> --><h3 id="toc8"><a name="Notation-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 />
<!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="Notation-Useful Moment-of-Symmetry Scales-Dodecatonic:"></a><!-- ws:end:WikiTextHeadingRule:18 --><span style="font-size: 1.1em;">Dodecatonic:</span></h3>
<br />
6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Notation-Application to Guitar"></a><!-- ws:end:WikiTextHeadingRule:20 --><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 "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).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:22 -->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> < 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 ><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;"><br />
</td>
<td>| 23 6 -14 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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:24:<h1> --><h1 id="toc12"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:24 -->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> => <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>