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 ||= 5L3s Notation ||||||= [[Ups and Downs Notation|up/down ]][[Ups and Downs Notation|notation]] based on 18b-edo, with sharp lower than flat || Nearest Ratio || Error ||< 17-Limit Ratios* || || 0 || 0 ||= C ||= C ||= unison ||= P1 || 1/1 || 0 ||< 1/1 || || 1 || 66.67 ||= Db ||= C^, Dv ||= up-unison, downmajor 2nd ||= ^P1, vM2 || 27/26 || +1.329 ||< 78/75, 75/72 || || 2 || 133.33 ||= C# ||= D ||= major 2nd ||= M2 || 27/25 || +0.096 ||< 51/55, 42/39 || || 3 || 200 ||= D ||= D^, Ev ||= mid 2nd ||= ~2 || 9/8 || -3.910 ||< 9/8 || || 4 || 266.67 ||= Eb ||= Db, E ||= minor 2nd, major 3rd ||= m2, M3 || 7/6 || -0.204 ||< 75/64 || || 5 || 333.33 ||= D# ||= E^ ||= mid 3rd ||= ~3 || 17/14 or 40/33 || -2.796 +0.293 ||< 39/32 || || 6 || 400 ||= E ||= Eb, F# ||= minor 3rd, augmented 4th ||= m3, A4 || 5/4 or 44/35 || +13.686 +3.822 ||< 64/55 || || 7 || 466.67 ||= F ||= Fv ||= upminor 3rd, down-fourth ||= ^m3, vP4 || 21/16 || -4.114 ||< 21/16 || || 8 || 533.33 ||= Gb ||= F ||= fourth ||= P4 || 15/11 || -3.617 ||< 102/75 || || 9 || 600 ||= F# ||= F^, Gv ||= upfourth, downfifth ||= ^P4, vP5 || 17/12 or 24/17 || -3.000 +3.000 ||< 17/12 || || 10 || 666.67 ||= G ||= G ||= fifthj ||= P5 || 22/15 || +3.617 ||< 75/51 || || 11 || 733.33 ||= Hb ||= G^ ||= upfifth, downmajor 6th ||= ^P5, vM6 || 32/21 || +4.114 ||< 32/21 || || 12 || 800 ||= G# ||= A ||= diminished 5th, major 6th ||= d5, M6 || 8/5 or 35/22 || -13.686 -3.822 ||< 51/32 || || 13 || 866.67 ||= H ||= A^, Bv ||= mid 6th ||= ~6 || 28/17 or 33/20 || +2.796 -0.293 ||< 64/39 || || 14 || 933.33 ||= A ||= B ||= minor 6th, major 7th ||= m6, M7 || 12/7 || +0.204 ||< 55/32 || || 15 || 1000 ||= Bb ||= B^ ||= mid 7th ||= ~7 || 16/9 || +3.910 ||< 16/9 || || 16 || 1066.67 ||= A# ||= Bb, C# ||= minor 7th ||= m7 || 50/27 || -0.096 ||< 39/21 || || 17 || 1133.33 ||= B ||= Cv ||= upminor 7th, down-octave ||= ^m7, vP8 || 52/27 || -1.329 ||< 75/39 || || 18 || 1200 ||= C ||= C ||= octave ||= 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 Like 16edo, 18b-edo can be notated two ways. The first is with sharp higher than flat, to preserve melodic contour: D * E * * * F * G * A * B * * * C * D D - D^/Ev - E - E^ - E#/Fb - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - B#/Cb - Cv - C - C^/Dv - D P1 - ^P1/vm2 - m2 - ~2 - mM2/m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vm6 - m6 - ~6 - M6/m7 - ~7 - M7 - ^M7/d8 - P8 The second way is with sharp lower than flat, to preserve interval arithmetic and chord names: D * E * * * F * G * A * B * * * C * D =[[#Notation]]Notation= 18edo can also 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. 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 approach 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". D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D P1 - ^1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/v4 - P4 - ^4/vP5 - P5 - ^5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m7/d8 - P8 ||= 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 wider than minor || ||= 0 ||= 0 ||= perfect unison ||= P1 ||= D ||= perfect unison ||= P1 ||= D || ||= 1 ||= 67 ||= up unison ||= ^1 ||= ||= up unison ||= ^1 ||= D^ || ||= 2 ||= 133 ||= minor 2nd ||= ||= ||= aug unison, major 2nd ||= A1, M2 ||= D#, E || ||= 3 ||= 200 ||= mid 2nd ||= ||= ||= mid 2nd ||= ~2 ||= E^ || ||= 4 ||= 267 ||= major 2nd, minor 3rd ||= ||= ||= minor 2nd, major 3rd ||= m2, M3 ||= Eb, F# || ||= 5 ||= 333 ||= mid 3rd ||= ||= ||= mid 3rd ||= ~3 ||= Fv || ||= 6 ||= 400 ||= major 3rd ||= ||= ||= minor 3rd ||= m3 ||= F || ||= 7 ||= 467 ||= upmajor 3rd, down 4th ||= ||= ||= upminor 3rd, down 4th ||= ^m3, v4 ||= F^, Gv || ||= 8 ||= 533 ||= perfect 4th ||= ||= G ||= perfect 4th ||= P4 ||= G || ||= 9 ||= 600 ||= up 4th, down 5th ||= ||= ||= up 4th, down 5th ||= ^4, v5 ||= G^, Av || ||= 10 ||= 667 ||= perfect 5th ||= ||= A ||= perfect 5th ||= P5 ||= A || ||= 11 ||= 733 ||= up 5th ||= ||= ||= up fifth, downmajor 6th ||= ^5, vM6 ||= A^, vB || ||= 12 ||= 800 ||= minor 6th ||= ||= ||= major 6th ||= M6 ||= B || ||= 13 ||= 867 ||= mid 6th ||= ||= ||= mid 6th ||= ~6 ||= Bv || ||= 14 ||= 933 ||= major 6th, minor 7th ||= ||= ||= minor 6th, major 7th ||= m6, M7 ||= Bb, C# || ||= 15 ||= 1000 ||= mid 7th ||= ||= ||= mid 7th ||= ~7 ||= Cv || ||= 16 ||= 1067 ||= major 7th ||= ||= ||= minor 7th, dim 8ve ||= m7, d8 ||= C, Db || ||= 17 ||= 1133 ||= down 8ve ||= ||= ||= down 8ve ||= v8 ||= Dv || ||= 18 ||= 1200 ||= perfect 8ve ||= ||= ||= 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, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated). 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 || ||= || | 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:24:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><a href="#Basic Properties">Basic Properties</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Notation">Notation</a><!-- 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: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: -->
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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. 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.<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 style="text-align: center;">5L3s Notation<br />
</td>
<td colspan="3" style="text-align: center;"><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">up/down </a><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">notation</a><br />
based on 18b-edo, with<br />
sharp lower than flat<br />
</td>
<td>Nearest Ratio<br />
</td>
<td>Error<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;">C<br />
</td>
<td style="text-align: center;">C<br />
</td>
<td style="text-align: center;">unison<br />
</td>
<td style="text-align: center;">P1<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 style="text-align: center;">C^, Dv<br />
</td>
<td style="text-align: center;">up-unison,<br />
downmajor 2nd<br />
</td>
<td style="text-align: center;">^P1, vM2<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 style="text-align: center;">D<br />
</td>
<td style="text-align: center;">major 2nd<br />
</td>
<td style="text-align: center;">M2<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 style="text-align: center;">D^, Ev<br />
</td>
<td style="text-align: center;">mid 2nd<br />
</td>
<td style="text-align: center;">~2<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 style="text-align: center;">Db, E<br />
</td>
<td style="text-align: center;">minor 2nd,<br />
major 3rd<br />
</td>
<td style="text-align: center;">m2, M3<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 style="text-align: center;">E^<br />
</td>
<td style="text-align: center;">mid 3rd<br />
</td>
<td style="text-align: center;">~3<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 style="text-align: center;">Eb, F#<br />
</td>
<td style="text-align: center;">minor 3rd,<br />
augmented 4th<br />
</td>
<td style="text-align: center;">m3, A4<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 style="text-align: center;">Fv<br />
</td>
<td style="text-align: center;">upminor 3rd,<br />
down-fourth<br />
</td>
<td style="text-align: center;">^m3, vP4<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 style="text-align: center;">F<br />
</td>
<td style="text-align: center;">fourth<br />
</td>
<td style="text-align: center;">P4<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 style="text-align: center;">F^, Gv<br />
</td>
<td style="text-align: center;">upfourth,<br />
downfifth<br />
</td>
<td style="text-align: center;">^P4, vP5<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 style="text-align: center;">G<br />
</td>
<td style="text-align: center;">fifthj<br />
</td>
<td style="text-align: center;">P5<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 style="text-align: center;">G^<br />
</td>
<td style="text-align: center;">upfifth,<br />
downmajor 6th<br />
</td>
<td style="text-align: center;">^P5, vM6<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 style="text-align: center;">A<br />
</td>
<td style="text-align: center;">diminished 5th,<br />
major 6th<br />
</td>
<td style="text-align: center;">d5, M6<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 style="text-align: center;">A^, Bv<br />
</td>
<td style="text-align: center;">mid 6th<br />
</td>
<td style="text-align: center;">~6<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 style="text-align: center;">B<br />
</td>
<td style="text-align: center;">minor 6th,<br />
major 7th<br />
</td>
<td style="text-align: center;">m6, M7<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 style="text-align: center;">B^<br />
</td>
<td style="text-align: center;">mid 7th<br />
</td>
<td style="text-align: center;">~7<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 style="text-align: center;">Bb, C#<br />
</td>
<td style="text-align: center;">minor 7th<br />
</td>
<td style="text-align: center;">m7<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 style="text-align: center;">Cv<br />
</td>
<td style="text-align: center;">upminor 7th,<br />
down-octave<br />
</td>
<td style="text-align: center;">^m7, vP8<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 style="text-align: center;">C<br />
</td>
<td style="text-align: center;">octave<br />
</td>
<td style="text-align: center;">P8<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 />
Like 16edo, 18b-edo can be notated two ways. The first is with sharp higher than flat, to preserve melodic contour:<br />
D * E * * * F * G * A * B * * * C * D<br />
D - D^/Ev - E - E^ - E#/Fb - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - B#/Cb - Cv - C - C^/Dv - D<br />
P1 - ^P1/vm2 - m2 - ~2 - mM2/m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vm6 - m6 - ~6 - M6/m7 - ~7 - M7 - ^M7/d8 - P8<br />
<br />
The second way is with sharp lower than flat, to preserve interval arithmetic and chord names:<br />
D * E * * * F * G * A * B * * * C * D<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:38:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Notation" title="Anchor: Notation"/> --><a name="Notation"></a><!-- ws:end:WikiTextAnchorRule:38 -->Notation</h1>
<br />
18edo can also 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. 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 <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 approach 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 />
D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D<br />
P1 - ^1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/v4 - P4 - ^4/vP5 - P5 - ^5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m7/d8 - P8<br />
<table class="wiki_table">
<tr>
<td style="text-align: center;">Degree<br />
</td>
<td style="text-align: center;">Cents<br />
</td>
<td colspan="3" style="text-align: center;"><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 />
</td>
<td colspan="3" style="text-align: center;">Up/down notation using the narrow 5th of 10\18,<br />
with major wider than minor<br />
</td>
</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<br />
</td>
<td style="text-align: center;">^1<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">up unison<br />
</td>
<td style="text-align: center;">^1<br />
</td>
<td style="text-align: center;">D^<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;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">aug unison, major 2nd<br />
</td>
<td style="text-align: center;">A1, M2<br />
</td>
<td style="text-align: center;">D#, 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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><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;"><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;"><br />
</td>
<td style="text-align: center;"><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;"><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<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><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^, vB<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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">mid 6th<br />
</td>
<td style="text-align: center;">~6<br />
</td>
<td style="text-align: center;">Bv<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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><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;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">minor 7th, dim 8ve<br />
</td>
<td style="text-align: center;">m7, d8<br />
</td>
<td style="text-align: center;">C, Db<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;">down 8ve<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;">down 8ve<br />
</td>
<td style="text-align: center;">v8<br />
</td>
<td style="text-align: center;">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;"><br />
</td>
<td style="text-align: center;"><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 />
<br />
<br />
<br />
<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, octotonic 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 />
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 -"Superflat" 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 -"Supersharp" 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: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 />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Notation-Application to Guitar"></a><!-- ws:end:WikiTextHeadingRule:18 --><span style="font-size: 1.3em;">Application to Guitar</span></h2>
18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!<br />
<br />
The "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:20:<h1> --><h1 id="toc10"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:20 -->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:22:<h1> --><h1 id="toc11"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:22 -->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>