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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:hstraub|hstraub]] and made on <tt>2015-08-13 02:41:05 UTC</tt>.<br>
| | | en = 18edo |
| : The original revision id was <tt>556605395</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">=<span style="display: block; text-align: right;">[[18平均律|日本語]] </span>=
| | {{ED intro}} |
| =18 Equal Divisions of the Octave=
| |
| 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 >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). |
|
| |
|
| 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.
| | 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. |
|
| |
|
| 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.
| | 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]]. |
|
| |
|
| ===Representations of Just Intervals===
| | 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. |
| || Degree || Cents
| |
| DMS ||= 5L3s Notation || Nearest Ratio || Error (cents)
| |
| (DMS) || 17-Limit Ratios* ||
| |
| || 0 || 0 ||= **C** || 1/1 || 0 || **1/1** ||
| |
| || 1 || 66.667
| |
| 20° ||= Db || 27/26 || +1.329
| |
| +23'56" ||> 78/75, 75/72 ||
| |
| || 2 || 133.333
| |
| 40° ||= C# || 27/25 || +0.096
| |
| +1'43" ||> 51/55, 42/39 ||
| |
| || 3 || 200
| |
| 60° ||= **D** || 9/8 || -3.910
| |
| -1°10'23" || **9/8** ||
| |
| || 4 || 266.667
| |
| 80° ||= Eb || 7/6 || -0.204
| |
| -1"50" || **75/64** ||
| |
| || 5 || 333.333
| |
| 100° ||= D# || 17/14 or 40/33 || -2.796 +0.293
| |
| -50'20" +5'16" || **39/32** ||
| |
| || 6 || 400
| |
| 120° ||= **E** || 5/4 or 44/35 || +13.686 +3.822
| |
| +4°6'21" +1°8'47" ||> 64/55 ||
| |
| || 7 || 466.667
| |
| 140° ||= **F** || 21/16 || -4.114
| |
| -1°14'3" || **21/16** ||
| |
| || 8 || 533.333
| |
| 160° ||= Gb || 15/11 || -3.617
| |
| -1°5'7" ||> 102/75 ||
| |
| || 9 || 600
| |
| 180° ||= F# || 17/12 or 24/17 || -3.000 +3.000
| |
| -54' +54' ||> 17/12 ||
| |
| || 10 || 666.667
| |
| 200° ||= **G** || 22/15 || +3.617
| |
| +1°5'7" ||> 75/51 ||
| |
| || 11 || 733.333
| |
| 220° ||= Hb || 32/21 || +4.114
| |
| +1°14'3" ||> 32/21 ||
| |
| || 12 || 800
| |
| 240° ||= G# || 8/5 or 35/22 || -13.686 -3.8222
| |
| -4°6'21" -1°8'47" || **51/32** ||
| |
| || 13 || 866.667
| |
| 260° ||= **H** || 28/17 or 33/20 || +2.796 -0.293
| |
| <span style="line-height: 15.6000003814697px;">+</span>50'20" <span style="line-height: 15.6000003814697px;">-</span><span style="line-height: 1.5;">5'16"</span> ||> 64/39 ||
| |
| || 14 || 933.333
| |
| 280° ||= **A** || 12/7 || +0.204
| |
| +1"50" || **55/32** ||
| |
| || 15 || 1000
| |
| 300° ||= Bb || 16/9 || +3.910
| |
| +1°10'23" ||> 16/9 ||
| |
| || 16 || 1066.667
| |
| 320° ||= A# || 50/27 || -0.096
| |
| -1'43" ||> 39/21 ||
| |
| || 17 || 1133.333
| |
| 340° ||= **B** || 52/27 || -1.329
| |
| -23'56" ||> 75/39 ||
| |
| || 18 || 1200
| |
| 360° ||= **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
| |
|
| |
|
| ==<span style="font-size: 1.3em;">Useful Moment-of-Symmetry Scales</span>==
| | 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. |
| 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>===
| |
| 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
| |
| ===<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>== | | === Odd harmonics === |
| 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! | | {{Harmonics in equal|18}} |
|
| |
|
| 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).
| | == Intervals == |
|
| |
|
| ==Commas==
| | [[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png|thumb]] |
| 18 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] < 18 29 42 51 62 67 |.)
| |
| ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
| |
| ||= 128/125 || | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||
| |
| ||= 1212717/1210381 || | 23 6 -14 > ||> 3.34 ||= Vishnuzma ||= Semisuper ||
| |
| ||= 50/49 || | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||
| |
| ||= 686/675 || | 1 -3 -2 3 > ||> 27.99 ||= Senga ||= ||
| |
| ||= 875/864 || | -5 -3 3 1 > ||> 21.90 ||= Keema ||= ||
| |
| ||= 1728/1715 || | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||
| |
| ||= 16875/16807 || | 0 3 4 -5 > ||> 6.99 ||= Mirkwai ||= ||
| |
| ||= 3136/3125 || | 6 0 -5 2 > ||> 6.08 ||= Hemimean ||= ||
| |
| ||= 99/98 || | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= ||
| |
| ||= 100/99 || | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||= ||
| |
| ||= 65536/65219 || | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||= ||
| |
| ||= 385/384 || | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||= ||
| |
| ||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0.18 ||= Kalisma ||= Gauss' Comma ||
| |
| ||= 91/90 || | -1 -2 -1 1 1 > ||> 19.13 ||= Superleap ||= ||
| |
|
| |
|
| ==Listen== | | {| class="wikitable center-all right-2" |
| * [[http://www.h-pi.com/mp3/18ETPrelude.mp3|18ETPrelude]] by [[Aaron Andrew Hunt]]
| | ! Degree |
| * [[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]]
| | ! Cents |
| * [[http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3|Flippertronics]] by Chris Vaisvil
| | ! Nearest Ratio |
| * [[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)]]
| | ! Error |
| * [[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]])</pre></div>
| | ! 17-Limit Ratios <ref>based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament</ref> |
| <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"><html><head><title>18edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x日本語"></a><!-- ws:end:WikiTextHeadingRule:0 --><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> </span></h1>
| | | 0 |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x18 Equal Divisions of the Octave"></a><!-- ws:end:WikiTextHeadingRule:2 -->18 Equal Divisions of the Octave</h1>
| | | 0.000 |
| AKA The Third-Tone System<br />
| | | 1/1 |
| <br />
| | | 0 |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:4 -->Basic Properties</h2>
| | | 1/1 |
| 18-EDO divides the octave into 18 equal parts of ~66.667 cents each. It does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6).<br />
| | |- |
| <br />
| | | 1 |
| In order to access the excellent consonances actually available, one must take a considerably &quot;non-common-practice&quot; approach, meaning to avoid the usual closed-voice &quot;root-3rd-5th&quot; type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of the 17-limit <a class="wiki_link" href="/k%2AN%20subgroups">4*18 subgroup</a> <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full <a class="wiki_link" href="/17-limit">17-limit</a>, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.<br />
| | | 66.667 |
| <br />
| | | 27/26 |
| However, less accurate approximations can be used. 18 equal does temper out 28/27, which makes three &quot;fifths&quot; (ie. 3/2) up, a 7/4. Thus 9/8 = a near just 7/6 (and what the relatively accurate 200 cents as 9/8, is in fact 8/7 - what do you make of that? Music.) This treatment applies to the scale generated by the large fifth, known as Father. One, if one really gets into it, can generate scales from the 3/2 and the half octave: with all the sharpness, what's 18e going to hurt? Call 600 cents 11/8 and 866 cents 13/8. Hey it's possible, lots of people like mavila.<br />
| | | +1.329 |
| <br />
| | | 26/25, 25/24 |
| 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.<br />
| | |- |
| <br />
| | | 2 |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x18 Equal Divisions of the Octave-Basic Properties-Representations of Just Intervals"></a><!-- ws:end:WikiTextHeadingRule:6 -->Representations of Just Intervals</h3>
| | | 133.333 |
|
| | | 27/25 |
| | | +0.096 |
| | | 55/51, 14/13 |
| | |- |
| | | 3 |
| | | 200.000 |
| | | 9/8 |
| | | -3.910 |
| | | 9/8 |
| | |- |
| | | 4 |
| | | 266.667 |
| | | 7/6 |
| | | -0.204 |
| | | 75/64 |
| | |- |
| | | 5 |
| | | 333.333 |
| | | 17/14 or 40/33 |
| | | -2.796 +0.293 |
| | | 39/32 |
| | |- |
| | | 6 |
| | | 400.000 |
| | | 5/4 or 44/35 |
| | | +13.686 +3.822 |
| | | 64/55 |
| | |- |
| | | 7 |
| | | 466.667 |
| | | 21/16 |
| | | -4.114 |
| | | 21/16 |
| | |- |
| | | 8 |
| | | 533.333 |
| | | 15/11 |
| | | -3.617 |
| | | 34/25 |
| | |- |
| | | 9 |
| | | 600.000 |
| | | 17/12 or 24/17 |
| | | -3.000 +3.000 |
| | | 17/12 |
| | |- |
| | | 10 |
| | | 666.667 |
| | | 22/15 |
| | | +3.617 |
| | | 25/17 |
| | |- |
| | | 11 |
| | | 733.333 |
| | | 32/21 |
| | | +4.114 |
| | | 32/21 |
| | |- |
| | | 12 |
| | | 800.000 |
| | | 8/5 or 35/22 |
| | | -13.686 -3.822 |
| | | 51/32 |
| | |- |
| | | 13 |
| | | 866.667 |
| | | 28/17 or 33/20 |
| | | +2.796 -0.293 |
| | | 64/39 |
| | |- |
| | | 14 |
| | | 933.333 |
| | | 12/7 |
| | | +0.204 |
| | | 55/32 |
| | |- |
| | | 15 |
| | | 1000.000 |
| | | 16/9 |
| | | +3.910 |
| | | 16/9 |
| | |- |
| | | 16 |
| | | 1066.667 |
| | | 50/27 |
| | | -0.096 |
| | | 13/7 |
| | |- |
| | | 17 |
| | | 1133.333 |
| | | 52/27 |
| | | -1.329 |
| | | 25/13 |
| | |- |
| | | 18 |
| | | 1200.000 |
| | | 2/1 |
| | | 0 |
| | | 2/1** |
| | |} |
| | <references /> |
| | {{Clear}} |
| | == Notation == |
| | === Ups and downs 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{{c}} worse that the best approximation, which becomes the up-fifth. |
| | {{Mavila}} |
| | {| class="wikitable center-all right-2" |
| | ! Degree |
| | ! Cents |
| | ! colspan="3" | [[Ups and downs notation|Up/down notation]] using the narrow 5th of 10\18, <br> with major wider than minor |
| | ! colspan="3" | Up/down notation using the narrow 5th of 10\18, <br> with major narrower than minor |
| | ! 5L3s Notation |
| | |- |
| | | 0 |
| | | 0 |
| | | perfect unison |
| | | P1 |
| | | D |
| | | perfect unison |
| | | P1 |
| | | D |
| | | C |
| | |- |
| | | 1 |
| | | 67 |
| | | up unison, downminor 2nd |
| | | ^1, vm2 |
| | | ^D, vE |
| | | up unison, downmajor 2nd |
| | | ^1, vM2 |
| | | ^D, vE |
| | | Db |
| | |- |
| | | 2 |
| | | 133 |
| | | minor 2nd |
| | | m2 |
| | | E |
| | | major 2nd |
| | | M2 |
| | | E |
| | | C# |
| | |- |
| | | 3 |
| | | 200 |
| | | mid 2nd |
| | | ~2 |
| | | ^E |
| | | mid 2nd |
| | | ~2 |
| | | ^E |
| | | D |
| | |- |
| | | 4 |
| | | 267 |
| | | major 2nd, minor 3rd |
| | | M2, m3 |
| | | E#, Fb |
| | | minor 2nd, major 3rd |
| | | m2, M3 |
| | | Eb, F# |
| | | Eb |
| | |- |
| | | 5 |
| | | 333 |
| | | mid 3rd |
| | | ~3 |
| | | vF |
| | | mid 3rd |
| | | ~3 |
| | | vF |
| | | D# |
| | |- |
| | | 6 |
| | | 400 |
| | | major 3rd |
| | | M3 |
| | | F |
| | | minor 3rd |
| | | m3 |
| | | F |
| | | E |
| | |- |
| | | 7 |
| | | 467 |
| | | upmajor 3rd, down 4th |
| | | ^M3, v4 |
| | | ^F, vG |
| | | upminor 3rd, down 4th |
| | | ^m3, v4 |
| | | ^F, vG |
| | | F |
| | |- |
| | | 8 |
| | | 533 |
| | | perfect 4th |
| | | P4 |
| | | G |
| | | perfect 4th |
| | | P4 |
| | | G |
| | | Gb |
| | |- |
| | | 9 |
| | | 600 |
| | | up 4th, down 5th |
| | | ^4, v5 |
| | | ^G, vA |
| | | up 4th, down 5th |
| | | ^4, v5 |
| | | ^G, vA |
| | | F# |
| | |- |
| | | 10 |
| | | 667 |
| | | perfect 5th |
| | | P5 |
| | | A |
| | | perfect 5th |
| | | P5 |
| | | A |
| | | G |
| | |- |
| | | 11 |
| | | 733 |
| | | up 5th, downminor 6th |
| | | ^5, vm6 |
| | | ^A, vB |
| | | up fifth, downmajor 6th |
| | | ^5, vM6 |
| | | ^A, vB |
| | | Hb |
| | |- |
| | | 12 |
| | | 800 |
| | | minor 6th |
| | | m6 |
| | | B |
| | | major 6th |
| | | M6 |
| | | B |
| | | G# |
| | |- |
| | | 13 |
| | | 867 |
| | | mid 6th |
| | | ~6 |
| | | ^B |
| | | mid 6th |
| | | ~6 |
| | | ^B |
| | | H |
| | |- |
| | | 14 |
| | | 933 |
| | | major 6th, minor 7th |
| | | M6, m7 |
| | | B#, Cb |
| | | minor 6th, major 7th |
| | | m6, M7 |
| | | 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 |
| | |} |
|
| |
|
| <table class="wiki_table">
| | 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. |
| <tr>
| |
| <td>Degree<br />
| |
| </td>
| |
| <td>Cents<br />
| |
| DMS<br />
| |
| </td>
| |
| <td style="text-align: center;">5L3s Notation<br />
| |
| </td>
| |
| <td>Nearest Ratio<br />
| |
| </td>
| |
| <td>Error (cents)<br />
| |
| (DMS)<br />
| |
| </td>
| |
| <td>17-Limit Ratios*<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>C</strong><br />
| |
| </td>
| |
| <td>1/1<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td><strong>1/1</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>66.667<br />
| |
| 20°<br />
| |
| </td>
| |
| <td style="text-align: center;">Db<br />
| |
| </td>
| |
| <td>27/26<br />
| |
| </td>
| |
| <td>+1.329<br />
| |
| +23'56&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">78/75, 75/72<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>133.333<br />
| |
| 40°<br />
| |
| </td>
| |
| <td style="text-align: center;">C#<br />
| |
| </td>
| |
| <td>27/25<br />
| |
| </td>
| |
| <td>+0.096<br />
| |
| +1'43&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">51/55, 42/39<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>200<br />
| |
| 60°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>D</strong><br />
| |
| </td>
| |
| <td>9/8<br />
| |
| </td>
| |
| <td>-3.910<br />
| |
| -1°10'23&quot;<br />
| |
| </td>
| |
| <td><strong>9/8</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>266.667<br />
| |
| 80°<br />
| |
| </td>
| |
| <td style="text-align: center;">Eb<br />
| |
| </td>
| |
| <td>7/6<br />
| |
| </td>
| |
| <td>-0.204<br />
| |
| -1&quot;50&quot;<br />
| |
| </td>
| |
| <td><strong>75/64</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>333.333<br />
| |
| 100°<br />
| |
| </td>
| |
| <td style="text-align: center;">D#<br />
| |
| </td>
| |
| <td>17/14 or 40/33<br />
| |
| </td>
| |
| <td>-2.796 +0.293<br />
| |
| -50'20&quot; +5'16&quot;<br />
| |
| </td>
| |
| <td><strong>39/32</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>400<br />
| |
| 120°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>E</strong><br />
| |
| </td>
| |
| <td>5/4 or 44/35<br />
| |
| </td>
| |
| <td>+13.686 +3.822<br />
| |
| +4°6'21&quot; +1°8'47&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">64/55<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>466.667<br />
| |
| 140°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>F</strong><br />
| |
| </td>
| |
| <td>21/16<br />
| |
| </td>
| |
| <td>-4.114<br />
| |
| -1°14'3&quot;<br /> | |
| </td>
| |
| <td><strong>21/16</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>533.333<br />
| |
| 160°<br />
| |
| </td>
| |
| <td style="text-align: center;">Gb<br />
| |
| </td>
| |
| <td>15/11<br />
| |
| </td>
| |
| <td>-3.617<br />
| |
| -1°5'7&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">102/75<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>600<br />
| |
| 180°<br />
| |
| </td>
| |
| <td style="text-align: center;">F#<br />
| |
| </td>
| |
| <td>17/12 or 24/17<br />
| |
| </td>
| |
| <td>-3.000 +3.000<br />
| |
| -54' +54'<br />
| |
| </td>
| |
| <td style="text-align: right;">17/12<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>666.667<br />
| |
| 200°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>G</strong><br />
| |
| </td>
| |
| <td>22/15<br />
| |
| </td>
| |
| <td>+3.617<br />
| |
| +1°5'7&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">75/51<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>733.333<br />
| |
| 220°<br />
| |
| </td>
| |
| <td style="text-align: center;">Hb<br />
| |
| </td>
| |
| <td>32/21<br />
| |
| </td>
| |
| <td>+4.114<br />
| |
| +1°14'3&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">32/21<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>800<br />
| |
| 240°<br />
| |
| </td>
| |
| <td style="text-align: center;">G#<br />
| |
| </td>
| |
| <td>8/5 or 35/22<br />
| |
| </td>
| |
| <td>-13.686 -3.8222<br />
| |
| -4°6'21&quot; -1°8'47&quot;<br />
| |
| </td>
| |
| <td><strong>51/32</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>866.667<br />
| |
| 260°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>H</strong><br />
| |
| </td>
| |
| <td>28/17 or 33/20<br />
| |
| </td>
| |
| <td>+2.796 -0.293<br />
| |
| <span style="line-height: 15.6000003814697px;">+</span>50'20&quot; <span style="line-height: 15.6000003814697px;">-</span><span style="line-height: 1.5;">5'16&quot;</span><br />
| |
| </td>
| |
| <td style="text-align: right;">64/39<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>933.333<br />
| |
| 280°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>A</strong><br />
| |
| </td>
| |
| <td>12/7<br />
| |
| </td>
| |
| <td>+0.204<br />
| |
| +1&quot;50&quot;<br />
| |
| </td>
| |
| <td><strong>55/32</strong><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>1000<br />
| |
| 300°<br />
| |
| </td>
| |
| <td style="text-align: center;">Bb<br />
| |
| </td>
| |
| <td>16/9<br />
| |
| </td>
| |
| <td>+3.910<br />
| |
| +1°10'23&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">16/9<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>1066.667<br />
| |
| 320°<br />
| |
| </td>
| |
| <td style="text-align: center;">A#<br />
| |
| </td>
| |
| <td>50/27<br />
| |
| </td>
| |
| <td>-0.096<br />
| |
| -1'43&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">39/21<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>1133.333<br />
| |
| 340°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>B</strong><br />
| |
| </td>
| |
| <td>52/27<br />
| |
| </td>
| |
| <td>-1.329<br />
| |
| -23'56&quot;<br />
| |
| </td>
| |
| <td style="text-align: right;">75/39<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>1200<br />
| |
| 360°<br />
| |
| </td>
| |
| <td style="text-align: center;"><strong>C</strong><br />
| |
| </td>
| |
| <td>2/1<br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td><strong>2/1</strong><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| *based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament<br />
| | '''<u>Pentatonic 5th-generated:</u> D * * * E * * G * * * A * * C * * * D''' (generator = wide 3/2 = 11\18 = perfect 5thoid) |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales"></a><!-- ws:end:WikiTextHeadingRule:8 --><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:10:&lt;h3&gt; --><h3 id="toc5"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 --><span style="font-size: 1.1em;">Pentatonic:</span></h3>
| |
| 3L2s Father Pentatonic: 4 4 3 4 3<br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:12 --><span style="font-size: 1.1em;">Hexatonic:</span></h3>
| |
| 6-Equal Whole-Tone Scale: 3 3 3 3 3 3<br />
| |
| 4L2s Bicycle: 4 4 1 4 4 1<br />
| |
| 2L4s Rice Hexatonic: 2 5 2 2 5 2<br />
| |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonic:"></a><!-- ws:end:WikiTextHeadingRule:14 --><span style="font-size: 1.1em;">Heptatonic:</span></h3>
| |
| 4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2<br />
| |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonic:"></a><!-- ws:end:WikiTextHeadingRule:16 --><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:18:&lt;h3&gt; --><h3 id="toc9"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonic:"></a><!-- ws:end:WikiTextHeadingRule:18 --><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:20:&lt;h2&gt; --><h2 id="toc10"><a name="x18 Equal Divisions of the Octave-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 &quot;Father Octatonic&quot; scale maps very simply to a 6-string guitar tuned in &quot;reverse-standard&quot; tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x18 Equal Divisions of the Octave-Commas"></a><!-- ws:end:WikiTextHeadingRule:22 -->Commas</h2>
| |
| 18 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the <a class="wiki_link" href="/val">val</a> &lt; 18 29 42 51 62 67 |.)<br />
| |
|
| |
|
| | D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D |
|
| |
|
| <table class="wiki_table">
| | P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d (s = sub-, d = -oid) |
| <tr>
| |
| <th>Comma<br />
| |
| </th>
| |
| <th>Monzo<br />
| |
| </th>
| |
| <th>Value (Cents)<br />
| |
| </th>
| |
| <th>Name 1<br />
| |
| </th>
| |
| <th>Name 2<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">128/125<br />
| |
| </td>
| |
| <td>| 7 0 -3 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">41.06<br />
| |
| </td>
| |
| <td style="text-align: center;">Diesis<br />
| |
| </td>
| |
| <td style="text-align: center;">Augmented Comma<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">1212717/1210381<br />
| |
| </td>
| |
| <td>| 23 6 -14 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">3.34<br />
| |
| </td>
| |
| <td style="text-align: center;">Vishnuzma<br />
| |
| </td>
| |
| <td style="text-align: center;">Semisuper<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">50/49<br />
| |
| </td>
| |
| <td>| 1 0 2 -2 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">34.98<br />
| |
| </td>
| |
| <td style="text-align: center;">Tritonic Diesis<br />
| |
| </td>
| |
| <td style="text-align: center;">Jubilisma<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">686/675<br />
| |
| </td>
| |
| <td>| 1 -3 -2 3 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">27.99<br />
| |
| </td>
| |
| <td style="text-align: center;">Senga<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">875/864<br />
| |
| </td>
| |
| <td>| -5 -3 3 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">21.90<br />
| |
| </td>
| |
| <td style="text-align: center;">Keema<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">1728/1715<br />
| |
| </td>
| |
| <td>| 6 3 -1 -3 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">13.07<br />
| |
| </td>
| |
| <td style="text-align: center;">Orwellisma<br />
| |
| </td>
| |
| <td style="text-align: center;">Orwell Comma<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">16875/16807<br />
| |
| </td>
| |
| <td>| 0 3 4 -5 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">6.99<br />
| |
| </td>
| |
| <td style="text-align: center;">Mirkwai<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">3136/3125<br />
| |
| </td>
| |
| <td>| 6 0 -5 2 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">6.08<br />
| |
| </td>
| |
| <td style="text-align: center;">Hemimean<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">99/98<br />
| |
| </td>
| |
| <td>| -1 2 0 -2 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">17.58<br />
| |
| </td>
| |
| <td style="text-align: center;">Mothwellsma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">100/99<br />
| |
| </td>
| |
| <td>| 2 -2 2 0 -1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">17.40<br />
| |
| </td>
| |
| <td style="text-align: center;">Ptolemisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">65536/65219<br />
| |
| </td>
| |
| <td>| 16 0 0 -2 -3 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">8.39<br />
| |
| </td>
| |
| <td style="text-align: center;">Orgonisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">385/384<br />
| |
| </td>
| |
| <td>| -7 -1 1 1 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">4.50<br />
| |
| </td>
| |
| <td style="text-align: center;">Keenanisma<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">9801/9800<br />
| |
| </td>
| |
| <td>| -3 4 -2 -2 2 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">0.18<br />
| |
| </td>
| |
| <td style="text-align: center;">Kalisma<br />
| |
| </td>
| |
| <td style="text-align: center;">Gauss' Comma<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">91/90<br />
| |
| </td>
| |
| <td>| -1 -2 -1 1 1 &gt;<br />
| |
| </td>
| |
| <td style="text-align: right;">19.13<br />
| |
| </td>
| |
| <td style="text-align: center;">Superleap<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx... |
| <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="x18 Equal Divisions of the Octave-Listen"></a><!-- ws:end:WikiTextHeadingRule:24 -->Listen</h2>
| | |
| <ul><li><a class="wiki_link_ext" href="http://www.h-pi.com/mp3/18ETPrelude.mp3" rel="nofollow">18ETPrelude</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3" rel="nofollow">Prelude in 18et</a> by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank">Chris Vaisvil</a> =&gt; <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=3" rel="nofollow" target="_blank">composer notes</a></li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3" rel="nofollow">Flippertronics</a> by Chris Vaisvil</li><li><a class="wiki_link_ext" href="http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3" rel="nofollow">Gerbils at the Wheel of Government</a> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1402" rel="nofollow" target="_blank">Chris Vaisvil (in 9 and 18 edo simultaneously)</a></li><li><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3" rel="nofollow">Do Androids Dream Of 18ED2?</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html" rel="nofollow">blog entry</a>)</li></ul></body></html></pre></div>
| | 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]] |