18edo
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2010-05-04 03:31:43 UTC.
- The original revision id was 139248817.
- The revision comment was: simplified structure :-) Are there any good examples for listening?
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=18 Equal Divisions of the Octave=
AKA The Third-Tone System
==Basic Properties==
=== Representations of Just Intervals ===
|| Degree || Cents || Nearest Ratio || Error (cents) ||
|| 0 || 0 || 1/1 || 0 ||
|| 1 || 66.667 || 27/26 || +1.329 ||
|| 2 || 133.333 || 27/25 || +0.096 ||
|| 3 || 200 || 9/8 || -3.910 ||
|| 4 || 266.667 || 7/6 || -0.204 ||
|| 5 || 333.333 || 17/14 or 40/33 || -2.796 +0.293 ||
|| 6 || 400 || 5/4 or 44/35 || +13.686 +3.822 ||
|| 7 || 466.667 || 21/16 || -4.114 ||
|| 8 || 533.333 || 15/11 || -3.617 ||
|| 9 || 600 || 17/12 or 24/17 || -3.000 +3.000 ||
|| 10 || 666.667 || 22/15 || +3.617 ||
|| 11 || 733.333 || 32/21 || +4.114 ||
|| 12 || 800 || 8/5 or 35/22 || -13.686 -3.8222 ||
|| 13 || 866.667 || 28/17 or 33/20 || +2.796 -0.293 ||
|| 14 || 933.333 || 12/7 || +0.204 ||
|| 15 || 1000 || 16/9 || +3.910 ||
|| 16 || 1066.667 || 50/27 || -0.096 ||
|| 17 || 1133.333 || 52/27 || -1.329 ||
|| 18 || 1200 || 2/1 || 0 ||
18-EDO does not approximate the 3rd Harmonic at all, unless a >30¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably "non-common-practice" approach.
=== Relationship to Other EDOs ===
18-EDO, aka the "third-tone" system, is related to [[12edo|12-tET]] by the whole-tone scale (which is [[6edo|6-EDO]]), since 18=6*3 and 12=6*2; hence a 12-tET "whole tone" is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of [[9edo|9-EDO]], offset from each other by a third-tone. 18-EDO is related to [[13edo|13-EDO]], [[21edo|21-EDO]], [[23edo|23-EDO]], and [[28edo|28-EDO]] in that all are [[Father Temperament|"Father" temperaments]] (they temper out 16/15--the difference between a major third and perfect fourth). It is related to [[11edo|11-EDO]], [[15edo|15-EDO]], [[25edo|25-EDO]], and [[19edo|29-EDO]] in that they are all [[Amity Temperament|"Amity" temperaments]] ("Amity" is derived from the acronym of "Acute Minor Thirds", meaning a minor third sharper than 6/5 but still flatter than a neutral third).
==Useful Moment-of-Symmetry Scales==
Note: This list excludes scales found in 9-EDO.
===Pentatonic:===
Father Pentatonic: 4 4 3 4 3
===Hexatonic:===
Whole-Tone Scale: 3 3 3 3 3 3
Bicycle: 4 4 1 4 4 1
Rice Hexatonic: 2 5 2 2 5 2
===Heptatonic:===
Amity/Mish Heptatonic: 3 2 3 2 3 3 2
===Octatonic:===
Father Octatonic: 3 1 3 3 1 3 3 1
Rice Octatonic: 2 2 3 2 2 2 3 2
===Decatonic:===
Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
==Application to Guitar==
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.Original HTML content:
<html><head><title>18edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x18 Equal Divisions of the Octave"></a><!-- ws:end:WikiTextHeadingRule:0 -->18 Equal Divisions of the Octave</h1>
AKA The Third-Tone System<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basic Properties</h2>
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x18 Equal Divisions of the Octave-Basic Properties-Representations of Just Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 --> Representations of Just Intervals </h3>
<table class="wiki_table">
<tr>
<td>Degree<br />
</td>
<td>Cents<br />
</td>
<td>Nearest Ratio<br />
</td>
<td>Error (cents)<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>1/1<br />
</td>
<td>0<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>66.667<br />
</td>
<td>27/26<br />
</td>
<td>+1.329<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>133.333<br />
</td>
<td>27/25<br />
</td>
<td>+0.096<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>200<br />
</td>
<td>9/8<br />
</td>
<td>-3.910<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>266.667<br />
</td>
<td>7/6<br />
</td>
<td>-0.204<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>333.333<br />
</td>
<td>17/14 or 40/33<br />
</td>
<td>-2.796 +0.293<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>400<br />
</td>
<td>5/4 or 44/35<br />
</td>
<td>+13.686 +3.822<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>466.667<br />
</td>
<td>21/16<br />
</td>
<td>-4.114<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>533.333<br />
</td>
<td>15/11<br />
</td>
<td>-3.617<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>600<br />
</td>
<td>17/12 or 24/17<br />
</td>
<td>-3.000 +3.000<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>666.667<br />
</td>
<td>22/15<br />
</td>
<td>+3.617<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>733.333<br />
</td>
<td>32/21<br />
</td>
<td>+4.114<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>800<br />
</td>
<td>8/5 or 35/22<br />
</td>
<td>-13.686 -3.8222<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>866.667<br />
</td>
<td>28/17 or 33/20<br />
</td>
<td>+2.796 -0.293<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>933.333<br />
</td>
<td>12/7<br />
</td>
<td>+0.204<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1000<br />
</td>
<td>16/9<br />
</td>
<td>+3.910<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>1066.667<br />
</td>
<td>50/27<br />
</td>
<td>-0.096<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>1133.333<br />
</td>
<td>52/27<br />
</td>
<td>-1.329<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>1200<br />
</td>
<td>2/1<br />
</td>
<td>0<br />
</td>
</tr>
</table>
18-EDO does not approximate the 3rd Harmonic at all, unless a >30¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, it does offer excellent approximations of 27/25, 9/8, 7/6, 17/14, 21/16, and 15/11 (and their respective reciprocal intervals), so it is still capable of playing consonant music; however, in order to access these consonances, one must take a considerably "non-common-practice" approach.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x18 Equal Divisions of the Octave-Basic Properties-Relationship to Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:6 --> Relationship to Other EDOs </h3>
18-EDO, aka the "third-tone" system, is related to <a class="wiki_link" href="/12edo">12-tET</a> by the whole-tone scale (which is <a class="wiki_link" href="/6edo">6-EDO</a>), since 18=6*3 and 12=6*2; hence a 12-tET "whole tone" is divided into 3 equal parts in 18-EDO. Since 18=9*2, 18-EDO contains two sets of <a class="wiki_link" href="/9edo">9-EDO</a>, offset from each other by a third-tone. 18-EDO is related to <a class="wiki_link" href="/13edo">13-EDO</a>, <a class="wiki_link" href="/21edo">21-EDO</a>, <a class="wiki_link" href="/23edo">23-EDO</a>, and <a class="wiki_link" href="/28edo">28-EDO</a> in that all are <a class="wiki_link" href="/Father%20Temperament">"Father" temperaments</a> (they temper out 16/15--the difference between a major third and perfect fourth). It is related to <a class="wiki_link" href="/11edo">11-EDO</a>, <a class="wiki_link" href="/15edo">15-EDO</a>, <a class="wiki_link" href="/25edo">25-EDO</a>, and <a class="wiki_link" href="/19edo">29-EDO</a> in that they are all <a class="wiki_link" href="/Amity%20Temperament">"Amity" temperaments</a> ("Amity" is derived from the acronym of "Acute Minor Thirds", meaning a minor third sharper than 6/5 but still flatter than a neutral third).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales"></a><!-- ws:end:WikiTextHeadingRule:8 -->Useful Moment-of-Symmetry Scales</h2>
Note: This list excludes scales found in 9-EDO.<br />
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Pentatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 -->Pentatonic:</h3>
Father Pentatonic: 4 4 3 4 3<br />
<!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:12 -->Hexatonic:</h3>
Whole-Tone Scale: 3 3 3 3 3 3<br />
Bicycle: 4 4 1 4 4 1<br />
Rice Hexatonic: 2 5 2 2 5 2<br />
<!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Heptatonic:"></a><!-- ws:end:WikiTextHeadingRule:14 -->Heptatonic:</h3>
Amity/Mish Heptatonic: 3 2 3 2 3 3 2<br />
<!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Octatonic:"></a><!-- ws:end:WikiTextHeadingRule:16 -->Octatonic:</h3>
Father Octatonic: 3 1 3 3 1 3 3 1<br />
Rice Octatonic: 2 2 3 2 2 2 3 2<br />
<!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Decatonic:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Decatonic:</h3>
Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="x18 Equal Divisions of the Octave-Application to Guitar"></a><!-- ws:end:WikiTextHeadingRule:20 -->Application to Guitar</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.</body></html>