18edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 239656913 - Original comment: ** |
Wikispaces>igliashon **Imported revision 241890068 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-19 01:23:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>241890068</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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==Basic Properties== | ==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. | |||
==Useful Moment-of-Symmetry Scales== | 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. | ||
===Representations of Just Intervals=== | |||
|| Degree || Cents || Nearest Ratio || Error (cents) || 17-Limit Ratios* || | |||
|| 0 || 0 || 1/1 || 0 || **1/1** || | |||
|| 1 || 66.667 || 27/26 || +1.329 ||> 78/75, 75/72 || | |||
|| 2 || 133.333 || 27/25 || +0.096 ||> 51/55, 42/39 || | |||
|| 3 || 200 || 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 || 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 ||> 102/75 || | |||
|| 9 || 600 || 17/12 or 24/17 || -3.000 +3.000 ||> 17/12 || | |||
|| 10 || 666.667 || 22/15 || +3.617 ||> 75/51 || | |||
|| 11 || 733.333 || 32/21 || +4.114 ||> 32/21 || | |||
|| 12 || 800 || 8/5 or 35/22 || -13.686 -3.8222 || **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 || 16/9 || +3.910 ||> 16/9 || | |||
|| 16 || 1066.667 || 50/27 || -0.096 ||> 39/21 || | |||
|| 17 || 1133.333 || 52/27 || -1.329 ||> 75/39 || | |||
|| 18 || 1200 || 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>== | |||
Note: This list excludes scales found in 9-EDO. | Note: This list excludes scales found in 9-EDO. | ||
===Pentatonic:=== | ===<span style="font-size: 1.1em;">Pentatonic:</span>=== | ||
Father Pentatonic: 4 4 3 4 3 | 3L2s Father Pentatonic: 4 4 3 4 3 | ||
===Hexatonic:=== | ===<span style="font-size: 1.1em;">Hexatonic:</span>=== | ||
Whole-Tone Scale: 3 3 3 3 3 3 | 6-Equal Whole-Tone Scale: 3 3 3 3 3 3 | ||
Bicycle: 4 4 1 4 4 1 | 4L2s Bicycle: 4 4 1 4 4 1 | ||
Rice Hexatonic: 2 5 2 2 5 2 | 2L4s Rice Hexatonic: 2 5 2 2 5 2 | ||
===Heptatonic:=== | ===<span style="font-size: 1.1em;">Heptatonic:</span>=== | ||
Amity/Mish Heptatonic: 3 2 3 2 3 3 2 | 4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2 | ||
===Octatonic:=== | ===<span style="font-size: 1.1em;">Octatonic:</span>=== | ||
Father Octatonic: 3 1 3 3 1 3 3 1 | 5L3s Father Octatonic: 3 1 3 3 1 3 3 1 | ||
Rice Octatonic: 2 2 3 2 2 2 3 2 | 2L6s Rice Octatonic: 2 2 3 2 2 2 3 2 | ||
===Decatonic:=== | ===<span style="font-size: 1.1em;">Decatonic:</span>=== | ||
Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2 | 8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2 | ||
==Application to Guitar== | ==<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! | 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 | 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== | ==Commas== | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basic Properties</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x18 Equal Divisions of the Octave-Basic Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basic Properties</h2> | ||
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 /> | <br /> | ||
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 /> | |||
<br /> | <br /> | ||
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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><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> | |||
<!-- ws:start:WikiTextHeadingRule: | |||
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</td> | </td> | ||
<td>Error (cents)<br /> | <td>Error (cents)<br /> | ||
</td> | |||
<td>17-Limit Ratios*<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>0<br /> | <td>0<br /> | ||
</td> | |||
<td><strong>1/1</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+1.329<br /> | <td>+1.329<br /> | ||
</td> | |||
<td style="text-align: right;">78/75, 75/72<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+0.096<br /> | <td>+0.096<br /> | ||
</td> | |||
<td style="text-align: right;">51/55, 42/39<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-3.910<br /> | <td>-3.910<br /> | ||
</td> | |||
<td><strong>9/8</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-0.204<br /> | <td>-0.204<br /> | ||
</td> | |||
<td><strong>75/64</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-2.796 +0.293<br /> | <td>-2.796 +0.293<br /> | ||
</td> | |||
<td><strong>39/32</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+13.686 +3.822<br /> | <td>+13.686 +3.822<br /> | ||
</td> | |||
<td style="text-align: right;">64/55<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 203: | Line 201: | ||
</td> | </td> | ||
<td>-4.114<br /> | <td>-4.114<br /> | ||
</td> | |||
<td><strong>21/16</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-3.617<br /> | <td>-3.617<br /> | ||
</td> | |||
<td style="text-align: right;">102/75<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-3.000 +3.000<br /> | <td>-3.000 +3.000<br /> | ||
</td> | |||
<td style="text-align: right;">17/12<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+3.617<br /> | <td>+3.617<br /> | ||
</td> | |||
<td style="text-align: right;">75/51<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+4.114<br /> | <td>+4.114<br /> | ||
</td> | |||
<td style="text-align: right;">32/21<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-13.686 -3.8222<br /> | <td>-13.686 -3.8222<br /> | ||
</td> | |||
<td><strong>51/32</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+2.796 -0.293<br /> | <td>+2.796 -0.293<br /> | ||
</td> | |||
<td style="text-align: right;">64/39<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+0.204<br /> | <td>+0.204<br /> | ||
</td> | |||
<td><strong>55/32</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>+3.910<br /> | <td>+3.910<br /> | ||
</td> | |||
<td style="text-align: right;">16/9<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-0.096<br /> | <td>-0.096<br /> | ||
</td> | |||
<td style="text-align: right;">39/21<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>-1.329<br /> | <td>-1.329<br /> | ||
</td> | |||
<td style="text-align: right;">75/39<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>0<br /> | <td>0<br /> | ||
</td> | |||
<td><strong>2/1</strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x18 Equal Divisions of the Octave-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:&lt;h3&gt; --><h3 id="toc4"><a name="x18 Equal Divisions of the Octave-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:&lt;h3&gt; --><h3 id="toc5"><a name="x18 Equal Divisions of the Octave-Useful Moment-of-Symmetry Scales-Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:10 --><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:12:&lt;h3&gt; --><h3 id="toc6"><a name="x18 Equal Divisions of the Octave-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:&lt;h3&gt; --><h3 id="toc7"><a name="x18 Equal Divisions of the Octave-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:&lt;h3&gt; --><h3 id="toc8"><a name="x18 Equal Divisions of the Octave-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:&lt;h2&gt; --><h2 id="toc9"><a name="x18 Equal Divisions of the Octave-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 &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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x18 Equal Divisions of the Octave-Commas"></a><!-- ws:end:WikiTextHeadingRule:20 -->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 /> | 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 /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x18 Equal Divisions of the Octave-Listen"></a><!-- ws:end:WikiTextHeadingRule:22 -->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></ul></body></html></pre></div> | <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></ul></body></html></pre></div> | ||