18edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 214539722 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 214715302 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-03-28 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-28 12:46:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>214715302</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== | ||
The //18 equal division// divides the octave into 18 equal parts of 66.667 cents each. It does not do a good job of representing low-limit harmony, but it does a superb job with the 2.7/3.13/3.17/3 subgroup. Hence the intervals 7/6, 13/12, 17/12, 13/7, 17/14 and 17/13 are natural to it, as is the 1-13/7-17/7 chord and its inversion. It tempers out 169/168, 16848/16807 and 289/288 on this subgroup, and provides the [[optimal patent val]] for the rank three temperament tempering out 169/168; the rank two temperament tempering it out belongs to [[9edo]]. A larger subgroup, containing 9/8 and 27/25 among other intervals, is 2.9.75.21.55.39.51, which allows for a much wider variety of chords. On this subgroup it also tempers out 221/220, 225/224, 243/242, 273/272, 275/274, 325/324 and 441/440. | The //18 equal division// divides the octave into 18 equal parts of 66.667 cents each. It does not do a good job of representing low-limit harmony, but it does a superb job with the 2.7/3.13/3.17/3 subgroup. Hence the intervals 7/6, 13/12, 17/12, 13/7, 17/14 and 17/13 are natural to it, as is the 1-13/7-17/7 chord and its inversion. It tempers out 169/168, 16848/16807 and 289/288 on this subgroup, and provides the [[optimal patent val]] for the rank three temperament tempering out 169/168; the rank two temperament tempering it out belongs to [[9edo]]. A larger subgroup, containing 9/8 and 27/25 among other intervals, is 2.9.75.21.55.39.51, which allows for a much wider variety of chords. On this subgroup it also tempers out 221/220, 225/224, 243/242, 273/272, 275/274, 325/324 and 441/440. | ||
If we take the commas listed above on the entire [[17-limit]], we find that they define [[72edo]] in the 17-limit. It is, in fact, the largest subgroup of the 17-limit for which 18 acts as 72. | |||
18edo 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, since it does offer excellent approximations to 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 centering on the chords available in the 2.9.75.21.55.39.51 subgroup. | 18edo 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, since it does offer excellent approximations to 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 centering on the chords available in the 2.9.75.21.55.39.51 subgroup. | ||
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<!-- 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> | ||
The <em>18 equal division</em> divides the octave into 18 equal parts of 66.667 cents each. It does not do a good job of representing low-limit harmony, but it does a superb job with the 2.7/3.13/3.17/3 subgroup. Hence the intervals 7/6, 13/12, 17/12, 13/7, 17/14 and 17/13 are natural to it, as is the 1-13/7-17/7 chord and its inversion. It tempers out 169/168, 16848/16807 and 289/288 on this subgroup, and provides the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank three temperament tempering out 169/168; the rank two temperament tempering it out belongs to <a class="wiki_link" href="/9edo">9edo</a>. A larger subgroup, containing 9/8 and 27/25 among other intervals, is 2.9.75.21.55.39.51, which allows for a much wider variety of chords. On this subgroup it also tempers out 221/220, 225/224, 243/242, 273/272, 275/274, 325/324 and 441/440.<br /> | The <em>18 equal division</em> divides the octave into 18 equal parts of 66.667 cents each. It does not do a good job of representing low-limit harmony, but it does a superb job with the 2.7/3.13/3.17/3 subgroup. Hence the intervals 7/6, 13/12, 17/12, 13/7, 17/14 and 17/13 are natural to it, as is the 1-13/7-17/7 chord and its inversion. It tempers out 169/168, 16848/16807 and 289/288 on this subgroup, and provides the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank three temperament tempering out 169/168; the rank two temperament tempering it out belongs to <a class="wiki_link" href="/9edo">9edo</a>. A larger subgroup, containing 9/8 and 27/25 among other intervals, is 2.9.75.21.55.39.51, which allows for a much wider variety of chords. On this subgroup it also tempers out 221/220, 225/224, 243/242, 273/272, 275/274, 325/324 and 441/440.<br /> | ||
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If we take the commas listed above on the entire <a class="wiki_link" href="/17-limit">17-limit</a>, we find that they define <a class="wiki_link" href="/72edo">72edo</a> in the 17-limit. It is, in fact, the largest subgroup of the 17-limit for which 18 acts as 72.<br /> | |||
<br /> | <br /> | ||
18edo does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, since it does offer excellent approximations to 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 &quot;non-common-practice&quot; approach centering on the chords available in the 2.9.75.21.55.39.51 subgroup.<br /> | 18edo does not approximate the 3rd harmonic at all, unless a &gt;30¢-error is considered acceptable. This makes it unsuitable for rendering common-practice music. However, since it does offer excellent approximations to 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 &quot;non-common-practice&quot; approach centering on the chords available in the 2.9.75.21.55.39.51 subgroup.<br /> | ||