18edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 139248817 - Original comment: simplified structure :-) Are there any good examples for listening?** |
Wikispaces>guest **Imported revision 147555503 - 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:guest|guest]] and made on <tt>2010-06-07 23:02:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>147555503</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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AKA The Third-Tone System | AKA The Third-Tone System | ||
==Basic Properties== | ==Basic Properties== | ||
=== Representations of Just Intervals === | ===Representations of Just Intervals=== | ||
|| Degree || Cents || Nearest Ratio || Error (cents) || | || Degree || Cents || Nearest Ratio || Error (cents) || | ||
|| 0 || 0 || 1/1 || 0 || | || 0 || 0 || 1/1 || 0 || | ||
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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. | 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 === | ===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 | 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 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== | ==Useful Moment-of-Symmetry Scales== | ||
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AKA The Third-Tone System<br /> | AKA The Third-Tone System<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> | ||
<!-- 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: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> | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
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18-EDO 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, 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 &quot;non-common-practice&quot; approach.<br /> | 18-EDO 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, 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 &quot;non-common-practice&quot; approach.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><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> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><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 &quot;third-tone&quot; 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 &quot;whole tone&quot; 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">&quot;Father&quot; 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 | 18-EDO, aka the &quot;third-tone&quot; 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 &quot;whole tone&quot; 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">&quot;Father&quot; 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 29-EDO in that they are all <a class="wiki_link" href="/Amity%20Temperament">&quot;Amity&quot; temperaments</a> (&quot;Amity&quot; is derived from the acronym of &quot;Acute Minor Thirds&quot;, meaning a minor third sharper than 6/5 but still flatter than a neutral third).<br /> | ||
<br /> | <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 -->Useful Moment-of-Symmetry Scales</h2> | <!-- 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 -->Useful Moment-of-Symmetry Scales</h2> | ||