13edo: Difference between revisions
Wikispaces>guest **Imported revision 241829697 - Original comment: ** |
Wikispaces>guest **Imported revision 241835257 - 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:guest|guest]] and made on <tt>2011-07-18 | : This revision was by author [[User:guest|guest]] and made on <tt>2011-07-18 17:26:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>241835257</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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<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">=13 tone equal temperament / 13edo= | <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">=13 tone equal temperament / 13edo= | ||
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). | 13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). | ||
|| Degree || Cents || Approximate Ratios* || | || Degree || Cents || Approximate Ratios* || | ||
|| 0 || 0 || 1/1 || | || 0 || 0 || 1/1 || | ||
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One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely. | One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely. | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings), and the most | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, triads of 0-2-4, 0-2-5, and 0-3-5 can be quite consonant in some musical circumstances. Another approach is to use chords where the intervals are widely-spaced, exceeding the octave; for instance, 0-10-15, 0-11-15, 0-12-15, and 0-11-16 can also sound reasonably consonant. Of course, the consonance available in 13-EDO is quite different than that available in 12-EDO, but that's kind of the point of using 13-EDO. It is harmonically on another planet from 12-EDO. | ||
==Scales in 13edo== | ==Scales in 13edo== | ||
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[[file:13edo horograms.pdf]] | [[file:13edo horograms.pdf]] | ||
~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson | ~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson | ||
Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected. | |||
**Compositions** | **Compositions** | ||
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[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]] by [[http://www.io.com/%7Ehmiller/music/|Herman Miller]] | [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]] by [[http://www.io.com/%7Ehmiller/music/|Herman Miller]] | ||
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=835265|Spikey Hair in 13tET]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]] by [[Andrew Heathwaite]] | [[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=835265|Spikey Hair in 13tET]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]] by [[Andrew Heathwaite]] | ||
[[@http://cityoftheasleep.bandcamp.com/track/broken-dream-jar|Broken Dream Jar]] by [[IgliashonJones|City of the Asleep]] | |||
[[@http://www.last.fm/music/City+of+the+Asleep/Map+of+an+Internal+Landscape/Blinding+White+Darkness|Blinding White Darkness]] by [[IgliashonJones|City of the Asleep]] | |||
==Commas== | ==Commas== | ||
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<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>13edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x13 tone equal temperament / 13edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->13 tone equal temperament / 13edo</h1> | <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>13edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x13 tone equal temperament / 13edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->13 tone equal temperament / 13edo</h1> | ||
<br /> | <br /> | ||
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp; major sixth are xenharmonic (not similar to anything available in 12edo). <br /> | 13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third &amp; major sixth are xenharmonic (not similar to anything available in 12edo).<br /> | ||
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One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely.<br /> | One way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. Another way to view it is to totally disregard JI approximations entirely.<br /> | ||
<br /> | <br /> | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings), and the most | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, triads of 0-2-4, 0-2-5, and 0-3-5 can be quite consonant in some musical circumstances. Another approach is to use chords where the intervals are widely-spaced, exceeding the octave; for instance, 0-10-15, 0-11-15, 0-12-15, and 0-11-16 can also sound reasonably consonant. Of course, the consonance available in 13-EDO is quite different than that available in 12-EDO, but that's kind of the point of using 13-EDO. It is harmonically on another planet from 12-EDO.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x13 tone equal temperament / 13edo-Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scales in 13edo</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x13 tone equal temperament / 13edo-Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scales in 13edo</h2> | ||
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~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br /> | ~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br /> | ||
<br /> | |||
Another neat facet of 13-EDO is the fact that any 12-EDO scale can be &quot;turned into&quot; a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.<br /> | |||
<br /> | <br /> | ||
<strong>Compositions</strong><br /> | <strong>Compositions</strong><br /> | ||
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<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow">Triskaidekaphobia</a> by <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow">Herman Miller</a><br /> | <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow">Triskaidekaphobia</a> by <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow">Herman Miller</a><br /> | ||
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=835265" rel="nofollow">Spikey Hair in 13tET</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> | <a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=835265" rel="nofollow">Spikey Hair in 13tET</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br /> | ||
<a class="wiki_link_ext" href="http://cityoftheasleep.bandcamp.com/track/broken-dream-jar" rel="nofollow" target="_blank">Broken Dream Jar</a> by <a class="wiki_link" href="/IgliashonJones">City of the Asleep</a><br /> | |||
<a class="wiki_link_ext" href="http://www.last.fm/music/City+of+the+Asleep/Map+of+an+Internal+Landscape/Blinding+White+Darkness" rel="nofollow" target="_blank">Blinding White Darkness</a> by <a class="wiki_link" href="/IgliashonJones">City of the Asleep</a><br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x13 tone equal temperament / 13edo-Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h2> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x13 tone equal temperament / 13edo-Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h2> | ||