13edo: Difference between revisions
Wikispaces>guest **Imported revision 242210435 - Original comment: ** |
Wikispaces>igliashon **Imported revision 243012613 - 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>2011-07-26 23:11:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>243012613</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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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* || Note Name** || | ||
|| 0 || 0 ||= 1/1 || | || 0 || 0 ||= 1/1 || C || | ||
|| 1 || 92.3077 ||= 55/52, 117/110, 26/25 || | || 1 || 92.3077 ||= 55/52, 117/110, 26/25 || C#/Db || | ||
|| 2 || 184.6154 ||= 10/9, 9/8, 11/10 || | || 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D || | ||
|| 3 || 276.9231 ||= 13/11 || | || 3 || 276.9231 ||= 13/11 || D#/Eb || | ||
|| 4 || 369.2308 ||= 5/4, 16/13, 11/9 || | || 4 || 369.2308 ||= 5/4, 16/13, 11/9 || E || | ||
|| 5 || 461.5385 ||= 13/10 || | || 5 || 461.5385 ||= 13/10 || E#/Fb || | ||
|| 6 || 553.84 ||= 11/8, 18/13 || | || 6 || 553.84 ||= 11/8, 18/13 || F || | ||
|| 7 || 646.15 ||= 16/11, 13/9 || | || 7 || 646.15 ||= 16/11, 13/9 || F#/Gb || | ||
|| 8 || 738.46 ||= 20/13 || | || 8 || 738.46 ||= 20/13 || G || | ||
|| 9 || 830.77 ||= 8/5, 13/8, 18/11 || | || 9 || 830.77 ||= 8/5, 13/8, 18/11 || G#/Ab || | ||
|| 10 || 923.08 ||= 22/13 || | || 10 || 923.08 ||= 22/13 || A || | ||
|| 11 || 1015.38 ||= 9/5, 16/9, 20/11 || | || 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb || | ||
|| 12 || 1107.69 ||= 25/13, 104/55 || | || 12 || 1107.69 ||= 25/13, 104/55 || B/Cb || | ||
|| 13 || 1200 ||= 2/1 || | || 13 || 1200 ||= 2/1 || C/B# || | ||
*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible. | *based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible. | ||
**based on the 6L1s heptatonic scale; see below. | |||
==Harmony in 13edo== | ==Harmony in 13edo== | ||
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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. | 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**=== | ||
[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]] by [[http://danielthompson.blogspot.com/|Daniel Thompson]] | [[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]] by [[http://danielthompson.blogspot.com/|Daniel Thompson]] | ||
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[[@http://cityoftheasleep.bandcamp.com/track/broken-dream-jar|Broken Dream Jar]] by [[IgliashonJones|City of the Asleep]] | [[@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]] | [[@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]] | ||
==Igliashon's 13-EDO diatonic approaches== | |||
From a temperament perspective, we can probably make the best use of 13-EDO as a 2.5.9.11.13 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, and 2.11.13, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping <1 -1| (for 5 and 13), corresponding to the 3rd horogram above. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping <2 3| (for 11 and 13). This corresponds to the 2nd horogram above. | |||
2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of <2 1| (for 5 and 9) and <2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating the entire 2.5.9.11.13 subgroup, which it achieves in the space of 5 generators via the octave-equivalent mapping <2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it "the" tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for naming the notes of 13-EDO, leading to a lettering very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart. | |||
==Commas== | ==Commas== | ||
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</td> | </td> | ||
<td style="text-align: center;">Approximate Ratios*<br /> | <td style="text-align: center;">Approximate Ratios*<br /> | ||
</td> | |||
<td>Note Name<strong><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 81: | Line 91: | ||
</td> | </td> | ||
<td style="text-align: center;">1/1<br /> | <td style="text-align: center;">1/1<br /> | ||
</td> | |||
<td>C<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">55/52, 117/110, 26/25<br /> | <td style="text-align: center;">55/52, 117/110, 26/25<br /> | ||
</td> | |||
<td>C#/Db<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">10/9, 9/8, 11/10<br /> | <td style="text-align: center;">10/9, 9/8, 11/10<br /> | ||
</td> | |||
<td>D<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">13/11<br /> | <td style="text-align: center;">13/11<br /> | ||
</td> | |||
<td>D#/Eb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">5/4, 16/13, 11/9<br /> | <td style="text-align: center;">5/4, 16/13, 11/9<br /> | ||
</td> | |||
<td>E<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">13/10<br /> | <td style="text-align: center;">13/10<br /> | ||
</td> | |||
<td>E#/Fb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">11/8, 18/13<br /> | <td style="text-align: center;">11/8, 18/13<br /> | ||
</td> | |||
<td>F<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">16/11, 13/9<br /> | <td style="text-align: center;">16/11, 13/9<br /> | ||
</td> | |||
<td>F#/Gb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">20/13<br /> | <td style="text-align: center;">20/13<br /> | ||
</td> | |||
<td>G<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">8/5, 13/8, 18/11<br /> | <td style="text-align: center;">8/5, 13/8, 18/11<br /> | ||
</td> | |||
<td>G#/Ab<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">22/13<br /> | <td style="text-align: center;">22/13<br /> | ||
</td> | |||
<td>A<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">9/5, 16/9, 20/11<br /> | <td style="text-align: center;">9/5, 16/9, 20/11<br /> | ||
</td> | |||
<td>A#/Bb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">25/13, 104/55<br /> | <td style="text-align: center;">25/13, 104/55<br /> | ||
</td> | |||
<td>B/Cb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">2/1<br /> | <td style="text-align: center;">2/1<br /> | ||
</td> | |||
<td>C/B#<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.<br /> | *based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible.<br /> | ||
</strong>based on the 6L1s heptatonic scale; see below.<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x13 tone equal temperament / 13edo-Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h2> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x13 tone equal temperament / 13edo-Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h2> | ||
<br /> | <br /> | ||
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Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br /> | Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:306:&lt;img src=&quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><!-- ws:end:WikiTextLocalImageRule:306 --><br /> | ||
<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:307:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/13edo%20horograms.pdf?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@13edo horograms.pdf&quot; title=&quot;File: 13edo horograms.pdf&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /></a><div><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf">13edo horograms.pdf</a><br /><ul><li><a href="/file/detail/13edo%20horograms.pdf">Details</a></li><li><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf">Download</a></li><li style="color: #666">242 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:307 --><br /> | ||
~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 /> | <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 /> | 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>< | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x13 tone equal temperament / 13edo-Scales in 13edo-Compositions"></a><!-- ws:end:WikiTextHeadingRule:6 --><strong>Compositions</strong></h3> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br /> | <a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br /> | ||
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow">Prelude in 13ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br /> | <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow">Prelude in 13ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br /> | ||
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<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 /> | <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: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x13 tone equal temperament / 13edo-Igliashon's 13-EDO diatonic approaches"></a><!-- ws:end:WikiTextHeadingRule:8 -->Igliashon's 13-EDO diatonic approaches</h2> | ||
<br /> | |||
From a temperament perspective, we can probably make the best use of 13-EDO as a 2.5.9.11.13 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, and 2.11.13, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping &lt;1 -1| (for 5 and 13), corresponding to the 3rd horogram above. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping &lt;2 3| (for 11 and 13). This corresponds to the 2nd horogram above.<br /> | |||
<br /> | |||
2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of &lt;2 1| (for 5 and 9) and &lt;2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating the entire 2.5.9.11.13 subgroup, which it achieves in the space of 5 generators via the octave-equivalent mapping &lt;2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it &quot;the&quot; tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for naming the notes of 13-EDO, leading to a lettering very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart.<br /> | |||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x13 tone equal temperament / 13edo-Commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->Commas</h2> | |||
13 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 val &lt; 13 21 30 36 45 48 |.)<br /> | 13 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 val &lt; 13 21 30 36 45 48 |.)<br /> | ||