16edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 278440932 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 282025322 - 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:keenanpepper|keenanpepper]] and made on <tt>2011-12-04 08:14:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>282025322</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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"Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a | "Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a | ||
unison vector from Fokker periodicity blocks.Only in 16-EDO, that unison vector is 135:128, instead of 81:80." | unison vector from Fokker periodicity blocks.Only in 16-EDO, that unison vector is 135:128, instead of 81:80." | ||
=Linear temperaments= | |||
||~ Periods | |||
per octave ||~ Generator ||~ Temperaments || | |||
|| 1 || 1\16 ||= || | |||
|| 1 || 3\16 ||= [[Gorgo]] || | |||
|| 1 || 5\16 ||= [[Magic]]/muggles || | |||
|| 1 || 7\16 ||= [[Mavila]]/armodue || | |||
|| 2 || 1\16 ||= [[Bipelog]] || | |||
|| 2 || 3\16 ||= [[Lemba]] || | |||
|| 4 || 1\16 ||= [[Diminished]] || | |||
|| 8 || 1\16 ||= || | |||
**Mavila** | **Mavila** | ||
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[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3|Comets Over Flatland 5]] by [[Randy Winchester]]</pre></div> | [[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3|Comets Over Flatland 5]] by [[Randy Winchester]]</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>16edo</title></head><body><!-- ws:start:WikiTextTocRule: | <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>16edo</title></head><body><!-- ws:start:WikiTextTocRule:16:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Hexadecaphonic Notation:">Hexadecaphonic Notation:</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Books/Literature">Books/Literature</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#External Links">External Links</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:25 --><hr /> | ||
<br /> | <br /> | ||
<strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br /> | <strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br /> | ||
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Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh).<br /> | Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's <strong>Mavila</strong> as a major seventh).<br /> | ||
If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br /> | If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).<br /> | ||
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<br /> | <br /> | ||
The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br /> | The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.<br /> | ||
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16-EDO notation can be easy utilizing Goldsmith's Circle of keys, Nominals, and respective Notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon<br /> | 16-EDO notation can be easy utilizing Goldsmith's Circle of keys, Nominals, and respective Notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon<br /> | ||
additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | ||
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<hr /> | <hr /> | ||
In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br /> | In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.<br /> | ||
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&quot;Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a<br /> | &quot;Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a<br /> | ||
unison vector from Fokker periodicity blocks.Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&quot;<br /> | unison vector from Fokker periodicity blocks.Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&quot;<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Linear temperaments</h1> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>Periods<br /> | |||
per octave<br /> | |||
</th> | |||
<th>Generator<br /> | |||
</th> | |||
<th>Temperaments<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>1\16<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>3\16<br /> | |||
</td> | |||
<td style="text-align: center;"><a class="wiki_link" href="/Gorgo">Gorgo</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>5\16<br /> | |||
</td> | |||
<td style="text-align: center;"><a class="wiki_link" href="/Magic">Magic</a>/muggles<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>7\16<br /> | |||
</td> | |||
<td style="text-align: center;"><a class="wiki_link" href="/Mavila">Mavila</a>/armodue<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>1\16<br /> | |||
</td> | |||
<td style="text-align: center;"><a class="wiki_link" href="/Bipelog">Bipelog</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>3\16<br /> | |||
</td> | |||
<td style="text-align: center;"><a class="wiki_link" href="/Lemba">Lemba</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4<br /> | |||
</td> | |||
<td>1\16<br /> | |||
</td> | |||
<td style="text-align: center;"><a class="wiki_link" href="/Diminished">Diminished</a><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8<br /> | |||
</td> | |||
<td>1\16<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | <br /> | ||
<strong>Mavila</strong><br /> | <strong>Mavila</strong><br /> | ||
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135/128 (the difference between 16/15 and 9/8) is tempered out....&quot;<br /> | 135/128 (the difference between 16/15 and 9/8) is tempered out....&quot;<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1> | ||
16 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 <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59 |.)<br /> | 16 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 <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59 |.)<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Armodue Theory (4-line staff)"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong>Armodue Theory (4-line staff)</strong></h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.<br /> | <a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.<br /> | ||
Translations of parts of the Armodue pages can be found <a class="wiki_link" href="/Armodue">here</a> on this wiki..<br /> | Translations of parts of the Armodue pages can be found <a class="wiki_link" href="/Armodue">here</a> on this wiki..<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Books/Literature"></a><!-- ws:end:WikiTextHeadingRule:10 -->Books/Literature</h1> | ||
<br /> | <br /> | ||
Sword, Ronald. &quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&quot; IAAA Press, USA. First Ed: August, 2011<br /> | Sword, Ronald. &quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&quot; IAAA Press, USA. First Ed: August, 2011<br /> | ||
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Sword, Ronald. &quot;Esadekaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br /> | Sword, Ronald. &quot;Esadekaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="External Links"></a><!-- ws:end:WikiTextHeadingRule:12 -->External Links</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.ronsword.com/16tonepianoproject.html" rel="nofollow">http://www.ronsword.com/16tonepianoproject.html</a> &quot;The 16-tone Piano Project&quot;<br /> | <a class="wiki_link_ext" href="http://www.ronsword.com/16tonepianoproject.html" rel="nofollow">http://www.ronsword.com/16tonepianoproject.html</a> &quot;The 16-tone Piano Project&quot;<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:14 -->Compositions</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow">Etude in 16-tone equal tuning</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3" rel="nofollow">play</a> by Herman Miller<br /> | <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow">Etude in 16-tone equal tuning</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3" rel="nofollow">play</a> by Herman Miller<br /> | ||