16edo: Difference between revisions
Wikispaces>hstraub **Imported revision 622401131 - Original comment: ** |
Wikispaces>hstraub **Imported revision 622730141 - 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:hstraub|hstraub]] and made on <tt>2017-11- | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2017-11-29 14:41:43 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>622730141</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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**Diminished** | **Diminished** | ||
[8]: 1 3 1 3 1 3 1 3 | || [8]: || 1 3 1 3 1 3 1 3 || [[media type="file" key="htgt16edo.mp3"]] || | ||
[12]: 1 1 2 1 1 2 1 1 2 1 1 2 | || [12]: || 1 1 2 1 1 2 1 1 2 1 1 2 || || | ||
**Magic** | **Magic** | ||
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</span></pre></div> | </span></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><span style="display: block; text-align: right;"><a class="wiki_link" href=" | <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><span style="display: block; text-align: right;"><a class="wiki_link" href="https://xenharmonie.wikispaces.com/16edo">Deutsch</a> - <a class="wiki_link" href="/16%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br /> | ||
</span><br /> | </span><br /> | ||
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<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:36:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#Images">Images</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Chord names">Chord names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Chord names-Selected just intervals by error">Selected just intervals by error</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Notation">Hexadecaphonic Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Mavila[7]/&quot;Anti-diatonic&quot; Scale">16 Tone Piano Layout Based on the Mavila[7]/&quot;Anti-diatonic&quot; Scale</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Rank two temperaments-Metallic Harmony in 16 EDO">Metallic Harmony in 16 EDO</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 3em;"><a href="#Rank two temperaments-Metallic Harmony in 16 EDO-MOS scales supporting Metallic Harmony in 16edo">MOS scales supporting Metallic Harmony in 16edo</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Books/Literature">Books/Literature</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc0"><a name="Images"></a><!-- ws:end:WikiTextHeadingRule:8 -->Images</h1> | ||
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<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1270:&lt;img src=&quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 325px; width: 325px;&quot; /&gt; --><img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" /><!-- ws:end:WikiTextLocalImageRule:1270 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:10 -->Intervals</h1> | ||
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16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.<br /> | 16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.<br /> | ||
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*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.<br /> | *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc2"><a name="Chord names"></a><!-- ws:end:WikiTextHeadingRule:12 -->Chord names</h1> | ||
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16edo chords can be named using us and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See <a class="wiki_link" href=" | 16edo chords can be named using us and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See <a class="wiki_link" href="https://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Other%20EDOs">xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs</a>).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc3"><a name="Chord names-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:14 -->Selected just intervals by error</h2> | ||
The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 16-EDO (ordered by absolute error).<br /> | The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 16-EDO (ordered by absolute error).<br /> | ||
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It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant.<br /> | It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant.<br /> | ||
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<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:1272:&lt;img src=&quot;https://www.wikispaces.com/site/embedthumbnail/file/16ed2-001.svg?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@16ed2-001.svg&quot; title=&quot;File: 16ed2-001.svg&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');"><img src="https://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="16ed2-001.svg" /></a><div><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');" class="filename" title="16ed2-001.svg">16ed2-001.svg</a><br /><ul><li><a href="/file/detail/16ed2-001.svg">Details</a></li><li><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg">Download</a></li><li style="color: #666">18 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:1272 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc4"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hexadecaphonic Octave Theory</h1> | ||
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The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br /> | The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br /> | ||
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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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc5"><a name="Hexadecaphonic Notation"></a><!-- ws:end:WikiTextHeadingRule:18 -->Hexadecaphonic Notation</h1> | ||
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16-EDO notation can be easy utilizing <a name="comment-606393483_1-open" class="range"></a>Goldsmith's Circle of keys, nominals, and respective notation<a name="comment-606393483_1-close" class="range"></a>. The nominals for a 6 line staff can be switched for <a name="comment-606393483_2-open" class="range"></a>Wilson's Beta and Epsilon additions to A-G<a name="comment-606393483_2-close" class="range"></a>. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | 16-EDO notation can be easy utilizing <a name="comment-606393483_1-open" class="range"></a>Goldsmith's Circle of keys, nominals, and respective notation<a name="comment-606393483_1-close" class="range"></a>. The nominals for a 6 line staff can be switched for <a name="comment-606393483_2-open" class="range"></a>Wilson's Beta and Epsilon additions to A-G<a name="comment-606393483_2-close" class="range"></a>. Armodue of Italy uses a 4-line staff for 16-EDO.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc6"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Mavila[7]/&quot;Anti-diatonic&quot; Scale"></a><!-- ws:end:WikiTextHeadingRule:20 -->16 Tone Piano Layout Based on the Mavila[7]/&quot;Anti-diatonic&quot; Scale</h2> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1271:&lt;img src=&quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 293px; width: 748px;&quot; /&gt; --><img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /><!-- ws:end:WikiTextLocalImageRule:1271 --><br /> | ||
This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower, as per the &quot;harmonic notation&quot; above. Simply swap sharps with flats for &quot;melodic notation&quot;.<br /> | This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower, as per the &quot;harmonic notation&quot; above. Simply swap sharps with flats for &quot;melodic notation&quot;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc7"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:22 -->Rank two temperaments</h1> | ||
<a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br /> | <a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br /> | ||
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<td>3 2 2 3 2 2 2<br /> | <td>3 2 2 3 2 2 2<br /> | ||
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<td><!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot; | <td><!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;https://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaAntidiatonic16edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;MavilaAntidiatonic16edo.mp3&amp;quot; width=&amp;quot;240&amp;quot; height=&amp;quot;20&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=https%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FMavilaAntidiatonic16edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:1 --><br /> | ||
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<td>1 2 2 2 1 2 2 2 2<br /> | <td>1 2 2 2 1 2 2 2 2<br /> | ||
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<td><!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot; | <td><!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot;https://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaSuperdiatonic16edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;MavilaSuperdiatonic16edo.mp3&amp;quot; width=&amp;quot;240&amp;quot; height=&amp;quot;20&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=https%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FMavilaSuperdiatonic16edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:2 --><br /> | ||
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<strong>Diminished</strong><br /> | <strong>Diminished</strong><br /> | ||
[8]: 1 3 1 3 1 3 1 3<br /> | |||
[12]: 1 1 2 1 1 2 1 1 2 1 1 2<br /> | |||
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<td>[8]:<br /> | |||
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<td>1 3 1 3 1 3 1 3<br /> | |||
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<td><!-- ws:start:WikiTextMediaRule:3:&lt;img src=&quot;https://www.wikispaces.com/site/embedthumbnail/file-audio/htgt16edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;htgt16edo.mp3&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=https%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fhtgt16edo.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:3 --><br /> | |||
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<td>[12]:<br /> | |||
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<td>1 1 2 1 1 2 1 1 2 1 1 2<br /> | |||
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<td><br /> | |||
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<strong>Magic</strong><br /> | <strong>Magic</strong><br /> | ||
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[10]: 2 1 2 1 2 2 1 2 1 2<br /> | [10]: 2 1 2 1 2 2 1 2 1 2<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc8"><a name="Rank two temperaments-Metallic Harmony in 16 EDO"></a><!-- ws:end:WikiTextHeadingRule:24 -->Metallic Harmony in 16 EDO</h2> | ||
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Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres.<br /> | Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres.<br /> | ||
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Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads &quot;hard&quot; and &quot;soft&quot;, respectively. In addition, two other &quot;symmetrical&quot; triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to &quot;diminished&quot; and &quot;augmented&quot; triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them &quot;Metallic triads&quot;.<br /> | Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads &quot;hard&quot; and &quot;soft&quot;, respectively. In addition, two other &quot;symmetrical&quot; triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to &quot;diminished&quot; and &quot;augmented&quot; triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them &quot;Metallic triads&quot;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc9"><a name="Rank two temperaments-Metallic Harmony in 16 EDO-MOS scales supporting Metallic Harmony in 16edo"></a><!-- ws:end:WikiTextHeadingRule:26 -->MOS scales supporting Metallic Harmony in 16edo</h3> | ||
The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.<br /> | The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.<br /> | ||
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See <a class="wiki_link" href="/Metallic%20Harmony">Metallic Harmony</a>.<br /> | See <a class="wiki_link" href="/Metallic%20Harmony">Metallic Harmony</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc10"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:28 -->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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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc11"><a name="Armodue Theory (4-line staff)"></a><!-- ws:end:WikiTextHeadingRule:30 --><strong>Armodue Theory (4-line staff)</strong></h1> | ||
<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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc12"><a name="Books/Literature"></a><!-- ws:end:WikiTextHeadingRule:32 -->Books/Literature</h1> | ||
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 /> | ||
Sword, Ronald. &quot;Hexadecaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br /> | Sword, Ronald. &quot;Hexadecaphonic Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br /> | ||
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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc13"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:34 -->Compositions</h1> | ||
<a class="wiki_link_ext" href="https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve" rel="nofollow">Huckleberry Regional Preserve</a> by City of the Asleep<br /> | <a class="wiki_link_ext" href="https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve" rel="nofollow">Huckleberry Regional Preserve</a> by City of the Asleep<br /> | ||
<a class="wiki_link_ext" href="https://cityoftheasleep.bandcamp.com/track/illegible-red-ink" rel="nofollow">Illegible Red Ink</a> by City of the Asleep<br /> | <a class="wiki_link_ext" href="https://cityoftheasleep.bandcamp.com/track/illegible-red-ink" rel="nofollow">Illegible Red Ink</a> by City of the Asleep<br /> | ||