Tetrachord: Difference between revisions
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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>2012-04-10 03: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2012-04-10 03:48:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>318988062</tt>.<br> | ||
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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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In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are [[superparticular]]. | In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are [[superparticular]]. | ||
=Ajnas (tetrachords in middle-eastern music)= | |||
The concept of the tetrachord is extensively used in [[Arabic, Turkish, Persian|middle eastern]] music theory. The arabic word for tetrachord is "jins" (singular form) or "ajnas" (plural form). | |||
See [[http://www.maqamworld.com/|http://www.maqamworld.com]] for details. | |||
=Tetrachords Generalized= | =Tetrachords Generalized= | ||
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Related pages: <a class="wiki_link" href="/22edo%20tetrachords">22edo tetrachords</a>, <a class="wiki_link" href="/17edo%20tetrachords">17edo tetrachords</a>, <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a><br /> | Related pages: <a class="wiki_link" href="/22edo%20tetrachords">22edo tetrachords</a>, <a class="wiki_link" href="/17edo%20tetrachords">17edo tetrachords</a>, <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a><br /> | ||
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<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:30:&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:30 --><!-- ws:start:WikiTextTocRule:31: --><a href="#Ancient Greek Genera">Ancient Greek Genera</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Ajnas (tetrachords in middle-eastern music)">Ajnas (tetrachords in middle-eastern music)</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Tetrachords Generalized">Tetrachords Generalized</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#Tetrachords in equal temperaments">Tetrachords in equal temperaments</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#Dividing Other-Than-Perfect Fourths">Dividing Other-Than-Perfect Fourths</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#Tetrachords And Non-Octave Scales">Tetrachords And Non-Octave Scales</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Ancient Greek Genera"></a><!-- ws:end:WikiTextHeadingRule:0 -->Ancient Greek Genera</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Ancient Greek Genera"></a><!-- ws:end:WikiTextHeadingRule:0 -->Ancient Greek Genera</h1> | ||
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In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are <a class="wiki_link" href="/superparticular">superparticular</a>.<br /> | In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are <a class="wiki_link" href="/superparticular">superparticular</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Ajnas (tetrachords in middle-eastern music)"></a><!-- ws:end:WikiTextHeadingRule:14 -->Ajnas (tetrachords in middle-eastern music)</h1> | |||
<br /> | |||
The concept of the tetrachord is extensively used in <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">middle eastern</a> music theory. The arabic word for tetrachord is &quot;jins&quot; (singular form) or &quot;ajnas&quot; (plural form).<br /> | |||
See <a class="wiki_link_ext" href="http://www.maqamworld.com/" rel="nofollow">http://www.maqamworld.com</a> for details.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Tetrachords Generalized"></a><!-- ws:end:WikiTextHeadingRule:16 -->Tetrachords Generalized</h1> | ||
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All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals <em>a</em> &amp; <em>b</em>, &amp; then write our generalized tetrachord like this:<br /> | All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals <em>a</em> &amp; <em>b</em>, &amp; then write our generalized tetrachord like this:<br /> | ||
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[tetrachord #2], 9/8, [tetrachord #1]<br /> | [tetrachord #2], 9/8, [tetrachord #1]<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Tetrachords Generalized-Modes of a [tetrachord], 9/8, [tetrachord] scale"></a><!-- ws:end:WikiTextHeadingRule:18 -->Modes of a [tetrachord], 9/8, [tetrachord] scale</h2> | ||
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These three tetrachords are all rotations of each other (they contain the same steps in a different order).<br /> | These three tetrachords are all rotations of each other (they contain the same steps in a different order).<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Tetrachords Generalized-Tetrachord rotations"></a><!-- ws:end:WikiTextHeadingRule:20 -->Tetrachord rotations</h2> | ||
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If you think of a tetrachord as three steps which total to a perfect fourth, then it makes sense that we can put those steps in any order. If we have a tetrachord with three step sizes, s, M, and L, then we have six rotations:<br /> | If you think of a tetrachord as three steps which total to a perfect fourth, then it makes sense that we can put those steps in any order. If we have a tetrachord with three step sizes, s, M, and L, then we have six rotations:<br /> | ||
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And, if you have only one step size (as is the case in Porcupine temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in <a class="wiki_link" href="/22edo">22edo</a> - see <a class="wiki_link" href="/22edo%20tetrachords">22edo tetrachords</a>.)<br /> | And, if you have only one step size (as is the case in Porcupine temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in <a class="wiki_link" href="/22edo">22edo</a> - see <a class="wiki_link" href="/22edo%20tetrachords">22edo tetrachords</a>.)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Tetrachords in equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:22 -->Tetrachords in equal temperaments</h1> | ||
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Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with <a class="wiki_link" href="/7edo">7edo</a>, which has one tetrachord:<br /> | Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with <a class="wiki_link" href="/7edo">7edo</a>, which has one tetrachord:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Tetrachords in equal temperaments-Tetrachords of 10edo"></a><!-- ws:end:WikiTextHeadingRule:24 -->Tetrachords of <a class="wiki_link" href="/10edo">10edo</a></h2> | ||
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Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step:<br /> | Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Dividing Other-Than-Perfect Fourths"></a><!-- ws:end:WikiTextHeadingRule:26 -->Dividing Other-Than-Perfect Fourths</h1> | ||
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A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: <a class="wiki_link" href="/6edo">6edo</a>, <a class="wiki_link" href="/8edo">8edo</a>, <a class="wiki_link" href="/9edo">9edo</a>, <a class="wiki_link" href="/11edo">11edo</a>, <a class="wiki_link" href="/13edo">13edo</a>, <a class="wiki_link" href="/16edo">16edo</a>. Also, one may divide a just other-than-perfect fourth, such as 21/16, 43/32, 26/19, 11/8. At what point does the concept of &quot;tetrachord&quot; stop being useful?<br /> | A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: <a class="wiki_link" href="/6edo">6edo</a>, <a class="wiki_link" href="/8edo">8edo</a>, <a class="wiki_link" href="/9edo">9edo</a>, <a class="wiki_link" href="/11edo">11edo</a>, <a class="wiki_link" href="/13edo">13edo</a>, <a class="wiki_link" href="/16edo">16edo</a>. Also, one may divide a just other-than-perfect fourth, such as 21/16, 43/32, 26/19, 11/8. At what point does the concept of &quot;tetrachord&quot; stop being useful?<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Tetrachords And Non-Octave Scales"></a><!-- ws:end:WikiTextHeadingRule:28 -->Tetrachords And Non-Octave Scales</h1> | ||
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An example with <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a>:<br /> | An example with <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a>:<br /> | ||
<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/GloriousGuitars.mp3" rel="nofollow">Glorious Guitars</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html" rel="nofollow">blog entry</a>)</body></html></pre></div> | <a class="wiki_link_ext" href="http://www.seraph.it/dep/det/GloriousGuitars.mp3" rel="nofollow">Glorious Guitars</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html" rel="nofollow">blog entry</a>)</body></html></pre></div> | ||