Otonality and utonality: Difference between revisions
Wikispaces>genewardsmith **Imported revision 303800346 - 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:genewardsmith|genewardsmith]] and made on <tt>2012-02-21 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-21 17:47:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>303823170</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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=Scales= | =Scales= | ||
These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, | These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, Euler genera, combination product sets, or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal. | ||
=Essentially tempered chords= | |||
This kind of reduction can also be used to analyze [[Dyadic chord|essentially tempered chords]]. Consider for example the [[sinbadmic tetrad]], which is the 1001/1000-tempering of 1-11/10-13/10-10/7. The reduction of the JI version of this chord is {25, 35, 77, 91}; discarding the lowest number, 25, and reducing again gives {5, 11, 13}. This tells us the chord can be analyzed as an otonbal 1-11/10-13/10 chord plus a 10/7 addition requiring essential tempering.</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>Otonality and utonality</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>Otonality and utonality</title></head><body><!-- ws:start:WikiTextTocRule:18:&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:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Precise definitions">Precise definitions</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Properties of types of chords">Properties of types of chords</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Ambitonal chord theorem">Ambitonal chord theorem</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Essentially tempered chords">Essentially tempered chords</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:28 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> | ||
<ul><li>For the basic concepts, see the Wikipedia article <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonality_and_Utonality" rel="nofollow">Otonality and Utonality</a>.</li></ul><br /> | <ul><li>For the basic concepts, see the Wikipedia article <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Otonality_and_Utonality" rel="nofollow">Otonality and Utonality</a>.</li></ul><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Scales</h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Scales</h1> | ||
These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, | These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, Euler genera, combination product sets, or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Essentially tempered chords"></a><!-- ws:end:WikiTextHeadingRule:16 -->Essentially tempered chords</h1> | |||
This kind of reduction can also be used to analyze <a class="wiki_link" href="/Dyadic%20chord">essentially tempered chords</a>. Consider for example the <a class="wiki_link" href="/sinbadmic%20tetrad">sinbadmic tetrad</a>, which is the 1001/1000-tempering of 1-11/10-13/10-10/7. The reduction of the JI version of this chord is {25, 35, 77, 91}; discarding the lowest number, 25, and reducing again gives {5, 11, 13}. This tells us the chord can be analyzed as an otonbal 1-11/10-13/10 chord plus a 10/7 addition requiring essential tempering.</body></html></pre></div> | |||