Tetrachord: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 86879901 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 87401237 - 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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-09- | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-09-06 14:29:42 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>87401237</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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==Superparticular Intervals== | ==Superparticular Intervals== | ||
In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular-- meaning of the form n/n-1 (eg. 5/4, 6/5, 11/10, 39/38...).</pre></div> | In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular-- meaning of the form n/n-1 (eg. 5/4, 6/5, 11/10, 39/38...). | ||
=Tetrachords Generalized= | |||
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 //a// & //b//, & then write our generalized tetrachord like this: | |||
1/1, a, b, 4/3 | |||
We can build a heptatonic scale by copying this tetrachord at the perfect fifth. Thus: | |||
1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1 | |||
Between 3/2 and 4/3, we have 9/8, so another way to write it would be: | |||
[tetrachord], 9/8, [tetrachord] | |||
When a tetrachord is paired with its copy, in this way, I call it a "heptatonic mirror." Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord (eg. 1/1, c, d, 4/3): | |||
1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1 | |||
==Modes of a heptatonic mirror== | |||
Going back to our generalized heptatonic mirror, let's take a look at what modes we get by starting on different scale degrees. | |||
|| mode 1 || 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1 || | |||
|| mode 2 || 1/1, || | |||
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<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>tetrachord</title></head><body>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 /> | <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>tetrachord</title></head><body>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:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Ancient Greek Genera-Superparticular Intervals"></a><!-- ws:end:WikiTextHeadingRule:12 -->Superparticular Intervals</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Ancient Greek Genera-Superparticular Intervals"></a><!-- ws:end:WikiTextHeadingRule:12 -->Superparticular Intervals</h2> | ||
<br /> | <br /> | ||
In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular-- meaning of the form n/n-1 (eg. 5/4, 6/5, 11/10, 39/38...).</body></html></pre></div> | In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular-- meaning of the form n/n-1 (eg. 5/4, 6/5, 11/10, 39/38...).<br /> | ||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Tetrachords Generalized"></a><!-- ws:end:WikiTextHeadingRule:14 -->Tetrachords Generalized</h1> | |||
<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 /> | |||
<br /> | |||
1/1, a, b, 4/3<br /> | |||
<br /> | |||
We can build a heptatonic scale by copying this tetrachord at the perfect fifth. Thus:<br /> | |||
<br /> | |||
1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1<br /> | |||
<br /> | |||
Between 3/2 and 4/3, we have 9/8, so another way to write it would be:<br /> | |||
<br /> | |||
[tetrachord], 9/8, [tetrachord]<br /> | |||
<br /> | |||
When a tetrachord is paired with its copy, in this way, I call it a &quot;heptatonic mirror.&quot; Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord (eg. 1/1, c, d, 4/3):<br /> | |||
<br /> | |||
1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Tetrachords Generalized-Modes of a heptatonic mirror"></a><!-- ws:end:WikiTextHeadingRule:16 -->Modes of a heptatonic mirror</h2> | |||
<br /> | |||
Going back to our generalized heptatonic mirror, let's take a look at what modes we get by starting on different scale degrees.<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>mode 1<br /> | |||
</td> | |||
<td>1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1<br /> | |||
</td> | |||
</tr> | |||
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<td>mode 2<br /> | |||
</td> | |||
<td>1/1,<br /> | |||
</td> | |||
</tr> | |||
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</body></html></pre></div> | |||