Tetrachord: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2009-09-06 15:04:54 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-09-06 16:32:25 UTC</tt>.<br>

: The original revision id was <tt>~~87403719~~</tt>.<br>

: The original revision id was <tt>87410093</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>

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1/1, b/a, 4/3a, 4/3 (mode 6)

1/1, 4/3b, 4a/3b, 4/3 (mode 7)</pre></div>

1/1, 4/3b, 4a/3b, 4/3 (mode 7)

==Tetrachord rotations==

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:

sML, MsL, sLM, MLs, LsM, LMs

I would consider these different rotations as belonging to the same "family," speaking loosely. If another term exists for this, I'd like to hear it.

If you have only two step sizes, s and L, then you have three possible rotations:

ssL, sLs, Lss

And, if you have only one step size (as is the case in Porcupine temperament), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]] - see [[22edo tetrachords]].)

The ancient Greeks seemed to have a preference for tetrachords where the large interval is on top (sML, MsL, ssL). I wonder how widespread that preference is today....

=Tetrachords in equal temperaments=

Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with [[7edo]], which has one tetrachord:

1 + 1 + 1

Some of those tetrachords will approximate just tetrachords (traditional or non-traditional) and some of them won't. See [[22edo tetrachords]], [[17edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]].</pre></div>

<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 />

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1/1, b/a, 4/3a, 4/3 (mode 6)<br />

1/1, 4/3b, 4a/3b, 4/3 (mode 7)</body></html></pre></div>

1/1, 4/3b, 4a/3b, 4/3 (mode 7)<br />

<br />

<h2 id="toc9"><a name="Tetrachords Generalized-Tetrachord rotations"></a>Tetrachord rotations</h2>

<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 />

<br />

sML, MsL, sLM, MLs, LsM, LMs<br />

<br />

I would consider these different rotations as belonging to the same &quot;family,&quot; speaking loosely. If another term exists for this, I'd like to hear it.<br />

<br />

If you have only two step sizes, s and L, then you have three possible rotations:<br />

<br />

ssL, sLs, Lss<br />

<br />

And, if you have only one step size (as is the case in Porcupine temperament), 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 />

<br />

The ancient Greeks seemed to have a preference for tetrachords where the large interval is on top (sML, MsL, ssL). I wonder how widespread that preference is today....<br />

<br />

<h1 id="toc10"><a name="Tetrachords in equal temperaments"></a>Tetrachords in equal temperaments</h1>

<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 />

<br />

1 + 1 + 1<br />

<br />

Some of those tetrachords will approximate just tetrachords (traditional or non-traditional) and some of them won't. See <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>.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2009-09-06 15:04:54 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-09-06 16:32:25 UTC</tt>.<br>
-: The original revision id was <tt>87403719</tt>.<br>
+: The original revision id was <tt>87410093</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>
@@ Line 101: / Line 101: @@
 /1, b/a, 4/3a, 4/3 (mode 6)
-/1, 4/3b, 4a/3b, 4/3 (mode 7)</pre></div>
+/1, 4/3b, 4a/3b, 4/3 (mode 7)
+==Tetrachord rotations==
+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:
+sML, MsL, sLM, MLs, LsM, LMs
+I would consider these different rotations as belonging to the same "family," speaking loosely. If another term exists for this, I'd like to hear it.
+If you have only two step sizes, s and L, then you have three possible rotations:
+ssL, sLs, Lss
+And, if you have only one step size (as is the case in Porcupine temperament), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]] - see [[22edo tetrachords]].)
+The ancient Greeks seemed to have a preference for tetrachords where the large interval is on top (sML, MsL, ssL). I wonder how widespread that preference is today....
+=Tetrachords in equal temperaments=
+Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with [[7edo]], which has one tetrachord:
++ 1 + 1
+Some of those tetrachords will approximate just tetrachords (traditional or non-traditional) and some of them won't. See [[22edo tetrachords]], [[17edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]].</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;tetrachord&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Related pages: &lt;a class="wiki_link" href="/22edo%20tetrachords"&gt;22edo tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo%20tetrachords"&gt;17edo tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract"&gt;Tricesimoprimal Tetrachordal Tesseract&lt;/a&gt;&lt;br /&gt;
@@ Line 389: / Line 413: @@
 &lt;br /&gt;
 /1, b/a, 4/3a, 4/3 (mode 6)&lt;br /&gt;
-/1, 4/3b, 4a/3b, 4/3 (mode 7)&lt;/body&gt;&lt;/html&gt;</pre></div>
+/1, 4/3b, 4a/3b, 4/3 (mode 7)&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Tetrachords Generalized-Tetrachord rotations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Tetrachord rotations&lt;/h2&gt;
+ &lt;br /&gt;
+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:&lt;br /&gt;
+&lt;br /&gt;
+sML, MsL, sLM, MLs, LsM, LMs&lt;br /&gt;
+&lt;br /&gt;
+I would consider these different rotations as belonging to the same &amp;quot;family,&amp;quot; speaking loosely. If another term exists for this, I'd like to hear it.&lt;br /&gt;
+&lt;br /&gt;
+If you have only two step sizes, s and L, then you have three possible rotations:&lt;br /&gt;
+&lt;br /&gt;
+ssL, sLs, Lss&lt;br /&gt;
+&lt;br /&gt;
+And, if you have only one step size (as is the case in Porcupine temperament), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; - see &lt;a class="wiki_link" href="/22edo%20tetrachords"&gt;22edo tetrachords&lt;/a&gt;.)&lt;br /&gt;
+&lt;br /&gt;
+The ancient Greeks seemed to have a preference for tetrachords where the large interval is on top (sML, MsL, ssL). I wonder how widespread that preference is today....&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Tetrachords in equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Tetrachords in equal temperaments&lt;/h1&gt;
+ &lt;br /&gt;
+Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, which has one tetrachord:&lt;br /&gt;
+&lt;br /&gt;
++ 1 + 1&lt;br /&gt;
+&lt;br /&gt;
+Some of those tetrachords will approximate just tetrachords (traditional or non-traditional) and some of them won't. See &lt;a class="wiki_link" href="/22edo%20tetrachords"&gt;22edo tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo%20tetrachords"&gt;17edo tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract"&gt;Tricesimoprimal Tetrachordal Tesseract&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>