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 16~~:32:25 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-09-09 12:58:37 UTC</tt>.<br>

: The original revision id was <tt>~~87410093~~</tt>.<br>

: The original revision id was <tt>87915917</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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===diatonic genus===

The CI (and the other intervals) approximates a "tone," measuring less than 250 cents.

==Ptolomy's Catalog==

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|| 16/15, 9/8, 10/9 || 112 + 182 + 204 || intense diatonic ||

|| 12/11, 11/10, 10/9 || 151 + 165 + 182 || equable diatonic ||

==Superparticular Intervals==

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[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):

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

[tetrachord #1], 9/8, [tetrachord #2]

Of course, you can also put them in opposite order:

1/1, c, d, 4/3, 3/2, 3a/2, 3b/2, 2/1

[tetrachord #2], 9/8, [tetrachord #1]

==Modes of a heptatonic mirror==

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|| mode 7 || 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1 ||

A heptatonic mirror ~~thus~~ contains ~~more than a single kind of~~ tetrachord. ~~In addition to 1/1, a, b, 4/3, it would also have:~~

Notice that a heptatonic mirror contains not only one tetrachord, but three.

1/1, a, b, 4/3 (mode 1, mode 5)

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

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

These three tetrachords are all rotations of each other (they contain the same steps in a different order).

==Tetrachord rotations==

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Line 125:

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=

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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>

From now on, I'll use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:

||~ tetrachord notation ||~ cents between steps ||~ cents from 0 ||

|| 1-1-1 || 171 + 171 + 171 || 0, 171, 343, 514 ||

==Tetrachords of [[10edo]]==

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:

||~ tetrachord notation ||~ cents between ||~ cents from 0 ||

|| 1-1-2 || 120 + 120 + 240 || 0, 120, 240, 480 ||

|| 1-2-1 || 120 + 240 + 120 || 0, 120, 360, 480 ||

|| 2-1-1 || 240 + 120 + 120 || 0, 240, 360, 480 ||

Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents).

See also: [[17edo tetrachords]], [[22edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]] (tetrachords of [[31edo]]). If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!

=Dividing Other-Than-Perfect Fourths=

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: [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]]. 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 "tetrachord" stop being useful?

=More Than Three Divisions=

If you chop a fourth up into more than three steps, is it still useful to call it a "tetrachord"? I don't know, and barely care. If you find it useful, do it!</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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<h3 id="toc4"><a name="Ancient Greek Genera--diatonic genus"></a>diatonic genus</h3>

The CI (and the other intervals) approximates a &quot;tone,&quot; measuring less than 250 cents.<br />

~~<br />~~

<br />

<h2 id="toc5"><a name="Ancient Greek Genera-Ptolomy's Catalog"></a>Ptolomy's Catalog</h2>

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</table>

~~<br />~~

<br />

<h2 id="toc6"><a name="Ancient Greek Genera-Superparticular Intervals"></a>Superparticular Intervals</h2>

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[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 />

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 />

[tetrachord #1], 9/8, [tetrachord #2]<br />

<br />

Of course, you can also put them in opposite order:<br />

<br />

1/1, c, d, 4/3, 3/2, 3a/2, 3b/2, 2/1<br />

[tetrachord #2], 9/8, [tetrachord #1]<br />

<br />

<h2 id="toc8"><a name="Tetrachords Generalized-Modes of a heptatonic mirror"></a>Modes of a heptatonic mirror</h2>

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<br />

A heptatonic mirror ~~thus~~ contains ~~more than a single kind of~~ tetrachord. ~~In addition to 1/1, a, b, 4/3, it would also have:~~<br />

Notice that a heptatonic mirror contains not only one tetrachord, but three.<br />

<br />

1/1, a, b, 4/3 (mode 1, mode 5)<br />

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

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

<br />

These three tetrachords are all rotations of each other (they contain the same steps in a different order).<br />

<br />

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

Line 430:

Line 470:

<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>

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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>

From now on, I'll use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:<br />

<br />

<tr>

<th>tetrachord notation<br />

</th>

<th>cents between steps<br />

</th>

<th>cents from 0<br />

</th>

</tr>

<tr>

</td>

</td>

</td>

</tr>

</table>

<br />

<h2 id="toc11"><a name="Tetrachords in equal temperaments-Tetrachords of 10edo"></a>Tetrachords of <a class="wiki_link" href="/10edo">10edo</a></h2>

<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 />

<br />

<tr>

<th>tetrachord notation<br />

</th>

<th>cents between<br />

</th>

<th>cents from 0<br />

</th>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</tr>

</table>

Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all &quot;diatonic&quot; (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents).<br />

<br />

See also: <a class="wiki_link" href="/17edo%20tetrachords">17edo tetrachords</a>, <a class="wiki_link" href="/22edo%20tetrachords">22edo tetrachords</a>, <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a> (tetrachords of <a class="wiki_link" href="/31edo">31edo</a>). If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!<br />

<br />

<h1 id="toc12"><a name="Dividing Other-Than-Perfect Fourths"></a>Dividing Other-Than-Perfect Fourths</h1>

<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 />

<br />

<h1 id="toc13"><a name="More Than Three Divisions"></a>More Than Three Divisions</h1>

<br />

If you chop a fourth up into more than three steps, is it still useful to call it a &quot;tetrachord&quot;? I don't know, and barely care. If you find it useful, do it!</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 16:32:25 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-09-09 12:58:37 UTC</tt>.<br>
-: The original revision id was <tt>87410093</tt>.<br>
+: The original revision id was <tt>87915917</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 31: / Line 31: @@
 ===diatonic genus===
 The CI (and the other intervals) approximates a "tone," measuring less than 250 cents.
 ==Ptolomy's Catalog==
@@ Line 61: / Line 60: @@
 || 16/15, 9/8, 10/9 || 112 + 182 + 204 || intense diatonic ||
 || 12/11, 11/10, 10/9 || 151 + 165 + 182 || equable diatonic ||
 ==Superparticular Intervals==
@@ Line 82: / Line 80: @@
 [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):
+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, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1
+[tetrachord #1], 9/8, [tetrachord #2]
+Of course, you can also put them in opposite order:
+/1, c, d, 4/3, 3/2, 3a/2, 3b/2, 2/1
+[tetrachord #2], 9/8, [tetrachord #1]
 ==Modes of a heptatonic mirror==
@@ Line 98: / Line 102: @@
 || mode 7 || 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1 ||
-A heptatonic mirror thus contains more than a single kind of tetrachord. In addition to 1/1, a, b, 4/3, it would also have:
+Notice that a heptatonic mirror contains not only one tetrachord, but three.
+/1, a, b, 4/3 (mode 1, mode 5)
 /1, b/a, 4/3a, 4/3 (mode 6)
 /1, 4/3b, 4a/3b, 4/3 (mode 7)
+These three tetrachords are all rotations of each other (they contain the same steps in a different order).
 ==Tetrachord rotations==
@@ Line 118: / Line 125: @@
 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=
@@ Line 125: / Line 133: @@
 + 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>
+From now on, I'll use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:
+||~ tetrachord notation ||~ cents between steps ||~ cents from 0 ||
+|| 1-1-1 || 171 + 171 + 171 || 0, 171, 343, 514 ||
+==Tetrachords of [[10edo]]==
+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:
+||~ tetrachord notation ||~ cents between ||~ cents from 0 ||
+|| 1-1-2 || 120 + 120 + 240 || 0, 120, 240, 480 ||
+|| 1-2-1 || 120 + 240 + 120 || 0, 120, 360, 480 ||
+|| 2-1-1 || 240 + 120 + 120 || 0, 240, 360, 480 ||
+Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents).
+See also: [[17edo tetrachords]], [[22edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]] (tetrachords of [[31edo]]). If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!
+=Dividing Other-Than-Perfect Fourths=
+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: [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]]. 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 "tetrachord" stop being useful?
+=More Than Three Divisions=
+If you chop a fourth up into more than three steps, is it still useful to call it a "tetrachord"? I don't know, and barely care. If you find it useful, do it!</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 152: / Line 185: @@
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Ancient Greek Genera--diatonic genus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;diatonic genus&lt;/h3&gt;
   The CI (and the other intervals) approximates a &amp;quot;tone,&amp;quot; measuring less than 250 cents.&lt;br /&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Ancient Greek Genera-Ptolomy's Catalog"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Ptolomy's Catalog&lt;/h2&gt;
@@ Line 333: / Line 365: @@
 &lt;/table&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Ancient Greek Genera-Superparticular Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Superparticular Intervals&lt;/h2&gt;
@@ Line 354: / Line 385: @@
 [tetrachord], 9/8, [tetrachord]&lt;br /&gt;
 &lt;br /&gt;
-When a tetrachord is paired with its copy, in this way, I call it a &amp;quot;heptatonic mirror.&amp;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):&lt;br /&gt;
+When a tetrachord is paired with its copy in this way, I call it a &amp;quot;heptatonic mirror.&amp;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):&lt;br /&gt;
 &lt;br /&gt;
 /1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1&lt;br /&gt;
+[tetrachord #1], 9/8, [tetrachord #2]&lt;br /&gt;
+&lt;br /&gt;
+Of course, you can also put them in opposite order:&lt;br /&gt;
+&lt;br /&gt;
+/1, c, d, 4/3, 3/2, 3a/2, 3b/2, 2/1&lt;br /&gt;
+[tetrachord #2], 9/8, [tetrachord #1]&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Tetrachords Generalized-Modes of a heptatonic mirror"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Modes of a heptatonic mirror&lt;/h2&gt;
@@ Line 410: / Line 447: @@
 &lt;br /&gt;
-A heptatonic mirror thus contains more than a single kind of tetrachord. In addition to 1/1, a, b, 4/3, it would also have:&lt;br /&gt;
+Notice that a heptatonic mirror contains not only one tetrachord, but three.&lt;br /&gt;
 &lt;br /&gt;
+/1, a, b, 4/3 (mode 1, mode 5)&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;br /&gt;
+&lt;br /&gt;
+These three tetrachords are all rotations of each other (they contain the same steps in a different order).&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;
@@ Line 430: / Line 470: @@
 &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;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;
@@ Line 437: / Line 478: @@
 + 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>
+From now on, I'll use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;tetrachord notation&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;cents between steps&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;cents from 0&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1-1-1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;171 + 171 + 171&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0, 171, 343, 514&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Tetrachords in equal temperaments-Tetrachords of 10edo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Tetrachords of &lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt;&lt;/h2&gt;
+ &lt;br /&gt;
+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:&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;tetrachord notation&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;cents between&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;cents from 0&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1-1-2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;120 + 120 + 240&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0, 120, 240, 480&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1-2-1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;120 + 240 + 120&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0, 120, 360, 480&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2-1-1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;240 + 120 + 120&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0, 240, 360, 480&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all &amp;quot;diatonic&amp;quot; (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents).&lt;br /&gt;
+&lt;br /&gt;
+See also: &lt;a class="wiki_link" href="/17edo%20tetrachords"&gt;17edo tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo%20tetrachords"&gt;22edo tetrachords&lt;/a&gt;, &lt;a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract"&gt;Tricesimoprimal Tetrachordal Tesseract&lt;/a&gt; (tetrachords of &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;). If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here!&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Dividing Other-Than-Perfect Fourths"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Dividing Other-Than-Perfect Fourths&lt;/h1&gt;
+ &lt;br /&gt;
+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: &lt;a class="wiki_link" href="/6edo"&gt;6edo&lt;/a&gt;, &lt;a class="wiki_link" href="/8edo"&gt;8edo&lt;/a&gt;, &lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt;, &lt;a class="wiki_link" href="/13edo"&gt;13edo&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;. 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 &amp;quot;tetrachord&amp;quot; stop being useful?&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="More Than Three Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;More Than Three Divisions&lt;/h1&gt;
+ &lt;br /&gt;
+If you chop a fourth up into more than three steps, is it still useful to call it a &amp;quot;tetrachord&amp;quot;? I don't know, and barely care. If you find it useful, do it!&lt;/body&gt;&lt;/html&gt;</pre></div>