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:~~seraph57~~|~~seraph57~~]] and made on <tt>~~2009~~-11-~~27 19~~:42:54 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-02-13 16:36:48 UTC</tt>.<br>

: The original revision id was <tt>~~105691261~~</tt>.<br>

: The original revision id was <tt>201361546</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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[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):

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

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

==Modes of a ~~heptatonic mirror~~==

==Modes of a [tetrachord], 9/8, [tetrachord] scale==

~~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 ||

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|| mode 6 || 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1 ||

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

This type of scale contains not only one tetrachord, but three.

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

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

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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:

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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]].)

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 [[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=

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1 + 1 + 1

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

We can use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:

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

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=Tetrachords And Non-Octave Scales=

An example with [[~~Carlos Gamma|~~Carlos Gamma]]:

An example with [[Carlos Gamma]]:

[[http://www.seraph.it/dep/det/GloriousGuitars.mp3|Glorious Guitars]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html|blog entry]])

[[http://www.seraph.it/dep/det/GloriousGuitars.mp3|Glorious Guitars]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html|blog entry]])</pre></div>

~~=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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[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 />

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 />

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[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>

<h2 id="toc8"><a name="Tetrachords Generalized-Modes of a [tetrachord], 9/8, [tetrachord] scale"></a>Modes of a [tetrachord], 9/8, [tetrachord] scale</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 />~~

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</table>

~~<br />~~

This type of scale contains not only one tetrachord, but three.<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 />

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<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 />

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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 />~~

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 />

~~<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 />

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

We can use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:<br />

<br />

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<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>)~~<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>

~~<br />~~

<h1 id="toc14"><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:seraph57|seraph57]] and made on <tt>2009-11-27 19:42:54 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-02-13 16:36:48 UTC</tt>.<br>
-: The original revision id was <tt>105691261</tt>.<br>
+: The original revision id was <tt>201361546</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 80: / 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):
+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
@@ Line 90: / Line 90: @@
 [tetrachord #2], 9/8, [tetrachord #1]
-==Modes of a heptatonic mirror==
+==Modes of a [tetrachord], 9/8, [tetrachord] scale==
-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 ||
@@ Line 101: / Line 99: @@
 || mode 6 || 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1 ||
 || mode 7 || 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1 ||
+This type of scale contains not only one tetrachord, but three.
-Notice that a heptatonic mirror contains not only one tetrachord, but three.
 /1, a, b, 4/3 (mode 1, mode 5)
@@ Line 115: / Line 112: @@
 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:
@@ Line 122: / Line 117: @@
 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]].)
+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 [[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=
@@ Line 133: / Line 125: @@
 + 1 + 1
-From now on, I'll use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:
+We can use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:
 ||~ tetrachord notation ||~ cents between steps ||~ cents from 0 ||
@@ Line 157: / Line 149: @@
 =Tetrachords And Non-Octave Scales=
-An example with [[Carlos Gamma|Carlos Gamma]]:
+An example with [[Carlos Gamma]]:
-[[http://www.seraph.it/dep/det/GloriousGuitars.mp3|Glorious Guitars]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html|blog entry]])
+[[http://www.seraph.it/dep/det/GloriousGuitars.mp3|Glorious Guitars]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html|blog entry]])</pre></div>
-=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 389: / Line 377: @@
 [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;
+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;
@@ Line 399: / Line 387: @@
 [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;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Tetrachords Generalized-Modes of a [tetrachord], 9/8, [tetrachord] scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Modes of a [tetrachord], 9/8, [tetrachord] scale&lt;/h2&gt;
   &lt;br /&gt;
-Going back to our generalized heptatonic mirror, let's take a look at what modes we get by starting on different scale degrees.&lt;br /&gt;
-&lt;br /&gt;
@@ Line 450: / Line 436: @@
 &lt;/table&gt;
-&lt;br /&gt;
+This type of scale contains not only one tetrachord, but three.&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;
@@ Line 464: / Line 449: @@
 &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;
@@ Line 471: / Line 454: @@
 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;
+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 &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;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 482: / Line 462: @@
 + 1 + 1&lt;br /&gt;
 &lt;br /&gt;
-From now on, I'll use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:&lt;br /&gt;
+We can use a notation with hyphens to specify tetrachords in equal temperaments. This tetrachord thus becomes:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 559: / Line 539: @@
   &lt;br /&gt;
 An example with &lt;a class="wiki_link" href="/Carlos%20Gamma"&gt;Carlos Gamma&lt;/a&gt;:&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/GloriousGuitars.mp3" rel="nofollow"&gt;Glorious Guitars&lt;/a&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/GloriousGuitars.mp3" rel="nofollow"&gt;Glorious Guitars&lt;/a&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;/body&gt;&lt;/html&gt;</pre></div>
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="More Than Three Divisions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&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>