Gallery of omnitetrachordal scales: 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:spt3125|spt3125]] and made on <tt>2016-08-28 21:38:42 UTC</tt>.<br>

: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2016-08-28 21:42:57 UTC</tt>.<br>

: The original revision id was <tt>~~590309760~~</tt>.<br>

: The original revision id was <tt>590309998</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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[[#Q]]**Q** is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.

Consider for example the 2L+8s pajara MODMOS, LsssssLsss -- at P ``=`` L/s ``=`` 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q ``=`` L/s ``=`` 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:

Consider for example the [[OTC 2L 8s|OTC 2L+8s]] pajara MODMOS, LsssssLsss -- at P ``=`` L/s ``=`` 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q ``=`` L/s ``=`` 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:

[[image:P_Q_C_LsssssLsss.png]]

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...which leads to the curious result that, although P is undefined when calculated by the normal method (due to division by zero) for a scale such as blackwood[10] (LsLsLsLsLs), C and Q can be calculated, and thus a value of P ``=`` -1 can be found anyway, even though it seems not to be of any use :)

**"L/s range"**: For any L+s scale pattern, the ratio L/s may range from 1 (L``=``s, in which case the scale is (L+s)edo ) to ~infinity (s``=``0, in which case the scale is (L)edo ). A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.

**"L/s range"**: For any L+s scale pattern, the ratio L/s may range from 1 (L``=``s, in which case the scale is (L+s)edo ) to ~infinity (s``=``0, in which case the scale is (L)edo ). A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.</pre></div>

</pre></div>

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<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>Gallery of omnitetrachordal scales</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Scales with two step sizes">Scales with two step sizes</a></div>

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<a name="Q"></a><strong>Q</strong> is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.<br />

<br />

Consider for example the 2L+8s pajara MODMOS, LsssssLsss -- at P = L/s = 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q = L/s = 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:~~<br />~~

Consider for example the <a class="wiki_link" href="/OTC%202L%208s">OTC 2L+8s</a> pajara MODMOS, LsssssLsss -- at P = L/s = 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q = L/s = 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:<br />

<br />

<img src="/file/view/P_Q_C_LsssssLsss.png/590309824/P_Q_C_LsssssLsss.png" alt="P_Q_C_LsssssLsss.png" title="P_Q_C_LsssssLsss.png" /><br />

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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:spt3125|spt3125]] and made on <tt>2016-08-28 21:38:42 UTC</tt>.<br>
+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2016-08-28 21:42:57 UTC</tt>.<br>
-: The original revision id was <tt>590309760</tt>.<br>
+: The original revision id was <tt>590309998</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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 [[#Q]]**Q**  is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.
-Consider for example the 2L+8s pajara MODMOS, LsssssLsss -- at P ``=`` L/s ``=`` 1.885, L+3s forms a just 4/3.  As L/s increases, L gets larger and s smaller; at Q ``=`` L/s ``=`` 4.827, a just 4/3 is not L+3s, but L+2.5s.  Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:
+Consider for example the [[OTC 2L 8s|OTC 2L+8s]] pajara MODMOS, LsssssLsss -- at P ``=`` L/s ``=`` 1.885, L+3s forms a just 4/3.  As L/s increases, L gets larger and s smaller; at Q ``=`` L/s ``=`` 4.827, a just 4/3 is not L+3s, but L+2.5s.  Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:
 [[image:P_Q_C_LsssssLsss.png]]
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 ...which leads to the curious result that, although P is undefined when calculated by the normal method (due to division by zero) for a scale such as blackwood[10] (LsLsLsLsLs), C and Q can be calculated, and thus a value of P ``=`` -1 can be found anyway, even though it seems not to be of any use :)
-**"L/s range"**: For any L+s scale pattern, the ratio L/s may range from 1 (L``=``s, in which case the scale is (L+s)edo ) to ~infinity (s``=``0, in which case the scale is (L)edo ).  A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.
+**"L/s range"**: For any L+s scale pattern, the ratio L/s may range from 1 (L``=``s, in which case the scale is (L+s)edo ) to ~infinity (s``=``0, in which case the scale is (L)edo ).  A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.</pre></div>
-</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;Gallery of omnitetrachordal scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:80:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Scales with two step sizes"&gt;Scales with two step sizes&lt;/a&gt;&lt;/div&gt;
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 &lt;!-- ws:start:WikiTextAnchorRule:111:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Q&amp;quot; title=&amp;quot;Anchor: Q&amp;quot;/&amp;gt; --&gt;&lt;a name="Q"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:111 --&gt;&lt;strong&gt;Q&lt;/strong&gt;  is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.&lt;br /&gt;
 &lt;br /&gt;
-Consider for example the 2L+8s pajara MODMOS, LsssssLsss -- at P &lt;!-- ws:start:WikiTextRawRule:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&gt; L/s &lt;!-- ws:start:WikiTextRawRule:016:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:016 --&gt; 1.885, L+3s forms a just 4/3.  As L/s increases, L gets larger and s smaller; at Q &lt;!-- ws:start:WikiTextRawRule:017:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:017 --&gt; L/s &lt;!-- ws:start:WikiTextRawRule:018:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:018 --&gt; 4.827, a just 4/3 is not L+3s, but L+2.5s.  Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:&lt;br /&gt;
+Consider for example the &lt;a class="wiki_link" href="/OTC%202L%208s"&gt;OTC 2L+8s&lt;/a&gt; pajara MODMOS, LsssssLsss -- at P &lt;!-- ws:start:WikiTextRawRule:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&gt; L/s &lt;!-- ws:start:WikiTextRawRule:016:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:016 --&gt; 1.885, L+3s forms a just 4/3.  As L/s increases, L gets larger and s smaller; at Q &lt;!-- ws:start:WikiTextRawRule:017:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:017 --&gt; L/s &lt;!-- ws:start:WikiTextRawRule:018:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:018 --&gt; 4.827, a just 4/3 is not L+3s, but L+2.5s.  Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:&lt;br /&gt;
-&lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:1525:&amp;lt;img src=&amp;quot;/file/view/P_Q_C_LsssssLsss.png/590309824/P_Q_C_LsssssLsss.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/P_Q_C_LsssssLsss.png/590309824/P_Q_C_LsssssLsss.png" alt="P_Q_C_LsssssLsss.png" title="P_Q_C_LsssssLsss.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1525 --&gt;&lt;br /&gt;
 &lt;br /&gt;