Vals and tuning space: 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:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 15:09:36 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 15:10:54 UTC</tt>.<br>

: The original revision id was <tt>~~242774429~~</tt>.<br>

: The original revision id was <tt>242774589</tt>.<br>

: The revision comment was: <tt></tt><br>

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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;white-space: pre-wrap ! important" class="old-revision-html">=Definition=

<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]

=Definition=

A val "maps" just intonation to a certain number of steps; by putting vals together we can define the mapping of a [[Regular temperaments|regular temperament]] and thereby define the temperament. A val is written in the form <a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a [[Just intonation subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

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which is approximately <31.000 30.916 31.009 30.990|. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt(3838.694), or 61.957. To use the RMS we divide that by sqrt(4)=2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31.</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>Vals and Tuning Space</title></head><body><h1 id="toc0"><a name="Definition"></a>Definition</h1>

<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>Vals and Tuning Space</title></head><body><a href="#Definition">Definition</a> | <a href="#Vals and Monzos">Vals and Monzos</a> | <a href="#Example">Example</a>

<br />

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

A val &quot;maps&quot; just intonation to a certain number of steps; by putting vals together we can define the mapping of a <a class="wiki_link" href="/Regular%20temperaments">regular temperament</a> and thereby define the temperament. A val is written in the form &lt;a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest. <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:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 15:09:36 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-25 15:10:54 UTC</tt>.<br>
-: The original revision id was <tt>242774429</tt>.<br>
+: The original revision id was <tt>242774589</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>
 <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">=Definition=
+<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
+=Definition=
 A val "maps" just intonation to a certain number of steps; by putting vals together we can define the mapping of a [[Regular temperaments|regular temperament]] and thereby define the temperament. A val is written in the form &lt;a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a [[Just intonation subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
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 which is approximately &lt;31.000 30.916 31.009 30.990|. The standard Euclidean norm would then be the square root of the sum of squares of this vector, which is approximately sqrt(3838.694), or 61.957. To use the RMS we divide that by sqrt(4)=2, giving 30.976 for the TE norm. Note that the TE norm for this val is approximately 31.</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;Vals and Tuning Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definition&lt;/h1&gt;
+<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;Vals and Tuning Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#Vals and Monzos"&gt;Vals and Monzos&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Example"&gt;Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;
+&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definition&lt;/h1&gt;
   A val &amp;quot;maps&amp;quot; just intonation to a certain number of steps; by putting vals together we can define the mapping of a &lt;a class="wiki_link" href="/Regular%20temperaments"&gt;regular temperament&lt;/a&gt; and thereby define the temperament. A val is written in the form &amp;lt;a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any set of generators are mapped to, where a set of generators for a &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intonation subgroup&lt;/a&gt; is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest. &lt;br /&gt;
 &lt;br /&gt;