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:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2011-09-03 19:56:44 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 19:59:30 UTC</tt>.<br>

: The original revision id was <tt>~~250543568~~</tt>.<br>

: The original revision id was <tt>250543814</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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<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]]

=Abstract=

A val~~, intuitively speaking,~~ provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.

A val provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number "n" of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the "kernel" of the val, meaning the intervals mapped to 0 ("tempered out".) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.

A 12-EDO val tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.

A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation <a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].

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[[math]]

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>

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><a href="#Abstract">Abstract</a> | <a href="#Definition">Definition</a> | <a href="#Vals and Monzos">Vals and Monzos</a> | <a href="#Example">Example</a>

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

A val~~, intuitively speaking,~~ provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.<br />

A val provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number &quot;n&quot; of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the &quot;kernel&quot; of the val, meaning the intervals mapped to 0 (&quot;tempered out&quot;.) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.<br />

<br />

A 12-EDO val tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.<br />

<br />

A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br />

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:56:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 19:59:30 UTC</tt>.<br>
-: The original revision id was <tt>250543568</tt>.<br>
+: The original revision id was <tt>250543814</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 8: / Line 8: @@
 <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]]
 =Abstract=
-A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.
+A val provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number "n" of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the "kernel" of the val, meaning the intervals mapped to 0 ("tempered out".) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.
+A 12-EDO val tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.
 A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some [[harmonic limit|prime limit]].
@@ Line 58: / Line 60: @@
 [[math]]
-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>
+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:WikiTextTocRule:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Abstract"&gt;Abstract&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Vals and Monzos"&gt;Vals and Monzos&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Example"&gt;Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Abstract"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Abstract&lt;/h1&gt;
-  A val, intuitively speaking, provides a way to map intervals in an EDO back to JI. It tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.&lt;br /&gt;
+  A val provides a way to map intervals in just intonation to a certain number of steps; that is, to an integer. In many cases of interest the val is associated to an EDO, and the val maps to steps of the EDO. For any number &amp;quot;n&amp;quot; of steps in the EDO, a set of JI intervals will be mapped to n; this is what mathematicians call a coset, denoted by n+K, where K is the &amp;quot;kernel&amp;quot; of the val, meaning the intervals mapped to 0 (&amp;quot;tempered out&amp;quot;.) Hence the val maps from JI to the integers, and also from integers back to sets of just intervals.&lt;br /&gt;
+&lt;br /&gt;
+A 12-EDO val tells us, when we look at an EDO like 12-equal, how exactly we'd like to describe the intervals in an EDO as being tempered versions of more fundamental JI intervals. It tells us which interval we're going to describe as the tempered 3/2, which interval we're going to describe as the tempered 5/4, etc.&lt;br /&gt;
 &lt;br /&gt;
 A val maps the intervals in an EDO back to JI by describing the mapping for each of the primes. By mapping the primes, you hence indirectly map all of the rationals, since every rational number can be described as a product of primes. It's usually written in the notation &amp;lt;a b c d e f ... |, where each column represents prime 2, 3, 5, 7, 11, 13... etc, in that order, up to some &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit&lt;/a&gt;.&lt;br /&gt;