Wedgie/Archived version: Difference between revisions

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:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:37:37 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:46:06 UTC</tt>.<br>

: The original revision id was <tt>~~294106864~~</tt>.<br>

: The original revision id was <tt>294108052</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 30:

This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[http://en.wikipedia.org/wiki/Greatest_common_divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call //reduced//, and reduced n-vals can be used to give unique names to [[Regular Temperaments|regular temperaments]].

These reduced n-vals, and particularly reduced bivals, are called **wedgies**, and the fact that they are reduced both makes the name unique and tells us that wedgies are [[http://en.wikipedia.org/wiki/Projective_space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called //contorted//. Wedgies do not name or signify contorted temperaments.</pre></div>

These reduced n-vals, and particularly reduced bivals, are called **[[wedgies]]**, and the fact that they are reduced both makes the name unique and tells us that wedgies are [[http://en.wikipedia.org/wiki/Projective_space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called //contorted//. Wedgies do not name or signify contorted temperaments.</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>Wedgies and Multivals</title></head><body>An alternating <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multilinear_map" rel="nofollow">multilinear map</a> which is a multilinear function taking a certain number n of <a class="wiki_link" href="/monzos">monzos</a> as arguments and returning an integer as a value we may call an <strong>n-map</strong>. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.<br />

Line 58:

This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Greatest_common_divisor" rel="nofollow">GCD</a> of all of the coordinates is 1. An n-map with these properties we may call <em>reduced</em>, and reduced n-vals can be used to give unique names to <a class="wiki_link" href="/Regular%20Temperaments">regular temperaments</a>.<br />

<br />

These reduced n-vals, and particularly reduced bivals, are called <strong>wedgies</strong>, and the fact that they are reduced both makes the name unique and tells us that wedgies are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projective_space" rel="nofollow">projective</a>, and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = &lt;24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called <em>contorted</em>. Wedgies do not name or signify contorted temperaments.</body></html></pre></div>

These reduced n-vals, and particularly reduced bivals, are called <strong><a class="wiki_link" href="/wedgies">wedgies</a></strong>, and the fact that they are reduced both makes the name unique and tells us that wedgies are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projective_space" rel="nofollow">projective</a>, and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = &lt;24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called <em>contorted</em>. Wedgies do not name or signify contorted temperaments.</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:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:37:37 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:46:06 UTC</tt>.<br>
-: The original revision id was <tt>294106864</tt>.<br>
+: The original revision id was <tt>294108052</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 30: / Line 30: @@
 This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[http://en.wikipedia.org/wiki/Greatest_common_divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call //reduced//, and reduced n-vals can be used to give unique names to [[Regular Temperaments|regular temperaments]].
-These reduced n-vals, and particularly reduced bivals, are called **wedgies**, and the fact that they are reduced both makes the name unique and tells us that wedgies are [[http://en.wikipedia.org/wiki/Projective_space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = &lt;24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called //contorted//. Wedgies do not name or signify contorted temperaments.</pre></div>
+These reduced n-vals, and particularly reduced bivals, are called **[[wedgies]]**, and the fact that they are reduced both makes the name unique and tells us that wedgies are [[http://en.wikipedia.org/wiki/Projective_space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = &lt;24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called //contorted//. Wedgies do not name or signify contorted temperaments.</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;Wedgies and Multivals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An alternating &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multilinear_map" rel="nofollow"&gt;multilinear map&lt;/a&gt; which is a multilinear function taking a certain number n of &lt;a class="wiki_link" href="/monzos"&gt;monzos&lt;/a&gt; as arguments and returning an integer as a value we may call an &lt;strong&gt;n-map&lt;/strong&gt;. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.&lt;br /&gt;
@@ Line 58: / Line 58: @@
 This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Greatest_common_divisor" rel="nofollow"&gt;GCD&lt;/a&gt; of all of the coordinates is 1. An n-map with these properties we may call &lt;em&gt;reduced&lt;/em&gt;, and reduced n-vals can be used to give unique names to &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperaments&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-These reduced n-vals, and particularly reduced bivals, are called &lt;strong&gt;wedgies&lt;/strong&gt;, and the fact that they are reduced both makes the name unique and tells us that wedgies are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projective_space" rel="nofollow"&gt;projective&lt;/a&gt;, and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = &amp;lt;24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called &lt;em&gt;contorted&lt;/em&gt;. Wedgies do not name or signify contorted temperaments.&lt;/body&gt;&lt;/html&gt;</pre></div>
+These reduced n-vals, and particularly reduced bivals, are called &lt;strong&gt;&lt;a class="wiki_link" href="/wedgies"&gt;wedgies&lt;/a&gt;&lt;/strong&gt;, and the fact that they are reduced both makes the name unique and tells us that wedgies are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projective_space" rel="nofollow"&gt;projective&lt;/a&gt;, and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = &amp;lt;24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called &lt;em&gt;contorted&lt;/em&gt;. Wedgies do not name or signify contorted temperaments.&lt;/body&gt;&lt;/html&gt;</pre></div>