Tenney–Euclidean temperament measures: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~x31eq~~|~~x31eq~~]] and made on <tt>2013-03-31 11:35:21 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2013-11-11 23:21:09 UTC</tt>.<br>

: The original revision id was <tt>~~419048294~~</tt>.<br>

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=TE Complexity=

Given a [[Wedgies and Multivals|wedgie]] M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the //complexity// of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been [[http://x31eq.com/temper/primerr.pdf|extensively studied]] by [[Graham Breed]], and we may call it ~~TE or~~ Tenney-Euclidean complexity~~, as~~ it ~~may also~~ be defined in terms of the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]].

Given a [[Wedgies and Multivals|wedgie]] M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the //complexity// of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been [[http://x31eq.com/temper/primerr.pdf|extensively studied]] by [[Graham Breed]], and we may call it Tenney-Euclidean complexity since it can be defined in terms of the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]].

In fact, while TE complexity is easily computed if a routine for computing multivectors is available, such a routine is not required, as it can also be computed using the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gramian]]. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the [[http://en.wikipedia.org/wiki/Dot_product|dot product]] of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the TE norm we have

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<br />

<h1 id="toc1"><a name="TE Complexity"></a>TE Complexity</h1>

Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the <em>complexity</em> of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been <a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow">extensively studied</a> by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>, and we may call it ~~TE or~~ Tenney-Euclidean complexity~~, as~~ it ~~may also~~ be defined in terms of the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean norm</a>.<br />

Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the <em>complexity</em> of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been <a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow">extensively studied</a> by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>, and we may call it Tenney-Euclidean complexity since it can be defined in terms of the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean norm</a>.<br />

<br />

In fact, while TE complexity is easily computed if a routine for computing multivectors is available, such a routine is not required, as it can also be computed using the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow">Gramian</a>. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dot_product" rel="nofollow">dot product</a> of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the TE norm we have<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:x31eq|x31eq]] and made on <tt>2013-03-31 11:35:21 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2013-11-11 23:21:09 UTC</tt>.<br>
-: The original revision id was <tt>419048294</tt>.<br>
+: The original revision id was <tt>468188572</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 11: / Line 11: @@
 =TE Complexity=
-Given a [[Wedgies and Multivals|wedgie]] M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the //complexity// of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been [[http://x31eq.com/temper/primerr.pdf|extensively studied]] by [[Graham Breed]], and we may call it TE or Tenney-Euclidean complexity, as it may also be defined in terms of the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]].
+Given a [[Wedgies and Multivals|wedgie]] M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the //complexity// of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been [[http://x31eq.com/temper/primerr.pdf|extensively studied]] by [[Graham Breed]], and we may call it Tenney-Euclidean complexity since it can be defined in terms of the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]].
 In fact, while TE complexity is easily computed if a routine for computing multivectors is available, such a routine is not required, as it can also be computed using the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gramian]]. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the [[http://en.wikipedia.org/wiki/Dot_product|dot product]] of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the TE norm we have
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="TE Complexity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;TE Complexity&lt;/h1&gt;
-  Given a &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt; M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the &lt;em&gt;complexity&lt;/em&gt; of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow"&gt;extensively studied&lt;/a&gt; by &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;, and we may call it TE or Tenney-Euclidean complexity, as it may also be defined in terms of the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean norm&lt;/a&gt;.&lt;br /&gt;
+  Given a &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt; M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the &lt;em&gt;complexity&lt;/em&gt; of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow"&gt;extensively studied&lt;/a&gt; by &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;, and we may call it Tenney-Euclidean complexity since it can be defined in terms of the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean norm&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 In fact, while TE complexity is easily computed if a routine for computing multivectors is available, such a routine is not required, as it can also be computed using the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow"&gt;Gramian&lt;/a&gt;. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dot_product" rel="nofollow"&gt;dot product&lt;/a&gt; of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the TE norm we have&lt;br /&gt;