Tenney–Euclidean temperament measures: Difference between revisions

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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]].

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

If V is a matrix whose rows are the weighted vals vi, we may write det([vi.vj]) as det(VV*), where V* denotes the transpose. If W is [[http://en.wikipedia.org/wiki/Diagonal_matrix|diagonal matrix]] with 1, 1/log2(3), ..., 1/log2(p) along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV*) = det(AW^2A*). This may be related to the [[Tenney-Euclidean metrics|TE tuning projection matrix]] P, which is V*(VV*)^(-1)V, and the corresponding matix for unweighted monzos **P** = A*(AW^2A*)^(-1)A.

=TE simple badness=  

If J = &lt;1 1 ... 1| is the JI point in weighted coordinates, then the simple badness of M, which we may also call the relative error of M, is defined as ||J∧M||. This may considered to be a sort of badness which heavily favors complex temperaments, or it may be considered error relativized to the complexity of the temperament: relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that ai = J.vi/n is the mean value of the entries of vi. Then note that J∧(v1-a1J)∧(v2-a2J)∧...∧(vr-arJ) = J∧v1∧v2∧...∧vr, since wedge products with more than one term are zero. The Gram matrix of the vectors J and v1-aiJ will have n as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:

&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;

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;

&lt;br /&gt;

&lt;br /&gt;

If V is a matrix whose rows are the weighted vals vi, we may write det([vi.vj]) as det(VV*), where V* denotes the transpose. If W is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Diagonal_matrix" rel="nofollow"&gt;diagonal matrix&lt;/a&gt; with 1, 1/log2(3), ..., 1/log2(p) along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV*) = det(AW^2A*). This may be related to the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;TE tuning projection matrix&lt;/a&gt; P, which is V*(VV*)^(-1)V, and the corresponding matix for unweighted monzos &lt;strong&gt;P&lt;/strong&gt; = A*(AW^2A*)^(-1)A.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="TE simple badness"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;TE simple badness&lt;/h1&gt;

  If J = &amp;lt;1 1 ... 1| is the JI point in weighted coordinates, then the simple badness of M, which we may also call the relative error of M, is defined as ||J∧M||. This may considered to be a sort of badness which heavily favors complex temperaments, or it may be considered error relativized to the complexity of the temperament: relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that ai = J.vi/n is the mean value of the entries of vi. Then note that J∧(v1-a1J)∧(v2-a2J)∧...∧(vr-arJ) = J∧v1∧v2∧...∧vr, since wedge products with more than one term are zero. The Gram matrix of the vectors J and v1-aiJ will have n as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:&lt;br /&gt;