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:genewardsmith|genewardsmith]] and made on <tt>2011-07-22 10:02:48 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:35:16 UTC</tt>.<br>

: The original revision id was <tt>~~242413891~~</tt>.<br>

: The original revision id was <tt>278338330</tt>.<br>

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=TE simple badness=

If J = <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:

If J = <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:

[[math]]

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<br />

<h1 id="toc2"><a name="TE simple badness"></a>TE simple badness</h1>

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:<br />

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:<br />

<br />

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@@ 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>2011-07-22 10:02:48 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:35:16 UTC</tt>.<br>
-: The original revision id was <tt>242413891</tt>.<br>
+: The original revision id was <tt>278338330</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 25: / Line 25: @@
 =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:
+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:
 [[math]]
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 &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;
+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;
 &lt;br /&gt;
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