Tenney–Euclidean metrics: 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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 14:34:41 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 14:40:01 UTC</tt>.

: The original revision id was <tt>~~209336160~~</tt>.

: The original revision id was <tt>209338730</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 20:

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for **P** in terms of the matrix of monzos B.

To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0> representing 2 to the matrix B. An alternative proceedure is to find the [[normal lists|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix **P** = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining **S**, and defining the seminorm on monzos as sqrt(b**S**b*) for monzos b.

To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0> representing 2 to the matrix B. An alternative proceedure is to find the [[normal lists|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix **P** = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining **S**, and defining the seminorm on monzos as sqrt(b**S**b*) for monzos b. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of thenp-limit regular temperament given by A.

=Logflat TE badness=

Line 67:

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for P in terms of the matrix of monzos B.

To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0&gt; representing 2 to the matrix B. An alternative proceedure is to find the <a class="wiki_link" href="/normal%20lists">normal val list</a>, and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix P = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining S, and defining the seminorm on monzos as sqrt(bSb*) for monzos b.

To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0&gt; representing 2 to the matrix B. An alternative proceedure is to find the <a class="wiki_link" href="/normal%20lists">normal val list</a>, and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix P = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining S, and defining the seminorm on monzos as sqrt(bSb*) for monzos b. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of thenp-limit regular temperament given by A.

<h1 id="toc3"><a name="Logflat TE badness"></a>Logflat TE badness</h1>

@@ 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>2011-03-10 14:34:41 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 14:40:01 UTC</tt>.<br>
-: The original revision id was <tt>209336160</tt>.<br>
+: The original revision id was <tt>209338730</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 20: / Line 20: @@
 Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for **P** in terms of the matrix of monzos B.
-To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0&gt; representing 2 to the matrix B. An alternative proceedure is to find the [[normal lists|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix **P** = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining **S**, and defining the seminorm on monzos as sqrt(b**S**b*) for monzos b.
+To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0&gt; representing 2 to the matrix B. An alternative proceedure is to find the [[normal lists|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix **P** = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining **S**, and defining the seminorm on monzos as sqrt(b**S**b*) for monzos b. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of thenp-limit regular temperament given by A.
 =Logflat TE badness=
@@ Line 67: / Line 67: @@
   Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for &lt;strong&gt;P&lt;/strong&gt; in terms of the matrix of monzos B.&lt;br /&gt;
 &lt;br /&gt;
-To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0&amp;gt; representing 2 to the matrix B. An alternative proceedure is to find the &lt;a class="wiki_link" href="/normal%20lists"&gt;normal val list&lt;/a&gt;, and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix &lt;strong&gt;P&lt;/strong&gt; = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining &lt;strong&gt;S&lt;/strong&gt;, and defining the seminorm on monzos as sqrt(b&lt;strong&gt;S&lt;/strong&gt;b*) for monzos b.&lt;br /&gt;
+To define the OETES, or Tenney-Euclidean octave equivalent seminorm, we simply add a row |1 0 0 ... 0&amp;gt; representing 2 to the matrix B. An alternative proceedure is to find the &lt;a class="wiki_link" href="/normal%20lists"&gt;normal val list&lt;/a&gt;, and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This is equivalent to finding the matrix &lt;strong&gt;P&lt;/strong&gt; = A*(AW^2A*)^(-1)A defined previously, replacing the first row and column with zeros, obtaining &lt;strong&gt;S&lt;/strong&gt;, and defining the seminorm on monzos as sqrt(b&lt;strong&gt;S&lt;/strong&gt;b*) for monzos b. This seminorm is a measure of the octave-equivalent complexity of a given p-limit rational interval in terms of thenp-limit regular temperament given by A.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Logflat TE badness"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Logflat TE badness&lt;/h1&gt;