Tenney–Euclidean tuning: Difference between revisions

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<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>2010-08-28 03:13:43 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08-28 03:17:35 UTC</tt>.<br>

: The original revision id was <tt>~~158799437~~</tt>.<br>

: The original revision id was <tt>158799561</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>

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Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that common factors ("torsion problems") cannot be found in some combination of the unweighted vals. If J is the JI point, <1 1 ... 1|, then JV` gives the TOP-RMS tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.

We may also obtain the RMS~~-TOP~~ tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

===Pure octaves RMS tuning===

If T = JP is the RMS~~-TOP~~ tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.

If T = JP is the TOP-RMS tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.

==The rational projection map==

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Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that common factors (&quot;torsion problems&quot;) cannot be found in some combination of the unweighted vals. If J is the JI point, &lt;1 1 ... 1|, then JV` gives the TOP-RMS tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.<br />

<br />

We may also obtain the RMS~~-TOP~~ tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.<br />

We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.<br />

<br />

<h3 id="toc3"><a name="x-RMS tuning-Pure octaves RMS tuning"></a>Pure octaves RMS tuning</h3>

If T = JP is the RMS~~-TOP~~ tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.<br />

If T = JP is the TOP-RMS tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.<br />

<br />

<h2 id="toc4"><a name="x-The rational projection map"></a>The rational projection map</h2>

We may also do the same things starting from unweighted vals. This leads to a different tuning which is perfectly functional but has less theoretical justification. The real value of unweighted vals is that the pseudoinverse and projection map have rational entries. The projection map therefore, like the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which automatically takes care of &quot;torsion problems&quot;. It also may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.</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>2010-08-28 03:13:43 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08-28 03:17:35 UTC</tt>.<br>
-: The original revision id was <tt>158799437</tt>.<br>
+: The original revision id was <tt>158799561</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 31: / Line 31: @@
 Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that common factors ("torsion problems") cannot be found in some combination of the unweighted vals. If J is the JI point, &lt;1 1 ... 1|, then JV` gives the TOP-RMS tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.
-We may also obtain the RMS-TOP tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
+We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
 ===Pure octaves RMS tuning===
-If T = JP is the RMS-TOP tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
+If T = JP is the TOP-RMS tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
 ==The rational projection map==
@@ Line 52: / Line 52: @@
 Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that common factors (&amp;quot;torsion problems&amp;quot;) cannot be found in some combination of the unweighted vals. If J is the JI point, &amp;lt;1 1 ... 1|, then JV` gives the TOP-RMS tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.&lt;br /&gt;
 &lt;br /&gt;
-We may also obtain the RMS-TOP tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.&lt;br /&gt;
+We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-RMS tuning-Pure octaves RMS tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Pure octaves RMS tuning&lt;/h3&gt;
-If T = JP is the RMS-TOP tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.&lt;br /&gt;
+If T = JP is the TOP-RMS tuning map, then a corresponding pure-octaves map can be found by scalar multiplication, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-The rational projection map"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;The rational projection map&lt;/h2&gt;
 We may also do the same things starting from unweighted vals. This leads to a different tuning which is perfectly functional but has less theoretical justification. The real value of unweighted vals is that the pseudoinverse and projection map have rational entries. The projection map therefore, like the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which automatically takes care of &amp;quot;torsion problems&amp;quot;. It also may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I-Q is the same projection map as would be found if starting from a set of vals defining the same temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>