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-~~29 15~~:59:33 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-16 19:09:31 UTC</tt>.<br>

: The original revision id was <tt>~~158939035~~</tt>.<br>

: The original revision id was <tt>163292193</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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==The rational projection map==

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 [[Wedgies and Multivals|wedgie]], 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 "torsion problems". 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.</pre></div>

We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification. However, if greater weight needs to be attached to the larger primes than RMS-TOP attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[Fractional monzos|fractional monzos]]. The projection map therefore, like the [[Wedgies and Multivals|wedgie]], 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 "torsion problems". 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.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tenney-Euclidean Tuning</title></head><body>If we have k linearly independent vals of dimension n, they will define a subspace of the n-dimensional <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> Q^n over the field of rational numbers, which extends to a subspace of <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">tuning space</a> over the field of real numbers by &quot;tensoring&quot; or extending coefficients. Either subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is RMS tuning.<br />

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<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>

We may also do the same things starting from unweighted vals. This leads to a different tuning, the <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>, which is perfectly functional but has less theoretical justification. However, if greater weight needs to be attached to the larger primes than RMS-TOP attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are <a class="wiki_link" href="/Fractional%20monzos">fractional monzos</a>. 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>2010-08-29 15:59:33 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-16 19:09:31 UTC</tt>.<br>
-: The original revision id was <tt>158939035</tt>.<br>
+: The original revision id was <tt>163292193</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 39: / Line 39: @@
 ==The rational projection map==
-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 [[Wedgies and Multivals|wedgie]], 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 "torsion problems". 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.</pre></div>
+We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification. However, if greater weight needs to be attached to the larger primes than RMS-TOP attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[Fractional monzos|fractional monzos]]. The projection map therefore, like the [[Wedgies and Multivals|wedgie]], 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 "torsion problems". 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.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tenney-Euclidean Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;If we have k linearly independent vals of dimension n, they will define a subspace of the n-dimensional &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow"&gt;vector space&lt;/a&gt; Q^n over the field of rational numbers, which extends to a subspace of &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;tuning space&lt;/a&gt; over the field of real numbers by &amp;quot;tensoring&amp;quot; or extending coefficients. Either subspace defines a regular temperament of rank k in the prime limit p, where p is the nth prime. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is RMS tuning.&lt;br /&gt;
@@ Line 62: / Line 62: @@
 &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>
+We may also do the same things starting from unweighted vals. This leads to a different tuning, the &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Frobenius tuning&lt;/a&gt;, which is perfectly functional but has less theoretical justification. However, if greater weight needs to be attached to the larger primes than RMS-TOP attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional monzos&lt;/a&gt;. 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>