Tp tuning: Difference between revisions

Wikispaces>genewardsmith
**Imported revision 347655240 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 347655880 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>2012-06-24 14:49:22 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-24 14:54:44 UTC</tt>.<br>
: The original revision id was <tt>347655240</tt>.<br>
: The original revision id was <tt>347655880</tt>.<br>
: The revision comment was: <tt></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>
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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=Dual norm=
=Dual norm=
We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]]. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).</pre></div>
We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]], which is the same as the PEps(T) previously defined. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).</pre></div>
<h4>Original HTML content:</h4>
<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;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;
<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;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;
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&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Dual norm"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Dual norm&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Dual norm"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Dual norm&lt;/h1&gt;
We can extend the Lp norm on monzos to a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;vector space norm&lt;/a&gt; on &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow"&gt;dual norm&lt;/a&gt;. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define &lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;smonzos&lt;/a&gt; for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &amp;lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).&lt;/body&gt;&lt;/html&gt;</pre></div>
We can extend the Lp norm on monzos to a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;vector space norm&lt;/a&gt; on &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow"&gt;dual norm&lt;/a&gt;, which is the same as the PEps(T) previously defined. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define &lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;smonzos&lt;/a&gt; for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &amp;lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).&lt;/body&gt;&lt;/html&gt;</pre></div>