TOP tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 242783877 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 347532726 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-23 17:03:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>347532726</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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=TOP tuning= | =TOP tuning= | ||
For any tuning T, we may define the proportional error of PE(T) of T as the [[http://mathworld.wolfram.com/Supremum.html|supremum]] (maximum) of the proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A **TOP tuning** for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal proportional error. There is always at least one TOP tuning, and may be only one, but in general the set of TOP tunings is a convex region in Tenney tuning space. This region has a [[http://en.wikipedia.org/wiki/Centroid|centroid]], which | For any tuning T, we may define the proportional error of PE(T) of T as the [[http://mathworld.wolfram.com/Supremum.html|supremum]] (maximum) of the proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A **TOP tuning** for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal proportional error. This minimal proportional error is a measure of the error of the temperament, which we might call the TOP error. There is always at least one TOP tuning, and may be only one, but in general the set of TOP tunings is a convex region in Tenney tuning space. This region has a [[http://en.wikipedia.org/wiki/Centroid|centroid]], which can be used as a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the [[Lp tuning]] as p tends to infinity. It should be noted that the definition works as well for any [[Just intonation subgroups|subgroup temperament]] as it does for a full prime limit temperament. | ||
The concept of a TOP tuning was first suggested by [[Paul Erlich]], who gave it its name, which stands for both Tenney OPtimal and Tempered Octaves Please, the latter due to the fact that usually the octaves are tempered. | |||
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<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"><html><head><title>TOP tuning</title></head><body><!-- ws:start:WikiTextTocRule:4:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Proportional error">Proportional error</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#TOP tuning">TOP tuning</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> | <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>TOP tuning</title></head><body><!-- ws:start:WikiTextTocRule:4:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Proportional error">Proportional error</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#TOP tuning">TOP tuning</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="TOP tuning"></a><!-- ws:end:WikiTextHeadingRule:2 -->TOP tuning</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="TOP tuning"></a><!-- ws:end:WikiTextHeadingRule:2 -->TOP tuning</h1> | ||
For any tuning T, we may define the proportional error of PE(T) of T as the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow">supremum</a> (maximum) of the proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A <strong>TOP tuning</strong> for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal proportional error. There is always at least one TOP tuning, and may be only one, but in general the set of TOP tunings is a convex region in Tenney tuning space. This region has a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Centroid" rel="nofollow">centroid</a>, which | For any tuning T, we may define the proportional error of PE(T) of T as the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow">supremum</a> (maximum) of the proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A <strong>TOP tuning</strong> for a regular temperament is a tuning supporting the temperament (ie, one which sends commas of the temperament to 0) with minimal proportional error. This minimal proportional error is a measure of the error of the temperament, which we might call the TOP error. There is always at least one TOP tuning, and may be only one, but in general the set of TOP tunings is a convex region in Tenney tuning space. This region has a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Centroid" rel="nofollow">centroid</a>, which can be used as a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the <a class="wiki_link" href="/Lp%20tuning">Lp tuning</a> as p tends to infinity. It should be noted that the definition works as well for any <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup temperament</a> as it does for a full prime limit temperament.<br /> | ||
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The concept of a TOP tuning was first suggested by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>, who gave it its name, which stands for both Tenney OPtimal and Tempered Octaves Please, the latter due to the fact that usually the octaves are tempered.</body></html></pre></div> | |||