TOP 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>~~2012~~-06-~~23 22~~:39:13 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-10 11:29:44 UTC</tt>.<br>

: The original revision id was <tt>~~347562570~~</tt>.<br>

: The original revision id was <tt>501776848</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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=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. 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 1. 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.

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 is one way to define a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the [[Lp tuning]] as p tends to 1, which is sometimes called TIPTOP. It has the advantage that after minimizing the maximum error, it goes on if possible to minimize the second maximum, and so forth, so long as this can be done. 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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<br />

<h1 id="toc1"><a name="TOP tuning"></a>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. 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 1. 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 />

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 is one way to define 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 1, which is sometimes called TIPTOP. It has the advantage that after minimizing the maximum error, it goes on if possible to minimize the second maximum, and so forth, so long as this can be done. 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 />

<br />

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>

@@ 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>2012-06-23 22:39:13 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-10 11:29:44 UTC</tt>.<br>
-: The original revision id was <tt>347562570</tt>.<br>
+: The original revision id was <tt>501776848</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 14: / Line 14: @@
 =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. 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 1. 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.
+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 is one way to define a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the [[Lp tuning]] as p tends to 1, which is sometimes called TIPTOP. It has the advantage that after minimizing the maximum error, it goes on if possible to minimize the second maximum, and so forth, so long as this can be done. 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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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="TOP tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;TOP tuning&lt;/h1&gt;
-For any tuning T, we may define the proportional error of PE(T) of T as the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow"&gt;supremum&lt;/a&gt; (maximum) of the proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A &lt;strong&gt;TOP tuning&lt;/strong&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Centroid" rel="nofollow"&gt;centroid&lt;/a&gt;, which can be used as a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the &lt;a class="wiki_link" href="/Lp%20tuning"&gt;Lp tuning&lt;/a&gt; as p tends to 1. It should be noted that the definition works as well for any &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup temperament&lt;/a&gt; as it does for a full prime limit temperament.&lt;br /&gt;
+For any tuning T, we may define the proportional error of PE(T) of T as the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow"&gt;supremum&lt;/a&gt; (maximum) of the proportional errors of all q belonging to the domain of T; that is, for which T provides a value. A &lt;strong&gt;TOP tuning&lt;/strong&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Centroid" rel="nofollow"&gt;centroid&lt;/a&gt;, which is one way to define a canonical TOP tuning. Another choice for a canonical TOP tuning is the limit of the &lt;a class="wiki_link" href="/Lp%20tuning"&gt;Lp tuning&lt;/a&gt; as p tends to 1, which is sometimes called TIPTOP. It has the advantage that after minimizing the maximum error, it goes on if possible to minimize the second maximum, and so forth, so long as this can be done. It should be noted that the definition works as well for any &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup temperament&lt;/a&gt; as it does for a full prime limit temperament.&lt;br /&gt;
 &lt;br /&gt;
 The concept of a TOP tuning was first suggested by &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;, 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.&lt;/body&gt;&lt;/html&gt;</pre></div>