TOP tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 547982446 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 548181552 - 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>2015-04- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-21 19:28:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>548181552</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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For any tuning T, we may define the absolute proportional error APE(T) of T as the [[http://mathworld.wolfram.com/Supremum.html|supremum]] (maximum) of the absolute 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 APE. 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. | For any tuning T, we may define the absolute proportional error APE(T) of T as the [[http://mathworld.wolfram.com/Supremum.html|supremum]] (maximum) of the absolute 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 APE. 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.</pre></div> | 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. | ||
=Maximal error semigroups== | |||
For a tuning T and absolute proportional error E = APE(T), consider the set S of all rational q>0 such that PE(q) = E. If a and b are elements of s, then PE(ab) = E. Hence S is a semigroup under multiplcation, with the structure of a finitely generated free abelian semigroup. A minimal set of generators consists of a finite set of primes or the inverses of primes, where one or the other is chosen so they are tuned sharply, which entails that PE(q)>1 in each case. This is the //sharp semigroup//; inverting the elements of S leads to a mirror image flat semigroup. This has the consequence that the tuning of S is defined entirely by the tuning of the primes in the sharp group S or in the corresponding flat group. From this we may conclude that E is the minimal weighted L-inf error and TOP tuning may also be defined as the minimal weighted L-inf error tuning.</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"><html><head><title>TOP tuning</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | <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:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | ||
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For any tuning T, we may define the absolute proportional error APE(T) of T as the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow">supremum</a> (maximum) of the absolute 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 APE. 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 /> | For any tuning T, we may define the absolute proportional error APE(T) of T as the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Supremum.html" rel="nofollow">supremum</a> (maximum) of the absolute 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 APE. 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 /> | <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> | 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.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Maximal error semigroups="></a><!-- ws:end:WikiTextHeadingRule:4 -->Maximal error semigroups=</h1> | |||
For a tuning T and absolute proportional error E = APE(T), consider the set S of all rational q&gt;0 such that PE(q) = E. If a and b are elements of s, then PE(ab) = E. Hence S is a semigroup under multiplcation, with the structure of a finitely generated free abelian semigroup. A minimal set of generators consists of a finite set of primes or the inverses of primes, where one or the other is chosen so they are tuned sharply, which entails that PE(q)&gt;1 in each case. This is the <em>sharp semigroup</em>; inverting the elements of S leads to a mirror image flat semigroup. This has the consequence that the tuning of S is defined entirely by the tuning of the primes in the sharp group S or in the corresponding flat group. From this we may conclude that E is the minimal weighted L-inf error and TOP tuning may also be defined as the minimal weighted L-inf error tuning.</body></html></pre></div> | |||