TOP tuning: Difference between revisions
Wikispaces>xenwolf **Imported revision 548250454 - Original comment: (heading syntax fixed)** |
Wikispaces>genewardsmith **Imported revision 548491704 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-04-23 23:32:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>548491704</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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=Maximal error semigroups= | =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 semigroup S or in the corresponding flat semigroup. 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> | 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 semigroup S or in the corresponding flat semigroup. 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. | ||
For any regular temperament, we may define an //intrinsic prime// to be a prime dividing the numerator or denominator of some comma of the temperament. If the set of intrinsic primes generates the group on which the temperament is defined, we may call the temperament an //intrinsic temperament//. If the temperament is defined on a group generated by primes, then a prime which is not intrinsic is //extrinsic//. </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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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Maximal error semigroups"></a><!-- ws:end:WikiTextHeadingRule:4 -->Maximal error semigroups</h1> | <!-- 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 semigroup S or in the corresponding flat semigroup. 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> | 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 semigroup S or in the corresponding flat semigroup. 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.<br /> | ||
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For any regular temperament, we may define an <em>intrinsic prime</em> to be a prime dividing the numerator or denominator of some comma of the temperament. If the set of intrinsic primes generates the group on which the temperament is defined, we may call the temperament an <em>intrinsic temperament</em>. If the temperament is defined on a group generated by primes, then a prime which is not intrinsic is <em>extrinsic</em>.</body></html></pre></div> | |||