TOP tuning: Difference between revisions
Wikispaces>hstraub **Imported revision 582888005 - Original comment: ** |
Wikispaces>hstraub **Imported revision 582888153 - 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:hstraub|hstraub]] and made on <tt>2016-05-12 04: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-05-12 04:46:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>582888153</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc | <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;white-space: pre-wrap ! important" class="old-revision-html"> | ||
[[toc]] | |||
=Proportional error= | =Proportional error= | ||
A //tuning// for a regular temperament is defined by a vector T in [[Vals and Tuning Space#Vals%20and%20Monzos|Tenney tuning space]] whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [[http://en.wikipedia.org/wiki/Bra-ket_notation|bra vector]], and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). | A //tuning// for a regular temperament is defined by a vector T in [[Vals and Tuning Space#Vals%20and%20Monzos|Tenney tuning space]] whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a [[http://en.wikipedia.org/wiki/Bra-ket_notation|bra vector]], and if M is a monzo then <T|M> is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q). | ||
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For an example of how this works, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, |0 -11 15 0 -5>; in the 13 limit, |0 0 46 0 -19 -11>. Putting these in Graham's app will show how closely these are associated with magic. In many cases, the association is even more emphatic.</pre></div> | For an example of how this works, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, |0 -11 15 0 -5>; in the 13 limit, |0 0 46 0 -19 -11>. Putting these in Graham's app will show how closely these are associated with magic. In many cases, the association is even more emphatic.</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:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@ | <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><br /> | ||
<!-- ws:end:WikiTextTocRule:16 --> | <!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><div style="margin-left: 1em;"><a href="#Proportional error">Proportional error</a></div> | ||
<!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><div style="margin-left: 1em;"><a href="#TOP tuning">TOP tuning</a></div> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | <!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#Maximal error semigroups">Maximal error semigroups</a></div> | ||
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 1em;"><a href="#Finding the tuning">Finding the tuning</a></div> | |||
<!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#TOP commas and TOP extensions">TOP commas and TOP extensions</a></div> | |||
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<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Proportional error"></a><!-- ws:end:WikiTextHeadingRule:0 -->Proportional error</h1> | |||
A <em>tuning</em> for a regular temperament is defined by a vector T in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space#Vals%20and%20Monzos">Tenney tuning space</a> whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">bra vector</a>, and if M is a monzo then &lt;T|M&gt; is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q).<br /> | A <em>tuning</em> for a regular temperament is defined by a vector T in <a class="wiki_link" href="/Vals%20and%20Tuning%20Space#Vals%20and%20Monzos">Tenney tuning space</a> whose entries are the size of the interval, in cents, which the k generators of the regular temperament (often the first k primes) are mapped to. T is denoted by a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">bra vector</a>, and if M is a monzo then &lt;T|M&gt; is the size, in cents, of the interval defined by M in the tuning T. If q is the rational number which M represents, then we may also write this quantity as T(q).<br /> | ||
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