Patent val: Difference between revisions
Wikispaces>phylingual **Imported revision 353009138 - Original comment: ** |
Wikispaces>phylingual **Imported revision 353009274 - 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:phylingual|phylingual]] and made on <tt>2012-07-13 13:25 | : This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-07-13 13:26:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>353009274</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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You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 -- in other words, 81/80 "vanishes". | You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 -- in other words, 81/80 "vanishes". | ||
=Patent vals from real numbers= | =Patent vals from real numbers (Generalized patent vals)= | ||
Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the [[The Riemann Zeta Function and Tuning#The%20Z%20function|Z-function]] maximum at 48.9451 leads to a 13-limit X-edo val of <49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of <49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is <49 78 114 138 170 181|. | Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the [[The Riemann Zeta Function and Tuning#The%20Z%20function|Z-function]] maximum at 48.9451 leads to a 13-limit X-edo val of <49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of <49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is <49 78 114 138 170 181|.</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>Patent val</title></head><body><!-- ws:start:WikiTextTocRule: | <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>Patent val</title></head><body><!-- ws:start:WikiTextTocRule:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Further explanation">Further explanation</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#A 12 EDO Example">A 12 EDO Example</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#An alternate and expanded example for 31 EDO">An alternate and expanded example for 31 EDO</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#How this defines a rank 1 temperament">How this defines a rank 1 temperament</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#How this relates to commas">How this relates to commas</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Patent vals from real numbers (Generalized patent vals)">Patent vals from real numbers (Generalized patent vals)</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> | ||
The patent val for some EDO is the val that you obtain by simply finding the closest rounded-off approximation to each prime in the tuning. For example, the patent val for 17-EDO is &lt;17 27 39|, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal. This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; for instance the 7-limit patent val for 16.9 is &lt;17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47.<br /> | The patent val for some EDO is the val that you obtain by simply finding the closest rounded-off approximation to each prime in the tuning. For example, the patent val for 17-EDO is &lt;17 27 39|, indicating that the closest mapping for 2/1 is 17 steps, the closest mapping for 3/1 is 27 steps, and the closest mapping for 5/1 is 39 steps. This means, if octaves are pure, that 3/2 is 706 cents, which is what you get if you round off 3/2 to the closest location in 17-equal, and that 5/4 is 353 cents, which is what you get is you round off 5/4 to the closest location in 17-equal. This val can be extended to the case where the number of steps in an octave is a real number rather than an integer; for instance the 7-limit patent val for 16.9 is &lt;17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47.<br /> | ||
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You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 -- in other words, 81/80 &quot;vanishes&quot;.<br /> | You're dividing 81 by 80, so (assuming we're starting at zero, though it works no matter where you start) you add the steps for 81 (+196) and subtract the steps for 80 (-196). 196-196 = 0. This means that it takes zero steps to reach 81/80 -- in other words, 81/80 &quot;vanishes&quot;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Patent vals from real numbers"></a><!-- ws:end:WikiTextHeadingRule:12 -->Patent vals from real numbers</h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Patent vals from real numbers (Generalized patent vals)"></a><!-- ws:end:WikiTextHeadingRule:12 -->Patent vals from real numbers (Generalized patent vals)</h1> | ||
Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function">Z-function</a> maximum at 48.9451 leads to a 13-limit X-edo val of &lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &lt;49 78 114 138 170 181|. | Instead of assuming the patent val for N-edo comes from an integer N, we could define a patent val for X-edo, where X is any real number, in just the same way. For instance, the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function">Z-function</a> maximum at 48.9451 leads to a 13-limit X-edo val of &lt;49 78 114 137 169 181|, whereas the minimum at 49.1412 leads to an X-edo val of &lt;49 78 114 138 170 182|. Meanwhile, the patent val, which is the X-edo val for X=49.0000 exactly, is &lt;49 78 114 138 170 181|.</body></html></pre></div> | ||