Patent val: Difference between revisions
Wikispaces>jdfreivald **Imported revision 251253748 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 253594248 - 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>2011-09-13 12:53:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>253594248</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|flat]] | <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|flat]] | ||
= | |||
=Introduction= | |||
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 <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 <17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47. | |||
You may prefer to use the <17 27 40| val for as the 5-limit 17-equal instead, which val rather than <17 27 39|; this treats 424 cents as 5/4 - and indeed this val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while <17 27 39| may not necessarily be the "best" val for 17-equal for all purposes, it is the obvious, or "patent" val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, <17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40. | |||
=Further explanation= | |||
A [[p-limit]] [[Vals and Tuning Space|val]] contains the number of steps it takes to get to each prime number up to p, in prime number order: | A [[p-limit]] [[Vals and Tuning Space|val]] contains the number of steps it takes to get to each prime number up to p, in prime number order: | ||
< [2/1] [3/1] [5/1] [7/1] ... [p/1] | | < [2/1] [3/1] [5/1] [7/1] ... [p/1] | | ||
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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> | 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="#xAn 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">Patent vals from real numbers</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:22 --><br /> | ||
<!-- 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 /> | |||
<br /> | |||
You may prefer to use the &lt;17 27 40| val for as the 5-limit 17-equal instead, which val rather than &lt;17 27 39|; this treats 424 cents as 5/4 - and indeed this val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while &lt;17 27 39| may not necessarily be the &quot;best&quot; val for 17-equal for all purposes, it is the obvious, or &quot;patent&quot; val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, &lt;17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Further explanation"></a><!-- ws:end:WikiTextHeadingRule:2 -->Further explanation</h1> | |||
A <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> contains the number of steps it takes to get to each prime number up to p, in prime number order:<br /> | A <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> contains the number of steps it takes to get to each prime number up to p, in prime number order:<br /> | ||
&lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |<br /> | &lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |<br /> | ||
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Thus, the way to get the p-limit patent val for N-EDO is to multiply &lt;1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers.<br /> | Thus, the way to get the p-limit patent val for N-EDO is to multiply &lt;1 1.585 2.322 2.807 ... log2(p) | by N. Then, since you can't take fractional steps in an EDO, you round the results to the nearest integers.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="A 12 EDO Example"></a><!-- ws:end:WikiTextHeadingRule:4 -->A 12 EDO Example</h1> | ||
Multiplying 12 times &lt;1 1.585 2.322 2.807 3.459|<br /> | Multiplying 12 times &lt;1 1.585 2.322 2.807 3.459|<br /> | ||
yields &lt;12 19.020 27.863 33.688 41.513|,<br /> | yields &lt;12 19.020 27.863 33.688 41.513|,<br /> | ||
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which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br /> | which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="xAn alternate and expanded example for 31 EDO"></a><!-- ws:end:WikiTextHeadingRule:6 -->An alternate and expanded example for 31 EDO</h1> | ||
As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order:<br /> | As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order:<br /> | ||
&lt; [2/1] [3/1] [5/1] [7/1] [etc.] |<br /> | &lt; [2/1] [3/1] [5/1] [7/1] [etc.] |<br /> | ||
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Note that these are the same answers you would get if you multiplied 31 times &lt;1 1.585 2.322 2.807 3.459 3.700 4.087 4.248 | and rounded the result.<br /> | Note that these are the same answers you would get if you multiplied 31 times &lt;1 1.585 2.322 2.807 3.459 3.700 4.087 4.248 | and rounded the result.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="How this defines a rank 1 temperament"></a><!-- ws:end:WikiTextHeadingRule:8 -->How this defines a rank 1 temperament</h1> | ||
A val defines a rank 1 temperament by defining the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2^(700/1200), or 1.4983070769.<br /> | A val defines a rank 1 temperament by defining the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2^(700/1200), or 1.4983070769.<br /> | ||
<br /> | <br /> | ||
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That doesn't make 31 EDO better or worse than 12; it just means there's more error in the 3/1 ratio in 31 EDO than in 12 EDO. If you run these calculations for 5/1 using the patent vals for 12 EDO and 31 EDO, you'll find that 5/1 has more error in 12 EDO than in 31 EDO: 5.0396842 vs. 5.002262078, respectively. 31 EDO may therefore be preferred by people who like sweeter thirds (5/4 ratios) and are willing to have flatter fifths (3/2 ratios).<br /> | That doesn't make 31 EDO better or worse than 12; it just means there's more error in the 3/1 ratio in 31 EDO than in 12 EDO. If you run these calculations for 5/1 using the patent vals for 12 EDO and 31 EDO, you'll find that 5/1 has more error in 12 EDO than in 31 EDO: 5.0396842 vs. 5.002262078, respectively. 31 EDO may therefore be preferred by people who like sweeter thirds (5/4 ratios) and are willing to have flatter fifths (3/2 ratios).<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="How this relates to commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->How this relates to commas</h1> | ||
These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:<br /> | These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:<br /> | ||
<br /> | <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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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> | ||
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> | 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> | ||