Patent val: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2011-09-~~13 12~~:53:25 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-14 00:45:09 UTC</tt>.<br>

: The original revision id was <tt>~~253594248~~</tt>.<br>

: The original revision id was <tt>253819988</tt>.<br>

: The revision comment was: <tt></tt><br>

: The revision comment was: <tt>grammar</tt><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>

<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.

You may prefer to use the <17 27 40| val as the 5-limit 17-equal val instead, which rather than <17 27 39| treats 424 cents as 5/4. 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=

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<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><a href="#Introduction">Introduction</a> | <a href="#Further explanation">Further explanation</a> | <a href="#A 12 EDO Example">A 12 EDO Example</a> | <a href="#xAn alternate and expanded example for 31 EDO">An alternate and expanded example for 31 EDO</a> | <a href="#How this defines a rank 1 temperament">How this defines a rank 1 temperament</a> | <a href="#How this relates to commas">How this relates to commas</a> | <a href="#Patent vals from real numbers">Patent vals from real numbers</a>

~~<br />~~

<h1 id="toc0"><a name="Introduction"></a>Introduction</h1>

<h1 id="toc0"><a name="Introduction"></a>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 />

You may prefer to use the &lt;17 27 40| val as the 5-limit 17-equal val instead, which rather than &lt;17 27 39| treats 424 cents as 5/4. 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 />

<h1 id="toc1"><a name="Further explanation"></a>Further explanation</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-09-13 12:53:25 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-14 00:45:09 UTC</tt>.<br>
-: The original revision id was <tt>253594248</tt>.<br>
+: The original revision id was <tt>253819988</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>grammar</tt><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>
 <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=
-=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 &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.
-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 "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, &lt;17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40.
+You may prefer to use the &lt;17 27 40| val as the 5-limit 17-equal val instead, which rather than &lt;17 27 39| treats 424 cents as 5/4. This val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while &lt;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, &lt;17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40.
 =Further explanation=
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 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Patent val&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Introduction"&gt;Introduction&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Further explanation"&gt;Further explanation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#A 12 EDO Example"&gt;A 12 EDO Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#xAn alternate and expanded example for 31 EDO"&gt;An alternate and expanded example for 31 EDO&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#How this defines a rank 1 temperament"&gt;How this defines a rank 1 temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#How this relates to commas"&gt;How this relates to commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Patent vals from real numbers"&gt;Patent vals from real numbers&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Introduction&lt;/h1&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Introduction&lt;/h1&gt;
+ 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 &amp;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 &amp;lt;17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47.&lt;br /&gt;
-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 &amp;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 &amp;lt;17 27 39 47|, since 16.9 * log2(7) = 47.444, which rounds down to 47.&lt;br /&gt;
 &lt;br /&gt;
-You may prefer to use the &amp;lt;17 27 40| val for as the 5-limit 17-equal instead, which val rather than &amp;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 &amp;lt;17 27 39| may not necessarily be the &amp;quot;best&amp;quot; val for 17-equal for all purposes, it is the obvious, or &amp;quot;patent&amp;quot; val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, &amp;lt;17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40.&lt;br /&gt;
+You may prefer to use the &amp;lt;17 27 40| val as the 5-limit 17-equal val instead, which rather than &amp;lt;17 27 39| treats 424 cents as 5/4. This val has lower Tenney-Euclidean error than the 17-EDO patent val. However, while &amp;lt;17 27 39| may not necessarily be the &amp;quot;best&amp;quot; val for 17-equal for all purposes, it is the obvious, or &amp;quot;patent&amp;quot; val, that you get by naively rounding primes off within the EDO and taking no further considerations into account. However, &amp;lt;17 27 40| is the patent val for 17.1, since 17.1 * log2(5) = 39.705, which rounds up to 40.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Further explanation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Further explanation&lt;/h1&gt;