Patent val: Difference between revisions

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>2011-09-03 19:49:26 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2011-09-03 22:48:05 UTC</tt>.<br>

: The original revision id was <tt>~~250542984~~</tt>.<br>

: The original revision id was <tt>250558724</tt>.<br>

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

: The revision comment was: <tt>Removed content is not an "abstract". And who is "you"? Tutorial content should be developed, but works better on a separate page.</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]]~~

<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">=Definition=

~~----~~

~~=Abstract=~~

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 as the 5-limit 17-equal 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.

=Definition=

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] |

Line 99:

Line 92:

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".</pre></div>

<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><a href="#Abstract">Abstract</a> | <a href="#Definition">Definition</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>

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

~~<hr />~~

<h1 id="toc0"><a name="~~Abstract~~"></a>Abstract</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 as the 5-limit 17-equal 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 />~~

<h1 id="toc1"><a name="Definition"></a>Definition</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 />

&lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |<br />

Line 119:

Line 105:

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 />

<h1 id="~~toc2~~"><a name="A 12 EDO Example"></a>A 12 EDO Example</h1>

<h1 id="toc1"><a name="A 12 EDO Example"></a>A 12 EDO Example</h1>

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 />

Line 125:

Line 111:

which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br />

<br />

<h1 id="~~toc3~~"><a name="xAn alternate and expanded example for 31 EDO"></a>An alternate and expanded example for 31 EDO</h1>

<h1 id="toc2"><a name="xAn alternate and expanded example for 31 EDO"></a>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 />

&lt; [2/1] [3/1] [5/1] [7/1] [etc.] |<br />

Line 145:

Line 131:

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 />

<h1 id="~~toc4~~"><a name="How this defines a rank 1 temperament"></a>How this defines a rank 1 temperament</h1>

<h1 id="toc3"><a name="How this defines a rank 1 temperament"></a>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 />

<br />

Line 171:

Line 157:

That doesn't make 31 EDO better or worse; 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 />

<h1 id="~~toc5~~"><a name="How this relates to commas"></a>How this relates to commas</h1>

<h1 id="toc4"><a name="How this relates to commas"></a>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 />

<br />

@@ 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>2011-09-03 19:49:26 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2011-09-03 22:48:05 UTC</tt>.<br>
-: The original revision id was <tt>250542984</tt>.<br>
+: The original revision id was <tt>250558724</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>Removed content is not an "abstract". And who is "you"? Tutorial content should be developed, but works better on a separate page.</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]]
+<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">=Definition=
-----
-=Abstract=
-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 as the 5-limit 17-equal 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.
-=Definition=
 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:
 &lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |
@@ Line 99: / Line 92: @@
 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".</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Patent val&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Abstract"&gt;Abstract&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#A 12 EDO Example"&gt;A 12 EDO Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&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:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#How this relates to commas"&gt;How this relates to commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
+<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:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;
-&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;hr /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Abstract"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Abstract&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;
-&lt;br /&gt;
-You may prefer to use the &amp;lt;17 27 40| val as the 5-limit 17-equal 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;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definition&lt;/h1&gt;
   A &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; contains the number of steps it takes to get to each prime number up to p, in prime number order:&lt;br /&gt;
 &amp;lt; [2/1] [3/1] [5/1] [7/1] ... [p/1] |&lt;br /&gt;
@@ Line 119: / Line 105: @@
 Thus, the way to get the p-limit patent val for N-EDO is to multiply &amp;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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="A 12 EDO Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;A 12 EDO Example&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="A 12 EDO Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;A 12 EDO Example&lt;/h1&gt;
   Multiplying 12 times &amp;lt;1 1.585 2.322 2.807 3.459|&lt;br /&gt;
 yields &amp;lt;12 19.020 27.863 33.688 41.513|,&lt;br /&gt;
@@ Line 125: / Line 111: @@
 which is the &lt;strong&gt;11-limit patent val for &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="xAn alternate and expanded example for 31 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;An alternate and expanded example for 31 EDO&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="xAn alternate and expanded example for 31 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;An alternate and expanded example for 31 EDO&lt;/h1&gt;
   As stated above, the val contains the number of steps it takes to get to a given prime number, in prime number order:&lt;br /&gt;
 &amp;lt; [2/1] [3/1] [5/1] [7/1] [etc.] |&lt;br /&gt;
@@ Line 145: / Line 131: @@
 Note that these are the same answers you would get if you multiplied 31 times &amp;lt;1 1.585 2.322 2.807 3.459 3.700 4.087 4.248 | and rounded the result.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="How this defines a rank 1 temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;How this defines a rank 1 temperament&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="How this defines a rank 1 temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;How this defines a rank 1 temperament&lt;/h1&gt;
   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.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 171: / Line 157: @@
 That doesn't make 31 EDO better or worse; 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).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="How this relates to commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;How this relates to commas&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="How this relates to commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;How this relates to commas&lt;/h1&gt;
   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:&lt;br /&gt;
 &lt;br /&gt;