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:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2011-09-03 18:26~~:30~~ UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 19:49:26 UTC</tt>.<br>

: The original revision id was <tt>~~250536000~~</tt>.<br>

: The original revision id was <tt>250542984</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>

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=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.

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~~ 17-equal ~~instead, which~~ 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.

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=

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<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.<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 />

<br />

You may prefer to use the &lt;17 27 40| val ~~for~~ 17-equal ~~instead, which~~ 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.<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>

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 18:26:30 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 19:49:26 UTC</tt>.<br>
-: The original revision id was <tt>250536000</tt>.<br>
+: The original revision id was <tt>250542984</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>
@@ Line 9: / Line 9: @@
 ----
 =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.
+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 17-equal instead, which 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.
+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=
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-  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.&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 17-equal instead, which 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.&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;