Patent val: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-08-22 09:19:33 UTC</tt>.<br>

: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-08-22 09:36:37 UTC</tt>.<br>

: The original revision id was <tt>~~247649877~~</tt>.<br>

: The original revision id was <tt>247653761</tt>.<br>

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

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Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.

Substitute in the values for 81/80 in 31 EDO and get this: (2.991035765^4) / (2^4)*(5.002262078) = 80.036193 / 80.036193 = 1/1.

The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.

~~The lesson here~~ is that ~~even though~~ the ~~errors in~~ the ~~primes are different for each EDO +~~ patent val ~~in these cases~~, 81/80 ~~is still tempered out~~. ~~However~~, that's ~~not true for all commas; for instance~~, ~~12 EDO tempers out 128~~/~~125~~, the ~~diesis~~, ~~while~~ 31 ~~EDO does not; and~~ 31 ~~EDO tempers out 393216/390625~~, the ~~Wuerschmidt comma~~, ~~while 12 EDO does not~~.</pre></div>

By the way, there's a faster way to find out if a comma is being tempered out. Recall that multiplying by a number means adding logarithms: for instance, to go up an octave you can multiply a ratio by 2/1 or you can add 1200 to the cents value. Since the patent val shows you how many EDO steps it takes to get to a prime number, you can add that many steps instead of multiplying things out like we did above. Similarly, you can subtract if you need to divide.

81/80 = 3*3*3*3 / (2*2*2*2 * 5).

To get 81, we need to multiply by three, four times. That means we add the number of steps to 3/1 in the patent val, four times. Using the 31 EDO patent val, that's 49+49+49+49, or 4*49, or 196.

To get 80, we need to multiply by two, four times, and multiply again by five, once. So add together the number of steps to 2/1, four times, and then add the number of steps to 5/1. Using the 31 EDO patent val, that's 31+31+31+31+72, or 4*31+72, or 196.

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="#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>

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Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.<br />

Substitute in the values for 81/80 in 31 EDO and get this: (2.991035765^4) / (2^4)*(5.002262078) = 80.036193 / 80.036193 = 1/1.<br />

The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.<br />

<br />

~~The lesson here~~ is that ~~even though~~ the ~~errors in~~ the ~~primes are different for each EDO +~~ patent val ~~in these cases~~, 81/80 ~~is still tempered out~~. ~~However~~, that's ~~not true for all commas~~; ~~for instance~~, ~~12 EDO tempers out 128~~/~~125~~, the ~~diesis~~, ~~while~~ 31 ~~EDO does not~~; ~~and 31 EDO tempers out 393216~~/~~390625~~, the ~~Wuerschmidt comma~~, ~~while 12 EDO does not~~.</body></html></pre></div>

By the way, there's a faster way to find out if a comma is being tempered out. Recall that multiplying by a number means adding logarithms: for instance, to go up an octave you can multiply a ratio by 2/1 or you can add 1200 to the cents value. Since the patent val shows you how many EDO steps it takes to get to a prime number, you can add that many steps instead of multiplying things out like we did above. Similarly, you can subtract if you need to divide.<br />

81/80 = 3*3*3*3 / (2*2*2*2 * 5). <br />

To get 81, we need to multiply by three, four times. That means we add the number of steps to 3/1 in the patent val, four times. Using the 31 EDO patent val, that's 49+49+49+49, or 4*49, or 196.<br />

To get 80, we need to multiply by two, four times, and multiply again by five, once. So add together the number of steps to 2/1, four times, and then add the number of steps to 5/1. Using the 31 EDO patent val, that's 31+31+31+31+72, or 4*31+72, or 196.<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;.</body></html></pre></div>

@@ 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:jdfreivald|jdfreivald]] and made on <tt>2011-08-22 09:19:33 UTC</tt>.<br>
+: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-08-22 09:36:37 UTC</tt>.<br>
-: The original revision id was <tt>247649877</tt>.<br>
+: The original revision id was <tt>247653761</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 78: / Line 78: @@
 Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.
 Substitute in the values for 81/80 in 31 EDO and get this: (2.991035765^4) / (2^4)*(5.002262078) = 80.036193 / 80.036193 = 1/1.
+The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.
-The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.</pre></div>
+By the way, there's a faster way to find out if a comma is being tempered out. Recall that multiplying by a number means adding logarithms: for instance, to go up an octave you can multiply a ratio by 2/1 or you can add 1200 to the cents value. Since the patent val shows you how many EDO steps it takes to get to a prime number, you can add that many steps instead of multiplying things out like we did above. Similarly, you can subtract if you need to divide.
+/80 = 3*3*3*3 / (2*2*2*2 * 5).
+To get 81, we need to multiply by three, four times. That means we add the number of steps to 3/1 in the patent val, four times. Using the 31 EDO patent val, that's 49+49+49+49, or 4*49, or 196.
+To get 80, we need to multiply by two, four times, and multiply again by five, once. So add together the number of steps to 2/1, four times, and then add the number of steps to 5/1. Using the 31 EDO patent val, that's 31+31+31+31+72, or 4*31+72, or 196.
+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:10:&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:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#A 12 EDO Example"&gt;A 12 EDO Example&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&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:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&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:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt; | &lt;a href="#How this relates to commas"&gt;How this relates to commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;
@@ Line 153: / Line 158: @@
 Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.&lt;br /&gt;
 Substitute in the values for 81/80 in 31 EDO and get this: (2.991035765^4) / (2^4)*(5.002262078) = 80.036193 / 80.036193 = 1/1.&lt;br /&gt;
+The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.&lt;br /&gt;
 &lt;br /&gt;
-The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.&lt;/body&gt;&lt;/html&gt;</pre></div>
+By the way, there's a faster way to find out if a comma is being tempered out. Recall that multiplying by a number means adding logarithms: for instance, to go up an octave you can multiply a ratio by 2/1 or you can add 1200 to the cents value. Since the patent val shows you how many EDO steps it takes to get to a prime number, you can add that many steps instead of multiplying things out like we did above. Similarly, you can subtract if you need to divide.&lt;br /&gt;
+/80 = 3*3*3*3 / (2*2*2*2 * 5). &lt;br /&gt;
+To get 81, we need to multiply by three, four times. That means we add the number of steps to 3/1 in the patent val, four times. Using the 31 EDO patent val, that's 49+49+49+49, or 4*49, or 196.&lt;br /&gt;
+To get 80, we need to multiply by two, four times, and multiply again by five, once. So add together the number of steps to 2/1, four times, and then add the number of steps to 5/1. Using the 31 EDO patent val, that's 31+31+31+31+72, or 4*31+72, or 196.&lt;br /&gt;
+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 &amp;quot;vanishes&amp;quot;.&lt;/body&gt;&lt;/html&gt;</pre></div>