Don Page comma: Difference between revisions
Wikispaces>genewardsmith **Imported revision 511459532 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 511477944 - 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:genewardsmith|genewardsmith]] and made on <tt>2014-05-27 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-27 12:52:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>511477944</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> | ||
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DPC(8/7, 3/2) = BMC(1/15, 1/5) = 1029/1024 | DPC(8/7, 3/2) = BMC(1/15, 1/5) = 1029/1024 | ||
DPC(5/4, 7/5) = BMC(1/9, 1/6) = 3136/3125 | DPC(5/4, 7/5) = BMC(1/9, 1/6) = 3136/3125 | ||
DPC(9/8, 10/7) = 5120/5103 | DPC(9/8, 10/7) = BMC(1/17, 3/17) = 5120/5103 | ||
DPC(27/25, 7/6) = 4375/4374 | DPC(27/25, 7/6) = BMC(1/26, 1/13) = 4375/4374 | ||
==11-limit commas== | |||
DPC(11/10, 4/3) = BMC(1/21, 1/7) = 4000/3993 | |||
DPC(11/10, 4/3) = 4000/3993 | DPC(10/9, 11/8) = BMC(1/19, 3/19) = 8019/8000 | ||
DPC(10/9, 11/8) = 8019/8000 | DPC(11/9, 3/2) = BMC(1/10, 1/5) = 243/242 | ||
DPC(11/9, 3/2) = 243/242 | DPC(5/4, 11/7) = BMC(1/9, 2/9) = 176/175 | ||
DPC(5/4, 11/7) = 176/175 | DPC(8/7, 11/9) = BMC(1/15, 1/10) = 41503/41472 | ||
DPC(8/7, 11/9) = 41503/41472 | |||
==13-limit commas== | |||
DPC(15/14, 16/13) = 43904/43875 | DPC(15/14, 16/13) = 43904/43875 | ||
DPC(14/13, 5/4) = 10985/10976 | DPC(14/13, 5/4) = 10985/10976 | ||
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<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>Don Page comma</title></head><body><br /> | <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>Don Page comma</title></head><body><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:12:&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; align=&quot;left&quot; /&gt; --><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" align="left" /><!-- ws:end:WikiTextLocalImageRule:12 --><br /> | ||
By a <em>Don Page comma</em> is meant a comma computed from two other intervals by the method suggested by the Don Page paper, <a class="wiki_link_ext" href="http://arxiv.org/abs/0907.5249" rel="nofollow">Why the Kirnberger Kernel Is So Small</a>. If a and b are two rational numbers &gt; 1, define r = ((a-1)(b+1)) / ((b-1)(a+1)). Suppose r reduced to lowest terms is p/q, and a and b are written in <a class="wiki_link" href="/Monzos">monzo</a> form as u and v. Then the Don Page comma is defined as DPC(a, b) = qu - pv, or else minus that if the size in cents is less than zero. The reason for introducing monzos is purely numerical; if monzos are not used the numerators and denominators of the Don Page comma quickly become so large they cannot easily be handled. In ratio form, the Don Page comma can be written a^q / b^p, or the reciprocal of that if that is less than 1.<br /> | By a <em>Don Page comma</em> is meant a comma computed from two other intervals by the method suggested by the Don Page paper, <a class="wiki_link_ext" href="http://arxiv.org/abs/0907.5249" rel="nofollow">Why the Kirnberger Kernel Is So Small</a>. If a and b are two rational numbers &gt; 1, define r = ((a-1)(b+1)) / ((b-1)(a+1)). Suppose r reduced to lowest terms is p/q, and a and b are written in <a class="wiki_link" href="/Monzos">monzo</a> form as u and v. Then the Don Page comma is defined as DPC(a, b) = qu - pv, or else minus that if the size in cents is less than zero. The reason for introducing monzos is purely numerical; if monzos are not used the numerators and denominators of the Don Page comma quickly become so large they cannot easily be handled. In ratio form, the Don Page comma can be written a^q / b^p, or the reciprocal of that if that is less than 1.<br /> | ||
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DPC(8/7, 3/2) = BMC(1/15, 1/5) = 1029/1024<br /> | DPC(8/7, 3/2) = BMC(1/15, 1/5) = 1029/1024<br /> | ||
DPC(5/4, 7/5) = BMC(1/9, 1/6) = 3136/3125<br /> | DPC(5/4, 7/5) = BMC(1/9, 1/6) = 3136/3125<br /> | ||
DPC(9/8, 10/7) = | DPC(9/8, 10/7) = BMC(1/17, 3/17) = 5120/5103<br /> | ||
DPC(27/25, 7/6) = BMC(1/26, 1/13) = 4375/4374<br /> | |||
DPC( | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Examples-11-limit commas"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit commas</h2> | |||
DPC(11/10, 4/3) = BMC(1/21, 1/7) = 4000/3993<br /> | |||
DPC(10/9, 11/8) = BMC(1/19, 3/19) = 8019/8000<br /> | |||
DPC(11/9, 3/2) = BMC(1/10, 1/5) = 243/242<br /> | |||
DPC(5/4, 11/7) = BMC(1/9, 2/9) = 176/175<br /> | |||
DPC(8/7, 11/9) = BMC(1/15, 1/10) = 41503/41472<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Examples-13-limit commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit commas</h2> | |||
DPC(15/14, 16/13) = 43904/43875<br /> | DPC(15/14, 16/13) = 43904/43875<br /> | ||
DPC(14/13, 5/4) = 10985/10976<br /> | DPC(14/13, 5/4) = 10985/10976<br /> | ||