Don Page comma: Difference between revisions

Revision as of 20:34, 30 October 2011

IMPORTED REVISION FROM WIKISPACES

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This revision was by author genewardsmith and made on 2011-10-30 20:34:24 UTC.

The original revision id was 270101514.

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Original Wikitext content:

By a //Don Page comma// is meant a comma computed from two other intervals by the method suggested by the Don Page paper, [[http://arxiv.org/abs/0907.5249|Why the Kirnberger Kernel Is So Small]]. If a and b are two rational numbers > 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 [[Monzos|monzo]] 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.

Successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a distinct tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, but these are by no means the only examples.

Here are some 5-limit Don Page commas:

DPC(5/3, 2) = 648/625
DPC(4/3, 9/5) = 81/80
DPC(5/4, 2) = 128/125
DPC(4/3, 5/3) = 16875/16384
DPC(3/2, 5/3) = 20000/19683
DPC(81/80, 128/125) = |161 -84 -12>, the atom
DPC(81/80, 25/24) = |71 -99 37>, raider

Here are some Don Page commas derived from 9-limit consonances:

DPC(7/5, 2) = 50/49
DPC(6/5, 7/4) = 875/864
DPC(7/5, 5/3) = 3125/3087
DPC(9/7, 5/3) = 245/243
DPC(7/6, 8/5) = 1728/1715
DPC(8/7, 3/2) = 1029/1024
DPC(5/4, 7/5) = 3136/3125
DPC(9/8, 10/7) = 5120/5103

From 11-limit consonances:

DPC(11/10, 4/3) = 4000/3993
DPC(10/9, 11/8) = 8019/8000
DPC(11/9, 3/2) = 243/242
DPC(5/4, 11/7) = 176/175
DPC(8/7, 11/9) = 41503/41472

Here are some complex Don Page commas derived from other commas:

DPC(525/512, 245/243) = |-153 277 -18 -87>
DPC(49/48, 50/49) = |-487 -97 -198 392>
DPC(10/9, 11/10) = |40 -38 40 0 -21>
DPC(11/10, 12/11) = |-67 -23 -21 0 44>
DPC(77/75, 245/243) = |0 286 -99 -103 19>
DPC(55/54, 56/55) = |-442 -327 220 -111 220>
DPC(176/175, 540/539) = |-58 -249 -137 139 110>

Original HTML content:

<html><head><title>Don Page comma</title></head><body>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 />
<br />
Successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a distinct tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, but these are by no means the only examples.<br />
<br />
Here are some 5-limit Don Page commas:<br />
<br />
DPC(5/3, 2) = 648/625<br />
DPC(4/3, 9/5) = 81/80<br />
DPC(5/4, 2) = 128/125<br />
DPC(4/3, 5/3) = 16875/16384<br />
DPC(3/2, 5/3) = 20000/19683<br />
DPC(81/80, 128/125) = |161 -84 -12&gt;, the atom<br />
DPC(81/80, 25/24) = |71 -99 37&gt;, raider<br />
<br />
Here are some Don Page commas derived from 9-limit consonances:<br />
<br />
DPC(7/5, 2) = 50/49<br />
DPC(6/5, 7/4) = 875/864<br />
DPC(7/5, 5/3) = 3125/3087<br />
DPC(9/7, 5/3) = 245/243<br />
DPC(7/6, 8/5) = 1728/1715<br />
DPC(8/7, 3/2) = 1029/1024<br />
DPC(5/4, 7/5) = 3136/3125<br />
DPC(9/8, 10/7) = 5120/5103<br />
<br />
From 11-limit consonances:<br />
<br />
DPC(11/10, 4/3) = 4000/3993<br />
DPC(10/9, 11/8) = 8019/8000<br />
DPC(11/9, 3/2) = 243/242<br />
DPC(5/4, 11/7) = 176/175<br />
DPC(8/7, 11/9) = 41503/41472<br />
<br />
Here are some complex Don Page commas derived from other commas:<br />
<br />
DPC(525/512, 245/243) = |-153 277 -18 -87&gt;<br />
DPC(49/48, 50/49) = |-487 -97 -198 392&gt;<br />
DPC(10/9, 11/10) = |40 -38 40 0 -21&gt;<br />
DPC(11/10, 12/11) = |-67 -23 -21 0 44&gt;<br />
DPC(77/75, 245/243) = |0 286 -99 -103 19&gt;<br />
DPC(55/54, 56/55) = |-442 -327 220 -111 220&gt;<br />
DPC(176/175, 540/539) = |-58 -249 -137 139 110&gt;</body></html>

@@ 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>2011-10-30 20:24:34 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-30 20:34:24 UTC</tt>.<br>
-: The original revision id was <tt>270099026</tt>.<br>
+: The original revision id was <tt>270101514</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>
 <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">By a //Don Page comma// is meant a comma computed from two other intervals by the method suggested by the Don Page paper, [[http://arxiv.org/abs/0907.5249|Why the Kirnberger Kernel Is So Small]]. If a and b are two commas, 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 [[Monzos|monzo]] 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.
+<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">By a //Don Page comma// is meant a comma computed from two other intervals by the method suggested by the Don Page paper, [[http://arxiv.org/abs/0907.5249|Why the Kirnberger Kernel Is So Small]]. 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 [[Monzos|monzo]] 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.
 Successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a distinct tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, but these are by no means the only examples.
@@ Line 49: / Line 49: @@
 DPC(176/175, 540/539) = |-58 -249 -137 139 110&gt;</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;Don Page comma&lt;/title&gt;&lt;/head&gt;&lt;body&gt;By a &lt;em&gt;Don Page comma&lt;/em&gt; is meant a comma computed from two other intervals by the method suggested by the Don Page paper, &lt;a class="wiki_link_ext" href="http://arxiv.org/abs/0907.5249" rel="nofollow"&gt;Why the Kirnberger Kernel Is So Small&lt;/a&gt;. If a and b are two commas, 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 &lt;a class="wiki_link" href="/Monzos"&gt;monzo&lt;/a&gt; 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.&lt;br /&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;Don Page comma&lt;/title&gt;&lt;/head&gt;&lt;body&gt;By a &lt;em&gt;Don Page comma&lt;/em&gt; is meant a comma computed from two other intervals by the method suggested by the Don Page paper, &lt;a class="wiki_link_ext" href="http://arxiv.org/abs/0907.5249" rel="nofollow"&gt;Why the Kirnberger Kernel Is So Small&lt;/a&gt;. If a and b are two rational numbers &amp;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 &lt;a class="wiki_link" href="/Monzos"&gt;monzo&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
 Successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a distinct tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, but these are by no means the only examples.&lt;br /&gt;

Don Page comma: Difference between revisions

Revision as of 20:34, 30 October 2011

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