Don Page comma: Difference between revisions

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It should be noted that a more general procedure for finding a comma from two intervals a and b is to use the convergents of the ratio of the logarithms; the Don Page construction might be regarded as a way pointing to examples where this is especially interesting. We might also note that 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, not surprising when we note that in this case the first non-constant term of the one-variable expansion is of order five, but these are by no means the only examples.

DPC(6/5, 5/4) = |-29 -11 20&gt;, gammic

DPC(7/5, 2) = 50/49

DPC(6/5, 7/4) = 875/864

<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;&lt;br /&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;

It should be noted that a more general procedure for finding a comma from two intervals a and b is to use the convergents of the ratio of the logarithms; the Don Page construction might be regarded as a way pointing to examples where this is especially interesting. We might also note that 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, not surprising when we note that in this case the first non-constant term of the one-variable expansion is of order five, but these are by no means the only examples.&lt;br /&gt;

&lt;br /&gt;

DPC(6/5, 5/4) = |-29 -11 20&amp;gt;, gammic&lt;br /&gt;

&lt;br /&gt;

DPC(7/5, 2) = 50/49&lt;br /&gt;

DPC(6/5, 7/4) = 875/864&lt;br /&gt;