Don Page comma: Difference between revisions

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.

If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that the corresponding Don Page comma is equal to an nth power of a^((b-1)/(b+1)) / b^((a-1)/(a+1)) for some n. If we set a = 1+x, b = 1+y, then r = r(x, y) is an analytic function of two complex variables with a power series expansion around x=0, y=0. This expansion begans as r(x, y) = 1 - (xy^3 - x^3y)/24 + (3xy^4 + x^2y^3 - x^3y^2 - 3x^4y)/48 + ..., with its first nonconstant term of total degree four, and so when x and y are small, r(x, y) will be close to 1. If n is not 1, the nth power will still be close to 1, and if there are common factors in the denominators of the exponents, the resulting comma will be less complex (lower in height), and this is the really interesting case. For example, if a = 7/6 and b = 27/25, we obtain (7/6)^(1/26) / (27/25)^(1/13), the lcm of 13 and 26 is 26, and taking the 26th power gives us 4375/4374. The common factor of 13 leads to the resulting comma (4375/4374) being relatively simple.  

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.

Here are some 5-limit Don Page commas:

DPC(5/3, 3) = 27/25

DPC(4/3, 5/2) = 135/128

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, negri

DPC(3/2, 5/3) = 20000/19683, tetracot

DPC(10/9, 32/25) = |8 14 -13>, parakleisma

DPC(5/4, 4/3) = |32 -7 -9>, escapade

DPC(6/5, 5/4) = |-29 -11 20>, gammic

DPC(10/9, 9/8) = |-70 72 -19>

DPC(81/80, 25/24) = |71 -99 37>, raider

DPC(81/80, 128/125) = |161 -84 -12>, the atom

Here are some 7-limit Don Page commas:

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

DPC(27/25, 7/6) = 4375/4374

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

From 15-limit consonances:

<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;

&lt;!-- ws:start:WikiTextLocalImageRule:0:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/mathhazard.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextLocalImageRule:0 --&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;