Don Page comma: Difference between revisions

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<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>2011-10-31 01:52:28 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-31 02:04:58 UTC</tt>.<br>

: The original revision id was <tt>~~270166646~~</tt>.<br>

: The original revision id was <tt>270168088</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>

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<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 > 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 it is equal to an nth ~~root~~ 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 + ..., and so when x and y are small, r(x, y) will be close to 1. If n is not 1, the nth ~~root~~ will still ~~be close~~ to 1, ~~but much less complex in terms of height;~~ a ~~comma is stronger in temperament terms than powers of a comma~~, and ~~this is~~ the ~~really interesting case~~.

If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it 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 + ..., 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 closer to 1. For example, if a = 7/6 and b = 27/25, then we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 13th power gives us 4375/4374.

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.

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<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>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 />

If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it is equal to an nth ~~root~~ 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 + ..., and so when x and y are small, r(x, y) will be close to 1. If n is not 1, the nth ~~root~~ will still ~~be close~~ to 1, ~~but much less complex in terms of height;~~ a ~~comma is stronger in temperament terms than powers of a comma~~, and ~~this is~~ the ~~really interesting case~~.<br />

If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it 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 + ..., 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 closer to 1. For example, if a = 7/6 and b = 27/25, then we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 13th power gives us 4375/4374.<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, 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.<br />

@@ 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>2011-10-31 01:52:28 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-31 02:04:58 UTC</tt>.<br>
-: The original revision id was <tt>270166646</tt>.<br>
+: The original revision id was <tt>270168088</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 8: / Line 8: @@
 <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.
-If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it is equal to an nth root 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 + ..., and so when x and y are small, r(x, y) will be close to 1. If n is not 1, the nth root will still be close to 1, but much less complex in terms of height; a comma is stronger in temperament terms than powers of a comma, and this is the really interesting case.
+If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it 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 + ..., 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 closer to 1. For example, if a = 7/6 and b = 27/25, then we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 13th power gives us 4375/4374.
 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.
@@ Line 54: / Line 54: @@
 <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;
-If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it is equal to an nth root 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 + ..., and so when x and y are small, r(x, y) will be close to 1. If n is not 1, the nth root will still be close to 1, but much less complex in terms of height; a comma is stronger in temperament terms than powers of a comma, and this is the really interesting case.&lt;br /&gt;
+If we write r as ((a-1)/(a+1)) / ((b-1)/(b+1)), then depending on common factors we have that it 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 + ..., 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 closer to 1. For example, if a = 7/6 and b = 27/25, then we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 13th power gives us 4375/4374.&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, 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;