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 02:40:15 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-31 02:48:36 UTC</tt>.<br>

: The original revision id was <tt>~~270172168~~</tt>.<br>

: The original revision id was <tt>270173040</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 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 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 base. For example, if a = 7/6 and b = 27/25, we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 26th power gives us 4375/4374. The ~~lcm~~ of 13 ~~and 26 is 26, and this~~ leads to the resulting comma (4375/4374) being relatively simple.

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 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 base. 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.

~~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.

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:

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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 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 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 base. For example, if a = 7/6 and b = 27/25, we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 26th power gives us 4375/4374. The ~~lcm~~ of 13 ~~and 26 is 26, and this~~ leads to the resulting comma (4375/4374) being relatively simple.<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 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 base. 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. <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 />

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.<br />

<br />

Here are some 5-limit Don Page commas:<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 02:40:15 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-31 02:48:36 UTC</tt>.<br>
-: The original revision id was <tt>270172168</tt>.<br>
+: The original revision id was <tt>270173040</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 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 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 base. For example, if a = 7/6 and b = 27/25, we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 26th power gives us 4375/4374. The lcm of 13 and 26 is 26, and this leads to the resulting comma (4375/4374) being relatively simple.
+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 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 base. 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.
-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.
+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:
@@ 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 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 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 base. For example, if a = 7/6 and b = 27/25, we obtain (7/6)^(1/26) / (27/25)^(1/13), and taking the 26th power gives us 4375/4374. The lcm of 13 and 26 is 26, and this leads to the resulting comma (4375/4374) being relatively simple.&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 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 base. 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. &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;
+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;
 Here are some 5-limit Don Page commas:&lt;br /&gt;