Don Page comma: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-26 16:09:17 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-26 18:47:28 UTC</tt>.<br>

: The original revision id was <tt>~~511283948~~</tt>.<br>

: The original revision id was <tt>511305282</tt>.<br>

: The revision comment was: <tt></tt><br>

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=Bimodular approximants=

If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the [[https://en.wikipedia.org/wiki/Pade_approximant|Padé approximant]] of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation|Möbius transformation]] and hence has an inverse, which we denote mib(x) = (1+x)/(1-x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3). We have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) 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 begins 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.

If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the [[https://en.wikipedia.org/wiki/Pade_approximant|Padé approximant]] of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation|Möbius transformation]] and hence has an inverse, which we denote mib(x) = (1+x)/(1-x), which is the (1, 1) Padé approximant around 0 for exp(2x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3) and ln(x)/2 = (x-1)/2 - (x-1)^2/4 + O(x^3), which is the same to the second order as bim(x). Using mib, we may also define BMC(a, b) = DPC(mib(a), mib(b)), where BMC is an acronym for "bimondular comma".

If r is as above we have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) = mib(u)^v/mib(v)^u 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 begins 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. 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.

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<br />

<h1 id="toc0"><a name="Bimodular approximants"></a>Bimodular approximants</h1>

If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pade_approximant" rel="nofollow">Padé approximant</a> of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation" rel="nofollow">Möbius transformation</a> and hence has an inverse, which we denote mib(x) = (1+x)/(1-x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3). We have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) 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 begins 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. <br />

If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pade_approximant" rel="nofollow">Padé approximant</a> of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation" rel="nofollow">Möbius transformation</a> and hence has an inverse, which we denote mib(x) = (1+x)/(1-x), which is the (1, 1) Padé approximant around 0 for exp(2x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3) and ln(x)/2 = (x-1)/2 - (x-1)^2/4 + O(x^3), which is the same to the second order as bim(x). Using mib, we may also define BMC(a, b) = DPC(mib(a), mib(b)), where BMC is an acronym for &quot;bimondular comma&quot;.<br />

<br />

If r is as above we have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) = mib(u)^v/mib(v)^u 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 begins 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. 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. <br />

<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 />

@@ 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>2014-05-26 16:09:17 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-26 18:47:28 UTC</tt>.<br>
-: The original revision id was <tt>511283948</tt>.<br>
+: The original revision id was <tt>511305282</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 11: / Line 11: @@
 =Bimodular approximants=
-If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the [[https://en.wikipedia.org/wiki/Pade_approximant|Padé approximant]] of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation|Möbius transformation]] and hence has an inverse, which we denote mib(x) = (1+x)/(1-x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3). We have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) 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 begins 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.
+If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the [[https://en.wikipedia.org/wiki/Pade_approximant|Padé approximant]] of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a [[https://en.wikipedia.org/wiki/M%C3%B6bius_transformation|Möbius transformation]] and hence has an inverse, which we denote mib(x) = (1+x)/(1-x), which is the (1, 1) Padé approximant around 0 for exp(2x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3) and ln(x)/2 = (x-1)/2 - (x-1)^2/4 + O(x^3), which is the same to the second order as bim(x). Using mib, we may also define BMC(a, b) = DPC(mib(a), mib(b)), where BMC is an acronym for "bimondular comma".
+If r is as above we have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) = mib(u)^v/mib(v)^u 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 begins 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. 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.
@@ Line 75: / Line 77: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Bimodular approximants&lt;/h1&gt;
-If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pade_approximant" rel="nofollow"&gt;Padé approximant&lt;/a&gt; of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation" rel="nofollow"&gt;Möbius transformation&lt;/a&gt; and hence has an inverse, which we denote mib(x) = (1+x)/(1-x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3). We have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) 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 begins 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. &lt;br /&gt;
+If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pade_approximant" rel="nofollow"&gt;Padé approximant&lt;/a&gt; of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation" rel="nofollow"&gt;Möbius transformation&lt;/a&gt; and hence has an inverse, which we denote mib(x) = (1+x)/(1-x), which is the (1, 1) Padé approximant around 0 for exp(2x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3) and ln(x)/2 = (x-1)/2 - (x-1)^2/4 + O(x^3), which is the same to the second order as bim(x). Using mib, we may also define BMC(a, b) = DPC(mib(a), mib(b)), where BMC is an acronym for &amp;quot;bimondular comma&amp;quot;.&lt;br /&gt;
+&lt;br /&gt;
+If r is as above we have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) = mib(u)^v/mib(v)^u 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 begins 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. 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. &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;