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**Imported revision 511305282 - Original comment: **
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**Imported revision 511308144 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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 18:47:28 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-26 19:17:48 UTC</tt>.<br>
: The original revision id was <tt>511305282</tt>.<br>
: The original revision id was <tt>511308144</tt>.<br>
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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.  
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.  


What is going on here becomes clearer if we shift to BMC rather than DPC. For instance, for n&gt;1 BMC(1/n, 1/(2n)) goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939..., with BMC(1/13, 1/26) being our example 4375/4374. Similarly, BMC(1/n, 1/(3n)) goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565... .
 
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.
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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&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;
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;
What is going on here becomes clearer if we shift to BMC rather than DPC. For instance, for n&amp;gt;1 BMC(1/n, 1/(2n)) goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939..., with BMC(1/13, 1/26) being our example 4375/4374. Similarly, BMC(1/n, 1/(3n)) goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565... .&lt;br /&gt;
&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;
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;
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