Don Page comma: Difference between revisions
Wikispaces>genewardsmith **Imported revision 511477944 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 511538698 - 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>2014-05-27 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-27 16:51:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>511538698</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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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 | What is going on here becomes clearer if we shift to BMC rather than DPC. If bim(x) was an exact logarithmic function rather than an approximation, then the Don Page commas would all be 1. They measure the deviation between an approximate relationship between intervals and an exact one. For example. bim(11/9) = 1/10 and bim(3/2) = 1/5, and two 11/9 intervals fall short of 3/2 by (3/2)/(11/9)^2 = BMC(1/10, 1/5) = 243/242. Not all relationships between intervals of this sort arise from bimodular approximation. The syntonic comma, 81/80, is how much two 9/8 intervals exceed 5/4, and how much two 10/9 intervals fall short of it. But bim(10/9) = 1/19 and bim(9/8) = 1/17, neither of which will add up to bim(5/4) = 1/9. Instead mib(1/18) = 19/17 will give BMC(1/18, 1/9) = 1445/1444, a whole other deal. To get 81/80, note that bim(4/3) = 1/7 and bim(9/5) = 2/7, and BMC(1/7, 2/7) = 81/80. | ||
For n>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... and BMC(2/n, 3/n) goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125... . | |||
We might also note that successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a some 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. | |||
=Examples= | =Examples= | ||
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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. <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 /> | <br /> | ||
What is going on here becomes clearer if we shift to BMC rather than DPC. For | What is going on here becomes clearer if we shift to BMC rather than DPC. If bim(x) was an exact logarithmic function rather than an approximation, then the Don Page commas would all be 1. They measure the deviation between an approximate relationship between intervals and an exact one. For example. bim(11/9) = 1/10 and bim(3/2) = 1/5, and two 11/9 intervals fall short of 3/2 by (3/2)/(11/9)^2 = BMC(1/10, 1/5) = 243/242. Not all relationships between intervals of this sort arise from bimodular approximation. The syntonic comma, 81/80, is how much two 9/8 intervals exceed 5/4, and how much two 10/9 intervals fall short of it. But bim(10/9) = 1/19 and bim(9/8) = 1/17, neither of which will add up to bim(5/4) = 1/9. Instead mib(1/18) = 19/17 will give BMC(1/18, 1/9) = 1445/1444, a whole other deal. To get 81/80, note that bim(4/3) = 1/7 and bim(9/5) = 2/7, and BMC(1/7, 2/7) = 81/80.<br /> | ||
<br /> | |||
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... and BMC(2/n, 3/n) goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125... .<br /> | |||
<br /> | <br /> | ||
We might also note that successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a some 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.<br /> | |||
<br /> | <br /> | ||
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