Don Page comma
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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. 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: DPC(5/3, 2) = 648/625 DPC(4/3, 9/5) = 81/80 DPC(5/4, 2) = 128/125 DPC(4/3, 5/3) = 16875/16384 DPC(3/2, 5/3) = 20000/19683 DPC(81/80, 128/125) = |161 -84 -12>, the atom DPC(81/80, 25/24) = |71 -99 37>, raider Here are some 7-limit Don Page commas: DPC(7/5, 2) = 50/49 DPC(6/5, 7/4) = 875/864 DPC(7/5, 5/3) = 3125/3087 DPC(9/7, 5/3) = 245/243 DPC(7/6, 8/5) = 1728/1715 DPC(8/7, 3/2) = 1029/1024 DPC(5/4, 7/5) = 3136/3125 DPC(9/8, 10/7) = 5120/5103 DPC(27/25, 7/6) = 4375/4374 From 11-limit consonances: DPC(11/10, 4/3) = 4000/3993 DPC(10/9, 11/8) = 8019/8000 DPC(11/9, 3/2) = 243/242 DPC(5/4, 11/7) = 176/175 DPC(8/7, 11/9) = 41503/41472 Here are some complex Don Page commas derived from other commas: DPC(525/512, 245/243) = |-153 277 -18 -87> DPC(49/48, 50/49) = |-487 -97 -198 392> DPC(10/9, 11/10) = |40 -38 40 0 -21> DPC(11/10, 12/11) = |-67 -23 -21 0 44> DPC(77/75, 245/243) = |0 286 -99 -103 19> DPC(55/54, 56/55) = |-442 -327 220 -111 220> DPC(176/175, 540/539) = |-58 -249 -137 139 110>
Original HTML content:
<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 > 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 /> <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 /> <br /> Here are some 5-limit Don Page commas:<br /> <br /> DPC(5/3, 2) = 648/625<br /> DPC(4/3, 9/5) = 81/80<br /> DPC(5/4, 2) = 128/125<br /> DPC(4/3, 5/3) = 16875/16384<br /> DPC(3/2, 5/3) = 20000/19683<br /> DPC(81/80, 128/125) = |161 -84 -12>, the atom<br /> DPC(81/80, 25/24) = |71 -99 37>, raider<br /> <br /> Here are some 7-limit Don Page commas:<br /> <br /> DPC(7/5, 2) = 50/49<br /> DPC(6/5, 7/4) = 875/864<br /> DPC(7/5, 5/3) = 3125/3087<br /> DPC(9/7, 5/3) = 245/243<br /> DPC(7/6, 8/5) = 1728/1715<br /> DPC(8/7, 3/2) = 1029/1024<br /> DPC(5/4, 7/5) = 3136/3125<br /> DPC(9/8, 10/7) = 5120/5103<br /> DPC(27/25, 7/6) = 4375/4374<br /> <br /> From 11-limit consonances:<br /> <br /> DPC(11/10, 4/3) = 4000/3993<br /> DPC(10/9, 11/8) = 8019/8000<br /> DPC(11/9, 3/2) = 243/242<br /> DPC(5/4, 11/7) = 176/175<br /> DPC(8/7, 11/9) = 41503/41472<br /> <br /> Here are some complex Don Page commas derived from other commas:<br /> <br /> DPC(525/512, 245/243) = |-153 277 -18 -87><br /> DPC(49/48, 50/49) = |-487 -97 -198 392><br /> DPC(10/9, 11/10) = |40 -38 40 0 -21><br /> DPC(11/10, 12/11) = |-67 -23 -21 0 44><br /> DPC(77/75, 245/243) = |0 286 -99 -103 19><br /> DPC(55/54, 56/55) = |-442 -327 220 -111 220><br /> DPC(176/175, 540/539) = |-58 -249 -137 139 110></body></html>