Don Page comma
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- This revision was by author genewardsmith and made on 2011-10-30 16:54:44 UTC.
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Original Wikitext content:
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 commas, 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 monzo form as u and v. Then the Don Page comma is defined as DPC(a, b) = qu - pb, 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 inverse of that if that is less than 1. 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, but these are by no means the only examples. Here are some Don Page commas derived from 9-limit consonances: 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 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(81/80, 128/125) = |161 -84 -12>, the atom DPC(81/80, 25/24) = |71 -99 37>, raider 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 commas, 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 monzo form as u and v. Then the Don Page comma is defined as DPC(a, b) = qu - pb, 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 inverse of that if that is less than 1.<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, but these are by no means the only examples.<br /> <br /> Here are some Don Page commas derived from 9-limit consonances:<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 /> <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(81/80, 128/125) = |161 -84 -12>, the atom<br /> DPC(81/80, 25/24) = |71 -99 37>, raider<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>