Taxicab distance
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author mbattaglia1 and made on 2011-09-03 04:43:20 UTC.
- The original revision id was 250469074.
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Original Wikitext content:
One measurement of the complexity of the comma could be the number of prime factors it has, regardless of their magnitude. This corresponds to an interval's unweighted L1 distance on the lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height. To calculate the number of prime factors in a [[monzo]], simply take sum of the absolute values of each coordinate. For example, 81/80 i.e. |-4 4 1> would have a factor limit of 4+4+1=9, or, with 2's taken for granted, 4+1=5. =Not yet the right name= I want to speak of a limit on the number of instances of prime factors, not the number of different prime factors. For example, 45 has factors 3, 3, and 5; here, we want to count each 3 separately. =With 2's taken for granted= ==2-factor-limit commas== 16/15 ( / 3 / 5) 33/32 (3 * 11) 65/64 (5 * 13) ==3-factor-limit commas== 25/24 (5 * 5 / 3) 128/125 (5 * 5 * 5) 21/20 (3 * 7 / 5) 26/25 (13 / 5 / 5) 49/48 (7 * 7 / 3) 64/63 ( / 3 / 7 / 7) 256/245 ( / 5 / 7 / 7) 80/77 (5 / 7 / 11) 22/21 (11 / 3 / 7) 40/39 (5 / 3 / 13) 96/91 (3 / 7 / 13) 55/52 (5 * 11 / 13) 1024/1001 (7 * 11 * 13) 512/507 (3 * 13 * 13) 169/160 (13 * 13 / 5) 176/169 (11 / 13 / 13)
Original HTML content:
<html><head><title>commas by taxicab distance</title></head><body>One measurement of the complexity of the comma could be the number of prime factors it has, regardless of their magnitude. This corresponds to an interval's unweighted L1 distance on the lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height.<br /> <br /> To calculate the number of prime factors in a <a class="wiki_link" href="/monzo">monzo</a>, simply take sum of the absolute values of each coordinate.<br /> <br /> For example, 81/80 i.e. |-4 4 1> would have a factor limit of 4+4+1=9, or, with 2's taken for granted, 4+1=5.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Not yet the right name"></a><!-- ws:end:WikiTextHeadingRule:0 -->Not yet the right name</h1> I want to speak of a limit on the number of instances of prime factors, not the number of different prime factors. For example, 45 has factors 3, 3, and 5; here, we want to count each 3 separately.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="With 2's taken for granted"></a><!-- ws:end:WikiTextHeadingRule:2 -->With 2's taken for granted</h1> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="With 2's taken for granted-2-factor-limit commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->2-factor-limit commas</h2> 16/15 ( / 3 / 5)<br /> 33/32 (3 * 11)<br /> 65/64 (5 * 13)<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="With 2's taken for granted-3-factor-limit commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->3-factor-limit commas</h2> 25/24 (5 * 5 / 3)<br /> 128/125 (5 * 5 * 5)<br /> 21/20 (3 * 7 / 5)<br /> 26/25 (13 / 5 / 5)<br /> 49/48 (7 * 7 / 3)<br /> 64/63 ( / 3 / 7 / 7)<br /> 256/245 ( / 5 / 7 / 7)<br /> 80/77 (5 / 7 / 11)<br /> 22/21 (11 / 3 / 7)<br /> 40/39 (5 / 3 / 13)<br /> 96/91 (3 / 7 / 13)<br /> 55/52 (5 * 11 / 13)<br /> 1024/1001 (7 * 11 * 13)<br /> 512/507 (3 * 13 * 13)<br /> 169/160 (13 * 13 / 5)<br /> 176/169 (11 / 13 / 13)</body></html>