Taxicab distance
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One measurement of the complexity of the comma could be the number of prime factors it has, regardless of their magnitude, but counting repetitions. When combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of. =How to calculate taxicab distance on a prime-number lattice= To calculate the taxicab distance between 1/1 and any interval, take the sum of the absolute values of the exponents of the prime factorization. For the example of 81/80: 81/80 = 2^-4 * 3^4 * 5^-1 |-4| + |4| + |-1| = 9 This corresponds to an interval's unweighted [[http://en.wikipedia.org/wiki/Lp_space|L1]] distance on a prime-factor lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height. One way to add weighting back in is to exclude powers of 2 (to assume that powers of 2 don't affect the complexity of the move). For 81/80 we then get 4+1=5. If you discard powers of both 2 and 3, you get an understanding of commas relevant to [[sagittal corner|Sagittal notation]], which notates higher-prime-limit ratios in terms of their deviation from a cycle of fifths. In this sense, 81/80 has a taxicab distance of 1, as it contains only a single instance of 5, which is why in Sagittal notation it is called the "5-comma". =With powers of 2 taken for granted= ==2-move commas== 16/15 ( / 3 / 5) 33/32 (3 * 11) 65/64 (5 * 13) ==3-move 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) =With powers of 2 and 3 taken for granted= The relation of powers of 3 to the other factor(s) is represented by "3's". The ones with names (all of them so far) have a corresponding sagittal accidental, though closeby commas share symbols. ==1-move commas== 81/80 ( 3's / 5 ) (5 comma) 32805/32768 ( 3's * 5 ) (5 schisma) 64/63 ( / 3's / 7) (7 comma) 729/704 ( 3's / 11 ) (11-L diesis) 33/32 ( 3's * 11 ) (11-M diesis) 27/26 ( 3's / 13 ) (13-L diesis) 1053/1024 ( 3's * 13 ) (13 M-diesis) 2187/2176 ( 3's / 17 ) (17 kleisma) 4131/4096 ( 3's * 17 ) (17 comma) 513/512 ( 3's * 19 ) (19 schisma) 19683/19456 ( 3's / 19 ) (19 comma) 736/729 ( 23 / 3's ) (23 comma) 261/256 ( 3's * 29 ) (29 comma) ==2-move commas== (ordered and grouped by size of comma in just intonation) 5103/5120 ( 3's * 7 / 5 ) (5:7 kleisma) 352/351 ( 11 / 3's / 13 ) (11:13 kleisma) 896/891 ( 7 / 3's / 11 ) (7:11 kleisma) 2048/2025 ( / 3's / 5 / 5 ) (25 comma/[[diaschisma]]) 55/54 ( 11 * 5 / 3's ) (55 comma) 45927/45056 ( 3's * 7 / 11 ) (7:11 comma) 52/51 ( 3's * 13 / 17 ) (13:17 comma) 45/44 ( 3's * 5 / 11 ) (5:11 S-diesis) 1701/1664 ( 3's * 7 / 13 ) (7:13 S-diesis) 1408/1377 ( 11 / 3's / 17 ) (11:17 S-diesis) 6561/6400 ( 3's / 5 / 5 ) (25 S-diesis) 40/39 ( 5 / 3's / 13 ) (5:13 S-diesis) 36/35 ( 3's / 5 / 7 ) (35 M-diesis) 8505/8192 ( 3's * 5 * 7 ) (35 L-diesis) ==3-move commas== 250/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis) 531441/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis)
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, but counting repetitions.<br /> <br /> When combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="How to calculate taxicab distance on a prime-number lattice"></a><!-- ws:end:WikiTextHeadingRule:0 -->How to calculate taxicab distance on a prime-number lattice</h1> <br /> To calculate the taxicab distance between 1/1 and any interval, take the sum of the absolute values of the exponents of the prime factorization. For the example of 81/80:<br /> <br /> 81/80 = 2^-4 * 3^4 * 5^-1<br /> |-4| + |4| + |-1| = 9<br /> <br /> This corresponds to an interval's unweighted <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lp_space" rel="nofollow">L1</a> distance on a prime-factor lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height.<br /> <br /> One way to add weighting back in is to exclude powers of 2 (to assume that powers of 2 don't affect the complexity of the move). For 81/80 we then get 4+1=5.<br /> <br /> If you discard powers of both 2 and 3, you get an understanding of commas relevant to <a class="wiki_link" href="/sagittal%20corner">Sagittal notation</a>, which notates higher-prime-limit ratios in terms of their deviation from a cycle of fifths. In this sense, 81/80 has a taxicab distance of 1, as it contains only a single instance of 5, which is why in Sagittal notation it is called the "5-comma".<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="With powers of 2 taken for granted"></a><!-- ws:end:WikiTextHeadingRule:2 -->With powers of 2 taken for granted</h1> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="With powers of 2 taken for granted-2-move commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->2-move 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 powers of 2 taken for granted-3-move commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->3-move 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)<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="With powers of 2 and 3 taken for granted"></a><!-- ws:end:WikiTextHeadingRule:8 -->With powers of 2 and 3 taken for granted</h1> The relation of powers of 3 to the other factor(s) is represented by "3's". The ones with names (all of them so far) have a corresponding sagittal accidental, though closeby commas share symbols.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="With powers of 2 and 3 taken for granted-1-move commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->1-move commas</h2> 81/80 ( 3's / 5 ) (5 comma)<br /> 32805/32768 ( 3's * 5 ) (5 schisma)<br /> 64/63 ( / 3's / 7) (7 comma)<br /> 729/704 ( 3's / 11 ) (11-L diesis)<br /> 33/32 ( 3's * 11 ) (11-M diesis)<br /> 27/26 ( 3's / 13 ) (13-L diesis)<br /> 1053/1024 ( 3's * 13 ) (13 M-diesis)<br /> 2187/2176 ( 3's / 17 ) (17 kleisma)<br /> 4131/4096 ( 3's * 17 ) (17 comma)<br /> 513/512 ( 3's * 19 ) (19 schisma)<br /> 19683/19456 ( 3's / 19 ) (19 comma)<br /> 736/729 ( 23 / 3's ) (23 comma)<br /> 261/256 ( 3's * 29 ) (29 comma)<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="With powers of 2 and 3 taken for granted-2-move commas"></a><!-- ws:end:WikiTextHeadingRule:12 -->2-move commas</h2> (ordered and grouped by size of comma in just intonation)<br /> <br /> 5103/5120 ( 3's * 7 / 5 ) (5:7 kleisma)<br /> 352/351 ( 11 / 3's / 13 ) (11:13 kleisma)<br /> <br /> 896/891 ( 7 / 3's / 11 ) (7:11 kleisma)<br /> 2048/2025 ( / 3's / 5 / 5 ) (25 comma/<a class="wiki_link" href="/diaschisma">diaschisma</a>)<br /> 55/54 ( 11 * 5 / 3's ) (55 comma)<br /> <br /> 45927/45056 ( 3's * 7 / 11 ) (7:11 comma)<br /> 52/51 ( 3's * 13 / 17 ) (13:17 comma)<br /> <br /> 45/44 ( 3's * 5 / 11 ) (5:11 S-diesis)<br /> 1701/1664 ( 3's * 7 / 13 ) (7:13 S-diesis)<br /> 1408/1377 ( 11 / 3's / 17 ) (11:17 S-diesis)<br /> <br /> 6561/6400 ( 3's / 5 / 5 ) (25 S-diesis)<br /> 40/39 ( 5 / 3's / 13 ) (5:13 S-diesis)<br /> <br /> 36/35 ( 3's / 5 / 7 ) (35 M-diesis)<br /> <br /> 8505/8192 ( 3's * 5 * 7 ) (35 L-diesis)<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="With powers of 2 and 3 taken for granted-3-move commas"></a><!-- ws:end:WikiTextHeadingRule:14 -->3-move commas</h2> 250/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis)<br /> 531441/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis)</body></html>