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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | One measurement of the complexity of the comma could be the number of prime factors it has, regardless of their magnitude, but counting repetitions. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:guest|guest]] and made on <tt>2011-09-05 17:52:49 UTC</tt>.<br>
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| : The original revision id was <tt>250952112</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">One measurement of the complexity of the comma could be the number of prime factors it has, regardless of their magnitude, but counting repetitions.
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| When combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of. | | When combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of. |
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| =How to calculate taxicab distance on a prime-number lattice= | | =How to calculate taxicab distance on a prime-number lattice= |
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| 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: | | 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: |
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| |-4| + |4| + |-1| = 9 | | |-4| + |4| + |-1| = 9 |
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| 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. | | 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. |
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| 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. | | 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. |
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| 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". | | 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". |
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| =With powers of 2 taken for granted= | | =With powers of 2 taken for granted= |
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| ==2-move commas== | | ==2-move commas== |
| 16/15 ( / 3 / 5) | | 16/15 ( / 3 / 5) |
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| 33/32 (3 * 11) | | 33/32 (3 * 11) |
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| 65/64 (5 * 13) | | 65/64 (5 * 13) |
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| ==3-move commas== | | ==3-move commas== |
| 25/24 (5 * 5 / 3) | | 25/24 (5 * 5 / 3) |
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| 128/125 (5 * 5 * 5) | | 128/125 (5 * 5 * 5) |
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| 21/20 (3 * 7 / 5) | | 21/20 (3 * 7 / 5) |
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| 26/25 (13 / 5 / 5) | | 26/25 (13 / 5 / 5) |
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| 49/48 (7 * 7 / 3) | | 49/48 (7 * 7 / 3) |
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| 64/63 ( / 3 / 7 / 7) | | 64/63 ( / 3 / 7 / 7) |
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| 256/245 ( / 5 / 7 / 7) | | 256/245 ( / 5 / 7 / 7) |
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| 80/77 (5 / 7 / 11) | | 80/77 (5 / 7 / 11) |
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| 22/21 (11 / 3 / 7) | | 22/21 (11 / 3 / 7) |
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| 40/39 (5 / 3 / 13) | | 40/39 (5 / 3 / 13) |
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| 96/91 (3 / 7 / 13) | | 96/91 (3 / 7 / 13) |
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| 55/52 (5 * 11 / 13) | | 55/52 (5 * 11 / 13) |
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| 1024/1001 (7 * 11 * 13) | | 1024/1001 (7 * 11 * 13) |
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| 512/507 (3 * 13 * 13) | | 512/507 (3 * 13 * 13) |
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| 169/160 (13 * 13 / 5) | | 169/160 (13 * 13 / 5) |
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| 176/169 (11 / 13 / 13) | | 176/169 (11 / 13 / 13) |
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| =With powers of 2 and 3 taken for granted= | | =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. | | 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. |
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| ==1-move commas== | | ==1-move commas== |
| 81/80 ( 3's / 5 ) (5 comma) | | 81/80 ( 3's / 5 ) (5 comma) |
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| 32805/32768 ( 3's * 5 ) (5 schisma) | | 32805/32768 ( 3's * 5 ) (5 schisma) |
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| 64/63 ( / 3's / 7) (7 comma) | | 64/63 ( / 3's / 7) (7 comma) |
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| 729/704 ( 3's / 11 ) (11-L diesis) | | 729/704 ( 3's / 11 ) (11-L diesis) |
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| 33/32 ( 3's * 11 ) (11-M diesis) | | 33/32 ( 3's * 11 ) (11-M diesis) |
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| 27/26 ( 3's / 13 ) (13-L diesis) | | 27/26 ( 3's / 13 ) (13-L diesis) |
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| 1053/1024 ( 3's * 13 ) (13 M-diesis) | | 1053/1024 ( 3's * 13 ) (13 M-diesis) |
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| 2187/2176 ( 3's / 17 ) (17 kleisma) | | 2187/2176 ( 3's / 17 ) (17 kleisma) |
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| 4131/4096 ( 3's * 17 ) (17 comma) | | 4131/4096 ( 3's * 17 ) (17 comma) |
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| 513/512 ( 3's * 19 ) (19 schisma) | | 513/512 ( 3's * 19 ) (19 schisma) |
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| 19683/19456 ( 3's / 19 ) (19 comma) | | 19683/19456 ( 3's / 19 ) (19 comma) |
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| 736/729 ( 23 / 3's ) (23 comma) | | 736/729 ( 23 / 3's ) (23 comma) |
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| 261/256 ( 3's * 29 ) (29 comma) | | 261/256 ( 3's * 29 ) (29 comma) |
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| ==2-move commas== | | ==2-move commas== |
| (ordered and grouped by size of comma in just intonation) | | (ordered and grouped by size of comma in just intonation) |
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| 5103/5120 ( 3's * 7 / 5 ) (5:7 kleisma) | | 5103/5120 ( 3's * 7 / 5 ) (5:7 kleisma) |
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| 352/351 ( 11 / 3's / 13 ) (11:13 kleisma) | | 352/351 ( 11 / 3's / 13 ) (11:13 kleisma) |
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| 896/891 ( 7 / 3's / 11 ) (7:11 kleisma) | | 896/891 ( 7 / 3's / 11 ) (7:11 kleisma) |
| 2048/2025 ( / 3's / 5 / 5 ) (25 comma/[[diaschisma]]) | | |
| | 2048/2025 ( / 3's / 5 / 5 ) (25 comma/[[diaschisma|diaschisma]]) |
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| 55/54 ( 11 * 5 / 3's ) (55 comma) | | 55/54 ( 11 * 5 / 3's ) (55 comma) |
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| 45927/45056 ( 3's * 7 / 11 ) (7:11 comma) | | 45927/45056 ( 3's * 7 / 11 ) (7:11 comma) |
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| 52/51 ( 3's * 13 / 17 ) (13:17 comma) | | 52/51 ( 3's * 13 / 17 ) (13:17 comma) |
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| 45/44 ( 3's * 5 / 11 ) (5:11 S-diesis) | | 45/44 ( 3's * 5 / 11 ) (5:11 S-diesis) |
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| 1701/1664 ( 3's * 7 / 13 ) (7:13 S-diesis) | | 1701/1664 ( 3's * 7 / 13 ) (7:13 S-diesis) |
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| 1408/1377 ( 11 / 3's / 17 ) (11:17 S-diesis) | | 1408/1377 ( 11 / 3's / 17 ) (11:17 S-diesis) |
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| 6561/6400 ( 3's / 5 / 5 ) (25 S-diesis) | | 6561/6400 ( 3's / 5 / 5 ) (25 S-diesis) |
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| 40/39 ( 5 / 3's / 13 ) (5:13 S-diesis) | | 40/39 ( 5 / 3's / 13 ) (5:13 S-diesis) |
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| 8505/8192 ( 3's * 5 * 7 ) (35 L-diesis) | | 8505/8192 ( 3's * 5 * 7 ) (35 L-diesis) |
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| ==3-move commas== | | ==3-move commas== |
| 250/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis) | | 250/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis) |
| 531441/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis)</pre></div> | | |
| <h4>Original HTML content:</h4>
| | 531441/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis) |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 />
| | [[Category:todo:link]] |
| <br />
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| When combined with excluding the smallest primes, this measurement can give an idea of how many &quot;strange harmonic moves&quot; a comma is comprised of.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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>
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| <br />
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| 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 />
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| <br />
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| 81/80 = 2^-4 * 3^4 * 5^-1<br />
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| |-4| + |4| + |-1| = 9<br />
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| <br />
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| 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 />
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| <br />
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| 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 />
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| <br />
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| 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 &quot;5-comma&quot;.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><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>
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="With powers of 2 taken for granted-2-move commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->2-move commas</h2>
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| 16/15 ( / 3 / 5)<br />
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| 33/32 (3 * 11)<br />
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| 65/64 (5 * 13)<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="With powers of 2 taken for granted-3-move commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->3-move commas</h2>
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| 25/24 (5 * 5 / 3)<br />
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| 128/125 (5 * 5 * 5)<br />
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| 21/20 (3 * 7 / 5)<br />
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| 26/25 (13 / 5 / 5)<br />
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| 49/48 (7 * 7 / 3)<br />
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| 64/63 ( / 3 / 7 / 7)<br />
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| 256/245 ( / 5 / 7 / 7)<br />
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| 80/77 (5 / 7 / 11)<br />
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| 22/21 (11 / 3 / 7)<br />
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| 40/39 (5 / 3 / 13)<br />
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| 96/91 (3 / 7 / 13)<br />
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| 55/52 (5 * 11 / 13)<br />
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| 1024/1001 (7 * 11 * 13)<br />
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| 512/507 (3 * 13 * 13)<br />
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| 169/160 (13 * 13 / 5)<br />
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| 176/169 (11 / 13 / 13)<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><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>
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| The relation of powers of 3 to the other factor(s) is represented by &quot;3's&quot;. The ones with names (all of them so far) have a corresponding sagittal accidental, though closeby commas share symbols.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><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>
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| 81/80 ( 3's / 5 ) (5 comma)<br />
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| 32805/32768 ( 3's * 5 ) (5 schisma)<br />
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| 64/63 ( / 3's / 7) (7 comma)<br />
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| 729/704 ( 3's / 11 ) (11-L diesis)<br />
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| 33/32 ( 3's * 11 ) (11-M diesis)<br />
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| 27/26 ( 3's / 13 ) (13-L diesis)<br />
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| 1053/1024 ( 3's * 13 ) (13 M-diesis)<br />
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| 2187/2176 ( 3's / 17 ) (17 kleisma)<br />
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| 4131/4096 ( 3's * 17 ) (17 comma)<br />
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| 513/512 ( 3's * 19 ) (19 schisma)<br />
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| 19683/19456 ( 3's / 19 ) (19 comma)<br />
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| 736/729 ( 23 / 3's ) (23 comma)<br />
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| 261/256 ( 3's * 29 ) (29 comma)<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><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>
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| (ordered and grouped by size of comma in just intonation)<br />
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| <br />
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| 5103/5120 ( 3's * 7 / 5 ) (5:7 kleisma)<br />
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| 352/351 ( 11 / 3's / 13 ) (11:13 kleisma)<br />
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| <br />
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| 896/891 ( 7 / 3's / 11 ) (7:11 kleisma)<br />
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| 2048/2025 ( / 3's / 5 / 5 ) (25 comma/<a class="wiki_link" href="/diaschisma">diaschisma</a>)<br />
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| 55/54 ( 11 * 5 / 3's ) (55 comma)<br />
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| <br />
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| 45927/45056 ( 3's * 7 / 11 ) (7:11 comma)<br />
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| 52/51 ( 3's * 13 / 17 ) (13:17 comma)<br />
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| <br />
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| 45/44 ( 3's * 5 / 11 ) (5:11 S-diesis)<br />
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| 1701/1664 ( 3's * 7 / 13 ) (7:13 S-diesis)<br />
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| 1408/1377 ( 11 / 3's / 17 ) (11:17 S-diesis)<br />
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| <br />
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| 6561/6400 ( 3's / 5 / 5 ) (25 S-diesis)<br />
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| 40/39 ( 5 / 3's / 13 ) (5:13 S-diesis)<br />
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| <br />
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| 36/35 ( 3's / 5 / 7 ) (35 M-diesis)<br />
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| 8505/8192 ( 3's * 5 * 7 ) (35 L-diesis)<br />
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| <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><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>
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| 250/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis)<br />
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| 531441/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis)</body></html></pre></div>
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