Taxicab distance: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2011-09-03 04:43:20 UTC</tt>.<br>

: This revision was by author [[User:xenjacob|xenjacob]] and made on <tt>2011-09-03 13:05:50 UTC</tt>.<br>

: The original revision id was <tt>~~250469074~~</tt>.<br>

: The original revision id was <tt>250505430</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

<h4>Original Wikitext content:</h4>

<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~~. 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~~.

<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.

~~To calculate~~ the ~~number~~ of ~~prime factors in~~ a ~~[[monzo]], simply take sum of the absolute values~~ of ~~each coordinate~~.

When combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of.

~~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.

=How to calculate taxicab distance on a prime-number lattice=

~~=Not yet~~ the ~~right name=~~

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:

~~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~~=

81/80 = 2^-4 * 3^4 * 5^-1

|-4| + |4| + |-1| = 9

==2-~~factor~~-limit commas==

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-~~factor-limit~~ commas==

==3-move commas==

25/24 (5 * 5 / 3)

128/125 (5 * 5 * 5)

Line 38:

Line 46:

512/507 (3 * 13 * 13)

169/160 (13 * 13 / 5)

176/169 (11 / 13 / 13)</pre></div>

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)</pre></div>

<h4>Original HTML content:</h4>

<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. ~~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 />

<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 />

<br />

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 />

<br />

<h1 id="toc0"><a name="How to calculate taxicab distance on a prime-number lattice"></a>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 />

~~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 />

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 />

For ~~example,~~ 81/80 ~~i.e. |-4 4 1&gt; would have a factor limit of 4+4+1=9, or, with 2's taken for granted,~~ 4+1=5.<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 />

<~~!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id~~="~~toc0~~"~~><a name~~="~~Not yet the right name~~"></a>~~<!~~-- ~~ws:end:WikiTextHeadingRule:0 -->Not yet the right name</h1>~~

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 />

~~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 />

<h1 id="toc1"><a name="With 2's taken for granted"></a>With 2's taken for granted</h1>

<h1 id="toc1"><a name="With powers of 2 taken for granted"></a>With powers of 2 taken for granted</h1>

<br />

<h2 id="toc2"><a name="With 2's taken for granted-2-~~factor-limit~~ commas"></a>2-~~factor-limit~~ commas</h2>

<h2 id="toc2"><a name="With powers of 2 taken for granted-2-move commas"></a>2-move commas</h2>

16/15 ( / 3 / 5)<br />

33/32 (3 * 11)<br />

65/64 (5 * 13)<br />

<br />

<h2 id="toc3"><a name="With 2's taken for granted-3-~~factor-limit~~ commas"></a>3-~~factor-limit~~ commas</h2>

<h2 id="toc3"><a name="With powers of 2 taken for granted-3-move commas"></a>3-move commas</h2>

25/24 (5 * 5 / 3)<br />

128/125 (5 * 5 * 5)<br />

Line 72:

Line 134:

512/507 (3 * 13 * 13)<br />

169/160 (13 * 13 / 5)<br />

176/169 (11 / 13 / 13)</body></html></pre></div>

176/169 (11 / 13 / 13)<br />

<br />

<h1 id="toc4"><a name="With powers of 2 and 3 taken for granted"></a>With powers of 2 and 3 taken for granted</h1>

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 />

<br />

<h2 id="toc5"><a name="With powers of 2 and 3 taken for granted-1-move commas"></a>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 />

<h2 id="toc6"><a name="With powers of 2 and 3 taken for granted-2-move commas"></a>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/diaschisma)<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 />

<h2 id="toc7"><a name="With powers of 2 and 3 taken for granted-3-move commas"></a>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></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 04:43:20 UTC</tt>.<br>
+: This revision was by author [[User:xenjacob|xenjacob]] and made on <tt>2011-09-03 13:05:50 UTC</tt>.<br>
-: The original revision id was <tt>250469074</tt>.<br>
+: The original revision id was <tt>250505430</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
-<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. 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.
+<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.
-To calculate the number of prime factors in a [[monzo]], simply take sum of the absolute values of each coordinate.
+When combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of.
-For example, 81/80 i.e. |-4 4 1&gt; would have a factor limit of 4+4+1=9, or, with 2's taken for granted, 4+1=5.
+=How to calculate taxicab distance on a prime-number lattice=
-=Not yet the right name=
+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:
-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=
+/80 = 2^-4 * 3^4 * 5^-1
+|-4| + |4| + |-1| = 9
-==2-factor-limit commas==
+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==
 /15 ( / 3 / 5)
 /32 (3 * 11)
 /64 (5 * 13)
-==3-factor-limit commas==
+==3-move commas==
 /24 (5 * 5 / 3)
 /125 (5 * 5 * 5)
@@ Line 38: / Line 46: @@
 /507 (3 * 13 * 13)
 /160 (13 * 13 / 5)
-/169 (11 / 13 / 13)</pre></div>
+/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==
+/80 ( 3's / 5 ) (5 comma)
+/32768 ( 3's * 5 ) (5 schisma)
+/63 ( / 3's / 7) (7 comma)
+/704 ( 3's / 11 ) (11-L diesis)
+/32 ( 3's * 11 ) (11-M diesis)
+/26 ( 3's / 13 ) (13-L diesis)
+/1024 ( 3's * 13 ) (13 M-diesis)
+/2176 ( 3's / 17 ) (17 kleisma)
+/4096 ( 3's * 17 ) (17 comma)
+/512 ( 3's * 19 ) (19 schisma)
+/19456 ( 3's / 19 ) (19 comma)
+/729 ( 23 / 3's ) (23 comma)
+/256 ( 3's * 29 ) (29 comma)
+==2-move commas==
+(ordered and grouped by size of comma in just intonation)
+/5120 ( 3's * 7 / 5 ) (5:7 kleisma)
+/351 ( 11 / 3's / 13 ) (11:13 kleisma)
+/891 ( 7 / 3's / 11 ) (7:11 kleisma)
+/2025 ( / 3's / 5 / 5 ) (25 comma/diaschisma)
+/54 ( 11 * 5 / 3's ) (55 comma)
+/45056 ( 3's * 7 / 11 ) (7:11 comma)
+/51 ( 3's * 13 / 17 ) (13:17 comma)
+/44 ( 3's * 5 / 11 ) (5:11 S-diesis)
+/1664 ( 3's * 7 / 13 ) (7:13 S-diesis)
+/1377 ( 11 / 3's / 17 ) (11:17 S-diesis)
+/6400 ( 3's / 5 / 5 ) (25 S-diesis)
+/39 ( 5 / 3's / 13 ) (5:13 S-diesis)
+/35 ( 3's / 5 / 7 ) (35 M-diesis)
+/8192 ( 3's * 5 * 7 ) (35 L-diesis)
+==3-move commas==
+/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis)
+/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis)</pre></div>
 <h4>Original HTML content:</h4>
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;commas by taxicab distance&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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.&lt;br /&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;commas by taxicab distance&lt;/title&gt;&lt;/head&gt;&lt;body&gt;One measurement of the complexity of the comma could be the number of prime factors it has, regardless of their magnitude, but counting repetitions.&lt;br /&gt;
+&lt;br /&gt;
+When combined with excluding the smallest primes, this measurement can give an idea of how many &amp;quot;strange harmonic moves&amp;quot; a comma is comprised of.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="How to calculate taxicab distance on a prime-number lattice"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;How to calculate taxicab distance on a prime-number lattice&lt;/h1&gt;
+ &lt;br /&gt;
+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:&lt;br /&gt;
+&lt;br /&gt;
+/80 = 2^-4 * 3^4 * 5^-1&lt;br /&gt;
+|-4| + |4| + |-1| = 9&lt;br /&gt;
 &lt;br /&gt;
-To calculate the number of prime factors in a &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt;, simply take sum of the absolute values of each coordinate.&lt;br /&gt;
+This corresponds to an interval's unweighted &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lp_space" rel="nofollow"&gt;L1&lt;/a&gt; distance on a prime-factor lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height.&lt;br /&gt;
 &lt;br /&gt;
-For example, 81/80 i.e. |-4 4 1&amp;gt; would have a factor limit of 4+4+1=9, or, with 2's taken for granted, 4+1=5.&lt;br /&gt;
+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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Not yet the right name"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Not yet the right name&lt;/h1&gt;
+If you discard powers of both 2 and 3, you get an understanding of commas relevant to &lt;a class="wiki_link" href="/sagittal%20corner"&gt;Sagittal notation&lt;/a&gt;, 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 &amp;quot;5-comma&amp;quot;.&lt;br /&gt;
- 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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="With 2's taken for granted"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;With 2's taken for granted&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="With powers of 2 taken for granted"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;With powers of 2 taken for granted&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="With 2's taken for granted-2-factor-limit commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;2-factor-limit commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="With powers of 2 taken for granted-2-move commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;2-move commas&lt;/h2&gt;
 /15 ( / 3 / 5)&lt;br /&gt;
 /32 (3 * 11)&lt;br /&gt;
 /64 (5 * 13)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="With 2's taken for granted-3-factor-limit commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;3-factor-limit commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="With powers of 2 taken for granted-3-move commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;3-move commas&lt;/h2&gt;
 /24 (5 * 5 / 3)&lt;br /&gt;
 /125 (5 * 5 * 5)&lt;br /&gt;
@@ Line 72: / Line 134: @@
 /507 (3 * 13 * 13)&lt;br /&gt;
 /160 (13 * 13 / 5)&lt;br /&gt;
-/169 (11 / 13 / 13)&lt;/body&gt;&lt;/html&gt;</pre></div>
+/169 (11 / 13 / 13)&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="With powers of 2 and 3 taken for granted"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;With powers of 2 and 3 taken for granted&lt;/h1&gt;
+ The relation of powers of 3 to the other factor(s) is represented by &amp;quot;3's&amp;quot;. The ones with names (all of them so far) have a corresponding sagittal accidental, though closeby commas share symbols.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="With powers of 2 and 3 taken for granted-1-move commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;1-move commas&lt;/h2&gt;
+/80 ( 3's / 5 ) (5 comma)&lt;br /&gt;
+/32768 ( 3's * 5 ) (5 schisma)&lt;br /&gt;
+/63 ( / 3's / 7) (7 comma)&lt;br /&gt;
+/704 ( 3's / 11 ) (11-L diesis)&lt;br /&gt;
+/32 ( 3's * 11 ) (11-M diesis)&lt;br /&gt;
+/26 ( 3's / 13 ) (13-L diesis)&lt;br /&gt;
+/1024 ( 3's * 13 ) (13 M-diesis)&lt;br /&gt;
+/2176 ( 3's / 17 ) (17 kleisma)&lt;br /&gt;
+/4096 ( 3's * 17 ) (17 comma)&lt;br /&gt;
+/512 ( 3's * 19 ) (19 schisma)&lt;br /&gt;
+/19456 ( 3's / 19 ) (19 comma)&lt;br /&gt;
+/729 ( 23 / 3's ) (23 comma)&lt;br /&gt;
+/256 ( 3's * 29 ) (29 comma)&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="With powers of 2 and 3 taken for granted-2-move commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;2-move commas&lt;/h2&gt;
+ (ordered and grouped by size of comma in just intonation)&lt;br /&gt;
+&lt;br /&gt;
+/5120 ( 3's * 7 / 5 ) (5:7 kleisma)&lt;br /&gt;
+/351 ( 11 / 3's / 13 ) (11:13 kleisma)&lt;br /&gt;
+&lt;br /&gt;
+/891 ( 7 / 3's / 11 ) (7:11 kleisma)&lt;br /&gt;
+/2025 ( / 3's / 5 / 5 ) (25 comma/diaschisma)&lt;br /&gt;
+/54 ( 11 * 5 / 3's ) (55 comma)&lt;br /&gt;
+&lt;br /&gt;
+/45056 ( 3's * 7 / 11 ) (7:11 comma)&lt;br /&gt;
+/51 ( 3's * 13 / 17 ) (13:17 comma)&lt;br /&gt;
+&lt;br /&gt;
+/44 ( 3's * 5 / 11 ) (5:11 S-diesis)&lt;br /&gt;
+/1664 ( 3's * 7 / 13 ) (7:13 S-diesis)&lt;br /&gt;
+/1377 ( 11 / 3's / 17 ) (11:17 S-diesis)&lt;br /&gt;
+&lt;br /&gt;
+/6400 ( 3's / 5 / 5 ) (25 S-diesis)&lt;br /&gt;
+/39 ( 5 / 3's / 13 ) (5:13 S-diesis)&lt;br /&gt;
+&lt;br /&gt;
+/35 ( 3's / 5 / 7 ) (35 M-diesis)&lt;br /&gt;
+&lt;br /&gt;
+/8192 ( 3's * 5 * 7 ) (35 L-diesis)&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="With powers of 2 and 3 taken for granted-3-move commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;3-move commas&lt;/h2&gt;
+/243 ( 5 * 5 * 5 / 3's ) (125 M-diesis)&lt;br /&gt;
+/512000 ( 3's / 5 / 5 / 5 ) (125 L-diesis)&lt;/body&gt;&lt;/html&gt;</pre></div>

Taxicab distance: Difference between revisions

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