Hahn distance: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-~~15 14~~:03:42 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-19 01:34:08 UTC</tt>.

: The original revision id was <tt>~~357979194~~</tt>.

: The original revision id was <tt>358550201</tt>.

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

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

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where y = signum(x2)ceil(|x2/2|); here "signum" is +1 or -1 depending on the sign of x2 and "ceil" is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula.

It should be noted that this formula ~~does~~ not ~~define~~ a norm and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to

It should be noted that this formula defines a [[http://en.wikipedia.org/wiki/Metric_space|metric space distance function]] but not a norm, and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to

[[math]]

|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || =

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where y = signum(x2)ceil(|x2/2|); here &quot;signum&quot; is +1 or -1 depending on the sign of x2 and &quot;ceil&quot; is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula.

It should be noted that this formula ~~does~~ not ~~define~~ a norm and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to

It should be noted that this formula defines a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow">metric space distance function</a> but not a norm, and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to

<!-- ws:start:WikiTextMathRule:5:

[[math]]&lt;br/&gt;

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-08-15 14:03:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-19 01:34:08 UTC</tt>.<br>
-: The original revision id was <tt>357979194</tt>.<br>
+: The original revision id was <tt>358550201</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>
@@ Line 33: / Line 33: @@
 where y = signum(x2)ceil(|x2/2|); here "signum" is +1 or -1 depending on the sign of x2 and "ceil" is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula.
-It should be noted that this formula does not define a norm and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to
+It should be noted that this formula defines a [[http://en.wikipedia.org/wiki/Metric_space|metric space distance function]] but not a norm, and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to
 [[math]]
 || |x_1\ x_2\ x_3\ x_4\ x_5\ x_6&gt; || =
@@ Line 74: / Line 74: @@
 where y = signum(x2)ceil(|x2/2|); here &amp;quot;signum&amp;quot; is +1 or -1 depending on the sign of x2 and &amp;quot;ceil&amp;quot; is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula. &lt;br /&gt;
 &lt;br /&gt;
-It should be noted that this formula does not define a norm and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to&lt;br /&gt;
+It should be noted that this formula defines a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow"&gt;metric space distance function&lt;/a&gt; but not a norm, and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:5:
 [[math]]&amp;lt;br/&amp;gt;