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:<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2012-08-08 04:35:19 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2012-08-08 04:38:51 UTC</tt>.<br>

: The original revision id was <tt>~~356818286~~</tt>.<br>

: The original revision id was <tt>356818512</tt>.<br>

: The revision comment was: <tt>~~converted using~~ "math" ~~- the linebreak made problems~~</tt><br>

: The revision comment was: <tt>again switched to "math" syntax</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>

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[[math]]

We may take this formula (or the similar formulas we would obtain for higher odd limits) and apply it to any triple of real numbers||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)

If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by||(a, b, c)||~~_sym~~ = √(a^2 + b^2 + c^2 + ab + bc + ca)

If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by

[[math]]

||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}

[[math]]

and discussed [[The Seven Limit Symmetrical Lattices|here]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.</pre></div>

<h4>Original HTML content:</h4>

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--><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><br />

We may take this formula (or the similar formulas we would obtain for higher odd limits) and apply it to any triple of real numbers||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br />

If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by||(a, b, c)||~~_sym~~ = √(a^2 + b^2 + c^2 + ab + bc + ca)<br />

If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by<br />

<!-- ws:start:WikiTextMathRule:2:

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

||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}&lt;br/&gt;[[math]]

--><script type="math/tex">||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</script><br />

and discussed <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">here</a>. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.</body></html></pre></div>

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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:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2012-08-08 04:35:19 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2012-08-08 04:38:51 UTC</tt>.<br>
-: The original revision id was <tt>356818286</tt>.<br>
+: The original revision id was <tt>356818512</tt>.<br>
-: The revision comment was: <tt>converted using "math" - the linebreak made problems</tt><br>
+: The revision comment was: <tt>again switched to "math" syntax</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>
@@ Line 18: / Line 18: @@
 [[math]]
 We may take this formula (or the similar formulas we would obtain for higher odd limits) and apply it to any triple of real numbers||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)
-If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)
+If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by
+[[math]]
+||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}
+[[math]]
 and discussed [[The Seven Limit Symmetrical Lattices|here]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.</pre></div>
 <h4>Original HTML content:</h4>
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   --&gt;&lt;script type="math/tex"&gt;= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
 We may take this formula (or the similar formulas we would obtain for higher odd limits) and apply it to any triple of real numbers||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;br /&gt;
-If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)&lt;br /&gt;
+If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:2:
+[[math]]&amp;lt;br/&amp;gt;
+||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
 and discussed &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;here&lt;/a&gt;. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.&lt;/body&gt;&lt;/html&gt;</pre></div>