Hahn distance: Difference between revisions
Wikispaces>genewardsmith **Imported revision 357324078 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 357345216 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2012-08-11 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-11 19:23:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>357345216</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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[[math]] | [[math]] | ||
We may take this formula 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|) | We may take this formula 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 | 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; it also defines a seminorm on 7-limit [[Monzos and Interval Space|interval space]]. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by | ||
[[math]] | [[math]] | ||
||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)} | ||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)} | ||
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(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2 | (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2 | ||
[[math]] | [[math]] | ||
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 | 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 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 | ||
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--><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
We may take this formula 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 /> | We may take this formula 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<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; it also defines a seminorm on 7-limit <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>. 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: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2&lt;br/&gt;[[math]] | (|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2&lt;br/&gt;[[math]] | ||
--><script type="math/tex">(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</script><!-- ws:end:WikiTextMathRule:4 --><br /> | --><script type="math/tex">(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
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 | 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. <br /> | ||
<br /> | <br /> | ||
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<br /> | 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<br /> | ||