Hahn distance: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 357979194 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 358550201 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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- | : 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> | : The original revision id was <tt>358550201</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> | ||
| 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. | 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 | 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]] | [[math]] | ||
|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = | || |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = | ||
| Line 74: | Line 74: | ||
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 /> | 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 | 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<br /> | ||
<!-- ws:start:WikiTextMathRule:5: | <!-- ws:start:WikiTextMathRule:5: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
Revision as of 01:34, 19 August 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-08-19 01:34:08 UTC.
- The original revision id was 358550201.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
In [[http://en.wikipedia.org/wiki/Graph_(mathematics)|graph theory]], the [[http://en.wikipedia.org/wiki/Distance_(graph_theory)|distance]] between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.
If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, excluding the unison and octaves, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.
Up to the 7-limit, Hahn distance has a very nice formula give by
[[math]]
||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2
[[math]]
[[math]]
= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)
[[math]]
We may take this formula and apply it to any triple of real numbers ||(a, b, c)||_hahn = (|a|+|b|+|c|+|a+b+c|)/2.
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]]
||(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.
In the 13-limit the formula for Hahn distance can be given as
[[math]]
|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} =
[[math]]
[[math]]
(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2
[[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 also be found from this formula.
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> || =
[[math]]
[[math]]
|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|
[[math]]
This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances.Original HTML content:
<html><head><title>Hahn distance</title></head><body>In <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Graph_(mathematics)" rel="nofollow">graph theory</a>, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Distance_(graph_theory)" rel="nofollow">distance</a> between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.<br />
<br />
If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, excluding the unison and octaves, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.<br />
<br />
Up to the 7-limit, Hahn distance has a very nice formula give by<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2<br/>[[math]]
--><script type="math/tex">||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2</script><!-- ws:end:WikiTextMathRule:0 --><br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br/>[[math]]
--><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 = (|a|+|b|+|c|+|a+b+c|)/2.<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:
[[math]]<br/>
||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}<br/>[[math]]
--><script type="math/tex">||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</script><!-- ws:end:WikiTextMathRule:2 --><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.<br />
<br />
In the 13-limit the formula for Hahn distance can be given as<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]<br/>
|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = <br/>[[math]]
--><script type="math/tex">|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = </script><!-- ws:end:WikiTextMathRule:3 --><br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]<br/>
(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2<br/>[[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 />
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. <br />
<br />
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<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]<br/>
|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = <br/>[[math]]
--><script type="math/tex">|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = </script><!-- ws:end:WikiTextMathRule:5 --><br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]<br/>
|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|<br/>[[math]]
--><script type="math/tex">|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|</script><!-- ws:end:WikiTextMathRule:6 --><br />
This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances.</body></html>