Hahn distance: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 151536375 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278341718 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:49:50 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278341718</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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||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|) | ||(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. 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, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca) | ||
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. | 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. | ||
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||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br /> | ||(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. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by<br /> | ||
||(a, b, c)||_sym = | ||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)<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> | 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> | ||
Revision as of 20:49, 22 November 2011
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 2011-11-22 20:49:50 UTC.
- The original revision id was 278341718.
- 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 graph theory, the 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. 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, 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 ||3^a 5^b 7^c||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|) 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) 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.
Original HTML content:
<html><head><title>Hahn distance</title></head><body><br /> In graph theory, the 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. 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, 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 /> ||3^a 5^b 7^c||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<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<br /> ||(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 /> ||(a, b, c)||_sym = √(a^2 + b^2 + c^2 + ab + bc + ca)<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>