Structure metric
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[[toc]]
=Definition=
The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[constant structure]] [[periodic scale]] within the period, which give to it the property of being a [[https://en.wikipedia.org/wiki/Metric_space|finite metric space]]. (In academic theory, constant structure is called the //partitioning property//.) If **s** is a periodic scale with quasiperiod **P**, and if **s**[i] with 0≤i<**P** is a note of **s** within the period **P**, then we may define the base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i], 0≤j<**P**}. These have the property that the interval between the base note **s**[j] and the note i steps away, **s**[j+i], is in class(i), the interval class to which **s**[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of **s**[i], and **P**-n which correspond to indicies of intervals other than **s**[i]. In other words, there are **P**-n intervals, counting multiplicities, in the class of **s**[i] other than **s**[i]. Then the //structure complexity// ||**s**[i]|| of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |**s**[i] - **s**[j]| ||.
=Properties=
The structure metric has the following properties:
1. d(**s**[i], **s**[i]) = 0
|| **s**[i] - **s**[i] || = ||0|| which equals 0.
2. d(**s**[i], **s**[j]) ≥ 0
This is so since the cardinality n of the base point set is less than or equal to **P**.
3. d(**s**[i], **s**[j]) = 0 implies **s**[i] equals **s**[j]
Suppose ||**s**[**I**]|| equals 0 with 0 < **I** < **P**. Then **s**[j+**I**] - **s**[j] = **s**[**I**], so that **s** is periodic with quasiperiod **I**. But by assumption, **P** is the least quasiperiod of **s**. Hence, ||**s**[**I**]|| equals 0 implies **I** equals 0. It follows that if d(**s**[i], **s**[j]) equals || **s**[i] - **s**[j] || equals 0, then **s**[i] - **s**[j] equals 0 and **s**[i] equals **s**[j].
4. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
d(**s**[i], **s**[j]) equals || |**s**[i] - **s**[j]| || equals || |**s**[j] - **s**[i]| || equals d(**s**[j], **s**[i])
5. d(**s**[i], **s**[k]) ≤ d(**s**[i], **s**[j]) + d(**s**[j], **s**[k])
First, || **s**[i + j] mod **O** || ≤ ||**s**[i]|| + ||**s**[j]|| where **O** is the interval of equivalence. If an interval in the interval class of **s**[i] equals **s**[i] and an interval in the interval class of **s**[j] equals **s**[j], then their product, reduced modulo the interval of equivalence **O** equals **s**[**P**], will be **s**[i + j] mod **O**. Hence to get an interval in the class of **s**[i + j] mod **O** other than **s**[i + j] mod **O** as a product, either the interval in the class of **s**[i] must be other than **s**[i], or the interval in the class of **s**[j] must be other than **s**[j]. If always only one of the intervals is different than the defining interval for its class, then || **s**[i + j] mod **O** || equals ||**s**[i]|| + ||**s**[j]||. However, there may be overlap, so that the first interval is not in the class for **s**[i] and the second not in the class for **s**[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(**s**[i], **s**[j]) + d(**s**[j], **s**[k]) = || |**s**[i] - **s**[j]| || + || |**s**[j] - **s**[k]| || ≥ || |**s**[i] - **s**[k]| || = d(**s**[i], **s**[k]).
These properties mean that the structure metric defines a //finite metric space//. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.
=Isometry=
An [[https://en.wikipedia.org/wiki/Isometry|isometry]] between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the [[https://en.wikipedia.org/wiki/Isometry_group|isometry group]].
In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix (d(i, j)), where "i" denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if S.D.S^(-1) = D, where the dot is matrix multiplication. In this case, D is permutation-similar to itself by S. An invariant under similarity, and hence permutation similarity in particular, is the characteristic polynomial, as well as related invariants such as the rank, eigenvalues and minimal polynomial. The characteristic polynomial tends to reflect the symmetries of the metric space and the isometry group.
An interesting example of this is given by the [[https://en.wikipedia.org/wiki/Hexany|hexany]], 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]], from which we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both [[27edo]] and [[31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.
=Invariants=
A metric invariant is a property of a metric space which is preserved under isometry. The metric invariants of the structure metric define properites of the scale from which it derives,
==Centrality==
The //eccentricity// of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center.
==Roundness==
The [[https://en.wikipedia.org/wiki/Gromov_product|Gromov product]] is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.
If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. Original HTML content:
<html><head><title>Structure metric</title></head><body><!-- ws:start:WikiTextTocRule:12:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 1em;"><a href="#Properties">Properties</a></div>
<!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#Isometry">Isometry</a></div>
<!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 1em;"><a href="#Invariants">Invariants</a></div>
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 2em;"><a href="#Invariants-Centrality">Centrality</a></div>
<!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#Invariants-Roundness">Roundness</a></div>
<!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --></div>
<!-- ws:end:WikiTextTocRule:19 --><br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1>
The <em>structure metric</em> is a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow">distance function</a> on the notes of a <a class="wiki_link" href="/constant%20structure">constant structure</a> <a class="wiki_link" href="/periodic%20scale">periodic scale</a> within the period, which give to it the property of being a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow">finite metric space</a>. (In academic theory, constant structure is called the <em>partitioning property</em>.) If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if <strong>s</strong>[i] with 0≤i<<strong>P</strong> is a note of <strong>s</strong> within the period <strong>P</strong>, then we may define the base points set base(<strong>s</strong>[i]) to be the set of integers {j | <strong>s</strong>[j+i] - <strong>s</strong>[j] = <strong>s</strong>[i], 0≤j<<strong>P</strong>}. These have the property that the interval between the base note <strong>s</strong>[j] and the note i steps away, <strong>s</strong>[j+i], is in class(i), the interval class to which <strong>s</strong>[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of <strong>s</strong>[i], and <strong>P</strong>-n which correspond to indicies of intervals other than <strong>s</strong>[i]. In other words, there are <strong>P</strong>-n intervals, counting multiplicities, in the class of <strong>s</strong>[i] other than <strong>s</strong>[i]. Then the <em>structure complexity</em> ||<strong>s</strong>[i]|| of <strong>s</strong>[i] is defined to be <strong>P</strong>-n, and the structure metric is defined as d(<strong>s</strong>[i], <strong>s</strong>[j]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| ||.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1>
The structure metric has the following properties:<br />
<br />
1. d(<strong>s</strong>[i], <strong>s</strong>[i]) = 0<br />
|| <strong>s</strong>[i] - <strong>s</strong>[i] || = ||0|| which equals 0.<br />
<br />
2. d(<strong>s</strong>[i], <strong>s</strong>[j]) ≥ 0<br />
This is so since the cardinality n of the base point set is less than or equal to <strong>P</strong>.<br />
<br />
3. d(<strong>s</strong>[i], <strong>s</strong>[j]) = 0 implies <strong>s</strong>[i] equals <strong>s</strong>[j]<br />
Suppose ||<strong>s</strong>[<strong>I</strong>]|| equals 0 with 0 < <strong>I</strong> < <strong>P</strong>. Then <strong>s</strong>[j+<strong>I</strong>] - <strong>s</strong>[j] = <strong>s</strong>[<strong>I</strong>], so that <strong>s</strong> is periodic with quasiperiod <strong>I</strong>. But by assumption, <strong>P</strong> is the least quasiperiod of <strong>s</strong>. Hence, ||<strong>s</strong>[<strong>I</strong>]|| equals 0 implies <strong>I</strong> equals 0. It follows that if d(<strong>s</strong>[i], <strong>s</strong>[j]) equals || <strong>s</strong>[i] - <strong>s</strong>[j] || equals 0, then <strong>s</strong>[i] - <strong>s</strong>[j] equals 0 and <strong>s</strong>[i] equals <strong>s</strong>[j]. <br />
<br />
4. d(<strong>s</strong>[i], <strong>s</strong>[j]) = d(<strong>s</strong>[j], <strong>s</strong>[i])<br />
d(<strong>s</strong>[i], <strong>s</strong>[j]) equals || |<strong>s</strong>[i] - <strong>s</strong>[j]| || equals || |<strong>s</strong>[j] - <strong>s</strong>[i]| || equals d(<strong>s</strong>[j], <strong>s</strong>[i])<br />
<br />
5. d(<strong>s</strong>[i], <strong>s</strong>[k]) ≤ d(<strong>s</strong>[i], <strong>s</strong>[j]) + d(<strong>s</strong>[j], <strong>s</strong>[k])<br />
First, || <strong>s</strong>[i + j] mod <strong>O</strong> || ≤ ||<strong>s</strong>[i]|| + ||<strong>s</strong>[j]|| where <strong>O</strong> is the interval of equivalence. If an interval in the interval class of <strong>s</strong>[i] equals <strong>s</strong>[i] and an interval in the interval class of <strong>s</strong>[j] equals <strong>s</strong>[j], then their product, reduced modulo the interval of equivalence <strong>O</strong> equals <strong>s</strong>[<strong>P</strong>], will be <strong>s</strong>[i + j] mod <strong>O</strong>. Hence to get an interval in the class of <strong>s</strong>[i + j] mod <strong>O</strong> other than <strong>s</strong>[i + j] mod <strong>O</strong> as a product, either the interval in the class of <strong>s</strong>[i] must be other than <strong>s</strong>[i], or the interval in the class of <strong>s</strong>[j] must be other than <strong>s</strong>[j]. If always only one of the intervals is different than the defining interval for its class, then || <strong>s</strong>[i + j] mod <strong>O</strong> || equals ||<strong>s</strong>[i]|| + ||<strong>s</strong>[j]||. However, there may be overlap, so that the first interval is not in the class for <strong>s</strong>[i] and the second not in the class for <strong>s</strong>[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(<strong>s</strong>[i], <strong>s</strong>[j]) + d(<strong>s</strong>[j], <strong>s</strong>[k]) = || |<strong>s</strong>[i] - <strong>s</strong>[j]| || + || |<strong>s</strong>[j] - <strong>s</strong>[k]| || ≥ || |<strong>s</strong>[i] - <strong>s</strong>[k]| || = d(<strong>s</strong>[i], <strong>s</strong>[k]).<br />
<br />
These properties mean that the structure metric defines a <em>finite metric space</em>. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Isometry"></a><!-- ws:end:WikiTextHeadingRule:4 -->Isometry</h1>
An <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Isometry" rel="nofollow">isometry</a> between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Isometry_group" rel="nofollow">isometry group</a>.<br />
<br />
In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix (d(i, j)), where "i" denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if S.D.S^(-1) = D, where the dot is matrix multiplication. In this case, D is permutation-similar to itself by S. An invariant under similarity, and hence permutation similarity in particular, is the characteristic polynomial, as well as related invariants such as the rank, eigenvalues and minimal polynomial. The characteristic polynomial tends to reflect the symmetries of the metric space and the isometry group.<br />
<br />
An interesting example of this is given by the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Hexany" rel="nofollow">hexany</a>, 1-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0, 4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]], from which we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both <a class="wiki_link" href="/27edo">27edo</a> and <a class="wiki_link" href="/31edo">31edo</a>, for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Invariants"></a><!-- ws:end:WikiTextHeadingRule:6 -->Invariants</h1>
A metric invariant is a property of a metric space which is preserved under isometry. The metric invariants of the structure metric define properites of the scale from which it derives,<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Invariants-Centrality"></a><!-- ws:end:WikiTextHeadingRule:8 -->Centrality</h2>
The <em>eccentricity</em> of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Invariants-Roundness"></a><!-- ws:end:WikiTextHeadingRule:10 -->Roundness</h2>
The <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Gromov_product" rel="nofollow">Gromov product</a> is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.<br />
<br />
If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = (d(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix.</body></html>