Structure metric

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 2015-11-09 12:54:10 UTC.

The original revision id was 565746425.

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:

[[toc]]

=Definition=
The //structure metric// is a [[https://en.wikipedia.org/wiki/Metric_(mathematics)|distance function]] on the notes of a [[periodic scale]] within a single period, which give to it the property of being a [[https://en.wikipedia.org/wiki/Metric_space|finite metric space]]. If **s** is a periodic scale with quasiperiod **P**, and if c is an interval **s**[i+j] - **s**[i] with 0≤i<**P**, then we may define the specific interval set S(c, j) to be {i|**s**[i+j] - **s**[i] = c} with 0≤i<**P**, that is, indicies for the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(**s**[a], **s**[b]), which we will abbreviate as d(a, b), to be **P** - #S(|**s**[a] - **s**[b]|, |a-b|). 

=Properties=
The structure metric has the following properties:

1. d(a, a) = 0
#S(|**s**[a] - **s**[a]|, |a - a|) = #S(0, 0) = **P**.

2. d(a, b) ≥ 0
The cardinality of #S(c, j) cannot exceed **P**, since 0≤i<**P**.

3. d(a, b) = 0 implies a equals b.
If d(a, b) = 0 then #S(|**s**[a] - **s**[b]|, |a - b|)) = **P**

4. d(**s**[i], **s**[j]) = d(**s**[j], **s**[i])
d(**s**[i], **s**[j]) equals || |i - j| || equals || |j - 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** which is  **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]) = || |i - j| || + || |j - k| || ≥ || |i - 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. A more refined measure than eccentricity is the //distance degree// of a point; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.

==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. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p/2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q<p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is "rounder", and with a lower one "flatter". Below is a listing of some scales (either JI or in some edo) by increasing roundness.

p = 1.1135814 [[duodene]],  [[novadene]], [[marveldene]]; these are not isometric
p = 1.1366768 [[http://xenharmonic.wikispaces.com/domdimpajinjmean|miller7]], [[wilson_class]], [[dekany-cs]]; these are isometric
p = 1.2651510 [[zeus8tri]], [[star]], [[nova]]; these are not isometric
p = 1.3404363 [[thirteendene]]
p = 1.3563125 [[wilson17]]
p = 1.3652790 [[centaur]]
p = 1.5709365 [[zarlino]]
p = 1.5865859 Cps([2,3,5,7,9,11], 3), the eikosany
p = 1.6426289 [[mandala]], the stellated hexany.
p = 1.8225500 [[zeus7tri]], [[diamond5]] the 5-limit tonality diamond; these are not isometric
p = 1.8501138 [[raven]]
p = 1.9855771 [[blue-ji|blue]]
p = 2 exactly all MOS scales, also [[diamond7]] the 7-limit tonality diamond
p = 2.1918973 [[shell5_3]]
p = 2.4079115 [[shell5-2]]
p = 2.7580875 Cps([2,3,5,7,11], 2) and Cps([2,3,5,7,11], 3), the 2)5 and 3)5 dekanys; these are isometric
p = 3.1062837 [[hexany]]
p = 4.4843144 otonal and utonal pentad; isometric
p = 6.9477267 otonal and utonal heptad; isometric
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric

Original HTML content:

<html><head><title>Structure metric</title></head><body><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><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:&lt;h1&gt; --><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="/periodic%20scale">periodic scale</a> within a single 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>. If <strong>s</strong> is a periodic scale with quasiperiod <strong>P</strong>, and if c is an interval <strong>s</strong>[i+j] - <strong>s</strong>[i] with 0≤i&lt;<strong>P</strong>, then we may define the specific interval set S(c, j) to be {i|<strong>s</strong>[i+j] - <strong>s</strong>[i] = c} with 0≤i&lt;<strong>P</strong>, that is, indicies for the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d(<strong>s</strong>[a], <strong>s</strong>[b]), which we will abbreviate as d(a, b), to be <strong>P</strong> - #S(|<strong>s</strong>[a] - <strong>s</strong>[b]|, |a-b|). <br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Properties"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties</h1>
The structure metric has the following properties:<br />
<br />
1. d(a, a) = 0<br />
#S(|<strong>s</strong>[a] - <strong>s</strong>[a]|, |a - a|) = #S(0, 0) = <strong>P</strong>.<br />
<br />
2. d(a, b) ≥ 0<br />
The cardinality of #S(c, j) cannot exceed <strong>P</strong>, since 0≤i&lt;<strong>P</strong>.<br />
<br />
3. d(a, b) = 0 implies a equals b.<br />
If d(a, b) = 0 then #S(|<strong>s</strong>[a] - <strong>s</strong>[b]|, |a - b|)) = <strong>P</strong><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 || |i - j| || equals || |j - 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> which is  <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]) = || |i - j| || + || |j - k| || ≥ || |i - 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:&lt;h1&gt; --><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 &quot;i&quot; 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:&lt;h1&gt; --><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:&lt;h2&gt; --><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. A more refined measure than eccentricity is the <em>distance degree</em> of a point; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><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. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type, and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p/2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q&lt;p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is &quot;rounder&quot;, and with a lower one &quot;flatter&quot;. Below is a listing of some scales (either JI or in some edo) by increasing roundness.<br />
<br />
p = 1.1135814 <a class="wiki_link" href="/duodene">duodene</a>,  <a class="wiki_link" href="/novadene">novadene</a>, <a class="wiki_link" href="/marveldene">marveldene</a>; these are not isometric<br />
p = 1.1366768 <a href="http://xenharmonic.wikispaces.com/domdimpajinjmean">miller7</a>, <a class="wiki_link" href="/wilson_class">wilson_class</a>, <a class="wiki_link" href="/dekany-cs">dekany-cs</a>; these are isometric<br />
p = 1.2651510 <a class="wiki_link" href="/zeus8tri">zeus8tri</a>, <a class="wiki_link" href="/star">star</a>, <a class="wiki_link" href="/nova">nova</a>; these are not isometric<br />
p = 1.3404363 <a class="wiki_link" href="/thirteendene">thirteendene</a><br />
p = 1.3563125 <a class="wiki_link" href="/wilson17">wilson17</a><br />
p = 1.3652790 <a class="wiki_link" href="/centaur">centaur</a><br />
p = 1.5709365 <a class="wiki_link" href="/zarlino">zarlino</a><br />
p = 1.5865859 Cps([2,3,5,7,9,11], 3), the eikosany<br />
p = 1.6426289 <a class="wiki_link" href="/mandala">mandala</a>, the stellated hexany.<br />
p = 1.8225500 <a class="wiki_link" href="/zeus7tri">zeus7tri</a>, <a class="wiki_link" href="/diamond5">diamond5</a> the 5-limit tonality diamond; these are not isometric<br />
p = 1.8501138 <a class="wiki_link" href="/raven">raven</a><br />
p = 1.9855771 <a class="wiki_link" href="/blue-ji">blue</a><br />
p = 2 exactly all MOS scales, also <a class="wiki_link" href="/diamond7">diamond7</a> the 7-limit tonality diamond<br />
p = 2.1918973 <a class="wiki_link" href="/shell5_3">shell5_3</a><br />
p = 2.4079115 <a class="wiki_link" href="/shell5-2">shell5-2</a><br />
p = 2.7580875 Cps([2,3,5,7,11], 2) and Cps([2,3,5,7,11], 3), the 2)5 and 3)5 dekanys; these are isometric<br />
p = 3.1062837 <a class="wiki_link" href="/hexany">hexany</a><br />
p = 4.4843144 otonal and utonal pentad; isometric<br />
p = 6.9477267 otonal and utonal heptad; isometric<br />
p = ∞ otonal and utonal tetrad; this implies the space is ultrametric</body></html>