Structure metric: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-08 11:18:27 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-08 12:25:09 UTC</tt>.

: The original revision id was <tt>~~565607599~~</tt>.

: The original revision id was <tt>565612683</tt>.

: The revision comment was: <tt></tt>

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=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~~<i+j~~<**P**, then we may define the specific interval set S(c, j) to be 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|). 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// || i || of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |i - j| ||.

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(**s**[i], **s**[i]~~) = 0~~

1. d(a, a) = 0

|| i - ~~i |~~| = ||0~~|| which equals~~ 0.

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

2. d(**s**[i], **s**[j]) ≥ 0

2. d(a, b) ≥ 0

~~This is so since the~~ cardinality n of ~~the base point set is less than or equal to~~ **P**.

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

3. d(**s**[i], **s**[j]) = 0 implies **s**[i] equals **s**[j]

3. d(a, b) = 0 implies a equals b.

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 || |i - j| || equals 0, then i - j equals 0 and so **s**[i] equals **s**[j].

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p = 6.9477267 otonal and utonal heptad

p = ∞ otonal and utonal tetrad; this implies the space is ultrametric

</pre></div>

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<h1 id="toc0"><a name="Definition"></a>Definition</h1>

The structure metric 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 s is a periodic scale with quasiperiod P, and if c is an interval s[i+j] - s[i] with 0≤i~~&lt;i+j~~&lt;P, then we may define the specific interval set S(c, j) to be 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|~~). base points set base(~~s[i~~]) to be the set of integers {j | s[~~j+i] - s[j] = ~~s[i], 0≤j~~&lt;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 || i || of s[i] is defined to be P-~~n, and the structure metric is defined as d~~(s[i], s[j]~~) =~~ || |i - ~~j| |~~|.

The structure metric 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 s is a periodic scale with quasiperiod P, and if c is an interval s[i+j] - s[i] with 0≤i&lt;P, then we may define the specific interval set S(c, j) to be {i|s[i+j] - s[i] = c} with 0≤i&lt;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|).

<h1 id="toc1"><a name="Properties"></a>Properties</h1>

The structure metric has the following properties:

1. d(s[i], s[i]) = 0

1. d(a, a) = 0

~~|| i - i || = ||0|| which equals 0~~.

#S(|s[a] - s[a]|, |a - b|) = #S(0, 0) = P.

2. d(<~~strong>s<~~/~~strong~~>~~[i]~~, s~~[j]) ≥ 0~~<~~br />~~

2. d(a, b) ≥ 0

~~This is so since the cardinality n of the base point set is less than or equal to~~ P.

The cardinality of #S(c, j) cannot exceed *P, since 0≤i&lt;P.

3. d(~~s[i]~~, ~~s[j]~~) = 0 implies ~~s[i]~~ equals ~~s[j]~~

3. d(a, b) = 0 implies a equals b.

Suppose ||s[I]|| equals 0 with 0 &lt; I &lt; 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 || |i - j| || equals 0, then i - j equals 0 and so s[i] equals s[j].

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2015-11-08 11:18:27 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-08 12:25:09 UTC</tt>.<br>
-: The original revision id was <tt>565607599</tt>.<br>
+: The original revision id was <tt>565612683</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>
@@ Line 9: / Line 9: @@
 =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&lt;i+j&lt;**P**, then we may define the specific interval set S(c, j) to be 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|). base points set base(**s**[i]) to be the set of integers {j | **s**[j+i] - **s**[j] = **s**[i], 0≤j&lt;**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// || i || of **s**[i] is defined to be **P**-n, and the structure metric is defined as d(**s**[i], **s**[j]) = || |i - j| ||.
+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&lt;**P**, then we may define the specific interval set S(c, j) to be {i|**s**[i+j] - **s**[i] = c} with 0≤i&lt;**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:
-. d(**s**[i], **s**[i]) = 0
+. d(a, a) = 0
-|| i - i || = ||0|| which equals 0.
+#S(|**s**[a] - **s**[a]|, |a - b|) = #S(0, 0) = **P**.
-. d(**s**[i], **s**[j]) ≥ 0
+. d(a, b) ≥ 0
-This is so since the cardinality n of the base point set is less than or equal to **P**.
+The cardinality of #S(c, j) cannot exceed ***P**, since 0≤i&lt;**P**.
-. d(**s**[i], **s**[j]) = 0 implies **s**[i] equals **s**[j]
+. d(a, b) = 0 implies a equals b.
 Suppose ||**s**[**I**]|| equals 0 with 0 &lt; **I** &lt; **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 || |i - j| || equals 0, then i - j equals 0 and so **s**[i] equals **s**[j].
@@ Line 64: / Line 64: @@
 p = 6.9477267 otonal and utonal heptad
 p = ∞ otonal and utonal tetrad; this implies the space is ultrametric
 </pre></div>
@@ Line 77: / Line 76: @@
 &lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;
-The &lt;em&gt;structure metric&lt;/em&gt; is a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow"&gt;distance function&lt;/a&gt; on the notes of a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; within a single period, which give to it the property of being a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow"&gt;finite metric space&lt;/a&gt;. If &lt;strong&gt;s&lt;/strong&gt; is a periodic scale with quasiperiod &lt;strong&gt;P&lt;/strong&gt;, and if c is an interval &lt;strong&gt;s&lt;/strong&gt;[i+j] - &lt;strong&gt;s&lt;/strong&gt;[i] with 0≤i&amp;lt;i+j&amp;lt;&lt;strong&gt;P&lt;/strong&gt;, then we may define the specific interval set S(c, j) to be 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(&lt;strong&gt;s&lt;/strong&gt;[a], &lt;strong&gt;s&lt;/strong&gt;[b]), which we will abbreviate as d(a, b), to be &lt;strong&gt;P&lt;/strong&gt; - #S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[b]|, |a-b|). base points set base(&lt;strong&gt;s&lt;/strong&gt;[i]) to be the set of integers {j | &lt;strong&gt;s&lt;/strong&gt;[j+i] - &lt;strong&gt;s&lt;/strong&gt;[j] = &lt;strong&gt;s&lt;/strong&gt;[i], 0≤j&amp;lt;&lt;strong&gt;P&lt;/strong&gt;}. These have the property that the interval between the base note &lt;strong&gt;s&lt;/strong&gt;[j] and the note i steps away, &lt;strong&gt;s&lt;/strong&gt;[j+i], is in class(i), the interval class to which &lt;strong&gt;s&lt;/strong&gt;[i] belongs. If the cardinality of this set is n, there are n indicies which correspond to intervals of &lt;strong&gt;s&lt;/strong&gt;[i], and &lt;strong&gt;P&lt;/strong&gt;-n which correspond to indicies of intervals other than &lt;strong&gt;s&lt;/strong&gt;[i]. In other words, there are &lt;strong&gt;P&lt;/strong&gt;-n intervals, counting multiplicities, in the class of &lt;strong&gt;s&lt;/strong&gt;[i] other than &lt;strong&gt;s&lt;/strong&gt;[i]. Then the &lt;em&gt;structure complexity&lt;/em&gt; || i || of &lt;strong&gt;s&lt;/strong&gt;[i] is defined to be &lt;strong&gt;P&lt;/strong&gt;-n, and the structure metric is defined as d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) = || |i - j| ||.&lt;br /&gt;
+The &lt;em&gt;structure metric&lt;/em&gt; is a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_(mathematics)" rel="nofollow"&gt;distance function&lt;/a&gt; on the notes of a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; within a single period, which give to it the property of being a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Metric_space" rel="nofollow"&gt;finite metric space&lt;/a&gt;. If &lt;strong&gt;s&lt;/strong&gt; is a periodic scale with quasiperiod &lt;strong&gt;P&lt;/strong&gt;, and if c is an interval &lt;strong&gt;s&lt;/strong&gt;[i+j] - &lt;strong&gt;s&lt;/strong&gt;[i] with 0≤i&amp;lt;&lt;strong&gt;P&lt;/strong&gt;, then we may define the specific interval set S(c, j) to be {i|&lt;strong&gt;s&lt;/strong&gt;[i+j] - &lt;strong&gt;s&lt;/strong&gt;[i] = c} with 0≤i&amp;lt;&lt;strong&gt;P&lt;/strong&gt;, 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(&lt;strong&gt;s&lt;/strong&gt;[a], &lt;strong&gt;s&lt;/strong&gt;[b]), which we will abbreviate as d(a, b), to be &lt;strong&gt;P&lt;/strong&gt; - #S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[b]|, |a-b|). &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Properties&lt;/h1&gt;
 The structure metric has the following properties:&lt;br /&gt;
 &lt;br /&gt;
-. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[i]) = 0&lt;br /&gt;
+. d(a, a) = 0&lt;br /&gt;
-|| i - i || = ||0|| which equals 0.&lt;br /&gt;
+#S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[a]|, |a - b|) = #S(0, 0) = &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
-. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) ≥ 0&lt;br /&gt;
+. d(a, b) ≥ 0&lt;br /&gt;
-This is so since the cardinality n of the base point set is less than or equal to &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
+The cardinality of #S(c, j) cannot exceed &lt;strong&gt;*P&lt;/strong&gt;, since 0≤i&amp;lt;&lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
-. d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) = 0 implies &lt;strong&gt;s&lt;/strong&gt;[i] equals &lt;strong&gt;s&lt;/strong&gt;[j]&lt;br /&gt;
+. d(a, b) = 0 implies a equals b.&lt;br /&gt;
 Suppose ||&lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;I&lt;/strong&gt;]|| equals 0 with 0 &amp;lt; &lt;strong&gt;I&lt;/strong&gt; &amp;lt; &lt;strong&gt;P&lt;/strong&gt;. Then &lt;strong&gt;s&lt;/strong&gt;[j+&lt;strong&gt;I&lt;/strong&gt;] - &lt;strong&gt;s&lt;/strong&gt;[j] = &lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;I&lt;/strong&gt;], so that &lt;strong&gt;s&lt;/strong&gt; is periodic with quasiperiod &lt;strong&gt;I&lt;/strong&gt;. But by assumption, &lt;strong&gt;P&lt;/strong&gt; is the least quasiperiod of &lt;strong&gt;s&lt;/strong&gt;. Hence, ||&lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;I&lt;/strong&gt;]|| equals 0 implies &lt;strong&gt;I&lt;/strong&gt; equals 0. It follows that if d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) equals || |i - j| || equals 0, then i - j equals 0 and so &lt;strong&gt;s&lt;/strong&gt;[i] equals &lt;strong&gt;s&lt;/strong&gt;[j]. &lt;br /&gt;
 &lt;br /&gt;