Height: Difference between revisions
Wikispaces>Sarzadoce **Imported revision 363321120 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 363456640 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-09-10 12:14:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>363456640</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Definition:= | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Definition:= | ||
A **Height** is a function on members of an | A **Height** is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes. | ||
A height function H(q) on the positive rationals q should fulfill the following criteria: | A height function H(q) on the positive rationals q should fulfill the following criteria: | ||
# Given any constant C, there are finitely many elements q such that H(q) | # Given any constant C, there are finitely many elements q such that H(q) ≤ C. | ||
# | # H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q. | ||
# H(q) = H(1/q) | # H(q) = H(1/q) | ||
# H(q^n) | # H(q^n) ≥ H(q) for any non-negative integer n | ||
If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation: | If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation: | ||
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[[math]] | [[math]] | ||
An **Improper Height** is a function which does not obey criteria #1 above in the strictest sense, | An **Improper Height** is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true: | ||
[[math]] | [[math]] | ||
2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q | 2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q | ||
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[[math]] | [[math]] | ||
n d = 2^{T1 \left( {q} \right)} | n d = 2^{T1 \left( {q} \right)} | ||
[[math]]</pre></div> | [[math]] | ||
Height functions can also be put on the points of [[http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html|projective varieties]]. Since [[abstract regular temperaments]] can be identified with rational points on [[http://en.wikipedia.org/wiki/Grassmannian|Grassmann varieties]], complexity measures of regular temperaments are also height functions.</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Height</title></head><body><!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:19 -->Definition:</h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Height</title></head><body><!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:19 -->Definition:</h1> | ||
A <strong>Height</strong> is a function on members of an | A <strong>Height</strong> is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.<br /> | ||
<br /> | <br /> | ||
A height function H(q) on the positive rationals q should fulfill the following criteria:<br /> | A height function H(q) on the positive rationals q should fulfill the following criteria:<br /> | ||
<ol><li>Given any constant C, there are finitely many elements q such that H(q) | <ol><li>Given any constant C, there are finitely many elements q such that H(q) ≤ C.</li><li>H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.</li><li>H(q) = H(1/q)</li><li>H(q^n) ≥ H(q) for any non-negative integer n</li></ol><br /> | ||
If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:<br /> | If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:<br /> | ||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
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--><script type="math/tex">H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
An <strong>Improper Height</strong> is a function which does not obey criteria #1 above in the strictest sense, | An <strong>Improper Height</strong> is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:<br /> | ||
<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
n d = 2^{T1 \left( {q} \right)}&lt;br/&gt;[[math]] | n d = 2^{T1 \left( {q} \right)}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">n d = 2^{T1 \left( {q} \right)}</script><!-- ws:end:WikiTextMathRule:18 --></body></html></pre></div> | --><script type="math/tex">n d = 2^{T1 \left( {q} \right)}</script><!-- ws:end:WikiTextMathRule:18 --><br /> | ||
<br /> | |||
Height functions can also be put on the points of <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html" rel="nofollow">projective varieties</a>. Since <a class="wiki_link" href="/abstract%20regular%20temperaments">abstract regular temperaments</a> can be identified with rational points on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmann varieties</a>, complexity measures of regular temperaments are also height functions.</body></html></pre></div> | |||