Height: Difference between revisions
Wikispaces>genewardsmith **Imported revision 363463254 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 363887748 - 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:Sarzadoce|Sarzadoce]] and made on <tt>2012-09-11 16:46:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>363887748</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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[[math]] || [[math]] | [[math]] || [[math]] | ||
T1 \left( {q} \right) + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right) | T1 \left( {q} \right) + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right) | ||
[[math]] || | |||
|| Harmonic Height || Improper || [[math]] | |||
\dfrac {n d} {n + d} | |||
[[math]] || [[math]] | |||
\dfrac {\sqrt{q}} {\left( {q + 1} \right)} \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right)} {2} \right) | |||
[[math]] || [[math]] | |||
T1 \left( {q} \right) - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right) | |||
[[math]] || | [[math]] || | ||
|| [[Kees Height]] || Improper || [[math]] | || [[Kees Height]] || Improper || [[math]] | ||
\max \left( {2^{-v_2 \left( {n} \right)} n , | \max \left( {2^{-v_2 \left( {n} \right)} n , | ||
2^{-v_2 \left( {d} \right)} d} \right) | 2^{-v_2 \left( {d} \right)} d} \right) | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
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T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) | | T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) | | ||
[[math]] || | [[math]] || | ||
Where T1(q) is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q in monzo form, and vp(x) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of x. | Where T1(q) is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q in monzo form, and vp(x) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of x. | ||
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Height functions can also be put on the points of [[http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html|projective varieties]]. Since [[Abstract regular temperament|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> | Height functions can also be put on the points of [[http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html|projective varieties]]. Since [[Abstract regular temperament|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: | <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:22:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:22 -->Definition:</h1> | ||
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 /> | 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 /> | ||
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<br /> | <br /> | ||
By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example.<br /> | By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example.<br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h6&gt; --><h6 id="toc1"><!-- ws:end:WikiTextHeadingRule:24 --> </h6> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc2"><a name="Examples of Height Functions:"></a><!-- ws:end:WikiTextHeadingRule:26 -->Examples of Height Functions:</h1> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>Harmonic Height<br /> | ||
</td> | </td> | ||
<td>Improper<br /> | <td>Improper<br /> | ||
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<td><!-- ws:start:WikiTextMathRule:13: | <td><!-- ws:start:WikiTextMathRule:13: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\ | \dfrac {n d} {n + d}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\dfrac {n d} {n + d}</script><!-- ws:end:WikiTextMathRule:13 --><br /> | |||
--><script type="math/tex">\ | |||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:14: | <td><!-- ws:start:WikiTextMathRule:14: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\ | \dfrac {\sqrt{q}} {\left( {q + 1} \right)} \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right)} {2} \right)&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\ | --><script type="math/tex">\dfrac {\sqrt{q}} {\left( {q + 1} \right)} \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right)} {2} \right)</script><!-- ws:end:WikiTextMathRule:14 --><br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:15: | <td><!-- ws:start:WikiTextMathRule:15: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
T1 | T1 \left( {q} \right) - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)&lt;br/&gt;[[math]] | ||
--><script type="math/tex">T1 | --><script type="math/tex">T1 \left( {q} \right) - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)</script><!-- ws:end:WikiTextMathRule:15 --><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><br /> | <td><a class="wiki_link" href="/Kees%20Height">Kees Height</a><br /> | ||
</td> | </td> | ||
<td><br /> | <td>Improper<br /> | ||
</td> | </td> | ||
<td><br /> | <td><!-- ws:start:WikiTextMathRule:16: | ||
[[math]]&lt;br/&gt; | |||
\max \left( {2^{-v_2 \left( {n} \right)} n ,&lt;br /&gt; | |||
2^{-v_2 \left( {d} \right)} d} \right)&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\max \left( {2^{-v_2 \left( {n} \right)} n , | |||
2^{-v_2 \left( {d} \right)} d} \right)</script><!-- ws:end:WikiTextMathRule:16 --><br /> | |||
</td> | </td> | ||
<td><br /> | <td><!-- ws:start:WikiTextMathRule:17: | ||
[[math]]&lt;br/&gt; | |||
\exp \left( {\ln \left( {2} \right) \dfrac {T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |} {2}}} \right)&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\exp \left( {\ln \left( {2} \right) \dfrac {T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |} {2}}} \right)</script><!-- ws:end:WikiTextMathRule:17 --><br /> | |||
</td> | </td> | ||
<td><br /> | <td><!-- ws:start:WikiTextMathRule:18: | ||
[[math]]&lt;br/&gt; | |||
T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |&lt;br/&gt;[[math]] | |||
--><script type="math/tex">T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</script><!-- ws:end:WikiTextMathRule:18 --><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
Some useful identities:<br /> | Some useful identities:<br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:19: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
n = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) + \log_2 \left( {q} \right)} {2} \right)&lt;br/&gt;[[math]] | n = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) + \log_2 \left( {q} \right)} {2} \right)&lt;br/&gt;[[math]] | ||
--><script type="math/tex">n = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) + \log_2 \left( {q} \right)} {2} \right)</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">n = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) + \log_2 \left( {q} \right)} {2} \right)</script><!-- ws:end:WikiTextMathRule:19 --><br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:20: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)&lt;br/&gt;[[math]] | d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)&lt;br/&gt;[[math]] | ||
--><script type="math/tex">d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)</script><!-- ws:end:WikiTextMathRule: | --><script type="math/tex">d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)</script><!-- ws:end:WikiTextMathRule:20 --><br /> | ||
<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:21: | ||
[[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: | --><script type="math/tex">n d = 2^{T1 \left( {q} \right)}</script><!-- ws:end:WikiTextMathRule:21 --><br /> | ||
<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%20temperament">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> | 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%20temperament">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> | ||