Height

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This revision was by author mbattaglia1 and made on 2013-08-31 06:49:09 UTC.
The original revision id was 447781960.
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Original Wikitext content:

=Definition:= 
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 consensus 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:
# 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^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:
[[math]]
H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)
[[math]]

A **semi-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]]
2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q
[[math]]

Or equivalently, if n has any integer solutions:
[[math]]
p = 2^n q
[[math]]

If the above condition is met, we may then establish the following equivalence relation:
[[math]]
p \equiv q
[[math]]

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.
====== ====== 
=Examples of Height Functions:= 
|| __Name:__ || __Type:__ || __H(n/d):__ || __H(q):__ || __H(q) simplified by equivalence relation:__ ||
|| [[Benedetti Height|Benedetti height]]
(or [[Tenney Height]]) || Height || [[math]]
n d
[[math]] || [[math]]
2^{\large{\|q\|_{T1}}}
[[math]] || [[math]]
\|q\|_{T1}
[[math]] ||
|| Weil Height || Height || [[math]]
\max \left( {n , d} \right)
[[math]] || [[math]]
2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}
[[math]] || [[math]]
\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid
[[math]] ||
|| Arithmetic Height || Height || [[math]]
n + d
[[math]] || [[math]]
\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}
[[math]] || [[math]]
\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)
[[math]] ||
|| Harmonic Height || Semi-Height || [[math]]
\dfrac {n d} {n + d}
[[math]] || [[math]]
\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}
[[math]] || [[math]]
\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)
[[math]] ||
|| [[Kees Height]] || Semi-Height || [[math]]
\max \left( {2^{-v_2 \left( {n} \right)} n ,
2^{-v_2 \left( {d} \right)} d} \right)
[[math]] || [[math]]
2^{\large{\left(\frac{1}{2}\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)}}
[[math]] || [[math]]
\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |
[[math]] ||
Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the [[xenharmonic/Generalized Tenney Norms and Tp Interval Space#The%20Tenney%20Norm%20(T1%20norm)|tenney norm]] of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of x.

Some useful identities:
[[math]]
n = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) + \log_2 \left( {q} \right)} {2} \right)
[[math]]
[[math]]
d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)
[[math]]
[[math]]
n d = 2^{T1 \left( {q} \right)}
[[math]]

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.

Original HTML content:

<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 consensus 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 />
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) ≤ 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 />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)&lt;br/&gt;[[math]]
 --><script type="math/tex">H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
A <strong>semi-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:
[[math]]&lt;br/&gt;
2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
Or equivalently, if n has any integer solutions:<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
p = 2^n q&lt;br/&gt;[[math]]
 --><script type="math/tex">p = 2^n q</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
If the above condition is met, we may then establish the following equivalence relation:<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]&lt;br/&gt;
p \equiv q&lt;br/&gt;[[math]]
 --><script type="math/tex">p \equiv q</script><!-- ws:end:WikiTextMathRule:3 --><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 />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h6&gt; --><h6 id="toc1"><!-- ws:end:WikiTextHeadingRule:24 --> </h6>
 <!-- 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>
 

<table class="wiki_table">
    <tr>
        <td><u>Name:</u><br />
</td>
        <td><u>Type:</u><br />
</td>
        <td><u>H(n/d):</u><br />
</td>
        <td><u>H(q):</u><br />
</td>
        <td><u>H(q) simplified by equivalence relation:</u><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/Benedetti%20Height">Benedetti height</a><br />
(or <a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)<br />
</td>
        <td>Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:4:
[[math]]&lt;br/&gt;
n d&lt;br/&gt;[[math]]
 --><script type="math/tex">n d</script><!-- ws:end:WikiTextMathRule:4 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:5:
[[math]]&lt;br/&gt;
2^{\large{\|q\|_{T1}}}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{\large{\|q\|_{T1}}}</script><!-- ws:end:WikiTextMathRule:5 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:6:
[[math]]&lt;br/&gt;
\|q\|_{T1}&lt;br/&gt;[[math]]
 --><script type="math/tex">\|q\|_{T1}</script><!-- ws:end:WikiTextMathRule:6 --><br />
</td>
    </tr>
    <tr>
        <td>Weil Height<br />
</td>
        <td>Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:7:
[[math]]&lt;br/&gt;
\max \left( {n , d} \right)&lt;br/&gt;[[math]]
 --><script type="math/tex">\max \left( {n , d} \right)</script><!-- ws:end:WikiTextMathRule:7 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:8:
[[math]]&lt;br/&gt;
2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</script><!-- ws:end:WikiTextMathRule:8 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:9:
[[math]]&lt;br/&gt;
\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid&lt;br/&gt;[[math]]
 --><script type="math/tex">\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</script><!-- ws:end:WikiTextMathRule:9 --><br />
</td>
    </tr>
    <tr>
        <td>Arithmetic Height<br />
</td>
        <td>Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:10:
[[math]]&lt;br/&gt;
n + d&lt;br/&gt;[[math]]
 --><script type="math/tex">n + d</script><!-- ws:end:WikiTextMathRule:10 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:11:
[[math]]&lt;br/&gt;
\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}&lt;br/&gt;[[math]]
 --><script type="math/tex">\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</script><!-- ws:end:WikiTextMathRule:11 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:12:
[[math]]&lt;br/&gt;
\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)&lt;br/&gt;[[math]]
 --><script type="math/tex">\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)</script><!-- ws:end:WikiTextMathRule:12 --><br />
</td>
    </tr>
    <tr>
        <td>Harmonic Height<br />
</td>
        <td>Semi-Height<br />
</td>
        <td><!-- ws:start:WikiTextMathRule:13:
[[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 />
</td>
        <td><!-- ws:start:WikiTextMathRule:14:
[[math]]&lt;br/&gt;
\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}&lt;br/&gt;[[math]]
 --><script type="math/tex">\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</script><!-- ws:end:WikiTextMathRule:14 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:15:
[[math]]&lt;br/&gt;
\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)&lt;br/&gt;[[math]]
 --><script type="math/tex">\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)</script><!-- ws:end:WikiTextMathRule:15 --><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/Kees%20Height">Kees Height</a><br />
</td>
        <td>Semi-Height<br />
</td>
        <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><!-- ws:start:WikiTextMathRule:17:
[[math]]&lt;br/&gt;
2^{\large{\left(\frac{1}{2}\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)}}&lt;br/&gt;[[math]]
 --><script type="math/tex">2^{\large{\left(\frac{1}{2}\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)}}</script><!-- ws:end:WikiTextMathRule:17 --><br />
</td>
        <td><!-- ws:start:WikiTextMathRule:18:
[[math]]&lt;br/&gt;
\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |&lt;br/&gt;[[math]]
 --><script type="math/tex">\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</script><!-- ws:end:WikiTextMathRule:18 --><br />
</td>
    </tr>
</table>

Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space#The%20Tenney%20Norm%20(T1%20norm)">tenney norm</a> of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow">p-adic valuation</a> of x.<br />
<br />
Some useful identities:<br />
<!-- ws:start:WikiTextMathRule:19:
[[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]]
 --><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:20:
[[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]]
 --><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:21:
[[math]]&lt;br/&gt;
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:21 --><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>
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