Height
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- This revision was by author guest and made on 2012-09-06 21:05:16 UTC.
- The original revision id was 362695182.
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Original Wikitext content:
=Definition:=
A **height** is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the (non-zero) rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.
A height function H(q) on the rationals q should fulfill the following criteria:
# Given any constant C, there are finitely many elements q such that H(q) <= C.
# There is a unique constant K such that H(q) >= K, for all q.
# H(q) = H(1/q)
Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.
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(q) \equiv F(H(q))
[[math]]
=Examples:=
|| Name: || H(n/d) || H(q) || H(q) simplified by equivalence relation ||
|| [[Benedetti Height|Benedetti height]]
(or [[Tenney Height]]) || [[math]]
nd
[[math]] || [[math]]
2^{T1(q)}
[[math]] || [[math]]
T1(q)
[[math]] ||
|| Weil Height || [[math]]
max(n,d)
[[math]] || [[math]]
2^{(T1(q)+|log_2(q)|)/2}
[[math]] || [[math]]
T1(q)+|log_2(q)|
[[math]] ||
|| Arithmetic Height || [[math]]
n+d
[[math]] || [[math]]
2^{T1(q)/2} (q+1)/q^{1/2}
[[math]] || [[math]]
T1(q)+2log_2(q+1)-log_2(q)
[[math]] ||
|| [[Kees Height]] || [[math]]
max(2^{-v_2(n)}n,
2^{-v_2(d)}d)
[[math]] || [[math]]
2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}
[[math]] || [[math]]
T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|
[[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.
Some useful identities:
[[math]]
n=2^{(T1(q)\pm|log_2(q)|)/2}
[[math]]
[[math]]
d=2^{(T1(q)\mp|log_2(q)|)/2}
[[math]]
[[math]]
nd=2^{T1(q)}
[[math]]Original HTML content:
<html><head><title>Height</title></head><body><!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:16 -->Definition:</h1>
A <strong>height</strong> is a function on an abelian group which maps elements to real numbers, yielding a type of complexity measurement. Since the (non-zero) rationals form an abelian group under multiplication, we can assign each element a height, and hence a complexity. While there is no concensus on the restrictions of a height, we will attempt to create a definition which is practical for musical purposes.<br />
<br />
A height function H(q) on the 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>There is a unique constant K such that H(q) >= K, for all q.</li><li>H(q) = H(1/q)</li></ol><br />
Since any rational q can be rewritten as a fraction n/d, we may sub this into the above equation to get H(n/d) = H(d/n). This relation is extremely useful - it tells us that we can switch n and d without any consequences on the outcome of the height.<br />
<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]]<br/>
H(q) \equiv F(H(q))<br/>[[math]]
--><script type="math/tex">H(q) \equiv F(H(q))</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc1"><a name="Examples:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Examples:</h1>
<table class="wiki_table">
<tr>
<td>Name:<br />
</td>
<td>H(n/d)<br />
</td>
<td>H(q)<br />
</td>
<td>H(q) simplified by equivalence relation<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><!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
nd<br/>[[math]]
--><script type="math/tex">nd</script><!-- ws:end:WikiTextMathRule:1 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:2:
[[math]]<br/>
2^{T1(q)}<br/>[[math]]
--><script type="math/tex">2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:2 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:3:
[[math]]<br/>
T1(q)<br/>[[math]]
--><script type="math/tex">T1(q)</script><!-- ws:end:WikiTextMathRule:3 --><br />
</td>
</tr>
<tr>
<td>Weil Height<br />
</td>
<td><!-- ws:start:WikiTextMathRule:4:
[[math]]<br/>
max(n,d)<br/>[[math]]
--><script type="math/tex">max(n,d)</script><!-- ws:end:WikiTextMathRule:4 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:5:
[[math]]<br/>
2^{(T1(q)+|log_2(q)|)/2}<br/>[[math]]
--><script type="math/tex">2^{(T1(q)+|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:5 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:6:
[[math]]<br/>
T1(q)+|log_2(q)|<br/>[[math]]
--><script type="math/tex">T1(q)+|log_2(q)|</script><!-- ws:end:WikiTextMathRule:6 --><br />
</td>
</tr>
<tr>
<td>Arithmetic Height<br />
</td>
<td><!-- ws:start:WikiTextMathRule:7:
[[math]]<br/>
n+d<br/>[[math]]
--><script type="math/tex">n+d</script><!-- ws:end:WikiTextMathRule:7 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:8:
[[math]]<br/>
2^{T1(q)/2} (q+1)/q^{1/2}<br/>[[math]]
--><script type="math/tex">2^{T1(q)/2} (q+1)/q^{1/2}</script><!-- ws:end:WikiTextMathRule:8 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:9:
[[math]]<br/>
T1(q)+2log_2(q+1)-log_2(q)<br/>[[math]]
--><script type="math/tex">T1(q)+2log_2(q+1)-log_2(q)</script><!-- ws:end:WikiTextMathRule:9 --><br />
</td>
</tr>
<tr>
<td><a class="wiki_link" href="/Kees%20Height">Kees Height</a><br />
</td>
<td><!-- ws:start:WikiTextMathRule:10:
[[math]]<br/>
max(2^{-v_2(n)}n,<br />
2^{-v_2(d)}d)<br/>[[math]]
--><script type="math/tex">max(2^{-v_2(n)}n,
2^{-v_2(d)}d)</script><!-- ws:end:WikiTextMathRule:10 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:11:
[[math]]<br/>
2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}<br/>[[math]]
--><script type="math/tex">2^{(T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:11 --><br />
</td>
<td><!-- ws:start:WikiTextMathRule:12:
[[math]]<br/>
T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|<br/>[[math]]
--><script type="math/tex">T1(2^{-v_2(q)}q)+|log_2(q)-v_2(q)|</script><!-- ws:end:WikiTextMathRule:12 --><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
Where T1(q) 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 vp(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:13:
[[math]]<br/>
n=2^{(T1(q)\pm|log_2(q)|)/2}<br/>[[math]]
--><script type="math/tex">n=2^{(T1(q)\pm|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:13 --><br />
<!-- ws:start:WikiTextMathRule:14:
[[math]]<br/>
d=2^{(T1(q)\mp|log_2(q)|)/2}<br/>[[math]]
--><script type="math/tex">d=2^{(T1(q)\mp|log_2(q)|)/2}</script><!-- ws:end:WikiTextMathRule:14 --><br />
<!-- ws:start:WikiTextMathRule:15:
[[math]]<br/>
nd=2^{T1(q)}<br/>[[math]]
--><script type="math/tex">nd=2^{T1(q)}</script><!-- ws:end:WikiTextMathRule:15 --></body></html>