Height
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Sarzadoce and made on 2012-09-06 17:21:35 UTC.
- The original revision id was 362646078.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
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 || [[math]]
nd
[[math]] || [[math]]
2^T^1^(^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]] ||
|| ?? || [[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]] ||
|| || || || ||
|| || || || ||
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.Original HTML content:
<html><head><title>Height</title></head><body><!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:10 -->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:12:<h1> --><h1 id="toc1"><a name="Examples:"></a><!-- ws:end:WikiTextHeadingRule:12 -->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>Benedetti Height<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^T^1^(^q^)<br/>[[math]]
--><script type="math/tex">2^T^1^(^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>??<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><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><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.</body></html>