Height: Difference between revisions

: The original revision id was <tt>363463254</tt>.<br>

<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 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:

# 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.

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:

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:

2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q

Or equivalently, if n has any integer solutions:

[[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.

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|| __Name:__ || __Type:__ || __H(n/d):__ || __H(q):__ || __H(q) simplified by equivalence relation:__ ||

|| [[Benedetti Height|Benedetti height]]

(or [[Tenney Height]]) || Proper || [[math]]

[[math]] || [[math]]

2^{T1 \left( {q} \right)}

[[math]] || [[math]]

T1 \left( {q} \right)

[[math]] ||

|| Weil Height || Proper || [[math]]

\max \left( {n , d} \right)

[[math]] || [[math]]

\exp \left( {\ln \left( {2} \right) {\dfrac{T1 \left( {q} \right) + | \log_2 \left( {q} \right) |} {2}}} \right)

[[math]] || [[math]]

T1 \left( {q} \right) + | \log_2 \left( {q} \right) |

[[math]] ||

|| Arithmetic Height || Proper || [[math]]

n + d

[[math]] || [[math]]

\dfrac {\left( {q + 1} \right)} {\sqrt{q}}} \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]] ||

|| [[Kees Height]] || Improper || [[math]]

\max \left( {2^{-v_2 \left( {n} \right)} n ,  

2^{-v_2 \left( {d} \right)} d} \right)

[[math]] || [[math]]

\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)

[[math]] || [[math]]

T1 \left( {2^{-v_2 \left( {q} \right)} q} \right) + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |

[[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.

[[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]]

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Height&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;Definition:&lt;/h1&gt;

        &lt;td&gt;&lt;u&gt;Name:&lt;/u&gt;&lt;br /&gt;

n = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) + \log_2 \left( {q} \right)} {2} \right)&amp;lt;br/&amp;gt;[[math]]

&lt;!-- ws:start:WikiTextMathRule:17:

[[math]]&amp;lt;br/&amp;gt;

d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)&amp;lt;br/&amp;gt;[[math]]

--&gt;&lt;script type="math/tex"&gt;d = \exp \left( \ln \left( {2} \right) \dfrac {T1 \left( {q} \right) - \log_2 \left( {q} \right)} {2} \right)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:17 --&gt;&lt;br /&gt;

[[math]]&amp;lt;br/&amp;gt;