Height: Difference between revisions

=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.

 

<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) iff q = 1.</li><li>H(q) = H(1/q)</li><li>H(q^n) ≥ H(q) for any non-negative integer n</li></ol>

 

A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. 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>

 

<math>p = 2^n q</math>

 

<math>p \equiv q</math>

 

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=Examples of Height Functions:=

 

| | <u>Name:</u>

| | <u>Type:</u>

| | <u>H(n/d):</u>

| | <u>H(q):</u>

| | <u>H(q) simplified by equivalence relation:</u>

|-

| | [[Benedetti_height|Benedetti height]]

 

(or [[Tenney_Height|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|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}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(q) - v_2(q) \mid \right)\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 [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|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.

<math>n = 2^{\large{\frac{1}{2}(\|q\|_{T1} + \log_2(q))}}</math>

 

<math>d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}</math>

<math>n d = 2^{\|q\|_{T1}}</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.