Height: Difference between revisions

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:

<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>

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 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:

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:

 

 

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.

 

 

! H(n/d)

! H(q)

! H(q) simplified by equivalence relation

| <math>n d</math>

| <math>\text{sopf}(n d)</math>

| <math>2^{\large{\text{sopf}(n d)}}</math>

| <math>\text{sopf}(q)</math>

| Weil height

| Harmonic height

| [[Kees height]]

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.

The function <math>\text{sopf}(nd)</math> is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] of n*d. Equivalently, this is the L1 norm on monzos, but where each prime is weighted by "p" rather than "log(p)". This is called "Wilson Complexity" in [[John Chalmers]] "Division of the Tetrachord."

Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [http://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.

[[Category:Measure]]