Height
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.
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.
- H(q) = H(1) iff q = 1.
- H(q) = H(1/q)
- H(q^n) ≥ H(q) for any non-negative integer n
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:
[math]2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q[/math]
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:
[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.
Examples of Height Functions:
Name: | Type: | H(n/d): | H(q): | H(q) simplified by equivalence relation: |
Benedetti height
(or Tenney Height) |
Height | [math]n d[/math] | [math]2^{\large{\|q\|_{T1}}}[/math] | [math]\|q\|_{T1}[/math] |
Wilson Height | Height | [math]\text{sopf}(n d)[/math] | [math]2^{\large{\text{sopf}(n d)}}[/math] | [math]\text{sopf}(q)[/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 | 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||T1 is the tenney norm of q in monzo form, and vp(x) is the p-adic valuation of x.
The function [math]\text{sopf}(nd)[/math] is the "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."
Some useful identities:
[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 projective varieties. Since abstract regular temperaments can be identified with rational points on Grassmann varieties, complexity measures of regular temperaments are also height functions.