Height
The height is a mathematical tool to measure the complexity of JI intervals.
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 (see Wikipedia: Height function). 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 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]\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)[/math]
Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.
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 q_{1} and q_{2} are considered equivalent if the following is true:
[math]\displaystyle 2^{-v_2 \left( {q_1} \right)} q_1 = 2^{-v_2 \left( {q_2} \right)} q_2[/math]
where v_{p}(q) is the p-adic valuation of q.
Or equivalently, if n has any integer solutions:
[math]\displaystyle q_1 = 2^n q_2[/math]
If the above condition is met, we may then establish the following equivalence relation:
[math]\displaystyle q_1 \equiv q_2[/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.
Height versus norm
Height functions are applied to ratios, whereas norms are measurements on interval lattices embedded in normed vector spaces. Some height functions are essentially norms, and they are numerically equal. For example, the Tenney height is also the Tenney norm.
However, not all height functions are norms, and not all norms are height functions. The Benedetti height is not a norm, since it does not satisfy the condition of absolute homogeneity. The taxicab distance is not a height, since there can be infinitely many intervals below a given bound.
Examples of height functions
Name | Type | H(n/d) | H(q) | H(q) simplified by equivalence relation |
---|---|---|---|---|
Benedetti height (or Tenney height) |
Height | [math]nd[/math] | [math]2^{\large{\|q\|_{T1}}}[/math] | [math]\|q\|_{T1}[/math] |
Wilson height | Height | [math]\text{sopfr}(n d)[/math] | [math]2^{\large{\text{sopfr}(q)}}[/math] | [math]\text{sopfr}(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 semi-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 semi-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 v_{p}(q) is the p-adic valuation of q.
The function sopfr (nd) is the "sum of prime factors with repetition" of n·d. Equivalently, this is the L_{1} norm on monzos, but where each prime is weighted by p rather than log (p). This is called "Wilson's Complexity" in John Chalmers's Divisions of the Tetrachord^{[1]}.
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.
History
The concept of height was introduced to xenharmonics by Gene Ward Smith in 2001^{[2]}; it comes from the mathematical field of number theory (for more information, see Wikipedia: Height function). It is not to be confused with the musical notion of pitch height (as opposed to pitch chroma)^{[3]}.
See also
References
- ↑ Division of the Tetrachord, page 55. John Chalmers.
- ↑ Yahoo! Tuning Group | Super Particular Stepsize
- ↑ Though it has also been used to refer to the size of an interval in cents. On page 23 of John Cage and the Theor of Harmony, Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it as opposed to his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".