Kees height
Given a ratio of positive integers p/q, the Kees height is found by first removing factors of two and all common factors from p/q, producing a ratio a/b of relatively prime odd positive integers. Then kees(p/q) = kees(a/b) = max(a, b). The Kees "expressibility" is then the logarithm base two of the Kees height.
Expressibility can be extended to all vectors in interval space, by means of the formula KE(|m2 m3 m5... mp>) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp> is a vector with weighted coordinates in interval space. It can also be thought of as the quotient norm of Weil height, mod 2/1. Additionally, it can be extended to tempered intervals using the quotient norm.
The set of JI intervals with Kees height less than or equal to an odd integer q comprises the q odd limit.
The point of Kees height is to serve as a metric/height on JI pitch classes corresponding to Benedetti height on pitches. The measure was proposed by Kees van Prooijen.
Examples
| intervals | kees height | Deduction steps |
|---|---|---|
| 7/4 | 7 | 7/4 → 7/1; max(7, 1) → 7 |
| 7/5 | 7 | max(7, 5) → 7 |
| 7/6 | 7 | 7/6 → 7/3; max(7, 3) → 7 |
| 8/7 | 7 | 8/7 → 1/7; max(1, 7) → 7 |
| 5/3 | 5 | max(3, 5) → 5 |
| 8/5 | 5 | 8/5 → 1/5; max(1, 5) → 5 |
| 5/4 | 5 | 5/4 → 5/1; max (5, 1) → 5 |
| 6/5 | 5 | 6/5 → 3/5; max(3, 5) → 5 |
| 4/3 | 3 | 4/3 → 1/3; max(1, 3) → 3 |
| 3/2 | 3 | 3/2 → 3/1; max(3, 1) → 3 |
| 2/1 | 1 | 2/1 → 1/1; max(1, 1) → 1 |
| 9/5 | 9 | max(9, 5) → 9 |
| 10/9 | 9 | 10/9 → 5/9; max(5, 9) → 9 |
| 15/14 | 15 | 15/14 → 15/7; max(15, 7) → 15 |
| 28/15 | 15 | 28/15 → 7/15; max(7, 15) → 15 |
| 25/26 | 25 | 25/26 → 25/13; max(25, 13) → 25 |
| 27/25 | 27 | max(27, 25) → 27 |
| 25/24 | 25 | 25/24 → 25/3; max(25, 3) → 25 |