Kees height

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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

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