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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.

Kees tuning pages

Examples

intervals
kees height
7/4, 7/5, 7/6, 8/7
7
5/3, 8/5, 5/4, 6/5
5
4/3, 3/2
3
2/1
1