Kees semi-height
Given a ratio of positive integers p/q, the Kees semi-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 semi-height.
Expressibility can be extended to all vectors in interval space, by means of the formula
[math]\displaystyle{ \lVert |m_2 \, m_3 \, m_5 \ldots m_p \rangle \rVert_{K1} = (|m_3 + m_5 + ... + m_p| + |m_3| + |m_5| + ... + |m_p|)/2 }[/math]
where "K1" denotes Kees expressibility and [m2 m3 m5 … mp⟩ is a vector with weighted coordinates in interval space.
The set of JI intervals with Kees semi-height less than or equal to an odd integer q comprises the q-odd-limit.
The Kees semi-height is only a semi-height function, rather than a true height function, because the set of all ratios with less than some Kees semi-height is infinite and unbounded. Thus it is only a seminorm (or a "semimetric," sometimes called "pseudometric") on the space of JI intervals. However, if one looks at it as a function bounding sets of octave-equivalent JI pitch classes, then there are only finitely many pitch classes with less than some specified Kees expressibility, making it sort of a height function on these "generalized rationals" which are octave equivalent.
In linear-algebraic terms, the Kees expressibility is a seminorm rather than a true norm; because the distance between two different intervals can be zero (if they are simply octave transpositions of one another). However, if one looks at the space of octave-equivalent intervals, which can be kind of thought of as "tempering" 2/1 as a "comma" and looking at the resulting equivalence classes, the Kees expressibility is a true norm on this space. The Kees expressibility 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 mod additional commas as a form of temperamental complexity.
The Kees semi-height is often used as a "default" measure of complexity for octave-equivalent pitch classes, similarly to the use of Benedetti height on pitches (although the Kees semi-height is not the same as "octave-equivalent Benedetti height", though it is related in a different way).
The use of max(a, b) as a complexity function, with or without octave equivalence, is very old; according to Paul Erlich, it may date back even to the Renaissance. In the 20th century the octave-equivalent version was used by Harry Partch, among others. The metric (and particularly the logarithmic version) has since become associated with Kees van Prooijen, who studied extensively its properties as a norm on the space of pitch classes.
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 |