View source for 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 [[Monzos and interval space|interval space]], by means of the formula

<math> \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 {{monzo| ''m''<sub>2</sub> ''m''<sub>3</sub> ''m''<sub>5</sub> … ''m''<sub>p</sub> }} 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 [[Odd limit|''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 [[Pitch class|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 [[wikipedia: Seminorm|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 ==

{| class="wikitable center-all"
|-
! Intervals
! Kees Height
! Deduction Steps
|-
| 7/4
| 7
| 7/4 &rarr;  7/1; max(7, 1) &rarr; 7
|-
| 7/5
| 7
| max(7, 5) &rarr; 7
|-
| 7/6
| 7
| 7/6 &rarr; 7/3; max(7, 3) &rarr; 7
|-
| 8/7
| 7
| 8/7 &rarr; 1/7; max(1, 7) &rarr; 7
|-
| 5/3
| 5
| max(3, 5) &rarr; 5
|-
| 8/5
| 5
| 8/5 &rarr; 1/5; max(1, 5) &rarr; 5
|-
| 5/4
| 5
| 5/4 &rarr; 5/1; max (5, 1) &rarr; 5
|-
| 6/5
| 5
| 6/5 &rarr; 3/5; max(3, 5) &rarr; 5
|-
| 4/3
| 3
| 4/3 &rarr; 1/3; max(1, 3) &rarr; 3
|-
| 3/2
| 3
| 3/2 &rarr; 3/1; max(3, 1) &rarr; 3
|-
| 2/1
| 1
| 2/1 &rarr; 1/1; max(1, 1) &rarr; 1
|-
| 9/5
| 9
| max(9, 5) &rarr; 9
|-
| 10/9
| 9
| 10/9 &rarr; 5/9; max(5, 9) &rarr; 9
|-
| 15/14
| 15
| 15/14 &rarr; 15/7; max(15, 7) &rarr; 15
|-
| 28/15
| 15
| 28/15 &rarr; 7/15; max(7, 15) &rarr; 15
|-
| 25/26
| 25
| 25/26 &rarr; 25/13; max(25, 13) &rarr; 25
|-
| 27/25
| 27
| max(27, 25) &rarr; 27
|-
| 25/24
| 25
| 25/24 &rarr; 25/3; max(25, 3) &rarr; 25
|}

== External links ==
* [http://www.kees.cc/tuning/perbl.html Kees tuning pages]

[[Category:Terms]]
[[Category:Interval complexity measures]]