View source for Kees semi-height

Given a ratio of positive integers p/q, the ''Kees [[Height|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 [[Monzos_and_Interval_Space|interval space]], by means of the formula KE(|m2 m3 m5... mp&gt;) = (|m3 + m5+ ... +mp| + |m3| + |m5| + ... + |mp|)/2, where "KE" denotes Kees expressibility and |m2 m3 m5 ... mp&gt; 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<span style="line-height: 1.5;"> be extended to tempered intervals using the quotient norm.</span>

The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd_limit|q odd limit]].

The point of Kees height is to serve as a metric/height on [[Pitch_class|JI pitch classes]] corresponding to [[Benedetti_height|Benedetti height]] on pitches. The measure was proposed by [[Kees_van_Prooijen|Kees van Prooijen]].

== Examples ==

{| class="wikitable" style="text-align:center;" 
|-
! intervals
! kees height
! style="color:#888" | Deduction steps
|-
| 7/4
| 7
| style="color:#aaa" | 7/4 &rarr;  7/1; max(7, 1) &rarr; 7
|-
| 7/5
| 7
| style="color:#aaa" | max(7, 5) &rarr; 7
|-
| 7/6
| 7
| style="color:#aaa" | 7/6 &rarr; 7/3; max(7, 3) &rarr; 7
|-
| 8/7
| 7
| style="color:#aaa" | 8/7 &rarr; 1/7; max(1, 7) &rarr; 7
|-
| 5/3
| 5
| style="color:#aaa" | max(3, 5) &rarr; 5
|-
| 8/5
| 5
| style="color:#aaa" | 8/5 &rarr; 1/5; max(1, 5) &rarr; 5
|-
| 5/4
| 5
| style="color:#aaa" | 5/4 &rarr; 5/1; max (5, 1) &rarr; 5
|-
| 6/5
| 5
| style="color:#aaa" | 6/5 &rarr; 3/5; max(3, 5) &rarr; 5
|-
| 4/3
| 3
| style="color:#aaa" | 4/3 &rarr; 1/3; max(1, 3) &rarr; 3
|-
| 3/2
| 3
| style="color:#aaa" | 3/2 &rarr; 3/1; max(3, 1) &rarr; 3
|-
| 2/1
| 1
| style="color:#aaa" | 2/1 &rarr; 1/1; max(1, 1) &rarr; 1
|-
| 9/5
| 9
| style="color:#aaa" | max(9, 5) &rarr; 9
|-
| 10/9
| 9
| style="color:#aaa" | 10/9 &rarr; 5/9; max(5, 9) &rarr; 9
|-
| 15/14
| 15
| style="color:#aaa" | 15/14 &rarr; 15/7; max(15, 7) &rarr; 15
|-
| 28/15
| 15
| style="color:#aaa" | 28/15 &rarr; 7/15; max(7, 15) &rarr; 15
|-
| 25/26
| 25
| style="color:#aaa" | 25/26 &rarr; 25/13; max(25, 13) &rarr; 25
|-
| 27/25
| 27
| style="color:#aaa" | max(27, 25) &rarr; 27
|-
| 25/24
| 25
| style="color:#aaa" | 25/24 &rarr; 25/3; max(25, 3) &rarr; 25
|}

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

[[Category:definition]]
[[Category:height]]
[[Category:ji]]
[[Category:measure]]
[[Category:psychoacoustics]]
[[Category:sonance]]
[[Category:sonance-measure]]
[[Category:theory]]