Kees semi-height: Difference between revisions
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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]]. | 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;" | |||
{| class="wikitable" | |||
|- | |- | ||
! | ! 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 | |||
|} | |} | ||
== External links == | |||
* [http://www.kees.cc/tuning/perbl.html Kees tuning pages] | |||
[[Category:definition]] | [[Category:definition]] | ||
[[Category:height]] | [[Category:height]] | ||
Revision as of 16:07, 18 May 2020
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 |
|---|---|
| 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 |