Kees semi-height: Difference between revisions

Given a ratio of positive integers p/q, the ''Kees [[Height|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 <math>|m_2 \, m_3 \, m_5 \ldots m_p \rangle</math> 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]] 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|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|Kees van Prooijen]], who studied extensively its properties as a norm on the space of pitch classes.

{| class="wikitable" style="text-align:center;"  

! intervals

! kees height

! style="color:#333" | Deduction steps

| style="color:#333" | 7/4 &rarr;  7/1; max(7, 1) &rarr; 7

| style="color:#333" | max(7, 5) &rarr; 7

| style="color:#333" | 7/6 &rarr; 7/3; max(7, 3) &rarr; 7

| style="color:#333" | 8/7 &rarr; 1/7; max(1, 7) &rarr; 7

| style="color:#333" | max(3, 5) &rarr; 5

| style="color:#333" | 8/5 &rarr; 1/5; max(1, 5) &rarr; 5

| style="color:#333" | 5/4 &rarr; 5/1; max (5, 1) &rarr; 5

| style="color:#333" | 6/5 &rarr; 3/5; max(3, 5) &rarr; 5

| style="color:#333" | 4/3 &rarr; 1/3; max(1, 3) &rarr; 3

| style="color:#333" | 3/2 &rarr; 3/1; max(3, 1) &rarr; 3

| style="color:#333" | 2/1 &rarr; 1/1; max(1, 1) &rarr; 1

| style="color:#333" | max(9, 5) &rarr; 9

| style="color:#333" | 10/9 &rarr; 5/9; max(5, 9) &rarr; 9

| style="color:#333" | 15/14 &rarr; 15/7; max(15, 7) &rarr; 15

| style="color:#333" | 28/15 &rarr; 7/15; max(7, 15) &rarr; 15

| style="color:#333" | 25/26 &rarr; 25/13; max(25, 13) &rarr; 25

| style="color:#333" | max(27, 25) &rarr; 27

| style="color:#333" | 25/24 &rarr; 25/3; max(25, 3) &rarr; 25

[[Category:Height]]