Kees semi-height: Difference between revisions

(13 intermediate revisions by 6 users not shown)

Line 1:

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.

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 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.

Expressibility can be extended to all vectors in [[Monzos and interval space|interval space]], by means of the formula

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

<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>

~~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]]~~.

where "K1" denotes Kees expressibility and {{monzo| ''m''2 ''m''3 ''m''5 … ''m''p }} is a vector with weighted coordinates in interval space.

[~~http://www.kees.cc/tuning/perbl~~.~~html Kees tuning pages]~~

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]].

~~==Examples==~~

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.

{| class="wikitable"

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 → 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

|-

| ~~style="text-align:center;"~~ | ~~'''intervals'''~~

| 9/5

| ~~style="text-align:center~~;~~" | '''kees height'''~~

| 9

| max(9, 5) → 9

|-

| ~~style="text-align:center;"~~ | 7/~~4, 7~~/5, ~~7/6, 8/7~~

| 10/9

~~| style="text-align:center~~;~~" | 7~~

| 9

| 10/9 → 5/9; max(5, 9) → 9

|-

| ~~style="text-align:center;"~~ | 5/~~3, 8~~/~~5, 5/4~~, ~~6/5~~

| 15/14

~~| style="text-align:center~~;~~" | 5~~

| 15

| 15/14 → 15/7; max(15, 7) → 15

|-

| ~~style="text-align:center~~;~~" | 4~~/3, ~~3/2~~

| 28/15

~~| style="text-align:center~~;~~" | 3~~

| 15

| 28/15 → 7/15; max(7, 15) → 15

|-

| ~~style="text~~-~~align:center~~;" | 2/1

| 25/26

| ~~style="text-align:center~~;~~" | 1~~

| 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

|}

~~[[Category:definition]]~~

~~[[Category:height]]~~

== External links ==

[~~[Category~~:~~ji]~~]

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

~~[[Category:measure]]~~

~~[[Category:psychoacoustics]]~~

[[Category:Terms]]

~~[[Category:sonance]]~~

[[Category:Interval complexity measures]]

[[Category:~~sonance-measure~~]]

[[Category:~~theory~~]]

@@ Line 1: / Line 1: @@
-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.
+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 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>
+Expressibility can be extended to all vectors in [[Monzos and interval space|interval space]], by means of the formula
-The set of JI intervals with Kees height less than or equal to an odd integer q comprises the [[Odd_limit|q odd limit]].
+<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>
-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]].
+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.
-[http://www.kees.cc/tuning/perbl.html Kees tuning pages]
+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]].
-==Examples==
+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.
-{| class="wikitable"
+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
 |-
-| style="text-align:center;" | '''intervals'''
+| 9/5
-| style="text-align:center;" | '''kees height'''
+| 9
+| max(9, 5) &rarr; 9
 |-
-| style="text-align:center;" | 7/4, 7/5, 7/6, 8/7
+| 10/9
-| style="text-align:center;" | 7
+| 9
+| 10/9 &rarr; 5/9; max(5, 9) &rarr; 9
 |-
-| style="text-align:center;" | 5/3, 8/5, 5/4, 6/5
+| 15/14
-| style="text-align:center;" | 5
+| 15
+| 15/14 &rarr; 15/7; max(15, 7) &rarr; 15
 |-
-| style="text-align:center;" | 4/3, 3/2
+| 28/15
-| style="text-align:center;" | 3
+| 15
+| 28/15 &rarr; 7/15; max(7, 15) &rarr; 15
 |-
-| style="text-align:center;" | 2/1
+| 25/26
-| style="text-align:center;" | 1
+| 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
 |}
-[[Category:definition]]
-[[Category:height]]
+== External links ==
-[[Category:ji]]
+* [http://www.kees.cc/tuning/perbl.html Kees tuning pages]
-[[Category:measure]]
-[[Category:psychoacoustics]]
+[[Category:Terms]]
-[[Category:sonance]]
+[[Category:Interval complexity measures]]
-[[Category:sonance-measure]]
-[[Category:theory]]