Kees semi-height - Revision history

FloraC: Recategorize

2023-11-26T11:14:21Z

Recategorize

← Older revision		Revision as of 11:14, 26 November 2023
Line 102:		Line 102:

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

FloraC: Recategorize

2023-08-29T13:38:19Z

Recategorize

← Older revision		Revision as of 13:38, 29 August 2023
Line 102:		Line 102:

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:~~Height~~]]		[[Category:Interval complexity measure]]

FloraC: Math formatting

2023-08-28T15:39:54Z

Math formatting

← Older revision		Revision as of 15:39, 28 August 2023
Line 1:		Line 1:
	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.		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 <~~math~~>~~\|m_2 \, m_3 \, m_5 \ldots m_p \rangle~~</~~math~~> is a vector with weighted coordinates in interval space.		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 set of JI intervals with Kees semi-height less than or equal to an odd integer q comprises the [[Odd limit\|''q''-odd-limit]].

FloraC: Cleanup

2023-08-28T15:38:01Z

Cleanup

← Older revision		Revision as of 15:38, 28 August 2023
Line 1:		Line 1:
	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.		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 <math>\|m_2 \, m_3 \, m_5 \ldots m_p \rangle</math> is a vector with weighted coordinates in interval space.		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 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.		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]].		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\|~~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 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\|~~Kees van Prooijen]], who studied extensively its properties as a norm on the space of pitch classes.		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 ==		== Examples ==

	{\| class="wikitable~~" style="text~~-~~align:center;~~"		{\| class="wikitable center-all"
	\|-		\|-
	! ~~intervals~~		! Intervals
	! ~~kees height~~		! Kees Height
	! ~~style="color:#333" \|~~ Deduction ~~steps~~		! Deduction Steps
	\|-		\|-
	\| 7/4		\| 7/4
	\| 7		\| 7
	~~\| style="color:#333"~~ \| 7/4 → 7/1; max(7, 1) → 7		\| 7/4 → 7/1; max(7, 1) → 7
	\|-		\|-
	\| 7/5		\| 7/5
	\| 7		\| 7
	~~\| style="color:#333"~~ \| max(7, 5) → 7		\| max(7, 5) → 7
	\|-		\|-
	\| 7/6		\| 7/6
	\| 7		\| 7
	~~\| style="color:#333"~~ \| 7/6 → 7/3; max(7, 3) → 7		\| 7/6 → 7/3; max(7, 3) → 7
	\|-		\|-
	\| 8/7		\| 8/7
	\| 7		\| 7
	~~\| style="color:#333"~~ \| 8/7 → 1/7; max(1, 7) → 7		\| 8/7 → 1/7; max(1, 7) → 7
	\|-		\|-
	\| 5/3		\| 5/3
	\| 5		\| 5
	~~\| style="color:#333"~~ \| max(3, 5) → 5		\| max(3, 5) → 5
	\|-		\|-
	\| 8/5		\| 8/5
	\| 5		\| 5
	~~\| style="color:#333"~~ \| 8/5 → 1/5; max(1, 5) → 5		\| 8/5 → 1/5; max(1, 5) → 5
	\|-		\|-
	\| 5/4		\| 5/4
	\| 5		\| 5
	~~\| style="color:#333"~~ \| 5/4 → 5/1; max (5, 1) → 5		\| 5/4 → 5/1; max (5, 1) → 5
	\|-		\|-
	\| 6/5		\| 6/5
	\| 5		\| 5
	~~\| style="color:#333"~~ \| 6/5 → 3/5; max(3, 5) → 5		\| 6/5 → 3/5; max(3, 5) → 5
	\|-		\|-
	\| 4/3		\| 4/3
	\| 3		\| 3
	~~\| style="color:#333"~~ \| 4/3 → 1/3; max(1, 3) → 3		\| 4/3 → 1/3; max(1, 3) → 3
	\|-		\|-
	\| 3/2		\| 3/2
	\| 3		\| 3
	~~\| style="color:#333"~~ \| 3/2 → 3/1; max(3, 1) → 3		\| 3/2 → 3/1; max(3, 1) → 3
	\|-		\|-
	\| 2/1		\| 2/1
	\| 1		\| 1
	~~\| style="color:#333"~~ \| 2/1 → 1/1; max(1, 1) → 1		\| 2/1 → 1/1; max(1, 1) → 1
	\|-		\|-
	\| 9/5		\| 9/5
	\| 9		\| 9
	~~\| style="color:#333"~~ \| max(9, 5) → 9		\| max(9, 5) → 9
	\|-		\|-
	\| 10/9		\| 10/9
	\| 9		\| 9
	~~\| style="color:#333"~~ \| 10/9 → 5/9; max(5, 9) → 9		\| 10/9 → 5/9; max(5, 9) → 9
	\|-		\|-
	\| 15/14		\| 15/14
	\| 15		\| 15
	~~\| style="color:#333"~~ \| 15/14 → 15/7; max(15, 7) → 15		\| 15/14 → 15/7; max(15, 7) → 15
	\|-		\|-
	\| 28/15		\| 28/15
	\| 15		\| 15
	~~\| style="color:#333"~~ \| 28/15 → 7/15; max(7, 15) → 15		\| 28/15 → 7/15; max(7, 15) → 15
	\|-		\|-
	\| 25/26		\| 25/26
	\| 25		\| 25
	~~\| style="color:#333"~~ \| 25/26 → 25/13; max(25, 13) → 25		\| 25/26 → 25/13; max(25, 13) → 25
	\|-		\|-
	\| 27/25		\| 27/25
	\| 27		\| 27
	~~\| style="color:#333"~~ \| max(27, 25) → 27		\| max(27, 25) → 27
	\|-		\|-
	\| 25/24		\| 25/24
	\| 25		\| 25
	~~\| style="color:#333"~~ \| 25/24 → 25/3; max(25, 3) → 25		\| 25/24 → 25/3; max(25, 3) → 25
	\|}		\|}

Fredg999 category edits: Categories

2023-02-20T03:18:19Z

Fredg999 category edits: Removing from Category:Psychoacoustics using Cat-a-lot

2023-02-20T02:55:15Z

Removing from Category:Psychoacoustics using Cat-a-lot

← Older revision		Revision as of 02:55, 20 February 2023
Line 101:		Line 101:
	[[Category:Just intonation]]		[[Category:Just intonation]]
	[[Category:Measure]]		[[Category:Measure]]
	~~[[Category:Psychoacoustics]]~~
	[[Category:Sonance]]		[[Category:Sonance]]
	[[Category:Sonance-measure]]		[[Category:Sonance-measure]]
	[[Category:Theory]]		[[Category:Theory]]

Mike Battaglia: quick edit clarifying semiheight, remove ambiguity w/ kees expressibility and kees euclidean, and add some history

2022-06-03T17:51:44Z

quick edit clarifying semiheight, remove ambiguity w/ kees expressibility and kees euclidean, and add some history

← Older revision		Revision as of 17:51, 3 June 2022
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 [[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 ~~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<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 <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 height less than or equal to an odd integer q comprises the [[Odd_limit\|q odd limit]].		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 ~~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 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.

	== Examples ==		== Examples ==

Mike Battaglia: Mike Battaglia moved page Kees height to Kees semi-height: This is really a semi height function

2022-06-03T17:29:15Z

Mike Battaglia moved page Kees height to Kees semi-height: This is really a semi height function

← Older revision	Revision as of 17:29, 3 June 2022
(No difference)

Sintel: Change color of hard-to-read column

2021-12-18T20:10:08Z

Change color of hard-to-read column

← Older revision		Revision as of 20:10, 18 December 2021
Line 13:		Line 13:
	! intervals		! intervals
	! kees height		! kees height
	! style="color:#~~888~~" \| Deduction steps		! style="color:#333" \| Deduction steps
	\|-		\|-
	\| 7/4		\| 7/4
	\| 7		\| 7
	\| style="color:#~~aaa~~" \| 7/4 → 7/1; max(7, 1) → 7		\| style="color:#333" \| 7/4 → 7/1; max(7, 1) → 7
	\|-		\|-
	\| 7/5		\| 7/5
	\| 7		\| 7
	\| style="color:#~~aaa~~" \| max(7, 5) → 7		\| style="color:#333" \| max(7, 5) → 7
	\|-		\|-
	\| 7/6		\| 7/6
	\| 7		\| 7
	\| style="color:#~~aaa~~" \| 7/6 → 7/3; max(7, 3) → 7		\| style="color:#333" \| 7/6 → 7/3; max(7, 3) → 7
	\|-		\|-
	\| 8/7		\| 8/7
	\| 7		\| 7
	\| style="color:#~~aaa~~" \| 8/7 → 1/7; max(1, 7) → 7		\| style="color:#333" \| 8/7 → 1/7; max(1, 7) → 7
	\|-		\|-
	\| 5/3		\| 5/3
	\| 5		\| 5
	\| style="color:#~~aaa~~" \| max(3, 5) → 5		\| style="color:#333" \| max(3, 5) → 5
	\|-		\|-
	\| 8/5		\| 8/5
	\| 5		\| 5
	\| style="color:#~~aaa~~" \| 8/5 → 1/5; max(1, 5) → 5		\| style="color:#333" \| 8/5 → 1/5; max(1, 5) → 5
	\|-		\|-
	\| 5/4		\| 5/4
	\| 5		\| 5
	\| style="color:#~~aaa~~" \| 5/4 → 5/1; max (5, 1) → 5		\| style="color:#333" \| 5/4 → 5/1; max (5, 1) → 5
	\|-		\|-
	\| 6/5		\| 6/5
	\| 5		\| 5
	\| style="color:#~~aaa~~" \| 6/5 → 3/5; max(3, 5) → 5		\| style="color:#333" \| 6/5 → 3/5; max(3, 5) → 5
	\|-		\|-
	\| 4/3		\| 4/3
	\| 3		\| 3
	\| style="color:#~~aaa~~" \| 4/3 → 1/3; max(1, 3) → 3		\| style="color:#333" \| 4/3 → 1/3; max(1, 3) → 3
	\|-		\|-
	\| 3/2		\| 3/2
	\| 3		\| 3
	\| style="color:#~~aaa~~" \| 3/2 → 3/1; max(3, 1) → 3		\| style="color:#333" \| 3/2 → 3/1; max(3, 1) → 3
	\|-		\|-
	\| 2/1		\| 2/1
	\| 1		\| 1
	\| style="color:#~~aaa~~" \| 2/1 → 1/1; max(1, 1) → 1		\| style="color:#333" \| 2/1 → 1/1; max(1, 1) → 1
	\|-		\|-
	\| 9/5		\| 9/5
	\| 9		\| 9
	\| style="color:#~~aaa~~" \| max(9, 5) → 9		\| style="color:#333" \| max(9, 5) → 9
	\|-		\|-
	\| 10/9		\| 10/9
	\| 9		\| 9
	\| style="color:#~~aaa~~" \| 10/9 → 5/9; max(5, 9) → 9		\| style="color:#333" \| 10/9 → 5/9; max(5, 9) → 9
	\|-		\|-
	\| 15/14		\| 15/14
	\| 15		\| 15
	\| style="color:#~~aaa~~" \| 15/14 → 15/7; max(15, 7) → 15		\| style="color:#333" \| 15/14 → 15/7; max(15, 7) → 15
	\|-		\|-
	\| 28/15		\| 28/15
	\| 15		\| 15
	\| style="color:#~~aaa~~" \| 28/15 → 7/15; max(7, 15) → 15		\| style="color:#333" \| 28/15 → 7/15; max(7, 15) → 15
	\|-		\|-
	\| 25/26		\| 25/26
	\| 25		\| 25
	\| style="color:#~~aaa~~" \| 25/26 → 25/13; max(25, 13) → 25		\| style="color:#333" \| 25/26 → 25/13; max(25, 13) → 25
	\|-		\|-
	\| 27/25		\| 27/25
	\| 27		\| 27
	\| style="color:#~~aaa~~" \| max(27, 25) → 27		\| style="color:#333" \| max(27, 25) → 27
	\|-		\|-
	\| 25/24		\| 25/24
	\| 25		\| 25
	\| style="color:#~~aaa~~" \| 25/24 → 25/3; max(25, 3) → 25		\| style="color:#333" \| 25/24 → 25/3; max(25, 3) → 25
	\|}		\|}

Xenwolf: recat

2020-12-05T14:47:40Z

recat

← Older revision		Revision as of 14:47, 5 December 2020
Line 91:		Line 91:
	* [http://www.kees.cc/tuning/perbl.html Kees tuning pages]		* [http://www.kees.cc/tuning/perbl.html Kees tuning pages]

	[[Category:~~definition~~]]		[[Category:Terms]]
	[[Category:~~height~~]]		[[Category:Height]]
	[[Category:ji]]		[[Category:Just intonation]]
	[[Category:~~measure~~]]		[[Category:Measure]]
	[[Category:~~psychoacoustics~~]]		[[Category:Psychoacoustics]]
	[[Category:~~sonance~~]]		[[Category:Sonance]]
	[[Category:~~sonance~~-measure]]		[[Category:Sonance-measure]]
	[[Category:~~theory~~]]		[[Category:Theory]]

← Older revision		Revision as of 03:18, 20 February 2023
Line 99:		Line 99:
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Height]]		[[Category:Height]]
	~~[[Category:Just intonation]]~~
	~~[[Category:Measure]]~~
	~~[[Category:Sonance]]~~
	~~[[Category:Sonance-measure]]~~
	~~[[Category:Theory]]~~

← Older revision		Revision as of 17:51, 3 June 2022
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 [[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 ~~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<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 <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 height less than or equal to an odd integer q comprises the [[Odd_limit\|q odd limit]].		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 ~~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 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.

	== Examples ==		== Examples ==