Benedetti height - Revision history

FloraC: Math display & misc. cleanup

2025-09-03T02:28:15Z

Math display & misc. cleanup

← Older revision		Revision as of 02:28, 3 September 2025
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	The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.		The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.

	The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm.		The [[logarithm base two]] of the Benedetti height is the Tenney height, or [[Tenney norm]].

	== Computation ==		== Computation ==
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	The Benedetti height of a ratio ''n''/''d'' is given by		The Benedetti height of a ratio ''n''/''d'' is given by

	~~<math>~~nd~~</math>~~		$$ nd $$

	=== Vector form ===		=== Vector form ===
	The Benedetti height of a [[Harmonic limit\|''p''-limit]] [[monzo]] '''m''' = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function\|prime-counting function]]) is given by		The Benedetti height of a [[Harmonic limit\|''p''-limit]] [[monzo]] '''m''' = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function\|prime-counting function]]) is given by

	~~<math>~~2^{\lVert H \vec m \rVert_1} \\		$$ 2^{\lVert H \vec m \rVert_1} = 2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|} $$
	= 2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|}~~</math>~~

	where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val\| 2 3 5 … ''p'' }},		where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val\| 2 3 5 … ''p'' }},

	~~<math>~~H = \operatorname {diag} (\log_2 (Q))~~</math>~~		$$ H = \operatorname {diag} (\log_2 (Q)) $$

	== Examples ==		== Examples ==
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	\|-		\|-
	\| [[1/1]]		\| [[1/1]]
	\| {{~~monzo~~\| 0 }}		\| {{Monzo\| 0 }}
	\| 1		\| 1
	\|-		\|-
	\| [[2/1]]		\| [[2/1]]
	\| {{~~monzo~~\| 1 }}		\| {{Monzo\| 1 }}
	\| 2		\| 2
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
	\| {{~~monzo~~\| -1 1 }}		\| {{Monzo\| -1 1 }}
	\| 6		\| 6
	\|-		\|-
	\| [[6/5]]		\| [[6/5]]
	\| {{~~monzo~~\| 1 1 -1 }}		\| {{Monzo\| 1 1 -1 }}
	\| 30		\| 30
	\|-		\|-
	\| [[9/7]]		\| [[9/7]]
	\| {{~~monzo~~\| 0 2 0 -1 }}		\| {{Monzo\| 0 2 0 -1 }}
	\| 63		\| 63
	\|-		\|-
	\| [[13/11]]		\| [[13/11]]
	\| {{~~monzo~~\| 0 0 0 0 -1 1 }}		\| {{Monzo\| 0 0 0 0 -1 1 }}
	\| 143		\| 143
	\|}		\|}

FloraC: Restylize and recategorize

2023-11-27T11:56:48Z

Restylize and recategorize

← Older revision		Revision as of 11:56, 27 November 2023
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	=== Vector form ===		=== Vector form ===
	The Benedetti height of a [[Harmonic limit\|''p''-limit]] [[monzo]] m = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function\|prime-counting function]]) is given by		The Benedetti height of a [[Harmonic limit\|''p''-limit]] [[monzo]] '''m''' = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function\|prime-counting function]]) is given by

	<math>2^{\lVert H \vec m \rVert_1} \\		<math>2^{\lVert H \vec m \rVert_1} \\
	= 2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|}</math>		= 2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|}</math>

	where H is the transformation matrix such that, for the prime basis Q = {{val\| 2 3 5 … ''p'' }},		where ''H'' is the transformation matrix such that, for the prime basis ''Q'' = {{val\| 2 3 5 … ''p'' }},

	<math>H = \operatorname {diag} (\log_2 (Q))</math>		<math>H = \operatorname {diag} (\log_2 (Q))</math>
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	[[Category:Interval complexity measures]]		[[Category:Interval complexity measures]]
	~~[[Category:Benedetti]]~~		[[Category:Tenney-weighted measures]]
	[[Category:Tenney]]

FloraC: Recategorize

2023-11-26T11:13:00Z

Recategorize

← Older revision		Revision as of 11:13, 26 November 2023
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	<references/>		<references/>

	[[Category:Interval complexity ~~measure~~]]		[[Category:Interval complexity measures]]
	[[Category:Benedetti]]		[[Category:Benedetti]]
	[[Category:Tenney]]		[[Category:Tenney]]

FloraC: Recategorize

2023-08-29T13:37:40Z

Recategorize

← Older revision		Revision as of 13:37, 29 August 2023
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	== See also ==		== See also ==
	* [[Kees height]]		* [[Kees semi-height]]
	* [[Wikipedia: Giambattista Benedetti]]		* [[Wikipedia: Giambattista Benedetti]]

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	<references/>		<references/>

	[[Category:~~Height~~]]		[[Category:Interval complexity measure]]
	[[Category:Benedetti]]		[[Category:Benedetti]]
	[[Category:Tenney]]		[[Category:Tenney]]

FloraC: Rework intro. Move the naming to the corresponding section

2023-08-28T13:49:55Z

Rework intro. Move the naming to the corresponding section

← Older revision		Revision as of 13:49, 28 August 2023
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	The '''Benedetti height''' of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.		The '''Benedetti height''' is a simple [[height]] function which measures the [[complexity]] of a [[JI]] [[interval]]. The Benedetti height of a positive rational number ''n''/''d'' reduced to lowest terms (no common factor between ''n'' and ''d'') is equal to ''nd'', the product of the numerator and denominator. In general mathematics it is known as ''product complexity''.

	The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm.		The [[logarithm base two]] of the Benedetti height is the [[Tenney height]], or Tenney norm.

	The name is based on the fact that the scientist, mathematician and music theorist [http://www.webcitation.org/6076Lm8r4 Giovanni Battista Benedetti] first proposed it as a [[measure of inharmonicity]]. It may be the first number-theoretic [[height]] function ever defined for any purpose.

	== Computation ==		== Computation ==
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	== History and terminology ==		== History and terminology ==
	Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956 Yahoo! Tuning Group \| ''Are "Tenney Height", "Benedetti Height", "Kees Height", etc actually height functions?'']</ref>		Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. The name is based on the fact that the scientist, mathematician and music theorist [http://www.webcitation.org/6076Lm8r4 Giovanni Battista Benedetti] first proposed it as a [[measure of inharmonicity]]. It may be the first number-theoretic height function ever defined for any purpose.

			Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956 Yahoo! Tuning Group \| ''Are "Tenney Height", "Benedetti Height", "Kees Height", etc actually height functions?'']</ref>

	== See also ==		== See also ==
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	<references/>		<references/>

			[[Category:Height]]
	[[Category:Benedetti]]		[[Category:Benedetti]]
	~~[[Category:Height]]~~
	[[Category:Tenney]]		[[Category:Tenney]]

FloraC: +computation from monzo; +monzo -tenney height from the example table

2023-08-28T13:37:41Z

+computation from monzo; +monzo -tenney height from the example table

← Older revision		Revision as of 13:37, 28 August 2023
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	The name is based on the fact that the scientist, mathematician and music theorist [http://www.webcitation.org/6076Lm8r4 Giovanni Battista Benedetti] first proposed it as a [[measure of inharmonicity]]. It may be the first number-theoretic [[height]] function ever defined for any purpose.		The name is based on the fact that the scientist, mathematician and music theorist [http://www.webcitation.org/6076Lm8r4 Giovanni Battista Benedetti] first proposed it as a [[measure of inharmonicity]]. It may be the first number-theoretic [[height]] function ever defined for any purpose.

			== Computation ==
			=== Ratio form ===
			The Benedetti height of a ratio ''n''/''d'' is given by

			<math>nd</math>

			=== Vector form ===
			The Benedetti height of a [[Harmonic limit\|''p''-limit]] [[monzo]] m = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }} (π being the [[Wikipedia: prime-counting function\|prime-counting function]]) is given by

			<math>2^{\lVert H \vec m \rVert_1} \\
			= 2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|}</math>

			where H is the transformation matrix such that, for the prime basis Q = {{val\| 2 3 5 … ''p'' }},

			<math>H = \operatorname {diag} (\log_2 (Q))</math>

	== Examples ==		== Examples ==
	{\| class="wikitable center-~~all~~"		{\| class="wikitable center-1 center-3"
	! ~~Interval~~		! Ratio
			! Monzo
	! Benedetti height		! Benedetti height
	~~! Tenney height~~
	\|-		\|-
	\| [[1/1]]		\| [[1/1]]
			\| {{monzo\| 0 }}
	\| 1		\| 1
	~~\| 0~~
	\|-		\|-
	\| [[2/1]]		\| [[2/1]]
			\| {{monzo\| 1 }}
	\| 2		\| 2
	~~\| 1~~
	\|-		\|-
	\| [[3/2]]		\| [[3/2]]
			\| {{monzo\| -1 1 }}
	\| 6		\| 6
	~~\| 2.585~~
	\|-		\|-
	\| [[6/5]]		\| [[6/5]]
			\| {{monzo\| 1 1 -1 }}
	\| 30		\| 30
	~~\| 4.907~~
	\|-		\|-
	\| [[9/7]]		\| [[9/7]]
			\| {{monzo\| 0 2 0 -1 }}
	\| 63		\| 63
	~~\| 5.977~~
	\|-		\|-
	\| [[13/11]]		\| [[13/11]]
			\| {{monzo\| 0 0 0 0 -1 1 }}
	\| 143		\| 143
	~~\| 7.160~~
	\|}		\|}

	== History ==		== History and terminology ==
	Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956 Yahoo! Tuning Group \| ''Are "Tenney Height", "Benedetti Height", "Kees Height", etc actually height functions?'']</ref>		Benedetti height was named by [[Gene Ward Smith]] sometime before 2011. Originally, both Benedetti height and Tenney height were called "Tenney height", and considered to be arithmetic and logarithmic variants of the same [[height]] function. Due to pushback from [[Paul Erlich]] (who ultimately preferred that "height" not be introduced to xenharmonics, and that the thing Gene called Tenney height should remain Tenney's "harmonic distance") the two were differentiated by eponym as well.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20956 Yahoo! Tuning Group \| ''Are "Tenney Height", "Benedetti Height", "Kees Height", etc actually height functions?'']</ref>

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

2023-03-06T21:52:06Z

Removing from Category:Definition using Cat-a-lot

← Older revision		Revision as of 21:52, 6 March 2023
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	[[Category:Benedetti]]		[[Category:Benedetti]]
	~~[[Category:Definition]]~~
	[[Category:Height]]		[[Category:Height]]
	[[Category:Tenney]]		[[Category:Tenney]]

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

2023-02-26T06:34:33Z

Removing from Category:Measure using Cat-a-lot

← Older revision		Revision as of 06:34, 26 February 2023
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	[[Category:Definition]]		[[Category:Definition]]
	[[Category:Height]]		[[Category:Height]]
	~~[[Category:Measure]]~~
	[[Category:Tenney]]		[[Category:Tenney]]

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

2023-02-20T04:04:10Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 04:04, 20 February 2023
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	[[Category:Measure]]		[[Category:Measure]]
	[[Category:Tenney]]		[[Category:Tenney]]
	~~[[Category:Theory]]~~

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
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	[[Category:Height]]		[[Category:Height]]
	[[Category:Measure]]		[[Category:Measure]]
	~~[[Category:Psychoacoustics]]~~
	[[Category:Tenney]]		[[Category:Tenney]]
	[[Category:Theory]]		[[Category:Theory]]