Tenney norm - Revision history

FloraC: - Wikipedia link. There's an article for James Tenney

2025-09-03T02:17:45Z

- Wikipedia link. There's an article for James Tenney

← Older revision		Revision as of 02:17, 3 September 2025
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	~~{{Wikipedia\| James Tenney }}~~
	{{Texops}}		{{Texops}}
	The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm.		The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm.

FloraC: Fix math display & use centered format

2025-09-03T02:16:53Z

Fix math display & use centered format

← Older revision		Revision as of 02:16, 3 September 2025
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	The Tenney norm of a ratio ''n''/''d'' is given by		The Tenney norm of a ratio ''n''/''d'' is given by

	~~<math>~~\log_2 (nd)~~</math>~~		$$\log_2 (nd) $$

	=== Vector form ===		=== Vector form ===
	The Tenney norm of a [[harmonic limit\|''p''-limit]] [[monzo]] {{nowrap\|'''m''' {{=}} {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w\|prime-counting function}}) is given by		The Tenney norm of a [[harmonic limit\|''p''-limit]] [[monzo]] {{nowrap\|'''m''' {{=}} {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>π (''p'')</sub> }}}} (π being the {{w\|prime-counting function}}) is given by

	~~<math>~~		$$
	\norm{H \vec m}_1 \\		\begin{align}
	= \abs{m_1} + \abs{m_2} ~~\abs{~~\log_2 (3)} + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\		\norm{H \vec m}_1 &= \abs{m_1} + \abs{m_2} \log_2 (3) + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\
	= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right)		&= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right)
	~~</math>~~		\end{align}
			$$

	where ''H'' is the transformation matrix such that, for the prime basis {{nowrap\|''Q'' {{=}} {{val\| 2 3 5 … ''p'' }}}},		where ''H'' is the transformation matrix such that, for the prime basis {{nowrap\| ''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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	* [[Generalized Tenney norms and Tp interval space\|Generalized Tenney norms and T<sub>''p''</sub> interval space]]		* [[Generalized Tenney norms and Tp interval space\|Generalized Tenney norms and T<sub>''p''</sub> interval space]]

	== ~~Notes~~ ==		== References ==
	<references />		<references />

FloraC: Height -> norm across the article

2025-06-01T11:22:42Z

Height -> norm across the article

← Older revision		Revision as of 11:22, 1 June 2025
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	{{Wikipedia\| James Tenney }}		{{Wikipedia\| James Tenney }}
	{{~~texops~~}}		{{Texops}}
	The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney ~~height~~.		The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney norm.

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

	<math>\log_2 (nd)</math>		<math>\log_2 (nd)</math>

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

	<math>		<math>
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	{\| class="wikitable center-2"		{\| class="wikitable center-2"
	\|-		\|-
	! Interval ~~Name~~		! Interval name
	! Ratio (''n''/''d'')		! Ratio (''n''/''d'')
	! Monzo		! Monzo
	! Tenney ~~Height~~		! Tenney norm
	\|-		\|-
	\| Unison		\| Unison

ArrowHead294 at 18:29, 8 May 2025

2025-05-08T18:29:23Z

← Older revision		Revision as of 18:29, 8 May 2025
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	{{Wikipedia\| James Tenney }}		{{Wikipedia\| James Tenney }}
			{{texops}}
	The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.		The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.

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

	<math>\~~lVert~~ H \vec m ~~\rVert_1~~ \\		<math>
	= \~~vert~~ m_1 ~~\vert~~ + \~~vert~~ m_2 \~~vert~~ \log_2 (3) + \ldots + \~~vert~~ m_{\pi (p)} ~~\vert~~ \log_2 (p) \\		\norm{H \vec m}_1 \\
	= \log_2 (2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|})</math>		= \abs{m_1} + \abs{m_2} \abs{\log_2 (3)} + \ldots + \abs{m_{\pi (p)}} \log_2 (p) \\
			= \log_2\left(2^{\abs{m_1}} \cdot 3^{\abs{m_2}} \cdot \ldots \cdot p^{\abs{m_{\pi (p)}}}\right)
			</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 {{nowrap\|''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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	== Examples ==		== Examples ==
	{\| class="wikitable center-2"		{\| class="wikitable center-2"
			\|-
	! Interval Name		! Interval Name
	! Ratio (''n''/''d'')		! Ratio (''n''/''d'')
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	== History and terminology ==		== History and terminology ==
	In general mathematics, this measurement is known as ''log-product complexity''. With respect to microtonal tuning, this measurement was first described by [[James Tenney]], who himself called it ''harmonic distance''.<ref>[https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theory of Harmony'']. James Tenney. </ref><ref>[https://zh.booksc.eu/book/68954431/f87a1d ''On the Conception and Measure of Consonance'']. Alex Wand. </ref><ref>[https://scholar.sun.ac.za/bitstream/handle/10019.1/98644/brand_signal_2016.pdf?sequence=2&isAllowed=y ''A Signal-Based Model of Teleology in Tonal Music'']. Mark André Brand. p. 28. "Tenney's measure of ''harmonic distance'' (Hd) is thus singled out as perhaps his most 'crucial development', affording him the means towards 'compactness'. His is a Manhattan, rather than Euclidean metric, defined as Hd (''a''/''b'') = ''k''log(''ab''), with ''a''/''b'' the maximally reduced ratio representing the frequency difference, and ''k'' = 1 indicating measure in octaves."</ref> This terminology was also used in [[Paul Erlich]]'s paper [[A Middle Path]]<ref>Wherein Erlich writes: "This is why, in Tenney’s terminology, the taxicab distance an interval traverses in his lattice is the 'Harmonic Distance' of that interval."</ref>.		In general mathematics, this measurement is known as ''log-product complexity''. With respect to microtonal tuning, this measurement was first described by [[James Tenney]], who himself called it ''harmonic distance''.<ref>[https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theory of Harmony'']. James Tenney. </ref><ref>[https://zh.booksc.eu/book/68954431/f87a1d ''On the Conception and Measure of Consonance'']. Alex Wand. </ref><ref>[https://scholar.sun.ac.za/bitstream/handle/10019.1/98644/brand_signal_2016.pdf?sequence=2&isAllowed=y ''A Signal-Based Model of Teleology in Tonal Music'']. Mark André Brand. p. 28. "Tenney's measure of ''harmonic distance'' (Hd) is thus singled out as perhaps his most 'crucial development', affording him the means towards 'compactness'. His is a Manhattan, rather than Euclidean metric, defined as {{nowrap\|Hd(''a''/''b'') {{=}} ''k'' log(''ab'')}}, with ''a''/''b'' the maximally reduced ratio representing the frequency difference, and {{nowrap\|''k'' {{=}} 1}} indicating measure in octaves."</ref> This terminology was also used in [[Paul Erlich]]'s paper [[A Middle Path]]<ref>Wherein Erlich writes: "This is why, in Tenney’s terminology, the taxicab distance an interval traverses in his lattice is the 'Harmonic Distance' of that interval."</ref>.

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

	[[Category:Consonance and dissonance]]		[[Category:Consonance and dissonance]]

Sintel: intro: order names

2025-05-03T21:01:03Z

intro: order names

← Older revision		Revision as of 21:01, 3 May 2025
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	{{Wikipedia\| James Tenney }}		{{Wikipedia\| James Tenney }}

	The ~~'''Tenney height''',~~ '''Tenney norm''', or otherwise known as '''harmonic distance''' ('''HD'''), is ~~widely~~ used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.		The '''Tenney norm''', otherwise known as '''harmonic distance''' ('''HD''') or '''Tenney height''', is commonly used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.

	== Computation ==		== Computation ==

Sintel: Sintel moved page Tenney height to Tenney norm over redirect: per discussion

2025-05-03T21:00:11Z

Sintel moved page Tenney height to Tenney norm over redirect: per discussion

← Older revision	Revision as of 21:00, 3 May 2025
(No difference)

Sintel: /* intro */ remove HE mention

2025-05-03T20:59:45Z

intro: remove HE mention

← Older revision		Revision as of 20:59, 3 May 2025
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	{{Wikipedia\| James Tenney }}		{{Wikipedia\| James Tenney }}

	The '''Tenney height''', '''Tenney norm''', or otherwise known as '''harmonic distance''' ('''HD'''), is widely used as a [[measure of ~~inharmonicity]] and/or~~ [[complexity]] for [[interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.		The '''Tenney height''', '''Tenney norm''', or otherwise known as '''harmonic distance''' ('''HD'''), is widely used as a measure of [[complexity]] for [[just interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.

	== Computation ==		== Computation ==

FloraC: Restylize and recategorize

2023-11-27T11:52:07Z

Restylize and recategorize

← Older revision		Revision as of 11:52, 27 November 2023
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	=== Vector form ===		=== Vector form ===
	The Tenney 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 Tenney 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>\lVert H \vec m \rVert_1 \\		<math>\lVert H \vec m \rVert_1 \\
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	= \log_2 (2^{\|m_1\|} \cdot 3^{\|m_2\|} \cdot \ldots \cdot p^{\|m_{\pi (p)}\|})</math>		= \log_2 (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>

	== Examples ==		== Examples ==
	{\| class="wikitable"		{\| class="wikitable center-2"
	! Interval Name		! Interval Name
	! Ratio (''n''/''d'')		! Ratio (''n''/''d'')
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	== Notes ==		== Notes ==

	~~[[Category:Interval complexity measures]]~~
	[[Category:Consonance and dissonance]]		[[Category:Consonance and dissonance]]
	[[Category:Harmonic entropy]]		[[Category:Harmonic entropy]]
	[[Category:~~Tenney~~]]		[[Category:Interval complexity measures]]
	[[Category:~~Benedetti~~]]		[[Category:Tenney-weighted measures]]

FloraC: Recategorize

2023-11-26T11:14:52Z

Recategorize

← Older revision		Revision as of 11:14, 26 November 2023
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	== Notes ==		== Notes ==

	[[Category:Interval complexity ~~measure~~]]		[[Category:Interval complexity measures]]
	[[Category:Consonance and dissonance]]		[[Category:Consonance and dissonance]]
	[[Category:Harmonic entropy]]		[[Category:Harmonic entropy]]
	[[Category:Tenney]]		[[Category:Tenney]]
	[[Category:Benedetti]]		[[Category:Benedetti]]

FloraC: Rework the intro to include more terms for this same concept

2023-08-31T13:00:04Z

Rework the intro to include more terms for this same concept

← Older revision		Revision as of 13:00, 31 August 2023
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	{{Wikipedia\| James Tenney }}		{{Wikipedia\| James Tenney }}

	If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the [[Benedetti height]], leading to ~~'''~~Tenney height~~'''. In either form, it is widely used as a [[measure of inharmonicity]] and/or [[complexity]] for intervals. Alternative terms used include '''harmonic distance''' ('''HD''')~~.		The '''Tenney height''', '''Tenney norm''', or otherwise known as '''harmonic distance''' ('''HD'''), is widely used as a [[measure of inharmonicity]] and/or [[complexity]] for [[interval]]s. If ''n''/''d'' is a positive rational number reduced to its lowest terms, then the [[Benedetti height]] is the integer ''nd''. Often it is more convenient instead to take the logarithm, usually base 2 ([[log2]]), of the Benedetti height, leading to the Tenney height.

	== Computation ==		== Computation ==
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	== See also ==		== See also ==
	* [[Generalized Tenney norms and Tp interval space]]		* [[Generalized Tenney norms and Tp interval space\|Generalized Tenney norms and T<sub>''p''</sub> interval space]]

	== Notes ==		== Notes ==