Generalized Tenney dual norms and Tp tuning space - Revision history

Sintel: -legacy, todo intro

2025-03-29T17:59:57Z

-legacy, todo intro

← Older revision		Revision as of 17:59, 29 March 2025
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	{{~~Legacy~~}}		{{todo\|intro\|inline=1}}
	== Dual norms ==		== Dual norms ==
	Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:		Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:

Lériendil at 16:58, 17 February 2025

2025-02-17T16:58:57Z

← Older revision		Revision as of 16:58, 17 February 2025
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	{{Legacy~~\|Generalized_Tenney_dual_norms_and_Tp_tuning_space~~}}		{{Legacy}}
	== Dual norms ==		== Dual norms ==
	Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:		Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:

Lériendil: deploying new cat

2025-02-17T16:14:56Z

deploying new cat

← Older revision		Revision as of 16:14, 17 February 2025
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			{{Legacy\|Generalized_Tenney_dual_norms_and_Tp_tuning_space}}
	== Dual norms ==		== Dual norms ==
	Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:		Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:

Lériendil: Changed protection level for "Generalized Tenney dual norms and Tp tuning space" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

2025-02-17T16:02:19Z

Changed protection level for "Generalized Tenney dual norms and Tp tuning space" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:02, 17 February 2025
(No difference)

FloraC: Recategorize

2023-11-27T11:54:52Z

Recategorize

← Older revision		Revision as of 11:54, 27 November 2023
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Tuning space]]		[[Category:Tuning space]]
	[[Category:Tenney]]		[[Category:Temperament complexity measures]]
			[[Category:Tenney-weighted measures]]

	{{Todo\| cleanup }}		{{Todo\| cleanup }}

FloraC: Fix heading

2023-08-30T07:17:11Z

Fix heading

← Older revision		Revision as of 07:17, 30 August 2023
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	Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16\|) are rated less complex than vals which are further from the origin (such as <171 271 397\|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity \|\|'''t''' - '''JIP'''\|\|. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.		Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16\|) are rated less complex than vals which are further from the origin (such as <171 271 397\|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity \|\|'''t''' - '''JIP'''\|\|. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.

	==Prime ~~Power Interval Groups~~==		== Prime power interval groups ==
	In the simplest case where '''G''' has as its chosen basis only primes and prime powers, \|\| · \|\|'''<sub>Tp</sub>''' is given by		In the simplest case where '''G''' has as its chosen basis only primes and prime powers, \|\| · \|\|'''<sub>Tp</sub>''' is given by

FloraC: Cleanup and recategorize

2023-08-30T07:16:28Z

Cleanup and recategorize

← Older revision		Revision as of 07:16, 30 August 2023
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	=Dual ~~Norms~~=		== Dual norms ==
	Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:		Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space\|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity:

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	For '''G''' with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm. Likewise, T1 interval space is dual to Tinf* tuning space, and T2 interval space is dual to T2* tuning space.		For '''G''' with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm. Likewise, T1 interval space is dual to Tinf* tuning space, and T2 interval space is dual to T2* tuning space.

	==Arbitrary ~~Interval Groups~~==		== Arbitrary interval groups ==
	For an arbitrary group '''G''' with its chosen basis containing intervals other than primes and prime powers, \|\| · \|\|'''<sub>Tp</sub>''' is given by		For an arbitrary group '''G''' with its chosen basis containing intervals other than primes and prime powers, \|\| · \|\|'''<sub>Tp</sub>''' is given by

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	Note that this is the quotient norm induced on the space '''Tq<sup>L</sup>'''/ker('''V<sub>G</sub>'''), where ker('''V<sub>G</sub>''') is the set of vals (or tuning maps) that are restricted away given the V-map. This result is due to a corollary of the [http://www.math.unl.edu/~s-bbockel1/928/node25.html Hahn-Banach theorem], which demonstrates that the dual space M to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V/ker(M), where ker(M) is the set of all f in V such that f(M) = 0. Since our vector space '''Tp<sup>L</sup>''' and our subspace is '''Tp<sup>G</sup>''', this proves that our dual space '''Tq<sup>G</sup>'''* must be isometrically isomorphic to '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>''').		Note that this is the quotient norm induced on the space '''Tq<sup>L</sup>'''/ker('''V<sub>G</sub>'''), where ker('''V<sub>G</sub>''') is the set of vals (or tuning maps) that are restricted away given the V-map. This result is due to a corollary of the [http://www.math.unl.edu/~s-bbockel1/928/node25.html Hahn-Banach theorem], which demonstrates that the dual space M to any subspace M of a Banach space V must be isometrically isomorphic to the quotient space V/ker(M), where ker(M) is the set of all f in V such that f(M) = 0. Since our vector space '''Tp<sup>L</sup>''' and our subspace is '''Tp<sup>G</sup>''', this proves that our dual space '''Tq<sup>G</sup>'''* must be isometrically isomorphic to '''Tq<sup>L</sup>'''*/ker('''V<sub>G</sub>''').

	[[~~category~~:~~math~~]]		[[Category:Math]]
			[[Category:Tuning space]]
			[[Category:Tenney]]

			{{Todo\| cleanup }}

FloraC: FloraC moved page Generalized Tenney Dual Norms and Tp Tuning Space to Generalized Tenney dual norms and Tp tuning space: WP:NCCAPS

2023-02-13T09:38:41Z

FloraC moved page Generalized Tenney Dual Norms and Tp Tuning Space to Generalized Tenney dual norms and Tp tuning space: WP:NCCAPS

← Older revision	Revision as of 09:38, 13 February 2023
(No difference)

Mike Battaglia: Protected "Generalized Tenney Dual Norms and Tp Tuning Space" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:12:32Z

Protected "Generalized Tenney Dual Norms and Tp Tuning Space" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:12, 15 November 2021
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Cmloegcmluin: add link to tuning map

2021-11-09T00:00:35Z

add link to tuning map

← Older revision		Revision as of 00:00, 9 November 2021
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	<math>\|\|f\|\|_\mathbf{Tq*} = \text{sup}\left \{\frac{\|f(\vec{v})\|}{\|\|\vec{v}\|\|_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math>		<math>\|\|f\|\|_\mathbf{Tq*} = \text{sup}\left \{\frac{\|f(\vec{v})\|}{\|\|\vec{v}\|\|_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math>

	for all f in '''Tq<sup>G</sup>'''. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq Tuning Space'''. Other vectors in this space may be interpreted as tuning ~~maps~~ that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.		for all f in '''Tq<sup>G</sup>'''. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq Tuning Space'''. Other vectors in this space may be interpreted as [[tuning map]]s that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.

	Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16\|) are rated less complex than vals which are further from the origin (such as <171 271 397\|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity \|\|'''t''' - '''JIP'''\|\|. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.		Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16\|) are rated less complex than vals which are further from the origin (such as <171 271 397\|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity \|\|'''t''' - '''JIP'''\|\|. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.

← Older revision		Revision as of 00:00, 9 November 2021
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	<math>\|\|f\|\|_\mathbf{Tq*} = \text{sup}\left \{\frac{\|f(\vec{v})\|}{\|\|\vec{v}\|\|_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math>		<math>\|\|f\|\|_\mathbf{Tq*} = \text{sup}\left \{\frac{\|f(\vec{v})\|}{\|\|\vec{v}\|\|_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math>

	for all f in '''Tq<sup>G</sup>'''. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq Tuning Space'''. Other vectors in this space may be interpreted as tuning ~~maps~~ that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.		for all f in '''Tq<sup>G</sup>'''. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq Tuning Space'''. Other vectors in this space may be interpreted as [[tuning map]]s that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance.

	Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16\|) are rated less complex than vals which are further from the origin (such as <171 271 397\|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity \|\|'''t''' - '''JIP'''\|\|. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.		Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16\|) are rated less complex than vals which are further from the origin (such as <171 271 397\|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity \|\|'''t''' - '''JIP'''\|\|. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals.