Generalized Tenney norms and Tp interval space - Revision history

Sintel: -legacy

2025-03-29T18:00:15Z

-legacy

← Older revision		Revision as of 18:00, 29 March 2025
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	~~{{Legacy}}~~{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T<sub>''p''</sub> interval space}}		{{texmap}}{{DISPLAYTITLE:Generalized Tenney norms and T<sub>''p''</sub> interval space}}
	It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any {{w\|free abelian group}} of ([[subgroup]]) [[Monzos and interval space\|monzos]] by {{w\|embedding}} the group in a {{w\|normed vector space}}, so that the {{w\|Norm (mathematics)\|norm}} of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space\|interval space]], with the monzos forming the {{w\|integer lattice}} of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.		It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any {{w\|free abelian group}} of ([[subgroup]]) [[Monzos and interval space\|monzos]] by {{w\|embedding}} the group in a {{w\|normed vector space}}, so that the {{w\|Norm (mathematics)\|norm}} of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space\|interval space]], with the monzos forming the {{w\|integer lattice}} of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.

ArrowHead294: /* Generalized Tenney norms (Tp norms) */

2025-02-26T14:16:46Z

Generalized Tenney norms (Tp norms)

← Older revision		Revision as of 14:16, 26 February 2025
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	In this scheme the ordinary Tenney norm now becomes the '''T<sub>1</sub> norm''', and in general we call an interval space that has been given a T<sub>''p''</sub> norm '''T<sub>''p''</sub> interval space'''. We may sometimes notate this as T<sub>''p''</sub><sup>G</sup>, where ''G'' is the associated group the interval space is built around.		In this scheme the ordinary Tenney norm now becomes the '''T<sub>1</sub> norm''', and in general we call an interval space that has been given a T<sub>''p''</sub> norm '''T<sub>''p''</sub> interval space'''. We may sometimes notate this as T<sub>''p''</sub><sup>G</sup>, where ''G'' is the associated group the interval space is built around.

	Note that the {{nowrap\|‖ · ‖<sub>T''p''</sub>}} norm on the left side of the equation now has a subscript of T<sub>''p''</sub> rather than T<sub>1</sub>, and that the {{~~nowrap‖~~ · ‖<sub>''p''</sub>}} norm on the right side of the equation now has a subscript of ''p'' rather than 1. The generalized Tenney norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary ''L''<sub>''p''</sub> norm rather than restricting our consideration to the ''L''<sub>1</sub> norm.		Note that the {{nowrap\|‖ · ‖<sub>T''p''</sub>}} norm on the left side of the equation now has a subscript of T<sub>''p''</sub> rather than T<sub>1</sub>, and that the {{nowrap\|‖ · ‖<sub>''p''</sub>}} norm on the right side of the equation now has a subscript of ''p'' rather than 1. The generalized Tenney norm can thus be thought of as applying the same three-step process that the Tenney norm does, but where the last step can be an arbitrary ''L''<sub>''p''</sub> norm rather than restricting our consideration to the ''L''<sub>1</sub> norm.

	T<sub>''p''</sub> norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but for which an arbitrary interval ''n''/''d'' may no longer have a complexity of log<sub>2</sub>(''nd''). Furthermore, generalized T<sub>''p''</sub> norms may sometimes differ from the T<sub>1</sub> norm in their ranking of intervals by T<sub>''p''</sub> complexity, although the T<sub>''p''</sub> norm of any interval is always bounded by its T<sub>1</sub> norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a T<sub>''p''</sub> norm other than T<sub>1</sub> which are theoretically justified; additionally, certain T<sub>''p''</sub> norms are worth using as an approximation to T<sub>1</sub> for their strong computational advantages. As such, T<sub>''p''</sub> spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T<sub>1</sub> norm.		T<sub>''p''</sub> norms are thus generalizations of the Tenney norm which continue to weight the primes and prime powers as the Tenney norm does, but for which an arbitrary interval ''n''/''d'' may no longer have a complexity of log<sub>2</sub>(''nd''). Furthermore, generalized T<sub>''p''</sub> norms may sometimes differ from the T<sub>1</sub> norm in their ranking of intervals by T<sub>''p''</sub> complexity, although the T<sub>''p''</sub> norm of any interval is always bounded by its T<sub>1</sub> norm. However, despite these supposed theoretical disadvantages, there can be many indirect implications of using a T<sub>''p''</sub> norm other than T<sub>1</sub> which are theoretically justified; additionally, certain T<sub>''p''</sub> norms are worth using as an approximation to T<sub>1</sub> for their strong computational advantages. As such, T<sub>''p''</sub> spaces are musically justified enough to merit further study, despite the surface advantages of sticking exclusively to the T<sub>1</sub> norm.

ArrowHead294 at 13:38, 24 February 2025

2025-02-24T13:38:57Z

← Older revision		Revision as of 13:38, 24 February 2025
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	<math>\displaystyle		<math>\displaystyle
	~~\left[~~\begin{~~array}{rrrrrl~~}		\begin{bmatrix}
	~~\tmonzo{~~1 & 0 & 0 & 0} \\		1 & 0 & 0 & 0 \\
	~~\tmonzo{~~0 & 2 & 0 & -1} \\		0 & 2 & 0 & -1 \\
	~~\tmonzo{~~0 & -1 & 1 & 0} \\		0 & -1 & 1 & 0 \\
	\end{~~array~~}~~\right]~~</math>		\end{bmatrix}</math>

	Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]].		Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]].
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	<math>\displaystyle		<math>\displaystyle
	\lVert \vec m \rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert		\left\lVert \vec m \right\rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert
	\begin{bmatrix}		\begin{bmatrix}
	\log_2(2) & 0 & 0 & 0\\		\log_2(2) & 0 & 0 & 0\\
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	0 & 0 & \log_2(5) & 0\\		0 & 0 & \log_2(5) & 0\\
	0 & 0 & 0 & \log_2(7)		0 & 0 & 0 & \log_2(7)
	\end{bmatrix} \cdot ~~\left[~~ \begin{~~array}{rrrrrl~~}		\end{bmatrix} \cdot \begin{bmatrix}
	~~\tmonzo{~~1 & 0 & 0 & 0} \\		1 & 0 & 0 & 0 \\
	~~\tmonzo{~~0 & 2 & 0 & -1} \\		0 & 2 & 0 & -1 \\
	~~\tmonzo{~~0 & -1 & 1 & 0} \\		0 & -1 & 1 & 0 \\
	\end{~~array~~} ~~\right]~~ \cdot ~~\left[~~ \begin{~~array}{rrrrl~~}		\end{bmatrix} \cdot \begin{bmatrix}
	~~\tmonzo{~~0 & -2 & 1} \\		0 & -2 & 1 \\
	\end{~~array~~} ~~\right]~~		\end{bmatrix}
	\right \rVert_1		\right \rVert_1
	</math>		</math>
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	which finally resolves to		which finally resolves to

	<math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math>		<math>\displaystyle \left\lVert \vec m \right\rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math>

	Note that {{nowrap\|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector.		Note that {{nowrap\|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector.

ArrowHead294 at 13:33, 24 February 2025

2025-02-24T13:33:28Z

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Lériendil at 16:58, 17 February 2025

2025-02-17T16:58:42Z

← Older revision		Revision as of 16:58, 17 February 2025
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	{{Legacy~~\|Generalized_Tenney_norms_and_Tp_interval_space~~}}		{{Legacy}}
	It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any [[wikipedia: Free abelian group\|free abelian group]] of ([[subgroup]]) [[Monzos and interval space\|monzos]] by [[wikipedia: Embedding\|embedding]] the group in a [[wikipedia: Normed vector space\|normed vector space]], so that the [[wikipedia: Norm (mathematics)\|norm]] of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space\|interval space]], with the monzos forming the [[wikipedia: Integer lattice\|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.		It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any [[wikipedia: Free abelian group\|free abelian group]] of ([[subgroup]]) [[Monzos and interval space\|monzos]] by [[wikipedia: Embedding\|embedding]] the group in a [[wikipedia: Normed vector space\|normed vector space]], so that the [[wikipedia: Norm (mathematics)\|norm]] of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space\|interval space]], with the monzos forming the [[wikipedia: Integer lattice\|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.

Lériendil: deploying new cat

2025-02-17T16:14:28Z

deploying new cat

← Older revision		Revision as of 16:14, 17 February 2025
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			{{Legacy\|Generalized_Tenney_norms_and_Tp_interval_space}}
	It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any [[wikipedia: Free abelian group\|free abelian group]] of ([[subgroup]]) [[Monzos and interval space\|monzos]] by [[wikipedia: Embedding\|embedding]] the group in a [[wikipedia: Normed vector space\|normed vector space]], so that the [[wikipedia: Norm (mathematics)\|norm]] of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space\|interval space]], with the monzos forming the [[wikipedia: Integer lattice\|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.		It can be useful to define a notion of the "[[complexity]]" of an [[interval]], so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any [[wikipedia: Free abelian group\|free abelian group]] of ([[subgroup]]) [[Monzos and interval space\|monzos]] by [[wikipedia: Embedding\|embedding]] the group in a [[wikipedia: Normed vector space\|normed vector space]], so that the [[wikipedia: Norm (mathematics)\|norm]] of any interval is taken to be its complexity. The monzos form a ℤ-module, with coordinates given by integers, and the vector space embedding can be constructed by simply allowing real coordinates, hence defining the module over ℝ instead of ℤ and giving it the structure of a vector space. The resulting space is called [[Monzos and interval space\|interval space]], with the monzos forming the [[wikipedia: Integer lattice\|integer lattice]] of vectors with integer coordinates, but where we will allow any vector space norm on ℝ<sup>''n''</sup>.

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

2025-02-17T16:02:14Z

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

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FloraC: Recategorize

2023-11-27T11:53:20Z

Recategorize

← Older revision		Revision as of 11:53, 27 November 2023
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	[[Category:Math]]		[[Category:Math]]
			[[Category:Interval space]]
	[[Category:Interval complexity measures]]		[[Category:Interval complexity measures]]
	~~[[Category:Interval space]]~~		[[Category:Tenney-weighted measures]]
	[[Category:Tenney]]

FloraC: Recategorize

2023-11-26T11:13:35Z

Recategorize

← Older revision		Revision as of 11:13, 26 November 2023
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Interval complexity ~~measure~~]]		[[Category:Interval complexity measures]]
	[[Category:Interval space]]		[[Category:Interval space]]
	[[Category:Tenney]]		[[Category:Tenney]]

FloraC: Another pass for style and notation. Rid of the archaic term V-map "V" in favor of subgroup basis matrix "S" and full-limit JI group is now "J"

2023-09-01T14:16:07Z

Another pass for style and notation. Rid of the archaic term V-map "V" in favor of subgroup basis matrix "S" and full-limit JI group is now "J"

Show changes