Temperament mapping matrix - Revision history

FloraC: Review on the last edit

2025-11-07T16:10:47Z

Review on the last edit

← Older revision		Revision as of 16:10, 7 November 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	A '''temperament mapping matrix''' (or for short, '''mapping matrix''' or '''mapping''') is an {{w\|integer matrix}} representing an abstract [[regular temperament]]. Because any [[subgroup]] of [[just intonation]] is a {{w\|free abelian group}} with a given rank (dimensionality) that we will call ''r,'' such a temperament can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a group ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates. Similarly, there are many possible mapping matrices for a single given temperament.		A '''temperament mapping matrix''' (or for short, '''mapping matrix''' or '''mapping''') is an {{w\|integer matrix}} representing an abstract [[regular temperament]]. Because any [[subgroup]] of [[just intonation]] is a {{w\|free abelian group}} with a given rank (dimensionality) that we will call ''r'', such a temperament can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a group ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates. Similarly, there are many possible mapping matrices for a single given temperament.

	The elements of the group K are not just intervals, but [[~~Mapped~~ interval~~\|mapped intervals~~]] - equivalence classes of intervals separated by the [[comma]] the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 7, "2" represents the equivalence class containing the integers 2, 9, 16, etc. ~~This means that K is a {{W\|quotient group}}.~~		The group ''K'' is a {{w\|quotient group}}. This means that the elements of the group ''K'' are not just intervals, but [[mapped interval]]s – equivalence classes of intervals separated by the [[comma]]s the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 7, "2" represents the equivalence class containing the integers 2, 9, 16, etc.

	Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if ~~if and only if~~ left-multiplying by ''M'' maps all the commas ~~to zero~~ that the temperament tempers out and no others ~~(that is, the right nullspace of ''M'' consists of the kernel of ''T''~~), ''M'' is of full row rank (that is, all of its rows are linearly independent), and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]] ~~(see [[Diatonic, chromatic, enharmonic, and subchromatic steps]])~~. There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.		Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if the right nullspace of ''M'' consists of the kernel of ''T'' (that is, left-multiplying by ''M'' maps all the commas that the temperament tempers out and no others to zero), ''M'' is of full row rank (that is, all of its rows are linearly independent), and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]]. There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.

	The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.		The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.

	Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal ~~lists~~ #Normal ~~val lists~~\|normal val list]], or more generally if they have the same Hermite normal form.		Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal forms #Normal forms for mappings\|normal val list]], or more generally if they have the same Hermite normal form.

	Temperament mapping matrices are dual, in a certain sense, to [[subgroup basis matrices]], which can be thought of as co-tempering [[vals and tuning space\|vals]] in the same way that temperament mapping matrices temper monzos.		Temperament mapping matrices are dual, in a certain sense, to [[subgroup basis matrix\|subgroup basis matrices]], which can be thought of as co-tempering [[vals and tuning space\|vals]] in the same way that temperament mapping matrices temper monzos.

	== Dual transformation ==		== Dual transformation ==

VectorGraphics: attempted to make the intro a bit more approachable

2025-11-07T10:10:06Z

attempted to make the intro a bit more approachable

← Older revision		Revision as of 10:10, 7 November 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	~~The~~ {{w\|~~multiplicative group~~}} ~~generated by~~ any ~~finite set~~ of {{w\|~~rational number\|rational numbers~~}} ~~is an~~ ''r''~~-rank {{w\|free abelian group}}. Thus, an [[abstract regular~~ temperament]] can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a ~~{{w\|quotient~~ group}} ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. What ''quotient group'' means is that the elements of this group are not intervals, but equivalence classes of intervals separated by the [[comma]] the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 10, "2" represents the equivalence class containing the integers 2, 12, 22, etc. This homomorphism can also be represented by an {{w\|integer matrix}}, called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.		A '''temperament mapping matrix''' (or for short, '''mapping matrix''' or '''mapping''') is an {{w\|integer matrix}} representing an abstract [[regular temperament]]. Because any [[subgroup]] of [[just intonation]] is a {{w\|free abelian group}} with a given rank (dimensionality) that we will call ''r,'' such a temperament can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a group ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates. Similarly, there are many possible mapping matrices for a single given temperament.

	~~These~~ are ~~dual, in a certain sense~~, to [[~~subgroup basis matrices~~]]~~, which can be thought~~ of ~~as co-tempering~~ [[~~vals and tuning space~~\|~~vals~~]] in the ~~same way~~ that ~~temperament mapping matrices temper monzos~~.		The elements of the group K are not just intervals, but [[Mapped interval\|mapped intervals]] - equivalence classes of intervals separated by the [[comma]] the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 7, "2" represents the equivalence class containing the integers 2, 9, 16, etc. This means that K is a {{W\|quotient group}}.

	Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if the right nullspace of ''M'' consists of the kernel of ''T'', ''M'' is of full row rank, and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]]. There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.		Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if if and only if left-multiplying by ''M'' maps all the commas to zero that the temperament tempers out and no others (that is, the right nullspace of ''M'' consists of the kernel of ''T''), ''M'' is of full row rank (that is, all of its rows are linearly independent), and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]] (see [[Diatonic, chromatic, enharmonic, and subchromatic steps]]). There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.

	The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.		The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.

	Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal lists #Normal val lists\|normal val list]], or more generally if they have the same Hermite normal form.		Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal lists #Normal val lists\|normal val list]], or more generally if they have the same Hermite normal form.

			Temperament mapping matrices are dual, in a certain sense, to [[subgroup basis matrices]], which can be thought of as co-tempering [[vals and tuning space\|vals]] in the same way that temperament mapping matrices temper monzos.

	== Dual transformation ==		== Dual transformation ==

FloraC: Style

2025-06-14T06:41:43Z

Style

Show changes

Sintel: -legacy

2025-03-29T18:11:43Z

-legacy

← Older revision		Revision as of 18:11, 29 March 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	~~{{Legacy}}~~
	The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals - that is, a function from J to K such that the group operation of [[stacking]] is preserved. What "quotient group" means is that the elements of this group are not "intervals", but equivalence classes of intervals separated by the comma the temperament tempers out, much like how in modular arithmetic, in a modulus of 10, "2" represents the equivalence class containing the integers 2, 12, 22, etc. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)		The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals - that is, a function from J to K such that the group operation of [[stacking]] is preserved. What "quotient group" means is that the elements of this group are not "intervals", but equivalence classes of intervals separated by the comma the temperament tempers out, much like how in modular arithmetic, in a modulus of 10, "2" represents the equivalence class containing the integers 2, 12, 22, etc. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)

VectorGraphics at 04:45, 23 March 2025

2025-03-23T04:45:51Z

← Older revision		Revision as of 04:45, 23 March 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	{{Legacy}}		{{Legacy}}
	The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)		The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals - that is, a function from J to K such that the group operation of [[stacking]] is preserved. What "quotient group" means is that the elements of this group are not "intervals", but equivalence classes of intervals separated by the comma the temperament tempers out, much like how in modular arithmetic, in a modulus of 10, "2" represents the equivalence class containing the integers 2, 12, 22, etc. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)

	These are dual, in a certain sense, to [[subgroup basis matrices]], which can be thought of as "co-tempering" [[Vals and tuning space\|vals]] in the same way that temperament mapping matrices "temper" monzos.		These are dual, in a certain sense, to [[subgroup basis matrices]], which can be thought of as "co-tempering" [[Vals and tuning space\|vals]] in the same way that temperament mapping matrices "temper" monzos.

Lériendil at 16:53, 17 February 2025

2025-02-17T16:53:44Z

← Older revision		Revision as of 16:53, 17 February 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	{{Legacy~~\|Temperament_mapping_matrix~~}}		{{Legacy}}
	The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)		The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)

Lériendil: deploying new cat

2025-02-17T16:07:39Z

deploying new cat

← Older revision		Revision as of 16:07, 17 February 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
			{{Legacy\|Temperament_mapping_matrix}}
	The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)		The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)

Lériendil: Changed protection level for "Temperament mapping matrix" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

2025-02-17T16:00:27Z

Changed protection level for "Temperament mapping matrix" ([Edit=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:00, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:00, 17 February 2025
(No difference)

FloraC: Fix a mistake I made in a previous cleanup session (accidentally changed bras to kets)

2023-12-19T06:06:35Z

Fix a mistake I made in a previous cleanup session (accidentally changed bras to kets)

← Older revision		Revision as of 06:06, 19 December 2023
Line 21:		Line 21:
	<math>		<math>
	\left[ \begin{array}{rrrrrrl}		\left[ \begin{array}{rrrrrrl}
	[ & 15 & 24 & 35 & 42 & 52 & \~~rangle~~\\		\langle & 15 & 24 & 35 & 42 & 52 & ]\\
	[ & 22 & 35 & 51 & 62 & 76 & ~~\rangle~~		\langle & 22 & 35 & 51 & 62 & 76 & ]
	\end{array} \right]		\end{array} \right]
	</math>		</math>
Line 30:		Line 30:
	<math>		<math>
	\left[ \begin{array}{rrrrrrl}		\left[ \begin{array}{rrrrrrl}
	[ & 1 & 2 & 3 & 2 & 4 & \~~rangle~~\\		\langle & 1 & 2 & 3 & 2 & 4 & ]\\
	[ & 0 & -3 & -5 & 6 & -4 & ~~\rangle~~		\langle & 0 & -3 & -5 & 6 & -4 & ]
	\end{array} \right]		\end{array} \right]
	</math>		</math>

FloraC: Adopt template: Expert

2023-11-27T15:51:10Z

Adopt template: Expert

← Older revision		Revision as of 15:51, 27 November 2023
Line 1:		Line 1:
	~~:''This page gives a formal mathematical approach to [[RTT]] mapping. For a page with a simpler introduction, see [[mapping]].''~~		{{Expert\|Mapping}}

	The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)		The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)
Line 75:		Line 75:
	</math>		</math>

	for which the rows are the patent vals for [[7edo]] and [[15edo]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the {{val\| 7 1}} and {{val\| 15 2 }} tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix '''V∙P''' is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting {{mapping\| 1 2 3 2 4 \|0 -3 -5 6 -4 }} as a result again.		for which the rows are the patent vals for [[7edo]] and [[15edo]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the {{val\| 7 1 }} and {{val\| 15 2 }} tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix '''V∙P''' is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting {{mapping\| 1 2 3 2 4 \| 0 -3 -5 6 -4 }} as a result again.

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Mapping]]		[[Category:Mapping]]

← Older revision		Revision as of 16:10, 7 November 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	A '''temperament mapping matrix''' (or for short, '''mapping matrix''' or '''mapping''') is an {{w\|integer matrix}} representing an abstract [[regular temperament]]. Because any [[subgroup]] of [[just intonation]] is a {{w\|free abelian group}} with a given rank (dimensionality) that we will call ''r,'' such a temperament can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a group ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates. Similarly, there are many possible mapping matrices for a single given temperament.		A '''temperament mapping matrix''' (or for short, '''mapping matrix''' or '''mapping''') is an {{w\|integer matrix}} representing an abstract [[regular temperament]]. Because any [[subgroup]] of [[just intonation]] is a {{w\|free abelian group}} with a given rank (dimensionality) that we will call ''r'', such a temperament can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a group ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates. Similarly, there are many possible mapping matrices for a single given temperament.

	The elements of the group K are not just intervals, but [[~~Mapped~~ interval~~\|mapped intervals~~]] - equivalence classes of intervals separated by the [[comma]] the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 7, "2" represents the equivalence class containing the integers 2, 9, 16, etc. ~~This means that K is a {{W\|quotient group}}.~~		The group ''K'' is a {{w\|quotient group}}. This means that the elements of the group ''K'' are not just intervals, but [[mapped interval]]s – equivalence classes of intervals separated by the [[comma]]s the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 7, "2" represents the equivalence class containing the integers 2, 9, 16, etc.

	Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if ~~if and only if~~ left-multiplying by ''M'' maps all the commas ~~to zero~~ that the temperament tempers out and no others ~~(that is, the right nullspace of ''M'' consists of the kernel of ''T''~~), ''M'' is of full row rank (that is, all of its rows are linearly independent), and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]] ~~(see [[Diatonic, chromatic, enharmonic, and subchromatic steps]])~~. There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.		Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if the right nullspace of ''M'' consists of the kernel of ''T'' (that is, left-multiplying by ''M'' maps all the commas that the temperament tempers out and no others to zero), ''M'' is of full row rank (that is, all of its rows are linearly independent), and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]]. There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.

	The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.		The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.

	Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal ~~lists~~ #Normal ~~val lists~~\|normal val list]], or more generally if they have the same Hermite normal form.		Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal forms #Normal forms for mappings\|normal val list]], or more generally if they have the same Hermite normal form.

	Temperament mapping matrices are dual, in a certain sense, to [[subgroup basis matrices]], which can be thought of as co-tempering [[vals and tuning space\|vals]] in the same way that temperament mapping matrices temper monzos.		Temperament mapping matrices are dual, in a certain sense, to [[subgroup basis matrix\|subgroup basis matrices]], which can be thought of as co-tempering [[vals and tuning space\|vals]] in the same way that temperament mapping matrices temper monzos.

	== Dual transformation ==		== Dual transformation ==

← Older revision		Revision as of 10:10, 7 November 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
	~~The~~ {{w\|~~multiplicative group~~}} ~~generated by~~ any ~~finite set~~ of {{w\|~~rational number\|rational numbers~~}} ~~is an~~ ''r''~~-rank {{w\|free abelian group}}. Thus, an [[abstract regular~~ temperament]] can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a ~~{{w\|quotient~~ group}} ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. What ''quotient group'' means is that the elements of this group are not intervals, but equivalence classes of intervals separated by the [[comma]] the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 10, "2" represents the equivalence class containing the integers 2, 12, 22, etc. This homomorphism can also be represented by an {{w\|integer matrix}}, called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.		A '''temperament mapping matrix''' (or for short, '''mapping matrix''' or '''mapping''') is an {{w\|integer matrix}} representing an abstract [[regular temperament]]. Because any [[subgroup]] of [[just intonation]] is a {{w\|free abelian group}} with a given rank (dimensionality) that we will call ''r,'' such a temperament can be represented by a {{w\|group homomorphism}} '''T''': ''J'' → ''K'' from the group ''J'' of [[JI]] rationals to a group ''K'' of tempered intervals – that is, a function from ''J'' to ''K'' such that the group operation of [[stacking]] is preserved. Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates. Similarly, there are many possible mapping matrices for a single given temperament.

	~~These~~ are ~~dual, in a certain sense~~, to [[~~subgroup basis matrices~~]]~~, which can be thought~~ of ~~as co-tempering~~ [[~~vals and tuning space~~\|~~vals~~]] in the ~~same way~~ that ~~temperament mapping matrices temper monzos~~.		The elements of the group K are not just intervals, but [[Mapped interval\|mapped intervals]] - equivalence classes of intervals separated by the [[comma]] the temperament [[tempering out\|tempers out]], much like how in modular arithmetic, in a modulus of 7, "2" represents the equivalence class containing the integers 2, 9, 16, etc. This means that K is a {{W\|quotient group}}.

	Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if the right nullspace of ''M'' consists of the kernel of ''T'', ''M'' is of full row rank, and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]]. There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.		Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix ''M'' is said to be a mapping matrix for a temperament ''T'' if and only if if and only if left-multiplying by ''M'' maps all the commas to zero that the temperament tempers out and no others (that is, the right nullspace of ''M'' consists of the kernel of ''T''), ''M'' is of full row rank (that is, all of its rows are linearly independent), and the rows of ''M'' generate a subgroup of the dual group of vals which is [[mathematical theory of saturation\|saturated]] (see [[Diatonic, chromatic, enharmonic, and subchromatic steps]]). There is generally not a unique matrix ''M'' satisfying this definition for arbitrary temperament ''T'', as for any ''M'' which is a valid mapping for ''T'', any matrix ''U''⋅''M'' where ''U'' is unimodular will also be a valid mapping for ''T''. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.

	The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.		The integer column span of any mapping matrix is the quotient group of ''T''-tempered intervals, also known as the quotient group of [[tempered monzos and vals\|tempered monzos]] for ''T''. The integer row span of any mapping matrix for a temperament ''T'' is the subgroup of vals that all support ''T''. Note also that this means that if ''T'' is of rank ''r'', then any rank-''r'' matrix in which the rows span the subgroup of vals supporting ''T'' will be a valid mapping for ''T''.

	Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal lists #Normal val lists\|normal val list]], or more generally if they have the same Hermite normal form.		Note also that since all mapping matrices for ''T'' will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[normal lists #Normal val lists\|normal val list]], or more generally if they have the same Hermite normal form.

			Temperament mapping matrices are dual, in a certain sense, to [[subgroup basis matrices]], which can be thought of as co-tempering [[vals and tuning space\|vals]] in the same way that temperament mapping matrices temper monzos.

	== Dual transformation ==		== Dual transformation ==

← Older revision		Revision as of 16:07, 17 February 2025
Line 1:		Line 1:
	{{Expert\|Mapping}}		{{Expert\|Mapping}}
			{{Legacy\|Temperament_mapping_matrix}}
	The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)		The [[wikipedia: Multiplicative group\|multiplicative group]] generated by any finite set of [[wikipedia: Rational number\|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group\|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism\|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group\|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix\|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is also sometimes just called a '''mapping matrix''' or even just a '''mapping''' for the temperament in question. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)