Temperament mapping matrix: Difference between revisions

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~~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 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 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 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''.

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 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 ==

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 {{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 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 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 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''.
-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 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 ==