Subgroup basis matrix - Revision history

Sintel: Fix the absolutely ridiculous notation

2025-06-14T15:00:40Z

Fix the absolutely ridiculous notation

← Older revision		Revision as of 15:00, 14 June 2025
Line 30:		Line 30:
	Say that our JI parent group ''J'' is in the 7-limit, and we want to look at temperaments on the 2.9/7.5/3 subgroup. We can create the subgroup mapping matrix ''S'' by forming a matrix in which the columns are the monzo representation of these intervals:		Say that our JI parent group ''J'' is in the 7-limit, and we want to look at temperaments on the 2.9/7.5/3 subgroup. We can create the subgroup mapping matrix ''S'' by forming a matrix in which the columns are the monzo representation of these intervals:

	<math>\displaystyle		:<math>\displaystyle
	~~\newcommand{dangle}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ \rangle}}~~		S =
	~~\newcommand{tbracket}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ [}}~~		\begin{bmatrix}
	S =		1 & 0 & 0 \\
	~~\left[~~ \begin{~~array}{rrr~~}		0 & 2 & -1 \\
	~~\tbracket & \tbracket & \tbracket\\[-20pt]~~		0 & 0 & 1 \\
	1 & 0 & 0 \\		0 & -1 & 0 \\
	0 & 2 & -1 \\		\end{bmatrix}
	0 & 0 & 1 \\
	0 & -1 & 0 \\~~[-20pt]~~
	~~\dangle & \dangle & \dangle\\[-20pt]~~
	\end{~~array~~} ~~\right]~~
	</math>		</math>

Line 50:		Line 46:
	The dual transformation can be found by the multiplication ''M'' = ''SM''<sub>''G''</sub>, which yields		The dual transformation can be found by the multiplication ''M'' = ''SM''<sub>''G''</sub>, which yields

	<math>\displaystyle		:<math>\displaystyle
	M =		M =
	~~\left[~~ \begin{~~array~~}~~{rrr}~~		\begin{bmatrix}
	~~\tbracket & \tbracket \\[-20pt]~~		0 & 0 \\
	0 & 0 \\		2 & -5 \\
	2 & -5 \\		0 & 1 \\
	0 & 1 \\		-1 & 2 \\
	-1 & 2 \\~~[-20pt]~~		\end{bmatrix}
	~~\dangle & \dangle\\[-20pt]~~
	\end{~~array~~} ~~\right]~~
	</math>		</math>

	~~These monzos are~~ the 7-limit representation of 9/7 and 245/243, respectively, in 2.3.5.7 coordinates.		The columns form the 7-limit representation of 9/7 and 245/243, respectively, in 2.3.5.7 coordinates.

	=== Dual transformation: subgroup restriction ===		=== Dual transformation: subgroup restriction ===
Line 68:		Line 62:
	To restrict a val to the subgroup defined by the subgroup basis matrix, we will left-multiply ''S'' by a val ''V''. In this case, our val ''V'' will be the 7-limit [[patent val]] for [[12edo]]:		To restrict a val to the subgroup defined by the subgroup basis matrix, we will left-multiply ''S'' by a val ''V''. In this case, our val ''V'' will be the 7-limit [[patent val]] for [[12edo]]:

	<math>\displaystyle		:<math>\displaystyle
	V =		V =
	~~\left[~~ \begin{~~array}{rrrrrl~~}		\begin{bmatrix}
	~~\langle~~ 12 & 19 & 28 & 34 ]		12 & 19 & 28 & 34
	\end{~~array~~} ~~\right]~~		\end{bmatrix}
	</math>		</math>

	The multiplication ''V''<sub>''G''</sub> = ''VS'' yields the result		The multiplication ''V''<sub>''G''</sub> = ''VS'' yields the result

	<math>\displaystyle		:<math>\displaystyle
	V_G =		V_G =
	~~\left[~~ \begin{~~array}{rrrrl~~}		\begin{bmatrix}
	~~\langle~~ 12 & 4 & 9 ]		12 & 4 & 9
	\end{~~array~~} ~~\right]~~		\end{bmatrix}
	</math>		</math>

	which tells us that the restriction of the 12edo patent val to the 2.9/7.5/3 subgroup is the subgroup val ''V''<sub>''G''</sub> = {{val\| 12 4 9 }}, with a mapping of 12 steps for 2/1, a mapping of 4 steps for 9/7, and a mapping of 9 steps for 5/3.		which tells us that the restriction of the 12edo patent val to the 2.9/7.5/3 subgroup is the subgroup val ''V''<sub>''G''</sub> = {{val\| 12 4 9 }}, with a mapping of 12 steps for 2/1, a mapping of 4 steps for 9/7, and a mapping of 9 steps for 5/3.

	We can also send temperament mapping matrices into the subgroup matrix. For instance, here is the matrix ''V'' for 7-limit [[sensi]] ~~– with the rows explicitly notated as vals, and the columns explicitly notated as tempered monzos~~:		We can also send temperament mapping matrices into the subgroup matrix. For instance, here is the matrix ''V'' for 7-limit [[sensi]]:

	<math>\displaystyle		:<math>\displaystyle
	V =		V =
	~~\left[~~ \begin{~~array}{rrrrrl~~}		\begin{bmatrix}
	~~\: \tbracket & \tbracket & \tbracket & \tbracket \:\: \\[-20pt]~~		1 & -1 & -1 & -2 \\
	~~\langle \:~~ 1 & -1 & -1 & -2 ~~\: ]~~\\		0 & 7 & 9 & 13 \\
	~~\langle \:~~ 0 & 7 & 9 & 13 \~~: ]~~\~~\[-20pt]~~		\end{bmatrix}
	~~\: \dangle & \dangle & \dangle & \dangle \:\: \\[-20pt]~~
	\end{~~array~~} ~~\right]~~
	</math>		</math>

	The matrix multiplication ''V''<sub>''G''</sub> = ''VS'' gives us the following result:		The matrix multiplication ''V''<sub>''G''</sub> = ''VS'' gives us the following result:

	<math>\displaystyle		:<math>\displaystyle
	V_G =		V_G =
	~~\left[~~ \begin{~~array}{rrrrrl~~}		\begin{bmatrix}
	~~\: \tbracket & \tbracket & \tbracket \:\: \\[-20pt]~~		1 & 0 & 0 \\
	~~\langle \:~~ 1 & 0 & 0 ~~\: ]~~ \\		0 & 1 & 2 \\
	~~\langle \:~~ 0 & 1 & 2 \~~: ]~~ \~~\[-20pt]~~		\end{bmatrix}
	~~\: \dangle & \dangle & \dangle \:\: \\[-20pt]~~
	\end{~~array~~} ~~\right]~~
	</math>		</math>

FloraC: Oops

2025-06-14T12:28:51Z

Oops

← Older revision		Revision as of 12:28, 14 June 2025
Line 1:		Line 1:
	{{Expert}}		{{Expert}}
	A '''subgroup basis matrix''' is matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list\| 1 0 0 0 \| 0 1 0 0 \| 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.		A '''subgroup basis matrix''' is a matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list\| 1 0 0 0 \| 0 1 0 0 \| 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.

	Subgroup basis matrices are dual to [[temperament mapping matrix\|temperament mapping matrices]]. Temperament mapping matrices are matrices that represent [[regular temperament]]s; they are {{w\|linear map\|linear maps}} that send monzos to [[tempered monzos and vals\|tempered monzos]]. The integer row span of any mapping matrix is the set of all [[vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals. Subgroup basis matrices are also linear maps, but they take [[subgroup monzos and vals\|subgroup monzos]] and map them to regular monzos on the parent JI group. And, dual to temperament mapping matrices, subgroup basis matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w\|group homomorphism\|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called ''restricting'' (or more rarely, ''co-tempering'') the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals\|tempered vals]] back to regular vals. Note the duality here – subgroup vals are a ''{{w\|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.		Subgroup basis matrices are dual to [[temperament mapping matrix\|temperament mapping matrices]]. Temperament mapping matrices are matrices that represent [[regular temperament]]s; they are {{w\|linear map\|linear maps}} that send monzos to [[tempered monzos and vals\|tempered monzos]]. The integer row span of any mapping matrix is the set of all [[vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals. Subgroup basis matrices are also linear maps, but they take [[subgroup monzos and vals\|subgroup monzos]] and map them to regular monzos on the parent JI group. And, dual to temperament mapping matrices, subgroup basis matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w\|group homomorphism\|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called ''restricting'' (or more rarely, ''co-tempering'') the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals\|tempered vals]] back to regular vals. Note the duality here – subgroup vals are a ''{{w\|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.

FloraC: Rework on the intro, making it more straight to the point

2025-06-14T06:30:27Z

Rework on the intro, making it more straight to the point

← Older revision		Revision as of 06:30, 14 June 2025
Line 1:		Line 1:
	{{Expert}}		{{Expert}}
	[[~~Temperament mapping matrices~~]] ~~are matrices that represent~~ [[~~regular temperament~~]]~~s; they are~~ {{w\|~~linear map~~\|~~linear maps~~}} ~~that send [[monzos and interval space\|monzos]] to [[tempered monzos and vals\|tempered monzos]]~~. ~~The integer row span~~ of ~~any mapping matrix is the set of all [[vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals~~.		A '''subgroup basis matrix''' is matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list\| 1 0 0 0 \| 0 1 0 0 \| 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.

	~~There is a "dual" set of '''subgroup~~ basis matrices~~''' (or "subgroup~~ matrices~~" for short when the context is clear), in which we look at~~ matrices ~~in which the columns~~ are ~~monzos. These~~ matrices ~~have relevance in representing~~ [[~~subgroup~~]]s~~, as their~~ integer ~~column spans~~ span ~~some subgroup~~ of [[JI]]~~. Each column represents an entry in~~ the ~~basis for~~ a ~~subgroup, e.g. {{monzo list\| 1 0 0 0 \| 0 1 0 0 \| 0 0 1 0 }} represents~~ the ~~2.3.5 subgroup~~ of 2.~~3.5.7. These~~ matrices take [[subgroup monzos and vals\|subgroup monzos]] and map them to regular monzos on the parent JI group.		Subgroup basis matrices are dual to [[temperament mapping matrix\|temperament mapping matrices]]. Temperament mapping matrices are matrices that represent [[regular temperament]]s; they are {{w\|linear map\|linear maps}} that send monzos to [[tempered monzos and vals\|tempered monzos]]. The integer row span of any mapping matrix is the set of all [[vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals. Subgroup basis matrices are also linear maps, but they take [[subgroup monzos and vals\|subgroup monzos]] and map them to regular monzos on the parent JI group. And, dual to temperament mapping matrices, subgroup basis matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w\|group homomorphism\|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called ''restricting'' (or more rarely, ''co-tempering'') the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals\|tempered vals]] back to regular vals. Note the duality here – subgroup vals are a ''{{w\|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.

	And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w\|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called ''restricting'' (or, more rarely, ''co-tempering'') the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals\|tempered vals]], back to regular vals.		Since the kernel of any temperament is a subgroup of JI, subgroup basis matrices can thus be used to represent kernels. They can also be used to compute the subgroup restriction of a val or mapping matrix to a smaller subgroup.

	~~(Note the duality here – subgroup vals are a ''{{w\|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)~~

	~~Subgroup basis matrices can be used as a generic representation for a basis of any subgroups of JI.~~ Since the kernel of any temperament is a subgroup of JI, ~~they~~ can thus be used to represent kernels. They can also be used to compute the "subgroup restriction" of a val or mapping matrix to a smaller subgroup.

	== Mathematical definition ==		== Mathematical definition ==

FloraC: Display style

2025-06-14T04:30:53Z

Display style

Show changes

FloraC: Internal style improvements

2025-06-14T04:18:47Z

Internal style improvements

← Older revision		Revision as of 04:18, 14 June 2025
Line 1:		Line 1:
	{{Expert}}		{{Expert}}
	[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are ~~[[wikipedia: Linear~~ map\|linear maps]] that send [[~~Monzos~~ and interval space\|monzos]] to tempered monzos ~~(→ [[Tmonzos~~ and ~~tvals~~\|~~tmonzos~~]]). The integer row span of any mapping matrix is the set of all [[~~Vals~~ and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.		[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are {{w\|linear map\|linear maps}} that send [[monzos and interval space\|monzos]] to [[tempered monzos and vals\|tempered monzos]]. The integer row span of any mapping matrix is the set of all [[vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.

	There is a "dual" set of '''subgroup basis matrices''' (or "subgroup matrices" for short when the context is clear), in which we look at matrices in which the columns are monzos. These matrices have relevance in representing [[subgroup]]s, as their integer column spans span some subgroup of [[JI]]. Each column represents an entry in the basis for a subgroup, e.g. {{monzo list\| 1 0 0 0 \| 0 1 0 0 \| 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7. These matrices take subgroup monzos (→ [[Smonzos and svals\|smonzos]]) and map them to regular monzos on the parent JI group.		There is a "dual" set of '''subgroup basis matrices''' (or "subgroup matrices" for short when the context is clear), in which we look at matrices in which the columns are monzos. These matrices have relevance in representing [[subgroup]]s, as their integer column spans span some subgroup of [[JI]]. Each column represents an entry in the basis for a subgroup, e.g. {{monzo list\| 1 0 0 0 \| 0 1 0 0 \| 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7. These matrices take subgroup monzos (→ [[Smonzos and svals\|smonzos]]) and map them to regular monzos on the parent JI group.

	And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or ~~[[wikipedia: Group homomorphism~~\|group homomorphisms]] on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called '''restricting''' (or, more rarely, "co-tempering") the vals. These are dual to how temperament mapping matrices send tempered vals ~~(→ [[tvals~~]]), back to regular vals.		And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w\|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called '''restricting''' (or, more rarely, "co-tempering") the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals\|tempered vals]], back to regular vals.

	(Note the duality here – subgroup vals are a ~~[[wikipedia: Quotient group\|~~''quotient group'']] of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)		(Note the duality here – subgroup vals are a ''{{w\|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)

	Subgroup basis matrices can be used as a generic representation for a basis of any subgroups of JI. Since the kernel of any temperament is a subgroup of JI, they can thus be used to represent kernels. They can also be used to compute the "subgroup restriction" of a val or mapping matrix to a smaller subgroup.		Subgroup basis matrices can be used as a generic representation for a basis of any subgroups of JI. Since the kernel of any temperament is a subgroup of JI, they can thus be used to represent kernels. They can also be used to compute the "subgroup restriction" of a val or mapping matrix to a smaller subgroup.

	== Mathematical definition ==		== Mathematical definition ==
	As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the ~~[[wikipedia: Free abelian group~~\|free abelian group]] J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is ~~[[wikipedia: Surjective~~ function\|surjective]]).		As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the {{w\|free abelian group}} J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is {{w\|surjective function\|surjective}}).

	We can similarly look at the matrices formed by monzos, in which the column vectors are monzos, which we call a '''subgroup basis matrix'''. In mathematical terms, these represent group homomorphisms '''S''': G → J, where G is some subgroup of J, being injected back into the parent JI group J. We can view this matrix as mapping the subgroup monzos back into the parent basis, and thus translating the coordinate system from the subgroup basis to the parent basis. The integer column span of these matrices represents all the monzos within the subgroup.		We can similarly look at the matrices formed by monzos, in which the column vectors are monzos, which we call a '''subgroup basis matrix'''. In mathematical terms, these represent group homomorphisms '''S''': G → J, where G is some subgroup of J, being injected back into the parent JI group J. We can view this matrix as mapping the subgroup monzos back into the parent basis, and thus translating the coordinate system from the subgroup basis to the parent basis. The integer column span of these matrices represents all the monzos within the subgroup.

	Typically, for a matrix S, with column vectors as monzos, to represent a true subgroup basis matrix, it must also be of full column rank, much like a temperament matrix must be of full row rank. Another way to look at this requirement is that it is ~~[[wikipedia: Injective~~ function\|injective]] into the parent group, dual to how we want mapping matrices to be surjective. However, we typically drop the restriction that this column span be [[saturated]], so that we can represent, for instance, the 2.9.5 subgroup, unlike with temperament mapping matrices, where unsaturated matrices have [[contorsion]] and are viewed as pathological.		Typically, for a matrix S, with column vectors as monzos, to represent a true subgroup basis matrix, it must also be of full column rank, much like a temperament matrix must be of full row rank. Another way to look at this requirement is that it is {{w\|injective function\|injective}} into the parent group, dual to how we want mapping matrices to be surjective. However, we typically drop the restriction that this column span be [[saturated]], so that we can represent, for instance, the 2.9.5 subgroup, unlike with temperament mapping matrices, where unsaturated matrices have [[contorsion]] and are viewed as pathological.

	Note that, much like with temperament mapping matrices, there is not a unique basis matrix corresponding to any subgroup: for instance, the two subgroup bases "3.2.5" and "2.3.5" represent the same subgroup, but will be represented by different matrices. Similarly, these two matrices will send vals to svals on the "2.3.5" and "3.2.5" bases respectively.		Note that, much like with temperament mapping matrices, there is not a unique basis matrix corresponding to any subgroup: for instance, the two subgroup bases "3.2.5" and "2.3.5" represent the same subgroup, but will be represented by different matrices. Similarly, these two matrices will send vals to svals on the "2.3.5" and "3.2.5" bases respectively.

FloraC: Fix math display

2025-06-14T04:15:35Z

Fix math display

← Older revision		Revision as of 04:15, 14 June 2025
Line 11:		Line 11:

	== Mathematical definition ==		== Mathematical definition ==
	As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the [[wikipedia: Free abelian group\|free abelian group]] J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\~~Bbb~~ Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is [[wikipedia: Surjective function\|surjective]]).		As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the [[wikipedia: Free abelian group\|free abelian group]] J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is [[wikipedia: Surjective function\|surjective]]).

	We can similarly look at the matrices formed by monzos, in which the column vectors are monzos, which we call a '''subgroup basis matrix'''. In mathematical terms, these represent group homomorphisms '''S''': G → J, where G is some subgroup of J, being injected back into the parent JI group J. We can view this matrix as mapping the subgroup monzos back into the parent basis, and thus translating the coordinate system from the subgroup basis to the parent basis. The integer column span of these matrices represents all the monzos within the subgroup.		We can similarly look at the matrices formed by monzos, in which the column vectors are monzos, which we call a '''subgroup basis matrix'''. In mathematical terms, these represent group homomorphisms '''S''': G → J, where G is some subgroup of J, being injected back into the parent JI group J. We can view this matrix as mapping the subgroup monzos back into the parent basis, and thus translating the coordinate system from the subgroup basis to the parent basis. The integer column span of these matrices represents all the monzos within the subgroup.

Sintel: -legacy

2025-03-29T18:10:51Z

-legacy

← Older revision		Revision as of 18:10, 29 March 2025
Line 1:		Line 1:
	{{Expert}}		{{Expert}}
	~~{{Legacy}}~~
	[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map\|linear maps]] that send [[Monzos and interval space\|monzos]] to tempered monzos (→ [[Tmonzos and tvals\|tmonzos]]). The integer row span of any mapping matrix is the set of all [[Vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.		[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map\|linear maps]] that send [[Monzos and interval space\|monzos]] to tempered monzos (→ [[Tmonzos and tvals\|tmonzos]]). The integer row span of any mapping matrix is the set of all [[Vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.

Lériendil at 16:55, 17 February 2025

2025-02-17T16:55:05Z

← Older revision		Revision as of 16:55, 17 February 2025
Line 1:		Line 1:
	{{Expert}}		{{Expert}}
	{{Legacy~~\|Subgroup_basis_matrix~~}}		{{Legacy}}
	[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map\|linear maps]] that send [[Monzos and interval space\|monzos]] to tempered monzos (→ [[Tmonzos and tvals\|tmonzos]]). The integer row span of any mapping matrix is the set of all [[Vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.		[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map\|linear maps]] that send [[Monzos and interval space\|monzos]] to tempered monzos (→ [[Tmonzos and tvals\|tmonzos]]). The integer row span of any mapping matrix is the set of all [[Vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.

Lériendil: deploying new cat

2025-02-17T16:10:18Z

deploying new cat

← Older revision		Revision as of 16:10, 17 February 2025
Line 1:		Line 1:
	{{Expert}}		{{Expert}}
			{{Legacy\|Subgroup_basis_matrix}}
	[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map\|linear maps]] that send [[Monzos and interval space\|monzos]] to tempered monzos (→ [[Tmonzos and tvals\|tmonzos]]). The integer row span of any mapping matrix is the set of all [[Vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.		[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map\|linear maps]] that send [[Monzos and interval space\|monzos]] to tempered monzos (→ [[Tmonzos and tvals\|tmonzos]]). The integer row span of any mapping matrix is the set of all [[Vals and tuning space\|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals.

Lériendil: Changed protection level for "Subgroup basis matrix" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

2025-02-17T16:01:05Z

Changed protection level for "Subgroup basis matrix" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:01, 17 February 2025
(No difference)