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Subgroup basis matrix: Difference between revisions

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Any subgroup basis matrix also thus has "left kernel", which is typically called the "left nullspace" in linear algebra (note the term "cokernel" is slightly different, so we do not use it here). The left kernel is a subgroup of vals that temper out everything in the subgroup generated by the subgroup basis matrix. So for instance, if matrix S generates a subgroup representing the kernel for some temperament, the left nullspace represents all the vals tempering out that kernel (and thus which support the temperament).
Any subgroup basis matrix also thus has "left kernel", which is typically called the "left nullspace" in linear algebra (note the term "cokernel" is slightly different, so we do not use it here). The left kernel is a subgroup of vals that temper out everything in the subgroup generated by the subgroup basis matrix. So for instance, if matrix S generates a subgroup representing the kernel for some temperament, the left nullspace represents all the vals tempering out that kernel (and thus which support the temperament).


S can also represent an arbitrary subgroup of JI, such as ones with monzos we would like to play (rather than just representing the kernel for some temperament). In this situation, it is useful to view S as a map from vals to svals on S's subgroup basis. With this interpretation, S still has a left kernel of vals, which is the set of vals that are '''restricted away''' (or "co-tempered out"), as their subgroup restriction under S is the zero sval. The vals in the left kernel have the property that, for any v and any other val k in the left kernel, we have (v + k)∙S = v∙S + k∙S = v∙S + 0 = v∙S. ''In other words, any two vals differing by an element in the left kernel will restrict to the same sval.''
S can also represent an arbitrary subgroup of JI, such as ones with monzos we would like to play (rather than just representing the kernel for some temperament). In this situation, it is useful to view S as a map from vals to svals on S's subgroup basis. With this interpretation, S still has a left kernel of vals, which is the set of vals that are '''restricted away''' (or "co-tempered out"), as their subgroup restriction under S is the zero sval. The vals in the left kernel have the property that, for any V and any other val V<sub>0</sub> in the left kernel, we have (V + V<sub>0</sub>)S = VS + V<sub>0</sub>S = VS + 0 = VS. ''In other words, any two vals differing by an element in the left kernel will restrict to the same sval.''


== Example ==
== Example ==
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 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>
<math>\displaystyle
\newcommand{dangle}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ \rangle}}
\newcommand{dangle}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ \rangle}}
\newcommand{tbracket}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ [}}
\newcommand{tbracket}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ [}}
</math>
S =
<math>
\left[ \begin{array}{rrr}
\left[ \begin{array}{rrr}
\tbracket & \tbracket & \tbracket\\[-20pt]
\tbracket & \tbracket & \tbracket\\[-20pt]
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\end{array} \right]
\end{array} \right]
</math>
</math>
This matrix will be called '''S''' in the examples below.


=== Main transformation: mapping from subgroup monzos to parent group monzos ===
=== Main transformation: mapping from subgroup monzos to parent group monzos ===


'''S''' can be viewed as a mapping from smonzos to monzos. As an example, we will consider the matrix of smonzos {{monzo list| 0 1 0 | 0 -2 1 }} on the 2.9/7.5/3 subgroup, which represent 9/7 and 245/243.
S can be viewed as a mapping from smonzos to monzos. As an example, we will consider the matrix of smonzos M<sub>G</sub> = {{monzo list| 0 1 0 | 0 -2 1 }}, which represent 9/7 and 245/243 on the 2.9/7.5/3 subgroup G.


If this matrix is X, then the dual transformation can be found by multiplying S∙X, which yields
The dual transformation can be found by the multiplication M = SM<sub>G</sub>, which yields


<math>
<math>\displaystyle
M =
\left[ \begin{array}{rrr}
\left[ \begin{array}{rrr}
\tbracket & \tbracket \\[-20pt]
\tbracket & \tbracket \\[-20pt]
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=== Dual transformation: subgroup restriction ===
=== Dual transformation: subgroup restriction ===


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


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


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


which tells us that the restriction of the 12edo patent val to the 2.9/7.5/3 subgroup is the sval {{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 sval 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 7-limit [[sensi]] – with the rows explicitly notated as vals, and the columns explicitly notated as tmonzos:
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 tmonzos:


<math>
<math>\displaystyle
V =
\left[ \begin{array}{rrrrrl}
\left[ \begin{array}{rrrrrl}
  \: \tbracket & \tbracket & \tbracket & \tbracket \:\: \\[-20pt]
  \: \tbracket & \tbracket & \tbracket & \tbracket \:\: \\[-20pt]
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</math>
</math>


If we call this matrix '''M''', then the matrix multiplication '''M'''∙'''S''' gives us the following result:
The matrix multiplication V<sub>G</sub> = VS gives us the following result:


<math>
<math>\displaystyle
V_G =
\left[ \begin{array}{rrrrrl}
\left[ \begin{array}{rrrrrl}
  \: \tbracket & \tbracket & \tbracket \:\: \\[-20pt]
  \: \tbracket & \tbracket & \tbracket \:\: \\[-20pt]
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