Subgroup basis matrix: Difference between revisions
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[[ | 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. | |||
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. | |||
== Mathematical definition == | == Mathematical definition == | ||
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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 | ||
S = | |||
\begin{bmatrix} | |||
S = | 1 & 0 & 0 \\ | ||
0 & 2 & -1 \\ | |||
0 & 0 & 1 \\ | |||
1 & 0 & 0 \\ | 0 & -1 & 0 \\ | ||
0 & 2 & -1 \\ | \end{bmatrix} | ||
0 & 0 & 1 \\ | |||
0 & -1 & 0 \\ | |||
\end{ | |||
</math> | </math> | ||
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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 = | ||
\begin{bmatrix} | |||
0 & 0 \\ | |||
0 & 0 \\ | 2 & -5 \\ | ||
2 & -5 \\ | 0 & 1 \\ | ||
0 & 1 \\ | -1 & 2 \\ | ||
-1 & 2 \\ | \end{bmatrix} | ||
\end{ | |||
</math> | </math> | ||
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 === | ||
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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 = | ||
\begin{bmatrix} | |||
12 & 19 & 28 & 34 | |||
\end{ | \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 = | ||
\begin{bmatrix} | |||
12 & 4 & 9 | |||
\end{ | \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]] | 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 = | ||
\begin{bmatrix} | |||
1 & -1 & -1 & -2 \\ | |||
0 & 7 & 9 & 13 \\ | |||
\end{bmatrix} | |||
\end{ | |||
</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 = | ||
\begin{bmatrix} | |||
1 & 0 & 0 \\ | |||
0 & 1 & 2 \\ | |||
\end{bmatrix} | |||
\end{ | |||
</math> | </math> | ||