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Subgroup basis matrix: Difference between revisions

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Fix the absolutely ridiculous notation
 
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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
\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>


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