Subgroup basis matrix: Difference between revisions
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=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 by forming a matrix in which the columns are the monzo representation of these intervals: | ||
<math> | |||
\newcommand{dangle}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ \rangle}} | |||
\newcommand{hpipe}[][]{\style{display: inline-block; transform-origin: 50% 50% 0px; transform: rotate(90deg); }{ |}} | |||
</math> | |||
<math> | <math> | ||
\left[ \begin{array}{rrr} | \left[ \begin{array}{rrr} | ||
\hpipe & \hpipe & \hpipe\\[-20pt] | |||
1 & 0 & 0\\ | 1 & 0 & 0\\ | ||
0 & 2 & -1\\ | 0 & 2 & -1\\ | ||
0 & 0 & 1\\ | 0 & 0 & 1\\ | ||
0 & -1 & 0 | 0 & -1 & 0\\[-20pt] | ||
\dangle & \dangle & \dangle\\[-20pt] | |||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
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<math> | <math> | ||
\left[ \begin{array}{rrr} | \left[ \begin{array}{rrr} | ||
\hpipe & \hpipe \\[-20pt] | |||
0 & 0\\ | 0 & 0\\ | ||
2 & -5\\ | 2 & -5\\ | ||
0 & 1\\ | 0 & 1\\ | ||
-1 & 2 | -1 & 2\\[-20pt] | ||
\dangle & \dangle\\[-20pt] | |||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
| Line 68: | Line 77: | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | \left[ \begin{array}{rrrrrl} | ||
12 & 19 & 28 & 34 | \langle 12 & 19 & 28 & 34 | | ||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
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<math> | <math> | ||
\left[ \begin{array}{rrrrl} | \left[ \begin{array}{rrrrl} | ||
12 & 4 & 9 | \langle 12 & 4 & 9 | | ||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
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which tells us that the restriction of the 12-EDO patent val to the 2.9/7.5/3 subgroup is the sval <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 12-EDO patent val to the 2.9/7.5/3 subgroup is the sval <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's 7-limit [[Starling_temperaments#Sensi temperament|sensi]]: | We can also send temperament mapping matrices into the subgroup matrix. For instance, here's 7-limit [[Starling_temperaments#Sensi temperament|sensi]] - with the rows explicitly notated as vals, and the columns explicitly notated as tmonzos: | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | \left[ \begin{array}{rrrrrl} | ||
1 & -1 & -1 & -2\\ | \: \hpipe & \hpipe & \hpipe & \hpipe \:\: \\[-20pt] | ||
0 & 7 & 9 & 13\\ | \langle \: 1 & -1 & -1 & -2 \: |\\ | ||
\langle \: 0 & 7 & 9 & 13 \: |\\[-20pt] | |||
\: \dangle & \dangle & \dangle & \dangle \:\: \\[-20pt] | |||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||
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<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | \left[ \begin{array}{rrrrrl} | ||
1 & 0 & 0 \\ | \: \hpipe & \hpipe & \hpipe \:\: \\[-20pt] | ||
0 & 1 & 2 \\ | \langle \: 1 & 0 & 0 \: | \\ | ||
\langle \: 0 & 1 & 2 \: | \\[-20pt] | |||
\: \dangle & \dangle & \dangle \:\: \\[-20pt] | |||
\end{array} \right] | \end{array} \right] | ||
</math> | </math> | ||