Subgroup basis matrix: Difference between revisions
Wikispaces>FREEZE No edit summary |
Malformed latex |
||
| Line 18: | Line 18: | ||
Say that our JI module 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 V-map by forming a matrix in which the columns are the monzo representation of these intervals: | Say that our JI module 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 V-map by forming a matrix in which the columns are the monzo representation of these intervals: | ||
<math> | <math> | ||
\left[ \begin{array}{rrr} | |||
1 & 0 & 0\\ | 1 & 0 & 0\\ | ||
0 & 2 & -1\\ | 0 & 2 & -1\\ | ||
0 & 0 & 1\\ | 0 & 0 & 1\\ | ||
0 & -1 & 0 | 0 & -1 & 0 | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
We can also write this matrix notationally as follows: | We can also write this matrix notationally as follows: | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | |||
| & 1 & 0 & 0 & 0 & \rangle\\ | | & 1 & 0 & 0 & 0 & \rangle\\ | ||
| & 0 & 2 & 0 & -1 & \rangle\\ | | & 0 & 2 & 0 & -1 & \rangle\\ | ||
| & 0 & -1 & 1 & 0 & \rangle | | & 0 & -1 & 1 & 0 & \rangle | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
where it's understood that the kets are representing that the rows in this matrix are really column vectors, just written as rows due to an abuse of notation. A shorthand notation of this matrix is [|1 0 0 0>, |0 2 0 -1>, |0 -1 1 0>]. This matrix will be called '''V'''. | where it's understood that the kets are representing that the rows in this matrix are really column vectors, just written as rows due to an abuse of notation. A shorthand notation of this matrix is [|1 0 0 0>, |0 2 0 -1>, |0 -1 1 0>]. This matrix will be called '''V'''. | ||
| Line 39: | Line 43: | ||
To restrict a val to the subgroup defined by the V-map, we'll left-multiply '''V''' by a val '''W'''. In this case, our val '''W''' will be the 7-limit patent val for [[12edo|12-EDO]]: | To restrict a val to the subgroup defined by the V-map, we'll left-multiply '''V''' by a val '''W'''. In this case, our val '''W''' will be the 7-limit patent val for [[12edo|12-EDO]]: | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | |||
| & 12 & 19 & 28 & 34 & \rangle | | & 12 & 19 & 28 & 34 & \rangle | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
Multiplying '''W'''∙'''V''' yields the result | Multiplying '''W'''∙'''V''' yields the result | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrl} | |||
| & 12 & 4 & 9 & \rangle | | & 12 & 4 & 9 & \rangle | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
which tells us that the restriction of the 12-EDO patent val to the 2.9/7.5/3 subgroup has 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 has 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. | ||
| Line 53: | Line 61: | ||
We can also send temperament mapping matrices into the V-map. For instance, here's 7-limit [[Starling_temperaments#Sensi temperament|sensi]]: | We can also send temperament mapping matrices into the V-map. For instance, here's 7-limit [[Starling_temperaments#Sensi temperament|sensi]]: | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | |||
\langle & 1 & -1 & -1 & -2 & |\\ | \langle & 1 & -1 & -1 & -2 & |\\ | ||
\langle & 0 & 7 & 9 & 13 & |\\ | \langle & 0 & 7 & 9 & 13 & |\\ | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
If we call this matrix '''M''', then the matrix multiplication '''M∙V''' gives us the following result: | If we call this matrix '''M''', then the matrix multiplication '''M∙V''' gives us the following result: | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrl} | |||
\langle & 1 & 0 & 0 & |\\ | \langle & 1 & 0 & 0 & |\\ | ||
\langle & 0 & 1 & 2 & |\\ | \langle & 0 & 1 & 2 & |\\ | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
This tells us that the subgroup restriction of sensi to the 2.9/7.5/3 subgroup is a new temperament mapping on the subgroup which sends 2/1 to one generator, 9/7 to the other generator, and 5/3 to two 9/7's. Additionally, since this is the multiplication of an M-map and a V-map, the resulting matrix also has the interpretation of having a set of columns representing the tmonzos that the 7-limit sensi M-map sends 2/1, 9/7, and 5/3 to, respectively. | This tells us that the subgroup restriction of sensi to the 2.9/7.5/3 subgroup is a new temperament mapping on the subgroup which sends 2/1 to one generator, 9/7 to the other generator, and 5/3 to two 9/7's. Additionally, since this is the multiplication of an M-map and a V-map, the resulting matrix also has the interpretation of having a set of columns representing the tmonzos that the 7-limit sensi M-map sends 2/1, 9/7, and 5/3 to, respectively. | ||
| Line 73: | Line 85: | ||
'''V''' implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0>, |0 -2 1>|]. If this matrix is X, then the dual transformation can be found by multiplying V∙X, which yields | '''V''' implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0>, |0 -2 1>|]. If this matrix is X, then the dual transformation can be found by multiplying V∙X, which yields | ||
<math> | <math> | ||
\left[ \begin{array}{rrrrrrl} | |||
| 0 & 2 & 0 & -1 & \rangle\\ | | 0 & 2 & 0 & -1 & \rangle\\ | ||
| 0 & -5 & 1 & 2 & \rangle | | 0 & -5 & 1 & 2 & \rangle | ||
\end{array} \right] | \end{array} \right] | ||
</math> | |||
These monzos are the 7-limit representation of 9/7 and 245/243, respectively. Again, the "rows" here are in kets to specify that they're still supposed to be monzos and hence columns. | These monzos are the 7-limit representation of 9/7 and 245/243, respectively. Again, the "rows" here are in kets to specify that they're still supposed to be monzos and hence columns. | ||