Kite'sthoughts on twin squares
Twin squares[idiosyncratic term] (term proposed by Kite Giedraitis) is a format for presenting a regular temperament. The gencom matrix G is shown side by side with its inverse, the mapping matrix M. M is transposed so that rows multiply with rows. 5-limit meantone temperament, with period 2/1, generator 3/2 and comma 81/80:
[math]\displaystyle{ G = \left[ \begin{array} {rrr} per \\ gen \\ com \\ \end{array} \right] = \left[ \begin{array} {rrr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -4 & 4 & -1 \\ \end{array} \right] M = \left[ \begin{array} {rrr} 1 & 1 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & -1 \\ \end{array} \right] }[/math]
[math]\displaystyle{ \begin{array} {rrr} per \\ gen \\ com \\ \end{array} \left[ \begin{array} {rrr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -4 & 4 & -1 \\ \end{array} \right] \left[ \begin{array} {rrr} 1 & 1 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & -1 \\ \end{array} \right] }[/math]
These two matrices can be considered to be a simple basis change from 2.3.5 to 2.3/2.81/80. Meantone can be thought of as a basis change, with one of the new basis members vanishing, thus projecting from 3-D to 2-D.
The dot product of any row in G with any row in M is 1 if the 2 rows are opposite each other (e.g. 2nd row of each matrix), and 0 if not (e.g. 1st row of G and 3rd row of M).
RTT can be explained in 3 steps:
- perform a basis change with one of the new generators being comma-sized
- temper out that comma
- discard the row in M that corresponds to that comma