Kite's thoughts 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. Here's 5-limit meantone temperament, with period 2/1, generator 3/2 and comma 81/80:

[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]

Pedagogical value

These two matrices can be thought of as a simple basis change from 2.3.5 to 2.3/2.81/80. Meantone temperament can be thought of as a basis change, with one of the new basis members vanishing, thus projecting from 3-D to 2-D. Thus creating a temperament consists of 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

The two matrices follow a simple rule: 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).