Kite'sthoughts on twin squares: Difference between revisions

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-'''Twin squares'''{{idiosyncratic}} (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:
+'''Twin squares'''{{idiosyncratic}} (term proposed by [[Kite Giedraitis]]) is a format for presenting a [[regular temperament]]. The [[gencom]] matrix is shown side by side with its inverse, the [[mapping matrix]], which has rows for not only each generator but also each comma. Here's 5-limit meantone temperament, with period 2/1, generator 3/2 and comma 81/80:
-<math>
-G =
-\left[ \begin{array} {rrr}
-per \\
-gen \\
-com \\
-\end{array} \right]
-=
-\left[ \begin{array} {rrr}
-& 0 & 0 \\
--1 & 1 & 0 \\
--4 & 4 & -1 \\
-\end{array} \right]
-M =
-\left[ \begin{array} {rrr}
-& 1 & 0 \\
-& 1 & 4 \\
-& 0 & -1 \\
-\end{array} \right]
-</math>
 <math>
 \begin{array} {rrr}
-per \\
+period \\
-gen \\
+generator \\
-com \\
+comma \\
 \end{array}
 \left[ \begin{array} {rrr}
 & 0 & 0 \\
 -1 & 1 & 0 \\
+\hline
 -4 & 4 & -1 \\
 \end{array} \right]
-&nbsp; &nbsp; &nbsp; &nbsp;
+\longleftrightarrow
 \left[ \begin{array} {rrr}
 & 1 & 0 \\
 & 1 & 4 \\
+\hline
 & 0 & -1 \\
 \end{array} \right]
 </math>
+== Pedagogical value ==
-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.
+These two matrices can be thought of as a simple basis change from 2.3.5 to 2.3/2.81/80. Thus any temperament can be thought of as a basis change, with one or more of the new basis members vanishing. The lefthand matrix expresses the new basis in terms of the old basis, and the righthand matrix expresses the old in terms of the new. Thus creating a temperament consists of 3 steps:
-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).
+* perform a basis change with some of the new generators being comma-sized
+* temper out those commas
+* discard the rows in the mapping matrix that corresponds to those commas
-RTT can be explained in 3 steps:
+The two matrices follow a simple rule: the dot product of any row in one with any row in another is 1 if the two rows are opposite each other (e.g. 2nd row of each matrix), and 0 if not (e.g. 1st row of one and 3rd row of the other). Thus one can easily verify that one is the inverse of the other. In fact, it's not too difficult to derive both matrices from either the comma list or the mapping. One proceeds step by step, checking as one goes, similar to solving a sudoku puzzle.
-* 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
 [[Category:Mapping]]
 {{todo|review}}