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 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>
-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>
@@ Line 42: / Line 20: @@
 </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. 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:
-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
+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).
 [[Category:Mapping]]
 {{todo|review}}