Math and the Musical Offering
3. The transformation g(t) = f(t-1) and Canon 2
Canon 2 in the Musical Offering is the simplest kind of canon,
just like ``Row, row, row your boat.'' The score is also 8 measures long,
but Voice 2 comes in after one measure only. An
additional wrinkle provided by Bach is a third voice: the ``Royal Theme''
plays in the bass in harmony with the two canonical voices above. Here is how
Bach presented this canon in the edition he had engraved for the King:
Graphic © 1996,
Timothy A. Smith, used by permission.
Here is the mathematical analogue of the relative position of the two
voices. The red graph represents f(t) (Voice 1); the blue
graph (Voice 2) is
||: this sign shows where the second voice comes in.|
g(t) = f(t-1).
This is the transformation that shifts a graph one unit to the right.
© copyright 1999, American Mathematical Society.