## Math and the Musical Offering

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 eight 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.

 : this sign shows where the second voice comes in.

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

g(t) = f(t-1).

This is the transformation that shifts a graph one unit to the right.

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