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

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

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