Math and the Musical Offering

4. The transformation  g(t) = f(t-1) + H  and Canon 5

In Canon 5 the second voice follows one measure behind, just as in Canon 2, but it is shifted up in pitch by a perfect fifth. In Bach's encryption of this canon, the shift is indicated by the two clefs in the lower staff.
Bach's score for Canon 5

Graphic 1996, Timothy A. Smith, used by permission.

The first voice is read using the bass clef, the second with the same notes but using the alto clef (centered on middle C). A modified Royal Theme plays in the top staff, in harmony with the two lower voices.

Function Example The mathematical analogue of a canon where Voice 2 starts one measure after Voice 1 and is shifted up a perfect fifth is a pair of functions f(t) and g(t) with

g(t) = f(t-1) + H,
if the t unit is a measure, and where H is the distance along the pitch axis representing ``up one fifth.''
This canon has another unusual feature, which is not duplicated in the graphs: it does not recycle periodically, but each successive run-through is modulated upward in pitch by a whole step.

© copyright 1999, American Mathematical Society.