## 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.

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.

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.