## 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.
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.

© copyright 1999, American Mathematical Society.