Canon 5 ## Math in the *Musical Offering*

*g*(*t*) = *f*(*t-*1) + *H* and Canon 5

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

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

The first voice is readusing the bass clef, the second with the same notes but using the altoclef (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 measureafter 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 thegraphs:It does not recycle periodically, but each successive run-through is modulatedupward in pitch by a whole step.

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