Math and the Musical Offering

6. The transformation  g(t) = -f((t-0.5)/2) + L  and Canon 4

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

The second clef is a treble clef, and positions Voice 2 (which enters halfway through the first measure) above the modified Royal Theme in the top staff. The Royal Theme now appears as a middle voice. But there is more: the ``per augmentationem'' in the title indicates that in Voice 2 each note has double the value that it had in Voice 1, so Voice 2 moves with half the speed of Voice 1.

To make the function g copy the function f but move half as fast, we define g(t) = f(t/2) so that g(2) = f(1), g(4) = f(2), etc. To make g also start half a measure later and higher and move in the reverse direction, we combine the previous modifications and set

As in Canon 3, the L is chosen to position our graph approximately where Voice 2 is positioned. Note that it takes 16 measures to hear the whole canon: Voice 1 has to play its tune twice before Voice 2 is finished.

• Introduction

• What is a Canon?

• g(t) = f(t-1) and Canon 2

• g(t) = f(t-1) + H and Canon 5

• g(t) = -f(t-0.5) + K and Canon 3

• g(t) = -f((t-0.5)/2) + L and Canon 4

• g(t) = f(18-t) and Canon 1