Math and the Musical Offering

2. What is a Canon?

Row row row your boat
Gently down the stream
Merrily merrily merrily merrily
Life is but a dream.

(Voice 1) Gently down the stream            Merrily merrily  merrily merrily  
(Voice 2) Row    row      row   your boat   Gently  down the stream

(Voice 1) Life is but  a  dream.             Row     row    row   your boat  
(Voice 2) Merrily merrily merrily merrily    Life is but a  dream.

A canon is musical when the two voices harmonize. It is an appealing form, because it gives a very easy introduction to part-singing: there is only one melody to learn, but the harmonies that come from the play of one voice against the other can be very pleasing.

Mathematically speaking, the operation that produces Voice 2 from Voice 1 is a translation in time. If the pitch sequence that describes the melody for Voice 1 is represented by a function f(t), and if we let t represent number of measures, then Voice 2 would be represented by the function g(t) = f(t-2). So for example at the beginning of the third measure (time t = 2), the pitch of Voice 2 would be g(2) = f(0), the pitch of Voice 1 at the beginning (time t = 0).

The translation operation that makes g out of f is studied in an Elementary Functions course, because it is one of the important ways of making new functions out of old, or of tailoring a given function to fit a new situation. Here is a typical example:

The red graph represents a function f(t), the blue graph is g(t) = f(t-2). The blue graph is a copy of the red graph, displaced 2 units to the right.

The analogue of the voices recycling through the material after 8 measures beats is to make the functions periodic of period 8:

The red graph represents a periodic function f(t), of period 8; the blue graph is g(t) = f(t-2).

• 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