Math and the Musical Offering
What is a canon?
In music a canon is a form in which two (or more) voices sing the same melodic line but start at different times. A familiar example is the canon ``Row, row, row your boat''. The text is
Row, row, row your boat Gently down the stream Merrily, merrily, merrily, merrily Life is but a dream.
In one kind of performance, the second voice starts at the beginning of the third measure, when the first voice reaches ``Gently'', so one hears:
(Voice 1) Gently down the stream Merrily, merrily, merrily, merrily (Voice 2) Row, row, row your boat Gently down the stream
The two voices continue; Voice 1 starts over when it gets to the end.
(Voice 1) Life is but a dream. Row, row, row your boat (Voice 2) Merrily, merrily, merrily, merrily Life is but a dream.
and so on. The canon naturally recycles.
A canon is musical when the two voices harmonize. It is an appealing form, because it gives a very easy introduction to partsinging: 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 the number of measures, then Voice 2 would be represented by the function g(t) = f(t2). 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(t2).
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 eight measure 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(t2).

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