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Johann Sebastian Bach
“A Musical Offering”
The Mathematics Behind the Canons
Jessica Williams
What is “A Musical
“A Musical Offering” is a composition written by
J.S. Bach for King Frederick II of Prussia in 1747.
During a meeting, the King challenged Bach to
write a six-voice fugue (similar to a canon), and
Bach asserted that he would need to work on it
and send it to the King later. This he did, but, as
was often done back then, he sent it to him not
completely finished, so that he could figure it
out like a sort of puzzle.
 The composition is a collection of canons,
fugues, and other pieces of music.
What is a canon?
A canon is a “copy” of a theme played by various participating
 Types of canons:
– Round = first voice comes in, after a fixed time delay a second
voice joins on the same key the first began on, same time delay,
third voice joins in, etc.
– Difference in starting keys or pitches for each voice, but notes
follow same pattern (still overlapping)
– Diminution or Augmentation = speeds of successive voices vary
(usually twice as slow (D) or twice as fast (A))
– Inversion = When the original theme jumps up, the “copy”
jumps down, and vice versa.
– Retrograde = The theme is played backwards in time. (Also
called a “crab canon”.)
Examples of Canons
Canon 1
Canon 2
“Row, Row, Row Your
“Three Blind Mice”
“Are You Sleeping / Frere
Any song that can be
sung as a round is
considered a canon when
sung in that manner. (one
type of canon)
How Canons Affect Functions
Round: If time (t) is expressed in measures, a delay of one measure
shifts a graph of the melody to the right one unit, changing the
original function from f(t) to f(t-1).
Difference in starting keys or pitches: A difference in the place
where a melody is begun shifts the graph of the melody up or down
by a distance representative of the difference in keys or pitches. The
original function is changed from f(t) to f(t) + h, where h represents
the shift.
Diminution or Augmentation: Varying the speed of a melody
changes the period of a graph. If the speed is doubled, the function
changes from f(t) to f(2t). If the speed is halved, it becomes f(t/2).
Inversion: The function of an inverted melody is changed from f(t)
to –f(t).
Retrograde: If a melody is played backwards, the function changes
from f(t) to f(m-t), where m is the length of the original melody in
Bach’s Canons
Canon 1: g(t) = f(18-t)
Canon 2: g(t) = f(t-1)
Canon 3: g(t) = -f(t-0.5) + h
Canon 4: g(t) = -f((t-0.5)/2) + h
Canon 5: f(t-1) + h
0 (Canons 1, 2, and 3)
Ms&feature=related (Canon 4)
Group Operations:
– T(n)