1. Johann Sebastian Bach “A Musical Offering”
The Mathematics Behind the Canons
Jessica Williams

2. What is “A Musical Offering”?
“A Musical Offering” is a composition written by J.S. Bach for King Frederick II of Prussia in 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.

3. What is a canon?
A canon is a “copy” of a theme played by various participating voices.
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”.)

4. Examples of Canons “Row, Row, Row Your Boat” “Three Blind Mice”
“Are You Sleeping / Frere Jacques”
Any song that can be sung as a round is considered a canon when sung in that manner. (one type of canon)
Canon 2

5. 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 measures.

6. 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
(Canons 1, 2, and 3)
(Canon 4)

7. Groups
Group Operations:
T(n) R I