Math and the Musical Offering

$e-MATH$
Math in the Musical Offering

Johan Sebastian Bach's Musical Offering contains 10 canons. In music, a canon is a form in which two (or more) voices sing the same melodic line, but starting 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 the simplest performance, the second voice starts at the beginning of the third measure, when the first voice reaches the first "Merrily, so one hears:

(Voice 1) Merrily merrily merrily merrily    Life is but   a  dream
(Voice 2) Row     row     row     your boat  Gently  down the stream

Then Voice 1 starts over as Voice 2 continues:

(Voice 1) Row     row     row     your boat  Gently  down the stream
(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 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 time t=2, the pitch of Voice 2 would be g(2) = f(0), the pitch of Voice 1 at 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 represents
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 represents
g(t)= f(t-2). The blue graph is a copy of the red graph, displaced 2 units to the right.

Canon 2 in the Musical Offering is the simplest kind of canon, just like ``Row, row, row your boat.'' The score is also 8 measures long, but Voice 2 comes in after one measure. An additional wrinkle provided by Bach is a third voice: the ``Royal Theme'' plays in the bass in harmony with the two canonical voices above. Here is how Bach presented this canon in the edition he had engraved for the King:

: this sign shows where the second voice comes in.

Here is the mathematical analogue of the relative position of the two voices. The red graph represents f(t) (Voice 1); the blue graph (Voice 2) is
g(t)= f(t-1). This is the transformation that shifts a graph one unit to the right.

Each of the canons in the Musical Offering uses a different way of constructing Voice 2 from Voice 1. I will sample three more of these, but all of them have equivalents in elementary mathematical constructions of one function from another.

In Canon 5 the second voice follows one measure behind, just as in Canon 2, but it is shifted up in pitch by a perfect fifth. In Bach's encryption of this canon, the shift is indicated by the two clefs in the lower staff.

The first voice is read using the bass clef, the second with the same notes but using the alto clef (centered on middle C). A modified Royal Theme plays in the top staff, in harmony with the two lower voices.

The mathematical analogue of a canon where Voice 2 starts one measure after Voice 1 and is shifted up a perfect fifth (3.5 whole steps in pitch) is a pair of functions f(t) and g(t) with

g(t) = f(t-1)+3.5, if the t unit is a measure, and the function are given in units of a whole step (a major second) in pitch. There is no natural 0 for the pitch coordinate; in fact all that matters in canons is relative pitch.
This canon has another unusual feature: it does not recycle periodically, but each successive run-through is modulated upward in pitch by a whole step.

Canon 3 is described by Bach as a 2 per Motum contarium.

Here the upside-down second signature in the lower staff indicates that the second voice is to be played upside-down. Voice 1 starts on C, and Voice 2 starts on G (a perfect fourth lower) at the sign and moves in the opposite direction from the first. Again, a modified Royal Theme plays in the top staff, in harmony with the two lower voices.

To make a function run upside-down only requires a minus sign. In this example the blue function g is defined from the red function f by

g(t)= -f(t-1)+2.5. Since there is no natural meaning for 0 pitch, the minus sign needs interpreting musically. I chose the ``2.5'' to make the starting points look plausible.

In Canon 4 (per Augmentationem contrario Motu) Voice 2 starts at the sign. The second clef is again upside-down, signalling that Voice 2 runs upside-down (as in Canon 3); it is a treble clef, and positions Voice 2 above the newly modified Royal Theme in the top staff, which now appears as a middle voice. 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 later and higher and move in the reverse direction, we combine the previous modifications and set
g(t) = -f((t-1)/2)+4. Again, the "4" is somewhat arbitrary, enough to position our graph approximately where Voice 2 is positioned. It takes 16 measures to hear the whole canon: Voice 1 has to play its tune twice before Voice 2 is finished.

Perhaps the most exotic canon in the collection is Canon 1 cancrizans. It is also the simplest.

``Cancrizans'' means ``crab-wise'' but in fact in this canon Voice 2 plays the score of Voice 1 backwards. This is indicated in Bach's cryptic presentation by the backwards signature at the end of the piece. Voice 1 plays the Royal Theme itself (8.5 measures), followed by 9.5 measures of counterpoint.

This score is 18 measures long. For a function g defined on that interval to be the ``backwards'' of f, we need g(0) = f(18), g(1) = f(17), etc. This is achieved by defining

g(t) = f(18-t). In this plot, as usual, f is the red graph and g is the blue, although on this case the operation is symmetric, and f runs g backards also.
Usually in performance each player plays his or her score forwards and then backwards (if the instruments are different this is not the same as running through the piece twice). This results in the pattern:

--Tony Phillips