February 2015

Shown below is the first voice of a new three-voice canon, based on the theme of Bach’s work  The Musical Offering. Your challenge is to figure out how to transform the first voice to make the second and third voices. To transform a voice you may delay when it starts by some number of bars, transpose it up or down by some interval, invert the intervals so the melody goes up where the original went down, reverse it in time (retrograde), or change the speed (augmentation or diminution, usually by a factor of two). The three voices, played simultaneously will make a harmonious whole, as they do in familiar canons like Frère Jacques. For a full introduction to canons, see Norman McLaren’s wonderful animation Canon or my talk for the Museum of Mathematics about symmetry in music and art. Please email me if you find the solution; I’m curious how hard it is to solve. I will post the solution on Bach’s birthday, March 31. (For those of you who think Bach’s birthday is on March 21, you are correct, but scholars analyzing how the calendar system changed between then and now have calculated that his birthday actually fell on what we would now call March 31.)

Musical Offering Canon 1: by augmentation

I have now written three canons on the theme of the Musical Offering. Here is the story of the first Musical Offering canon.

In 1975 I had the pleasure of meeting Doug Hofstadter at Stanford University. I was an undergraduate enamored of Escher’s art, Bach’s music, and Gödel’s mathematics (I had read Nagel and Newman’s book Gödel’s Proof the previous summer, in Hawaii). The encounter was an uncanny meeting of minds; as Doug recalls we were a bit suspicious of each other, having encountered wacky people with similarly eccentric interests. 

But the connection was real and we became fast friends. I spent many happy evenings working on my homework at a computer terminal alongside Doug, who was writing his magnum opus Gödel, Escher, Bach. To help him write his book he employed a classic professor trick: teach a course in order to motivate you to write the book. Each week new dialogs and chapters would appear hot off the dot matrix printer.

A couple years later he taught the course again, and I helped out as a teaching assistant. We decided to open the class with a re-enactment of the fateful meeting of Bach and King Frederick the Great of Prussia, in which the king sat down at a new-fangled pianoforte and challenged “Old” Bach to improvise a fugue on a long chromatic theme he had composed. Bach not only improvised a 3-voice fugue on the spot; three months later he sent the king an incredible manuscript called The Musical Offering, which included a six-voice fugue, a trio sonata, and ten canons, all on the royal theme. The full story is told in the dramatic the book Evening in the Palace of Reason. Here is the original theme, a sinuously chromatic melody with a somewhat clumsy closing cadence.

I had already learned the six-voice fugue when Doug’s 1977 GEB class began. Over the preceding winter break I learned the three-voice fugue as well. Not content with that accomplishment, I told Doug I would surprise him with a new canon on the Musical Offering theme. It was with great pleasure that I performed an original canon by augmentation (the faster voice plays the score twice in the same time as the slower voice plays it once) for the class. Here is the first voice. 

Voice 1 of Canon by Augmentation on the Theme of the Musical Offering

Note that the first half of the composition is an embellished version of the original Musical Offering theme. On the next two pages are the full canon written out for two voices. Both voices start at the same time. The second voice is an octave lower and twice as slow, which means by the time the first voice ends, the second voice is only half done. To fill in the remaining time, the first voice then repeats, until the two voices catch up with each other at the very end (actually they end a bar or so shy of completion). Note that every note in the canon appears three times, and must play three different roles — a bit of musical algebra that I found quite absorbing to solve.

Some of Doug’s friends took my canon further, playing it in four voices with two people seated at one piano. The additional two parts play the theme even slower at quarter speed and eighth speed in lower octaves. The canon was not intended to work this way, so there are a few clashing dissonances, but the overall effect is quite entertaining.

Musical Offering Canon 2: by 50% delay at the octave

Years later I presented Doug with a second Musical Offering canon. In this brief eight-bar composition one voice starts four bars after the other, so the two voices are 50% out of phase with each other. I gave it to Doug originally as a “puzzle” canon, meaning I gave him one voice, and asked him how to transform the theme to produce a second accompanying voice. Here is the first voice of the canon.

And here is the full two-voice canon.

To compose this canon, I first played the theme against itself at the desired 4-bar delay. Here is the combined effect using the original theme without modification.

Rough first approximation of the 180° Canon on the Theme of the Musical Offering

The light gray areas of the canon work decently without modification; I added passing notes to give them more movement. The first two dark gray areas fall flat because both voices sit on the same note. To solve this problem I shifted some notes in time — in the first region I shifted the E flat in the bass voice earlier by a quarter beat, and introduced a scale-wise descent, so that by the time E flat sounds in the treble voice, the bass has moved down to a harmonious C. To enliven the chromatic descent in the third dark gray region, I delayed the treble voice by a quarter beat, creating a pleasing staggered pattern.

Because of the way this canon is structured, the last four bars of the piece are the same as the first four bars but with the voices reversed, so modifications to the first four bars work almost automatically in the last four bars. As I do when creating an ambigram, I created half the composition, then copied it to make the other half. In fact, the whole process of creating a canon is quite similar to creating an ambigrams: you start with a theme (word), choose a transformation (musical or geometric), modify the theme (word) so it both obeys the strict transformation and sounds (looks) harmonious (legible). 

The finishing touch in composing both canons and ambigrams is to rationalize the quirks of the composition by making them part of a consistent style. In this canon, for instance, the alternating pattern of intervals in the first bar creates an angular marching rhythm that I continued throughout the piece. The goal is to compose a piece that works both mathemtatically and musically. I enjoy composing canons because the music seems to write itself, shaped by the pressures of the formal constraints.

Finally, here is a comparison of the original Musical Offering theme and the variations that appear in each of the canons, so you can hear how the theme has been altered.