Bach: Canon per Augmentationem in Contrario Motu

Back a few years I did a lot of posts on Bach and specifically his achievements in the area of counterpoint. Music composition used to be a very learnéd activity and, while for some composers it still is, the trend is for even the most intellectual composers to create their own "language" and rules from scratch which means that it is rather hard to judge their compositional virtuosity. From the 15th through the 18th centuries there were widely accepted rules of musical "grammar" that composers generally adhered to, so when they achieved a tour de force, it was one that could be widely understood and appreciated.

There were certainly great achievements in counterpoint before Bach, by Ockeghem and Josquin des Prez among others, but perhaps the most famous of all is the Art of Fugue, written towards the end of his life by J. S. Bach. I once took a graduate seminar on fugue and recall the moment of absolute astonishment when I realized that the so-called "mirror" fugues are ones in which an entire four-voice texture is invertible. I can hear you yawning, but what that means is that the challenge of writing two voices, which can be reversed and still follow all the rules of dissonance and voice-leading, is multiplied many times if you try and do it in four voices. I wrote a post on that here. Go have a look to see how that works.

But today I want to look at a piece that seems, at first glance at least, to be much simpler: the Canon per Augmentationem in Contrario Motu, also from the Art of Fugue. Here is the first page of the score:

Before we go any further, let's have a listen.

What does that title mean? First of all, a canon is a very strict musical form in which voices, in this case two, are "bound" or "obligated", that is, each note in the consequent voice is determined by the originating voice. "Canon" means "rule" and the rule is that the following voice is an exact imitation of the originating voice. The rudimentary example we are all likely familiar with is the schoolroom round "Row, row, row your boat". A simple canon is just like that: the following voice waits a certain time interval and reproduces the first voice exactly.

But for the great age of canon, in the fifteenth century and up to Bach, this was a bit too ordinary. Canons were such a regular part of musical life that they were published as "puzzle" canons, that is, just the originating voice was printed and you had to figure out how the other voices worked. For example, a following voice doesn't have to be at the same pitch, it can come in at a different interval, above or below. It can even be in a different time signature: Josquin was a particular master of this kind of canon, called a "mensuration"  or "prolation" canon. A following voice can even be in different note values: if larger than the original, it is by augmentation, if smaller, by diminution. You could even do something very fancy and have the following voice invert the original: every time the first voice goes up the second goes down and vice versa. The phrase "in Contrario Motu" in the title refers to the inversion. This leads to the entirely imaginary and hypothetical possibility of a canon in which the following voice enters at a different pitch, uses different note values and is in inversion. Completely nutty, hey? But this is exactly what Bach has done in this canon. The following voice enters five measures later, down a fourth, in double note values ("per Augmentationem") and in inversion: where the originating voice begins with minor 6th up, down minor second, down whole tone, the following voice is down minor 6th, up minor second, up whole tone. Do you see it?

Let's have another listen, this time with the score:

You can see that the two eighth notes, half note, quarter note in the original become two quarter notes, whole note, half note in the following voice. If you follow along, comparing the following voice with the original you will see that this is reproduced exactly all the way.

But wait, after a few pages we see this double bar:

Click to enlarge

This is half way through the piece and the upper voice does a simple cadence to end. This overlaps with the lower voice, which has been the follower, taking up the original voice, but down an octave. Then, as we see in the second line, the upper voice, now takes over the role of follower. What this means is that, not only has Bach written an incredibly complex canon, BUT, the whole thing is invertible! At this point most self-respecting composers would start gibbering into their muffins! This is another of those astonishing feats of compositional virtuosity that no composer since has even tried to equal.

Now I will tell you a secret: it is not as hard to write a canon as it might seem. If you imagine that Bach sat down and wrote all of the original voice, imagining in his mind how the whole of it could be matched up with a following voice, four measures later, down a fourth, in double note values and inverted (and also at the same time keeping track of intervals that do not work if the two voices are inverted) then the whole project is nigh-on inconceivable. But there is a trick to writing canons. Quoting from Peter Schubert's excellent book Modal Counterpoint, Renaissance Style:

Composing a canon is not difficult in principle ... You begin by composing a short fragment as long as the time interval of imitation. Then you copy it into the other part, transposing it at the pitch interval of your choice. Then you write a countermelody to the first fragment. Then you copy the countermelody into the consequent voice. Repeat as desired. [p. 154]

So you actually work both backwards and forwards with both voices. You might be thinking at this point that the canon is a dead technique, but Steve Reich has said in a few places that his basic technique involves writing a lot of unison canons. I intend to analyze some of his music in upcoming posts, so we will look to see how he does that.

Let's listen to the Bach canon one more time. I like this version because it is so slow and it is easier to hear what is going on. Pierre-Laurent Aimard, piano:

Music like this poses some interesting problems for the listener. Are you comparing the following voice with what you heard in the originating voice five measures back? Is it possible to listen on two levels like that? Is that kind of complex mirroring what gives this piece its eerie mood?