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:

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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?

Mirror Fugue/euguF rorriM

A while back I mentioned a college textbook on counterpoint: Modal and Tonal Counterpoint: From Josquin to Stravinsky by Harold Owen. It is intended for undergraduates, obviously. Just now I tried to look up "mirror fugues" to see what he had to say and there was nothing. Mirror fugue was for me the occasion of one of those stunning moments of realization that one has in one's life. Let me set the stage.


By a quirk of fate I never took a single class of counterpoint in my undergraduate years. I transferred from one university to another after first year theory. In the first university, they spent first year on harmony, reserving second year for counterpoint. In the second university, they did it the other way around. When I applied to the second university (McGill in Montreal), they told me I had to do some placement tests to see exactly what they would give me credit for and what I would have to enroll in. One of the placement tests involved counterpoint, so I spent the summer teaching myself the basics. As a result, I passed the placement and did not have to do the counterpoint course. I only ever did one counterpoint course, Fugue, the most advanced of all, and I didn't do it until I was in the doctoral program, many years later. It was in this course that I had one of those stunning moments of realization.


Fugue involves a number of interesting techniques, some of which involve transforming the subject. You can make the subject twice as long, by doubling the note values, (quarter notes become half notes and so on) or by inverting the subject: a step up becomes a step down and a leap down becomes a leap up and so on. Bach has lots of great examples. This one, from the Art of Fugue, manages to combine three different versions of the subject in the first three measures. First, there is the subject, which is itself a transformation of the basic subject of the whole Art of Fugue. Then, in measure two the subject again, but upside down and in double note values. Finally, in measure three, the subject in the original note values, but upside down. I omitted the continuation of the first, tenor, voice for clarity.

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Just a bit later, in measure five, the bass enters with the subject upside down and in quadruple note values with the quarter note becoming a whole note! Let's hear that whole piece, the Contrapunctus VII:

Another technique is called "invertible counterpoint" and involves putting the upper voice below the lower, or vice versa. The problem is that not everything works upside down. When you invert voices at the octave, the most obvious interval, the intervals change. Octaves become unisons, 7ths  become 2nds, 6ths become thirds, and so on. The problem, oddly enough, is with the 5ths, which become 4ths. The dissonant intervals like the 7th, stay dissonant in inversion, but the 5th, a consonant inverval, becomes the 4th, which is usually a dissonant interval needing resolution. So just avoid 5ths! Heh. You can invert at other intervals, such as the 10th, but they are more difficult because parallel thirds and tenths become parallel unisons or octaves and therefore must be avoided.


The kind of counterpoint that is written so that it may be inverted is called either invertible counterpoint, or double counterpoint. Now imagine if someone were to write an entire fugue, in three or four voices, and wrote it so that it could all be inverted. Almost unimaginable. But Bach did it three times in the Art of Fugue. Once in three voices and twice with four voices! Here, chosen somewhat at random, are three measures from Contrapunctus XVIII (in my score: the recorded versions call this Contrapunctus 12a and 12b). The top staff is from the right side up version, and the bottom staff is the mirror image:

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As you can see, the soprano becomes the bass, the alto the tenor, the tenor the alto and the bass the soprano. And everything works! I think you have to try writing invertible counterpoint a bit before you can appreciate how difficult this is. Put it this way, in the Fugue course we had been struggling with invertible counterpoint for a few weeks. And then the professor said, "oh yes, and Bach wrote some fugues in which the whole thing, all four voices, were invertible." I nearly fell off my chair. So let's hear the two versions of the fugue. First, right side up. The brief excerpt above occurs about 50 seconds in:

Then, upside down or "inversus":

The whole idea of writing a whole fugue so as to be invertible is at the very stratosphere of counterpoint. I can't think of a single composer who has managed it since Bach...

Invertible Counterpoint

This is a subject so esoteric that Blogger doesn't even recognize "invertible" as a word, but underlines it in red. Trust me, it is a word and correctly spelled. I want to put up a post on counterpoint as a sequel to the one yesterday about multiculturalism. When the narrator describes the music of the Aka pygmies as "extremely complex contrapuntal polyphony" what he is really doing is attacking the worth of what Bach did, by, at the very least, implying that what Bach did was contrapuntal polyphony, sure, but since it was all rigid and didn't allow for spontaneous expression, it's not quite as good as what the Aka do, flowing as it does from their communal wonderfulness.

Now if someone can argue, explain and describe just what is going on with the Aka, then great. Go to it in the comments. From listening and doing some web research, I conclude that what the Aka are doing is an interesting kind of polyrhythmic structure similar to ones found in other places in Africa and in the gamelan music of Java and Bali. What it is not, emphatically, is counterpoint, which is a word describing pitch structures. The Aka sit on the same pitches because they are unfolding a rhythmic structure, not a tonal one. Go have a listen here, sans narrator, and you will see what I mean.

What happened in Western music history in the last thousand years is something of quite another order. True, many of the rhythmic complexities of non-Western music were ironed out, but the reason was so that the possibilities of counterpoint and harmony could be developed. One of the most interesting of these is invertible counterpoint. Here is a great article on the subject. Here is the piece they chose as example, the F minor Invention:

People often ask what is it about Bach's music that makes it so powerful, so fresh even after many listenings. Bach was a master of what is sometimes called "variety in unity". If you go look at the article on invertible counterpoint you will see that the first four measures of the invention are repeated, exactly, as measures five through eight. But they don't sound the same. The reason is that Bach has written them so that the lower voice can be placed above the upper one: this is invertible counterpoint.

Much of the history of Western music can be seen as the quest by composers of ways of structuring music so as to produce both unity and variety. This can be done in a multitude of ways as each composer strives to find a new way with each piece. One of the earliest discoveries was imitation. One voice can be imitated by another on a different pitch level:

Listen to how the eight notes of the theme of the C major invention keep returning and returning in both voices. Of course, imitation doesn't have to be quite so literal. In this canon from the Art of Fugue, the lower voice imitates exactly the upper one, but a fifth lower, in notes twice as long and upside down. Then, in the second half, the voices invert so that the whole piece is also invertible counterpoint!

These are just a couple of the basic ways composers have discovered to knit together pieces of music.