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.

Click to enlarge

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:

Click to enlarge

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...