The 33rd piece in my canon album is an invertible crab canon. If we use a sequence of numbers to represent the progression of a musical phrase, one line in the canon would be of a form like this:
1 2 3 4 5 5 4 3 2 1
Having reached its midpoint after 1 2 3 4 5, the line reverses and plays its own first half backwards, 5 4 3 2 1. At the same time, the accompanying line plays the backwards or retrograde version followed by the forward version:
5 4 3 2 1 1 2 3 4 5.
So at any given time, you’ll be hearing 1 2 3 4 5 played alongside 5 4 3 2 1, with those fragments exchanging position — moving between the top and bottom lines — midway through the piece. (There’s no significance in the way this example goes to 5 and back; any number could have been used to make the point.)
The structure could be visualized like this:
I first learned about crab canons as a teenager reading Douglas Hofstadter’s Gödel, Escher, Bach. As much as I loved Hofstadter’s treatment of this and so many other subjects, I maintained a sense that crab canons are rather artificial as musical structures. I’m interested in writing canons because I enjoy listening to them, and part of that enjoyment depends on an ability to follow a canon’s structure as it is heard. But crab canons have always seemed to me a bit beyond the capacity of a mortal ear to follow. The idea of playing a musical line in reverse is simple to describe, but not a very natural thing to do. It can be hard for the ear to discern the relationship between a line and its retrograde, and if the relationship isn’t apparent to the ear, what’s the point of basing a piece on it? Now having worked on some crab canons of my own I can respond to that question by saying, first, that the relationship may not be blatant but it may still enter into perception in subtle ways, and second that it does become more apparent when you listen to a crab canon many times and begin to memorize the lines. The highly restrictive nature of crab canons is also valuable in that it demands, and hence inspires, creativity.
Canon 33 is the second crab canon I’ve written. The first one, Canon 31, is a non-strict kind of crab canon where the durations of some notes are allowed to vary between the forwards and backwards versions of the theme. In contrast, Canon 33, is strict crab canon where the note durations are identical in both directions. Another difference between Canon 31 and Canon 33 is that the first is based on the whole-tone scale and has somewhat unconventional harmonic design, while Canon 33 sticks to fairly straightforward tonality. There’s one structural liberty taken in Canon 33: there’s a certain note instance that’s played sharp when the line is moving in one direction but natural when the line is moving in the other direction. Here’s a visualization of the entire Canon 33 in piano roll notation:
In the clip below, you can hear each line of the canon played separately. First you’ll hear the bottom line; then, after a pause, you’ll hear the top:
Now listen to both lines played together in the official track on Bandcamp:
And here’s a video of it: