My Chicago Roots

Ziehn.jpgI’ve always had a fascination with canons, even long before I wrote a book about a composer (Nancarrow) whose major works were mostly canons. In the late 1980s, when I was in the habit of lecturing on the history of Chicago’s new-music scene at the School of the Art Institute and other places, I ran across, in a Chicago used bookstore, a little book called Canonical Studies, by Bernhard Ziehn (1845-1912, pictured). I recognized the name. Ziehn was one of two German composer-theorists who were living in Chicago when Ferruccio Busoni toured through. Busoni was trying to solve the puzzle of how the four fugue subjects
fit together in the unfinished fugue from Bach’s The Art of Fugue, and Ziehn solved it for him, enabling Busoni to write his Fantasia Contrappuntistica, which has long been one of my very favorite works in the world. His tour over, Busoni wrote an article about Ziehn and his colleague Wilhelm Middelschulte, titled “The Gothics of Chicago,” by which term he meant that they were masters and fanatics in the ancient art of counterpoint. Ziehn and Middelschulte taught a lot of the early Chicago composers, including John J. Becker (one of the “American Five”), whose widow I knew in Evanston. So I had multiple connections to Ziehn, and snapped the book up at once.

All but forgotten today (there’s a brief entry about him on German Wikipedia, none in the English one, and the second reference that came up on Google was a page of my own), Ziehn was ahead of his time. Books he published in the 1880s anticipated and classified chords (such as those based on the whole-tone scale) that the impressionists and Schoenberg would use considerably later. In the intro to Canonical Studies, Ziehn writes,

A canon is by definition strict. Our greatest authorities assert “strict” canons can be carried out in the Octave of Prime only. The examples given in this book demonstrate that real canons are possible in any interval…

And he gives examples of chord progressions that modulate to every possible interval away from the tonic, showing how one can continue repeating those progressions in ever-moving transposition to write canons not based on the octave or unison.

I was intrigued, and in 1987 wrote what I call a “spiral canon” as the third movement of my violin piece Cyclic Aphorisms, a canon at the major second. Then, more ambitiously, in 1990 I wrote Chicago Spiral, a nine-part triple canon also at the major 2nd, putting a postminimalist spin on Ziehn’s idea. A canon is easy to perceive as such at the unison, octave, or even fifth; it’s more
challenging at a more distantly related interval. A canon is also easier to process aurally if the beat-interval of rhythmic imitation is something symmetrical like 4 or 8 beats, more difficult if it’s 13 or 31. One thing I’ve realized is central to my music is that I love to fuse the simple with the incommensurable, making the listener think it ought to be easy to figure out what’s going on, but keeping it just out of reach. My Ziehn-inspired spiral canons ought to be simple to figure out by ear – they’re only canons, after all – but the complexity of the imitation intervals, both rhythm and pitch, keep the ear, I think, from ever quite settling into them. I also use the technique as kind of a postminimalist gradual-texture-metamorphosis generator, which is a little beyond what old Ziehn had in mind, I imagine. Paradoxically, the longer the rhythmic interval of imitation, the less gradual the changes can be made.

And now in recent months I’ve written two more such canons, Hudson Spiral and Concord Spiral, both for string quartet. Along with the middle section of my orchestra piece The Disappearance of All Holy Things from this Once So Promising World, I’ve produced five spiral canons altogether, at the following rhythmic and pitch intervals:

Cyclic Aphorism 3: 5 beats, major 2nd ascending
Chicago Spiral: 7 beats, major 2nd descending
Disappearance: 17 beats, minor 3rd descending
Hudson Spiral: 83 beats, major 6th ascending
Concord Spiral: 19 beats, minor 7th descending

The major 6th and minor 7th are the optimal intervals for a string quartet canon; using a major 6th, the cello can play down to its low E-flat (echoed by the viola’s low C string and second violin’s low A), and the first violin can play down to the F# above middle C, whereas with the 7th the cello can descend to D and the first violin only to A-flat in the treble clef. Concord Spiral generated some nice passages of what sounds like tonal Webern:


The scores are on my web site if you’re interested, and no performances are yet forthcoming. Spiral canons and Snake Dances are the two personal genres I feel I’ve invented for myself, along with my more generic tuning studies and Disklavier studies. And I hope Ziehn would have been happy to know that, 98 years after his death, his idea is still out there making the rounds.


  1. mclaren says

    Canons have a wonderful history and go way back. However, the question of whether a canon is strict is not so simple. There’s a broad gray area twixt canon and imitation and variation.
    For example, the baroque tradition of ignanno stretches the definition of a canon. That’s imitation with slight variation. Typical Baroque ignanno uses slight pitch variation in the follower but the same rhythmic pattern as the leader. However, gothic mensuration canons reverse that variation, but using the same pitch pattern, using the same pitch as the leader but a different rhythmic pattern. In gothic mensuration canon the variations arise from differences in splitting the longa into 2 or 3 parts. But this can be (and has been) extended by current composers. Suppose you split the whole note into 5 parts as opposed to 7 parts? Or 11 parts as opposed to 13 parts? Lots of opportunity there to update mensuration canons. Is it actually a canon if, say, the leader splits the whole note into 3 parts and the follower splits it into 2 parts? Well, the gothic composers said yes.
    We’d have to call these quasicanons. And there’s a rich field there to explore. For example, is a Nancarrow acceleration canon a strict canon? Most of us would say yes, judging by he general agreement on classifying Nancarrow’s acceleration canons as true canons, even though the rhythmic values in the follower are radically different from those in the leader. Suppose you change the shape of the tempo curve? For instance, suppose the leader is a simple acceleration canon but the follower is an accelerated acceleration canon (easy to do this by setting up tempo changes in a MIDI sequencer and playing back the result into another computer and recording it, then setting up the same tempo changes again). Is that a strict canon? How about the 3rd derivative (accelerated accelerate accelerated canon)? Or the 4th derivative?
    Or how about applying tempo changes separately to each phrase of a canon? For instance, suppose the first phrase uses a 3:1 speedup, but the second phrase uses a 5:1 speedup, and the third phrase uses a 7:1 speedup and the fourth phrase uses an 11:1 speedup? (It gets even more elegant if you just just intonation pitches where the pitch ratios relate to the tempo accelerations.) Is this still strict imitation? If we call a Nancarrow acceleration canon strict, as most folks would, we’ve have to say yes…but now it’s getting a little more dicey. We’re well into the Paradox of Theseus’ Ship at this point.
    This brings up the issue of pitch and rhythmic sequences, which nowadays need not be powers of two (rhythms) or nth roots of 12 (pitches). What happens when you take one sequence and map it nonlinearly into another sequence? Well, that’s what the gothic composers did with mensuration canons where the leader might divided the longest note into 2 parts but the follower divided it into 3 parts. Gothic composer regarded these kinds of mappings as strict canons, so why shouldn’t we?
    We can extend that gothic practice today — indeed, music typically renews itself going forward by looking back. Suppose, for example, you map a rhythmic sequence like whole-note half-note whole-note quarter-note quarter-note into tuplets which aren’t quite as neatly related as the leader, but close: for example, whole-note 9:5 whole-note 9:5 11:2 11:2. In that case the mapped replacements for the half notes have 5/9 or 55.55% of the length of the whole notes, not quite the same as the half notes in the original, while the replacements for the quarter notes have 2/11 or about 19.19% of the length of the quarter notes, as opposed to the 25% of the original quarter notes in proportion to the whole notes.
    When you play mapped sequences like that, you get the damndest syncopations. Now, is that a strict canon? There’s a strict mapping twixt leader and follower, so in that sense you’d have to say yes. On the other hand, the strict proportion in the leader twixt the longest and shortest notes are not quite preserved in the mapping. The paradox of Theseus’ ship with a vengeance.
    Composers have done this sort of not-quite-strict mapping from leader to follower for a number of centuries now. For instance, it’s long been standard practice when inverting a follower to invert it diatonically rather than chromatically. That keeps the follower in the same key, which was fairly important if you were using meantone tuning.
    Mathematically speaking, this kind of not-quite-exact mapping is known as a tensor transformation. What we’re really talking about here is a representation of a vector field. In a typical canon all the members of the follower’s vector field get mapped by exactly the same amount. In a Nancarrow acceleration canon, all the members of the follower’s vector field get mapped by a function that changes by exactly the same amount with each note. But in gothic mensuration canon the follower gets mapped by values which vary according to the value of the note. We can extend this today with a variety of new pitch and rhythm modifications, up to an including changing the intonation of the follower and changing the tempo curve of the follower — proving, of course, that medieval composers had much more sophistication than anyone likes to give ’em credit for.
    KG replies: Well, canon means rule, so I’d say that as long as there’s a rule that’s unvaryingly followed, that’s strict. As for the quasi-canons, I would hope the term prolation canon is still around.