How To Build A Rhythmic Tiling Canon (draft)

As I’ve mentioned in previous posts, a rhythmic tiling canon is a composition in which each player repeats the same rhythmic pattern, with each player beginning at a different time.  The rhythmic pattern and entry points are crafted so that once all the players have begun to play, there will be exactly one player striking each beat (that’s to say, each beat is hit by one of the players but no two players ever coincide on the same beat).  Why should musicians be interested in these canons?  The short answer is that many of them sound good.  (Yes, “good” is a technical term that I will leave undefined for now.)

Much of the literature on rhythmic tiling canons is aimed at mathematical readers who want to understand how to enumerate all possible canons with certain properties.  In my reading on the topic so far, I haven’t come across a simple explanation, aimed at musicians, of how one could go about constructing a rhythmic tiling canon by hand.  It turns out that some kinds of tiling canons are quite easy to build even if you haven’t delved into the mathematics of canon enumeration.

So let’s build a canon!  Let’s say we want our canon to have three voices.  Let’s make the entrances of the voices equally spaced, meaning that the lag between the first and second voices is the same as the lag between the second and third voices, and so on.

Now let’s choose how long the rhythmic pattern will be.  Let’s choose some number of beats that’s a multiple of the number of voices.  We have three voices; let’s use a multiplier of four.  We’ll make the rhythmic pattern last for 3*4=12 beats, and we’ll think of it as being comprised of three sections of four beats each.  We’ll call these three sections a, b, and c.  We can label all twelve beats in the pattern like this: a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4.

Now we need to decide which beats will have a hit or onset, and which beats will be silent.  Notice that when our pattern is played as a canon, the first voice will enter and play section a (that is: a1, a2, a3, a4).  When the first voice moves onto section b, the second voice will enter and play section a simultaneously with the first voice’s section b.  When the second voice moves to section b, the first voice will move to section c, and the third voice will enter with section a.  After all the voices have entered, there will be exactly one voice playing each section at any given time.  As the canon progresses, we’ll always be hearing sections a, b, and c together; the only difference will be that if we check in at one time, we might hear section a in voice 1 while if we check in at another time, we might hear it in voice 3, and so on.

We’ll have a correct tiling canon if our sections a, b, and c are structured so that when the three of them are played simultaneously throughout their common span of four beats, each of those four beats has a hit in exactly one section.  That means if there’s a hit at a1, there can’t also be a hit at b1, otherwise we’d have two voices hitting simultaneously.  It also means that if there’s no hit at b1 or c1, there must be a hit at a1, or else we’d have a beat with no hit in any of the voices.

So we can construct our pattern by iterating through the number of beats in a section and choosing one section to have a hit on each particular beat.  It might go like this:

For beat 1, let’s choose section a as the one with the hit.  If there’s a hit at a1, then b1 and c1 must be silent.  We’ll make the beat with a hit bold, and we’ll strike out the beats that we know must be silent:

a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4

For beat 2, let’s choose section b.  We’ll put a hit at b2, which means a2 and c2 must be silent:

a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4

For beat 3, let’s put a hit at c3, which means a3 and b3 must be silent:

a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4

For beat 4, let’s put a hit at a4, which means b4 and c4 must be silent:

a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4

This gives the following pattern:

100101000010

A mathematician would immediately point out that you could follow this process in multiple ways and end up with essentially the same canon.  You need to work through some math to understand how many unique canons this process can generate.  However if you’re a musician who just wants to get your hands dirty building some canons (with n equally spaced voices where the pattern length is a multiple of n), you’ve just seen a way to do it.

Another way for a musician to think of this is: zoom in on a blank part of your score where the three voices are to play simultaneously for four beats.  Now fill in the details: add hits to the score so that  each beat has a hit in exactly one of the voices.  Now take this three-voice passage and flatten it out by splicing the material from voice 1, voice 2, and voice 3 together into a sequence of twelve beats: the material from voice 1 becomes section a, the material from voice 2 becomes section b, and the material from voice 3 becomes section c.  Now use that flattened sequence as the rhythmic pattern that all voices play in a canon with equally spaced entrances.

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