or Why Carriage Wheels in Westerns Appear to Rotate Backwards and What That Has to Do with Global-Southern Music and Digital Signal Processing

There is a great deal of debate in music-theory circles about what the term ‘polymeter’ means. I’m going to ignore this and use my (and a few other people’s) definition. Polymeter, to me, is when two musical streams have the same smallest subdivision in common but have different numbers of that subdivision making up one cycle.

I was just listening to Fodé Seydou Bangoura’s music when I heard yet another delightful polymeter, which I’ll call crossrhythm so that fewer people stop reading this post in anger.

In this crossrhythm, the background meter is some multiple of 3. (Whether it’s exactly 3, 6, or 12 doesn’t matter for the present purpose; it only changes the arithmetic a little, with essentially the same result.

Over this background is a solo that, given the tones and accents, has phrases in groupings of four subdivisions. I call this a polymeter; I think many people call it one type of crossrhythm. More importantly, it is what we call aliasing in digital signal processing (DSP).

Here’s how (and why):

If one stream has groupings of three, like so:

1___2___3___1___2___3___1___2___3___1___2___3___1___2___3___1

and the other has groupings of four:

1___2___3___4___1___2___3___4___1___2___3___4___1___2___3___4

and the former only “looks up” to check where the latter is each time it’s back at 1, then here’s what it sees of the second line:

1___2___3___1___2___3___1___2___3___1___2___3___1___2___3___1

1___2___3___4___1___2___3___4___1___2___3___4___1___2___3___4

which, given only when the base rhythm “looks up” at the other, is like this:

1___2___3___1___2___3___1___2___3___1___2___3___1___2___3___1

1___________4___________3___________2___________1___________4

The act of “looking up” each time the base rhythm gets to one is equivalent to the human eye’s (probably, the neural circuitry’s, rather) rate of perception (sampling). Since we can’t see as fast as the spokes of the wheels on a horse-drawn carriage spin, we see the spokes at each time point we’re able to take a visual sample, just like the first rhythm taking a sort of “downbeat sample” and seeing the second rhythm going slowly backwards: 1, 4, 3, 2, 1, 4, …

If you know DSP, you must have noticed that my example is the opposite of actual aliasing. In signal processing, when the signal we are trying to capture has higher-frequency components than how fast we can sample, we see those high-frequency components folded down to lower frequencies. To match this, the music example should look more like the following.

1___2___3___4___5___1___2___3___4___5___1___2___3___4___5___1

1___2___3___4___1___2___3___4___1___2___3___4___1___2___3___4

However, in this case, the perceived second stream is simple going very slowly, not going backwards the way the spokes do in the westerns. For that, we just need a bigger difference between the reference rhythm and the overlaid one.

1___2___3___4___5___6___7___1___2___3___4___5___6___7___1

1___2___3___4___1___2___3___4___1___2___3___4___1___2___3___4

Now we see the “spokes” slowly turning in the opposite direction: 1, 4, 3, …

When I set out the write this post, I didn’t realize that there would be some cases in which the slow (aliased) rhythmic stream would meet the reference meter without the reversal of direction. Does this happen in aliasing in DSP? It does: Aliased components show up in both positive- and negative-frequency basebands. But that doesn’t seem to answer my question because they combine to form one real-valued low-frequency signal.

I think the answer is that whenever signals (rhythms, turning wheels) are periodic and there is a phase relationship between two such entities, that phase relationship is modulo-2π: If you switch where you look up or down the cyclic waveform, you’ll see the phase shift moving forward or moving backward.

A better answer, perhaps, is that the crossrhythm examples are akin to passband sampling, where a communications signal is modulated up to a band with a lower and an upper bandlimit but the Nyquist requirement is a lot nicer than the usual more-than-doble-the-highest-frequency but simply more than double the bandwidth. In that case, I expect to see the wagon wheels going forward as well as backward, depending on which end of the band we are close to.