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 steer clear of the debate and use the definition that I and a few others prefer. Polymeter, to me, is when two musical streams have the smallest subdivision in common but have different numbers of that subdivision constituting a cycle. I will also refer to this as ‘crossrhythm’, both for variety and in an effort to placate some potential readers.

What motivated this post was hearing yet another delightful example of polymeter, this time while listening to Fodé Seydou Bangoura.

In this example, 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 aurally the same result.)

Over this background is a solo that, given the tones and accents, has phrases in groupings of four equal subdivisions. I call this a polymeter; some call it crossrhythm. More importantly, it corresponds to 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 down” to check where the latter is each time it (the triple-grouped one) is back to the downbeat (beat 1), the bolded entries are 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

i.e., “1..4..3..2..1..4.”

“Looking down” each time the base rhythm gets to 1 is equivalent to the human eye’s (our neural circuitry’s, rather) rate of perception (sampling rate). Since our visual system does not sample as fast as the spokes of the wheels on a horse-drawn carriage spin through one cycle, each time point we’re able to take a visual sample we catch a given spoke having not quite come all the way around, just like the first rhythm taking a “downbeat sample” and seeing the second rhythm not having completed its cycle yet and appearing to go slowly backwards: 1, 4, 3, 2, 1, 4, …

A caveat: In the case of visual perception, the perceiver’s sampling rate is a little lower than the moving images’ cycle rate. In the musical-rhythm case, the reference meter in my example cycles faster. If you know DSP, you must have noticed that this is the opposite setup to that of 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, appearing slower for a different reason. 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 only going slowly, not 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 musical “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 whether you look up or down the pair of cyclic waveforms, you’ll see the phase shift progressing forward or backward.

A better answer, perhaps, is that the polymeter examples are akin to passband sampling, where a communications signal is modulated up to a band whose width is much smaller than its center frequency. In this case, the Nyquist requirement is a lot nicer than the usual (more than double the highest frequency); it only needs to be greater than double the bandwidth, f_high – f_low instead of twice f_high. In that case, I expect to see the wagon wheels going forward or backward depending on which end of the narrow band we observe.