Rhythmic variations: a mathematical-musical explanation (part 2: syncopation)

This post is a follow-up part 1 here, in which I laid out a musical theme and the different possible rhythmic combinations one could make by "choosing" different "empty" beats. To recap, here's the musical theme, which you can play back by hitting the play button in the upper left corner, along with all examples embedded in this post! (Although warning, they load pretty slowly, it seems.)

Since it has 9 notes over 12 beats, we can vary the rhythm by changing which 3 beats will be without a note, which is equivalent to choosing which 9 beats will have a note. Therefore there are "12 choose 9" or "12 choose 3," or 220, possible rhythmic variations for the theme. At the end of the post, I claimed that 191 of those possibilities are syncopated. So what does it mean for a melody to be syncopated, how did I arrive at that number, and why does it matter that there are so many more syncopated possibilities than non-syncopated?

In this post I'll define syncopation more carefully with supporting examples, explain the basic algorithm I wrote to detect syncopations in a rhythmic sequence, and explain why it's important that basically any rhythmic pattern has more syncopated variants than not.


What is Syncopation: inverting the hierarchy of beats
In music, the passage of time is organized into beats. Or really, it might be more accurate to say, as humans, we naturally and inevitably organize musical events into beats. Repetitive cycles of beats set up a perceptual hierarchy where some beats are "strong" and some beats are "weak." Syncopation, loosely defined, is when a musical event in one manner or another inverts that perception, placing an emphasis on the weak beat over the strong one.

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