Motivation: Rhythm gets short shrift!
Lately, I've been thinking a lot about rhythm! In a traditional study of music theory, one learns a ton about harmony but very little about rhythm beyond some very basics. Similarly, in studying music history, there's an overemphasis on the harmonic trajectory of Western music, but almost no attention to its rhythmic evolution. And yet often the most salient difference between musical styles is rhythmic. Especially when one considers folk and popular music of the past century, there's a surprising continuity of harmony! Rhythm, however, is another matter.
The point: Math is useful (and fun!) for building on intuition about rhythm and syncopation
This post and the next approach rhythm from a mathematical standpoint, starting with an example of a piece I wrote recently in comparison to one of Bach. In using variations of the main rhythmic theme, I realized most of them were syncopated; this post explains some of the math that backs up that intuitive realization, compares the theme to its obvious Bach counterpart, and motivates further exploration of the topic.
Here's the new piece:
There are many purely melodic (that is, intervallic) changes to the theme in the Bach that leave it in tact and recognizable, but essentially no rhythmic changes. In my piece, there's both types of variation. For instance, instead of repeating the first half of the piece (as is typical in Baroque music and happens in the Bach gigue), the second section of my piece starts with the same notes as the main theme over the same duration (two measures), but distributed differently:
This raises the question: how many rhythmic variations are there of this melodic idea, keeping the order of the notes, the total duration and (for now) restricting notes to the falling on the beats? The answer will probably surprise you (unless you're already well-versed in the math / combinatorics of the situation): 220. That's a lot!
Here is an explanation (for a more thorough one, scroll down to "combinations" here, or just skip to the next section): there are 9 notes in the idea, and they occur over 12 beats (2 measures with 6 beats each). That means there are going to be 3 "empty" beats without a note beginning on them. Making a choice of a purely rhythmic variation for the 9 notes is equivalent to choosing which 3 beats will be empty. For instance, in example 1 above, the "empty" beats are 2, 5, and 11 (where beat 11 = beat 5 of the second measure). The arrows below point to the empty beats.
If we had instead chosen where to put our 9 notes instead of our 3 empty beats, we would have ended up with the same answer (12 choose 9 = 12 choose 3), only it would have taken a lot longer to explain!
As one more example, let's say we did choose 2, 6, 10 (or 10, 2, and 6) for our empty beats. What would our melody look like?
The next question is, how many of the rhythmic variations are syncopated vs un-syncopated? Here I'll tell you the answer and follow up with explanation of what it means, why it matters, and how I calculated it in part 2. The answer is: 191 syncopated, 29 un-syncopated. That's a lot more syncopated than not!