Note: all musical examples in this post can be played back by opening the page here. 

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.

The details:
Here's the new piece:

Here's the main theme at the beginning (ex. 1):

Here's the fifth of the recently-completed six-movement "Syncopated Suite"! It starts out like a Gavotte from a Bach suite, but the rhythmic style again evolves into something different! More about this suite coming soon!

It's another "Mary Had a Little Lamb." Don't be fooled by the opening; the piece evolves gradually (and I hope seamlessly) in rhythmic style as it progresses, adding more and more syncopation.