Different, Yet the Same? Musical Experiments with “Modular Symmetry”

I like music — enough to explore new musical ideas, at least. A few months ago, I was playing around with chord progressions when I stumbled across a new concept (well, I think it’s new). The idea involves rapid key changes that come together to form a larger, overarching modulating pattern, giving off a “drifting” feel. Let me take you through my thoughts that guided me to this discovery. I’ve included a song I wrote based on this concept at the end of this post.

When playing around with chord progressions, I arrived at these 2 bars:

We can see that it starts on an F chord and ends with a V-I resolution to D♭. Easy enough, right?

Let’s think of ways to expand upon this idea. We can continue the pattern by replacing the 2nd bar with the 1st bar, modulated to start on D♭, and by extending this pattern further, we get the following chord progression, which we will call $X$:

We can see that after 3 bars, we resolve back to F, where we started!

Using rapid key changes in this manner isn't new by any stretch. It's pretty common in Jazz.

But what if we started at a different bar? The way we constructed $X$ enforces the following property:

Choose any bar in $X$ as a starting point. No matter where we pick, we get the same chord progression, just in a different key. We will say that $X$ has modular symmetry of 1 bar.

Let’s think about looping $X$ over 4-bar melodic phrases, which would give us a “4:3 polymeter between melody and key center” — or whatever term you prefer to use. Notably, after every phrase, the chord progression shifts back by 1 bar. Now here’s the cool part: $X$’s modular symmetry implies that:

For all $n \in \mathbb N$, the $(n + 1)$th 4-bar sequence in $X$ is a modulation of the $n$th 4-bar sequence.

So we’ve changed keys without breaking our chord loop! Even better, the chord structure remains the same across phrases. This leads us to interesting musical ideas like:

Voila ! The 2nd phrase has the same structure as the first one, which, to me, feels “drifty”, as if you’re floating out in the sea without a destination. We can also look at this from a symbolic standpoint — from the 1st phrase to the 2nd, we move to somewhere else (i.e. we change keys), but we make no actual progress (i.e. the melody stays the same).

I also get an “aimless” impression from the underlying chords, which I suspect comes from the key center changes every bar; a feeling of home is never established, so the chord progression can never “settle down”.

I don’t think I’ve heard this “aimless wander” quality in music before — and I think it’s a really cool effect — so I’m sure there’s a lot of potential in further exploring this concept. As I promised, here’s my song that I mentioned at the beginning of this post:

I hope you enjoy it .

What did we learn? That applying math concepts to music can lead to interesting results? Or maybe that you don’t care about all of this? Either way, thanks for reading.

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