I recently came across the idea of applying set theory to musical analysis (which apparently has been around for some time, although I’d never heard of it before). For most people, who have a stronger intuitive grasp of music than mathematics, this must seem a pointless exercise, but for anyone like me who’s the other way around it’s really very illuminating.
Take symmetry, for example. In most areas of the arts and sciences, symmetry is seen as a good thing – but in music, that’s not the case. All the most popular chords are asymmetric in terms of interval content. You can see that in the left-hand image above, which shows the three notes of the C major chord on the chromatic circle. They’re separated by intervals of 3, 4, and 5 semitones.
In contrast, an augmented C chord, shown on the right, is perfectly symmetric, with all three intervals equal to 4 semitones. The problem (as far as musicians are concerned) is that it’s not very firmly tied to C major. It could equally well be A flat or E major. In the same way, the four-note symmetric chord C – E♭ – F♯ – A can be interpreted in four different ways: as Cdim7, E♭dim7, F♯dim7 or Adim7.
There’s even a completely symmetric two-note interval, in the form of the tritone, consisting of two notes 6 semitones apart (or 3 whole tones, which is how it gets its name). That’s exactly half an octave, for example from C to F sharp. But it’s also the distance from F sharp to C, so you really don’t know which key you’re in. That’s why composers spent centuries trying to avoid it. They called it diabolus in musica, or “the devil in music”.
Being a symmetry-loving scientist rather than a musician, I decided to try writing something that consisted only of symmetric chords. It’s a sort of canon, in the key of everything.
Here’s a link to a YouTube video, with added graphics depicting the various chords on the chromatic circle. Hopefully you’ll enjoy the graphics even if you don’t like the music!