I mentioned musical set theory in a previous post, and now that I understand it better I’m getting very enthusiastic about it. It’s a really powerful technique for analysing and composing music. The mathematical connection may give the impression that it “dehumanizes” music by imposing mechanistic constraints and artificial rules – but the exact opposite is true. It’s traditional music theory that forces arbitrary rules and constraints on you – set theory liberates you from them. It’s a framework for organizing your own creativity – with no rules whatsoever.
I’ll explain how it works in a moment, but first a few words about my sources. The bible of the subject is Allen Forte’s The Structure of Atonal Music, which is divided into two roughly equal parts. The first is packed with useful stuff, although the second part was much too advanced for me. But Forte’s book is really about musical analysis, and what I was interested in was composition. On that front, I found a great little book by Stanley Funicelli called Basic Atonal Counterpoint (which is a CreateSpace book, but very professionally done). I also found a lot of practical tips on Frans Absil’s YouTube channel – he also produced the Pitch-Class Set Graphical Toolkit you can see on my iPad in the photograph above.
Musical set theory starts from a few basic observations:
- The notes of the chromatic scale can be represented by integer “pitch-classes”: C = 0, C# = 1, D = 2 etc. After B = 11 you get back to C = 0, so additions and subtractions have to be done with mod-12 arithmetic.
- Intervals between pitch-classes are much more important than absolute pitches. So C major [0, 4, 7] and E flat major [3, 7, 10] are just different transpositions of the same set (it’s called 3-11).
- Inverting an interval (i.e. subtracting it from 12) doesn’t change its basic nature. So interval 7 (perfect fifth) can be grouped with 5 (perfect fourth), interval 8 (minor sixth) with 4 (major third) etc. This leaves us with just six “interval classes”: 1, 2, 3, 4, 5, 6.
- The characteristic sound of a set is mainly determined by its interval vector. For example, the major chord 3-11 = [0, 4, 7] has an interval vector 001110 (one minor third, one major third, one perfect fifth and nothing else).
Traditional Western music depends heavily on set 7-35 [0, 2, 4, 5, 7, 9, 11] – the white notes on a piano, aka the major or minor scale (remember you can transpose these notes up by any integer between 1 and 11 to get all the other major and minor scales). Within that 7-element set, there are a number of strongly favoured subsets – most notably the aforementioned 3-11 (the major triad and its inversion, the minor triad).
The purpose of set theory should be obvious now. It gives you access to dozens of other sets, all with their own unique sound. You might think “but they’re going to sound terrible”, and in some cases they do. Set theory helps you to avoid the terrible-sounding ones! But there are some great-sounding sets that simply don’t exist in traditional music theory, such as 4z-29 = [0, 1, 3, 7], with an eyecatching interval vector of 111111.
To teach myself how the system works, I wrote a short “symphony” using the above ideas. It’s my first ever musical composition, and the result sounds a lot more interesting than if I’d struggled with all that traditional stuff about sharps and flats, majors and minors, dominants and subdominants etc. That wouldn’t have told me how to get close to the kind of spooky, spacey, quirky music I wanted to write.
Here is a link to the YouTube video: