In the previous post, we looked at the very basics of musical set theory. Namely, we defined the pitch classes and considered the operations Transposition & Inversion. Where given an integer n, Transposition is defined by: T(x) = x + n (mod 12) & Inversion is defined by: I(x) = n – x (mod 12).
All pretty basic stuff, we didn’t use any complicated mathematics and the ideas were all quite simple. However, we can really start to use some interesting mathematics if we define these pitch classes as what is known as a ‘group’.
A group is a type of algebraic structure that satisfies certain conditions. Put formally, given a set, G, and an operation, *, that acts on any two members of G. ( G , * ) is defined as group if the following four axioms hold:
- For all a, b in G, the result of the operation, a * b, is also in G.
- For all a, b and c in G, (a * b) * c = a * (b * c).
- There exists an element e in G, such that for every element a in G, the equation e * a = a * e = a holds.
- For each a in G, there exists an element b in G such that a * b = b * a = e, where e is the identity element.
(Definition taken from this wikipedia article).
There are many different groups that have been defined (in fact all finite simple groups have been classified). We’ll look at the group ‘Z mod 12′ which contains the integers from 0 to 11 as its set and the operation being addition modular 12. Sound familiar? This is because the group of pitch classes with operation modular 12 is ‘Isomorphic’ to ‘Z mod 12′. Isomorphic basically means that the group of pitch classes acts exactly the same as ‘Z mod 12′.
Perhaps more interestingly, groups can be defined using the operations we have defined previously. In fact both transposition, T(x), and inversion, I(x), can defined as group. Notably, these are both isomorphic to different permutation groups.
Many other interesting properties about musical set theory, it’s relevant group theory and the applications in atonal music composition can be found in this fantastic paper.
I’m still a novice when it comes to this field, so i’ll end this post with a quote from Hubski user ‘coffeesp00ns‘, someone with greater knowledge than I. The following is an excerpt from a really insightful comment left on the link to the previous post.
[Musical set theory] basically takes the breakdown of “Traditional” western harmony that occurred over the 19th century and early 20th century, then brings it back into the Baroque period where music was in many ways more complicated. faster harmonic rhythm ( often changing every beat), more complex musical styles (specifically the fugue) combined with the tonality (or more specifically, the lack thereof) and the difficulty of the intervals to tune correctly (we’re not used to hearing tritones and semitones as easily as 8ves, 5ths and 4ths) makes for music that is difficult to hear, difficult to play, and requires a lot of knowledge to appreciate.