Saturday, March 30, 2013

Music Theory As An Algebra

In the introduction to Charles Pinter's A Book of Abstract Algebra, he covers a very brief history of algebra, examines differences between matrix algebra, numeric algebra, and boolean algebra, and then says:

Other exotic algebras arose in a variety of contexts, often in connection with scientific problems... Today it is estimated that over 200 different kinds of algebraic systems have been studied, each of which arose in connection with some application or specific need.

As legions of new algebras began to occupy the attention of mathematicians, the awareness grew that algebra can no longer be conceived merely as the science of solving equations. It had to be viewed much more broadly as a branch of mathematics capable of revealing general principles which apply equally to all known and all possible algebras.

What is it that all algebras have in common? What trait do they share which lets us refer to them as "algebras"? In the most general sense, every algebra consists of a set (a set of numbers, a set of matrices, a set of switching components, or any other kind of set) and certain operations on that set. An operation is simply a way of combining any two members of a set to produce a third member of the same set.

Thus, we are led to the modern notion of algebraic structure. An algebraic structure is understood to be an arbitrary set, with one or more operations defined on it. And algebra, then, is defined to be the study of algebraic structures...

Any set, with a rule (or rules) for combining its elements, is already an algebraic structure. There does not need to be any connection with known mathematics. For example, consider the set of all colors (pure colors as well as color combinations) and the operation of mixing any two colors to produce a new color. This may be conceived of as an algebraic structure. It obeys certain rules, such as the commutative law (mixing red and blue is the same as mixing blue and red). In a similar vein, consider the set of all musical sounds with the operation of combining any two sounds to produce a new (harmonious or disharmonious) combination.


Pinter's definition is of course almost insanely broad. "Combining any two members of a set to produce a third member of the same set" describes sexual reproduction, the construction of sentences, software development, Borges's infinite library, and a huge variety of other things. As long as you can discover any form of predictable combinatorial patterns, you can infer rules exist. But I like this definition very much, because it is a really useful way to understand music theory, especially Western musical notation, which definitely qualifies as a sophisticated and deep algebra under Pinter's definition.

Of course, music theory is not everything. Orchestral musicians have a saying that "90% of the music is not in the score." (The term "score" refers to the written representation of the music.) The same is extremely true of my own long-term obsession, electronic dance music, where the number might be nearer 99%. It's very, very common in both techno and trance to play the same melody or bassline repeatedly, changing the timbre (or sound design) while leaving the motif unchanged.

Here are a couple of very illustrative examples. It occurs quite a lot in other subgenres as well, of course; the first example is from drum and bass. There's also good coverage of this in Unlocking The Groove, which despite its seemingly casual title is a thick academic book from a music theory PhD.

Anyway, although music theory is not everything, dance music is as hopeless without it as any other form of music. And speaking of challenging books, I'm only about 20 pages in — three appendices and half an introduction — but Pinter's book is so far the most entertaining book on math I've read in a very long time.