This may be pretty niche... For a project I'm working on I've been reading this paper on MusES; my goal is to find a simple musical representation system (or build my own) for a program I'm working on.
On the second page they talk about dealing with the algebra of pitch-classes, and state
There is a non-trivial algebra of alterations, which includes the following pseudo-equations:
# o b = b o # = identity x o natural = natural, for any x in (#, b, natural)
This algebra is non trivial because not everything is allowed, at least in the classical theory, e.g. triple sharps.
First, is it safe to assume that
o is basically function composition? It also seems that the convention
(a o b)(x) := b(a(x)), somewhat common in group theory, since otherwise the second equation makes no sense. Am I understanding this notation correctly?
Second, why limit this algebra to things that are denotable in regular music theory? I'm a bit confused why they would couch this in the abstractions offered by an algebra, where the
b operations can easily be represented as
-1 functions, and where
natural is the zero function, if they are going to insist on conforming to a notational system not built for such abstraction. I get that triple sharps don't show up on the page, but they certainly exist, even if we don't use them. Is there some larger context to this that I'm missing? I'm coming from more of a math/comp sci background so it is entirely possible that this is a convention in music theory that has some nice properties that I'm not aware of.