MusES musical representation system -- questions on the "algebra of pitch-classes" and some conventions

Question

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 # and b operations can easily be represented as +1 and -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.

Answer 1

I've never heard of this system, but it has a lot of similarities to other concepts I've come across.

The second question is much easier to explain than the first and maybe it will clear up what's. When talking about modifying a single note from it's base position, yes putting sharp as +1, flat as -1, and natural as 0 totally makes sense and I've done it in projects I've worked on in the past. There is a problem with that logic which is sharps and flats aren't additive, they override each other instead so assigning +1 to a sharp and -1 to a flat may lead to the wrong conclusion. In the same vein of thought, even the same symbols aren't additive! For example if you see a flat in the key signature and have a flat next to the note on the staff, the note is flat not double flat.

Questions on the site about these consents:

• Are accidentals in the key signature and measue additive?
• Do accidentals override key signature and previous accidentals?

Now back to the first. The best I can come up with for algebraic expressions they are rotations of the 7 PCs across one of the other 5 or else I can't really make sense. The author mention 7 PCs in correctional to the 7 notes A-G bring the total to 35 due to the 5 accidentals double flats to double sharps. So he mentions the sets like this:

Abb Bbb Cbb Dbb Ebb Fbb Gbb
Ab  Bb  Cb  Db  Eb  Fb  Gb
A   B   C   D   E   F   G
A#  B#  C#  D#  E#  F#  G#
A## B## C## D## E## F## G##

How the pitch equivalents though line up is shown below:

    0   1   2   3   4   5   6   7   8   9   10  11
0   A   A#  A##  
1   Bbb Bb  B   B#  B##
2       Cbb Cb  C   C#  C##
3               Dbb Db  D   D#  D##
4                       Ebb Eb  E   E#  E##
5                           Fbb Fb  F   F#  F##
6   Gb                              Gbb Gb  G   G#
0                                           Abb Ab 

The transformation only really makes sense on the first chart when you look how the vertical rotations happen across the same letter names. The second chart, saying # and b always have some kind of identity is wrong as seen in row 7 and 8 where sharps are not inverses for flats and vice versa as you get into double sharp and double flat territory.

In closing, I will note the the last 3 bullet bullet points of section 2.1 which you are quoting are incorrect (or at least very misguided). Triple flats and triple sharps exits and the key of G# Major does exist. While they are at the extreams of musical knowledge they are well defined concepts.