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.

  • Out of curiosity, is there a reason you're staying away from pitch-class set theory? It sounds like this is a more complicated version of it.
    – Dom
    Commented Aug 1, 2018 at 1:53
  • Not really. I'm trying to find a simple music representation system that I can use to implement some of Fux's counterpoint as a system of constraints. I've been googling "music representation systems" and this is one of the few things I've found that isn't behind a pay wall. Commented Aug 1, 2018 at 1:58
  • 1
    From my general experience (and a math degree) the quality of the "pseudo-mathematics" in a lot of modern music theory is abysmal, judged as mathematics. But I don't have a (publishable!) opinion on whether this is deliberate obfuscation of the trivially obvious, or just plain mathematical incompetence.
    – user19146
    Commented Aug 1, 2018 at 3:45
  • I'm not sure if this trivial observation blows away the basic axioms of the referenced paper - but in any tuning system other than ET F sharp is NOT the same note a G flat - they don't even have the same pitch. And unless your theory can deal with that, it won't "understand" tonal harmony very well - 12ET intonation is only an approximation to the Platonic "key system" underpinning tonal harmony IMO! Why else would Bach have written a prelude in Eb minor followed by a fugue in D# minor, in the Well-Tempered Clavier, unless he understood that fact?
    – user19146
    Commented Aug 1, 2018 at 3:51
  • … and as an aside, the Eb minor prelude includes a structurally important chord of Fb major, while the D# minor fugue does the same thing with an E# major chord. Just happenstance? I think not...
    – user19146
    Commented Aug 1, 2018 at 3:55

1 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:

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.

  • Great response. I agree that in standard notation a flat on a C# gives a Cb, in the paper they are defining algebraic operations b and # which, by their axioms of # o b = b o # = natural, codify the inverse property. In the paper they seem to sharpen and flatten rather than yield a sharp or flat note. But, as you say, this goes against standard notation. Commented Aug 1, 2018 at 6:05
  • Viewing a pitch class as a tuple (base-note, accidental), with base-note in A, B, ..., G and accidental an integer, this additive style notation is I think the same (up to some modulus) of your rotational notation. Commented Aug 1, 2018 at 6:07
  • I'm not sure about the second axiom, but if I'm not mistaken I think that it is saying that natural just sends (base-note, accidental) to (base-note, 0), where any non-zero accidental is cleared? Commented Aug 1, 2018 at 6:10

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