Music and Ring Theory: Automorphisms

Recently, I’ve been looking into how ring theory is able to model certain aspects of my music theory studies. Math truly is in everything! I’ll be focusing a little more on automorphisms (preserving algebraic structure) in this post.

In Western practice, we say that we are in a twelve-tone system, each of
the twelve notes can be denoted by a value. If each element represents a pitch class, we can say that we are in ℝ₁₂. If we wish to transpose a note or a couple notes by kk semitones, we can say that a transformation where T_k: ℝ₁₂ → ℝ₁₂ can be written as: T_k(x) = (x + k_k) mod 12.

Now, in the case that we want an inversion (flips the intervals around, e.g. C to E is a major 3rd, while the inversion of this will be E to C, which is a minor sixth), we can say this is I_p: ℝ₁₂ → ℝ₁₂, around pitch p, where I_p(x) = 2p - x mod 12. This also means that I_p is an automorphism of ℝ₁₂ since I_p is a bijection and a homomorphism.

Now in the case we’re modeling chords, we turn them into subsets of ℝ₁₂. If we imagine we have such a chord C such that C = {c₁, c₂, …, c_n} where each c₁ ∈ ℝ₁₂, operations can be applied. For transposing, we have T_k(C) that T_k(C) = { T_k(c₁), T_k(c₂), …, T_k(c_n) }. For inversions, we have I_p such that I_p(C) = { I_p(c₁), I_p(c₂), …, I_p(c_n) }.

From this, we can say that automorphisms of ℝ₁₂, that include these transpositions and inversions become a group Aut(ℝ₁₂). Basing off of these transpositions and inversions, this can create an affine transformation that can be expressed as a linear function. Specifically, when expressed as ax+b, with both a and b in ℝ₁₂, a can equate to 1 or -1 for transpositions and inversions.

This is a brief look into music notation when expressed in a ring. Of course, there are so many other ways to look at musical notation, but I’ll definitely go more in depth on this later. The group has been working on a mini writing on an introduction to algebra, please email me if you’d like to join us!

I’d also like to share a podcast I’ve enjoyed for a while. It’s a very interesting listen, and I’ve found so many little aspects about Mahler that makes me enjoy his music so much more: