The Mathematics of Music at Chicago
Posted by John Baez
As a cardcarrying Pythagorean, I’m fascinated by the mathematics of music… even though I’ve never studied it very deeply. So, my fascination was piqued when I learned a bit of ‘neoRiemannian theory’ from Tom Fiore, a topology postdoc who works on double categories at the University of Chicago.
NeoRiemannian theory is not an updated version of Riemannian geometry… it goes back to the work of the musicologist Hugo Riemann. The basic idea is that it’s fun to consider things like the 24element group generated by transpositions (music jargon for what mathematicians call translations in $\mathbb{Z}/12$) and inversion (music jargon for negation in $\mathbb{Z}/12$). And then it’s fun to study operations on triads that commute with transposition and inversion. These operations are generated by three musically significant ones called P, L, and R. Even better, these operations form a 24element group in their own right! I explained why in week234 of This Week’s Finds. For more details try this:
 Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra, Musical actions of dihedral groups.
Yes, that’s my student Alissa Crans, of Lie 2algebra fame!
On June 11th, Thomas Noll is giving some interesting talks on music theory at Chicago, which will delve deeper into such issues. They’ll even get into some topos theory!

Thomas Noll (Escola Superior de Musica de Catalunya), The Triad as Place and Action: a Transformational Perspective on Stability, Thursday, June 11, 1:30 pm, Department of Mathematics, University of Chicago.
Abstract: Recent transformational approaches to the study of triads are based on group actions on the set of the major and minor triads. A particular musictheoretical interest in this subject is driven by the possibility of parsimonious voice leadings between certain triads. To each triad X, say X = {C, E, G}, (considered modulo octave) there are three triads P(X)= {C, Eb, G}, L(X)= {E, G, B}, R(X)= {A, C, E}, each sharing two tones with X. What distinguishes triads from arbitrary 3chords is the small amount by which the third tone has to be displaced: In the case of P(X) it is an augmented prime (E → Eb), in the case of L(X) it is a minor second (C → B) and in the case of R(X) it is a major second (G → A). Richard Cohn therefore speaks of the “overdetermined triad”, as  traditionally  the musictheoretical prominence of the triad is explained in terms of consonance.
My pretalk is dedicated to yet another property which can be added to the list of overdetermining decorations of the triad. This property provides a conceptual link between (the discussion about) Hugo Riemann’s concept of consonance on the one hand and the NeoHugoRiemannian transformations P, L, R as mentioned above, on the other. This approach is based on a transformational investigation of the intervallic constitution of the triads. Each triad is studied as a subaction of a monoid action of an 8element monoid on $\mathbb{Z}/12$. Each transformation is a TwelveToneOperation (an affine endomorphism of $\mathbb{Z}/12$) which stabilizes the triad in question and which extrapolates an association of an internal interval of the triad with its fifth. With this approach I hope to make a contribution to an abandoned discourse between Hugo Riemann and Carl Stumpf and in particular to an elaboration of Stumpf’s concept of the triad as a concord of consonances.
Mathematically the approach is an application of elementary topos theory. The NeoRiemannian transformations P, L, R can be studied as equivariant maps between monoid actions (i.e. as arrows in an associated topos). The structure of the subobject classifier and its Lawvere–Tierney topologies allow to draw links between the different qualities of tones in the complement of a triad as such, and the roles of these tones as images of proper triad tones under the transformations P, L, R on the other. In a way, this approach is an attempt to actualize Hugo Riemann’s vision of the theory of harmony as a “musical logic”.
I will refer to two musical examples and related discussions in music theory: the first movement of Schubert’s sonata in Bb (D 960) as discussed by Richard Cohn and by Balz Trümpy and the last study of Alexander Skriabin Op. 65 No. 3 as discussed by Clifton Callender.

Thomas Noll (Escola Superior de Musica de Catalunya), Diatonic and Tetractys Modes as instances of Christoffel Duality, Thursday, June 11, 3:00 pm, Department of Mathematics, University of Chicago.
Abstract: A recent development in the mathematical and musictheoretical study of so called wellformed scales is closely related to research directions within the field of algebraic combinatorics on words, namely Sturmian words and their finite analogues, i.e. Christoffel words and their conjugates. I’m going to report on joint work with David Clampitt (Ohio), Karst de Jong (Barcelona) and Manuel Dominguez (Madrid). There is a general musictheoretical desire to understand the principles which guide or constrain the constitution of musical tone relations. Wellformed scales are generated by a fixed interval modulo some period (typically the octave) and thereby embody a concept of tone kinship  given by the generation order of the scale tones. A second concept of tone relation is given by the pitch height order of the scale (step order). The wellformedness condition (as introduced and studied by Carey and Clampitt 1989) requires that the conversion from step order to generation order is a linear automorphism of $\mathbb{Z}/n$ (where $n$ is the number of tones). A refinement of this condition for musical modes (instead of scales) leads to Christoffel words and their conjugates and to a related concept of duality.
This wordtheoretic approach is tightly connected with traditional accounts to tone kinship. This connection is given trough the projection from the free noncommutative group $F_2$ with two generators to the commutative group $\mathbb{Z}^2$. Traditionally, the mathematical investigation of tone kinship is based on the free commutative group over two (or more) musical intervals as linearly independent generators. In the case of the two generators fifth and octave this group is called the Pythagorean tone lattice $P$. Pitch height enters into this picture as a linear form $h: P \to \mathbb{R}$. It is insightful to embed $P$ into the real plane $\mathbb{R}^2$ with respect to a basis, which is constituted by the gradient (pitch height axis) and the kernel (pitch width axis) of this linear form. Word theory enters here through the approximation of the pitch width axis though polygons in $P$, i.e. though elements of $F_2$. For each Christoffel prefix of the infinite “Pythagorean word” (along the pitch width axis) there is a natural candidate for a metric on $P$, respective $\mathbb{R}^2$.
I will summarize actual results about the interdependence of regions (such as the Guidonian hexachord aabaa) and standard modes (such as the authentic Ionian mode aabaaab). The underlying duality has a clear expression in terms of Standard morphisms. The generalization of the diatonic case does not only include larger and/or musically counterfactual modes, but also simpler ones, such as the modes of the three note tetractys scale. I will conclude my talk with remarks on a “NeoRiemannian” approach to the analysis of fundamental progressions.
I think anyone who encounters these ideas is likely to wonder: is topos theory being put to good use in the study of music, or is it just a form of ‘showing off’?
I don’t know! I have no opinion yet. I don’t think topos theory is inherently too abstract to shed light on harmony. And, after years of slow study, I finally understand enough topos theory to follow what’s being said above, at least in principle. But I haven’t understood it yet. One way to dig deeper would be to read this book:
 Guerino Mazzola, The Topos of Music: Geometric Logic of Concepts, Theory, and Performance, Birkhäuser, 2002.
But reading and understanding Tom Fiore’s review of this paper:
 Thomas Noll, The topos of triads.
would be much quicker… at least for anyone who already understands Lawvere–Tierney topologies!
Does anyone here dare to venture some comments?
Re: The Mathematics of Music at Chicago
I tried once to read the Mazzola book. I sincerely hope that my impressions of the book are inaccurate, as that would restore some of my lost hopes for the use of topoi in interesting areas.
I found the book to be ghastly  to fluffy for mathematicians, and probably too mathematical for music theorists  and wrote a review for the Jenabased mathematics magazine that said as much. The review ended up not being published, because it was too harsh, but being widely circulated among the magazine staff, because it was amusingly so.