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I know it has been quite some time since the last post on music theory, and I’d like to continue with transformational theory. But in the meantime, I will share various things which are being investigated as of now regarding the relationships between music theory and topology.

Let’s begin with a very nice book by Dmitri Tymoczko:

• A geometry of music“, D. Tymoczko, Oxford University Press.
• The book follows two previous articles which were published by Tymoczko in Science: the first one is “The Geometry of Musical Chords”, D. Tymoczko, Science, 313: 72-74 (2006) and the second one is “Generalized Voice-Leading Spaces”, C. Callender, I. Quinn and D. Tymoczko, Science, 320: 346-348 (2008).

And I won’t talk about these works here ! At least, not now. But I encourage you to get a look at Tymoczko’s theories, which are very nicely presented. He maintains a website with most of his papers, in case you are interested.

Coming to a simpler subject, do you remember this picture ?

That’s right, this is the Tonnetz for major and minor triads. The picture above is an unfolded version, so that it lies in the plane, but the Tonnetz really has the topology of a torus. You can try to picture this by gluing correctly the fundamental domain which is highlited in violet.

Remember that I’ve introduced the Tonnetz in the context of the neo-Riemannian transformations $P$, $L$, and $R$. But in fact, we don’t need these operations, or even group theory, to draw some conclusions about the topology of the Tonnetz. You may recall that the three neo-Riemannian operations of the $PLR$ group each preserve two notes between triads (check the picture above). So, if we have a collection of triangles which represent the different major and minor triads, whose vertices are notes, the construction of this space comes simply from gluing the triangle along the edges which have the same vertices.

Let’s be more formal about this. We use here the language of simplicial complexes. A 0-simplex is a note, a 1-simplex is a dyad and a 2-simplex is a triad, like this:

We then select the triads of interest, for example all major and minor triads, and we form a simplicial complex (a kind of topological space) by gluing 2-simplices together whenever they share common notes or common dyads. For example, a C major chord and a C minor chord are glued together like this :

By the way, I’m not introducing anything new here. In fact, Louis Bigo has been working on this simplicial approach in IRCAM. His Ph.D. thesis is available somewhere on the Net, and you can check some of his articles :

• “Computation and Visualization of Musical Structures in Chord-based Simplicial Complexes”, L. Bigo, M. Andreatta, J.-L. Giavitto, O. Michel, and A. Spicher, Proceedings of Mathematics and Computation in Music 2013 – Montreal.
• “Building Topological Spaces for Musical Objects”, L. Bigo, J.-L. Giavitto, A. Spicher, Proceedings of Mathematics and Computation in Music 2011, Paris.

Things get interesting when you want to know more about the structure of the resulting space. What about holes for example ? As with any torus, the Tonnetz of major and minor triads has two 1-dimensional holes and one 2-dimensional hole. The really cool thing is that since we have built a simplicial complex from gluing 2-simplices, we can immediately apply simplicial homology to determine the homology groups of our space. If you don’t know what simplicial homology is, I urge you to learn it: it’s really easy, and it finds all kinds of applications, from music theory to sensor networks.

The homology groups, and their rank, which are also known as Betti numbers, give information about holes but does not determine entirely the space. For a classification of the possible spaces, you should check out this paper by M. J. Catanzaro :

• Generalized Tonnetze“, M. J. Catanzaro, Journal of Mathematics and Music, 5(2), 2011, pp. 117-139

In it, Catanzaro classifies the Tonnetze generated by a triad and its inversion. There are plenty of other topological spaces than the torus ! For example, the Tonnetz generated by the chords {C,C#,F#}, {C,F,F#} and their transpositions is a circle of 6 tetrahedra. I took some time to draw this space, so here it is

Note that you have to glue the dyads 5-11 to see the circle.

Still, the homological approach and the determination of the Betti numbers are pretty useful. I wrote some Matlab code to determine these Betti numbers. You can download the code from this link (the code is provided as is, it could certainly be improved): the homChord() function takes a list of triad prime forms as input, and returns the Betti numbers. For example, taking [0,1,6;0,5,6] as the input, which corresponds to the above-pictured Tonnetz, returns [1,1,6]: we have indeed one connected component, one 1-dimensional hole (the circle of tetrahedra), and six 2-dimensional holes (delimited by the 6 tetrahedra boundaries).

Here is the list of Betti numbers for the spaces constructed from all the transpositions of one or two generic triads.

Could we extend this to tetrachords, pentachords, etc. ? I haven’t looked at it yet, but I suspect we would have to build CW-complexes, instead of simplicial ones, and use cellular homology to get some information about the resulting topological spaces.

Finally, there is one last article I’d like to mention, by Luis A. Piovan about the construction of surfaces in which one can represent the Tonnetz of pentachords

• “A Tonnetz model for pentachords”, L. A. Piovan, Journal of Mathematics and Music, 7(1), pp. 29-53. There is an older, free, version available on Arxiv.