Strange spaces of networks and chords

From June, 28th, 2021 to July, 1st, 2021, the University of Pavia organized a mathematics and music workshop, in which I presented a summary of the work on the categorical formalization of musical networks (the videos of the different talks will probably be put on the site in a near future). An interesting part of this workshop is that it held an online concert at the end, asking for participants to contribute with a short piece of music based on mathematical ideas.

I decided to participate in this, and submitted a short piece for piano based on networks. Here it is:

Let me explain a little bit what is going on in this piece. As you know from previous blog posts, one can study networks of musical elements in which their interrelations are indicated by arrows labelled in an appropriate group (or category) of transformations. One example that I’ve shown before on this blog, and which I like very much, is the following passage in Webern’s op.11/2.

These three groups of three notes can be described with the following networks.

All these networks share the same I_8, I_9, and T_1 transformations, even though these pitch-class sets are not transpositionally or inversionally related. Going further, we can create all twelve networks based on the same set of transformations, which form the harmonic material of the piece above.

As seen before with network transformations, we can go from one of these chords to another by parsimonious voice-leading, as described below.

But there is more ! You may have noticed that some of these chords have common tones. Thus, we can imagine to stitch these chords whenever they have two tone in common, just as we would do for ordinary major and minor chords to get the neo-Riemannian Tonnetz. Except that in this particular case, we get the following space.

If we combine both the voice-leading transformations (an action of the \mathbb{Z}_{12} group, really) and the common-tone relations, we arrive at the following diagram, where the former relations are represented with thick plain lines, and the latter are represented with dashed lines.

The piece starts with the Bb-(Bb)-B chord and ends on the E-(E)-F chord by following a specific path represented below.

This path alternates between voice-leading transformations by semitone and common-tone relations. The transition between G-C#-Ab and C#-G-D is particularly emphasized in the piece, because these networks are located a tritone apart and yet share two common tones. Since Bb and E play important roles (they are the fixed points of I_8), they recur in the piece, often interacting with the pitch classes of the network to produce additional harmonies.

So far, we have seen only one network structure, but we can take any other kind of network. However, we will not get the same spaces. For example, if we take the following network structure (color coded here in orange),

take the corresponding chords, and stitch them if they have two pitch classes in common, we will get the following disconnected space.

Fun stuff happens when we start mixing more than one network structure. For example, if we do the same operation with the two network structures below,

we end up with the following space.

This has the topology of a torus (notice how it wraps on both sides), but this is very different from a neo-Riemannian Tonnetz because of the inversion transformations !

Not all network structures will give a torus however. Take for example the two network structures below.

Stitching the networks of the first one alone will give you a single band, whereas stitching the networks of the second one alone will give you three disconnected tetrahedrons. With the two together, we get the following space,

also represented as shown below in its unfolded form.

It surely would be interesting to compose some music based on these chords/networks and the associated properties of the spaces they generate !

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