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Less math and more music ! Today I’ll talk about an application of neo-Riemannian theory, namely the Tonnetz.

Originally, the Tonnetz was some sort of diagram introduced by Euler to represent tonal space and the relationships between tones, mainly in terms of fifths and thirds. The Tonnetz has been reused in a different way by neo-Riemannian theorists to describe the relationships between major and minor triads. In order to explain what this is about, we need to recall the PLR group of operations that we have seen in previous posts.

Recall that the PLR group is generated by three operations, $P$, $L$ and $R$ (actually it’s generated by just $L$ and $R$ but we include $P$ which is as important as the two others). These operations have a very specific action on major and minor triads which is pictured below. This is the action of $P$

which changes a major chord to a minor chord with the same root. This is the action of $L$

and this is the action of $R$

Of course, since these operations are involutions ($P^2=1$, $R^2=1$, $L^2=1$), you can find their actions on minor triads from the pictures above by simply reversing the order of the arrow.

We’ve seen that these operations can be obtained from a purely algebraic point of view by considering group extensions. However, this approach does not tell us about a very important point: all these transformations always operate on triads by keeping two tones identical. In a sense, we have here the “smoothest” transformations possible between major and minor triads.

Construction of the Tonnetz

Now, there is a way to relate the major and minor triads in a lattice by considering the $P$, $L$ and $R$ transformations. We start with a C major triad, which is composed of pitch-classes C, G and E. We can picture those pitch-classes as the vertices of a triangle, whose center is the C major chord, like this:

We will add more triangles of this type to this diagram by considering the transformations of the PLR group. For example, the $P$ operations takes the C major chord to a C minor chord which comprises the pitch-classes C, Eb and G. Thus, we can glue a triangle on the left since both C and G are common to the two chords. We can do the same with the $L$ and $R$ operations, and we obtain this diagram

There are more triangles, since we can do the same with the new chords we’ve introduced. Ultimately, we will use all 24 triads and we will obtain the following diagram, which is the Tonnetz in neo-Riemannian theory

Since all triads are used, the Tonnetz is not infinite and folds back on itself as you can clearly see on the above diagram. The tones represented in purple are in fact a fundamental domain of this lattice. What is then the topology of this lattice ? It has been proved that the topology of this lattice is the same as that of a torus ! You can get more information about this from the following references

• Douthett, J., Steinbach, P.; “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition”, Journal of Music Theory, 42(2), 1998, pp. 241-263
• Cohn, R.; “Neo-Riemannian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations”, Journal of Music Theory, 41(1), 1997, pp.1-66
• Cohn R.; “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic”, Music Analysis, 15(1), 1996, pp. 9-40

So what can we do with this lattice ?

The use of the Tonnetz for musical analysis

Well, this lattice is incredibly useful for visualizing the transformations between chords in terms of operations in the PLR group. Let’s take an example. Luckily, I’ve discovered that we can embed sound examples in this blog so here’s one. Can you guess what’s the reference of that small extract ?

In case you haven’t guessed, this is the Tarnhelm leitmotif in Wagner’s tetralogy Der Ring des Nibelungen. “Hmm, but what’s a Tarnhelm ?”, I’m sure you’re going to ask. Well, Alberich is a Nibelung (basically, a dwarf in german myths) who managed to steal the Rhine gold after renouncing love. From this gold, he cast a ring in his forge (reminds you of anything ?) and a helmet, the Tarnhelm. One who puts the Tarnhelm on his head can transform into anything he wants, for example a dragon as Alberich (and later the giant Fafner) does. As depicted on the score below, the Tarnhelm leitmotif oscillates between a G# minor chord and a E minor chord, before ending on an ambiguous F# chord which is neither major or minor since it has no thirds

The relation between G# minor and E minor is quite immediate in the Tonnetz

This is simply a $PL$ operation ! If you have a piano nearby, you can check that applying $PL$ to any minor chord will give a similar effect.

(Note: we call this a $PL$ operation as we operate on chords on the left. Some authors call it a $LP$ operation, as they operate on the right.)

John Williams, film music and the Tonnetz

In fact, you’ve already heard this $PL$ progression before. Yes, yes, I know you haven’t heard Wagner’s tetralogy before. But have you heard this ?

And here goes the score of the Imperial March from Star Wars by John Williams

This time, we oscillate between a G minor chord and a Eb minor chord, but the relationship between them is the same: a $PL$ operation as pictured above. The Tarnhelm is a nasty object, made by gold which will later be cursed by Alberich and will bring nothing but destruction throughout the whole opera. The Empire is similarly evil and this is exemplified by a $PL$ operation in both cases.

Let’s take another example. Can you recognize this extract ?

This is the theme for the ark of the covenant in “Indiana Jones: Raiders of the Lost Ark” (music by John Williams again). Here’s a reminder of what it sounds like in the movie

And here is how the two chords C minor and F# minor chord (urg, a tritone…) relates in the Tonnetz

The operation which takes C minor to F# minor is the $RPRP$ operation. The Ark is quite a nasty object as well, at least in the wrong hands (go check the movie in case you’ve forgotten…), and the $RPRP$ operation does a good job of showing it. By the way, that same operation also appears in the Imperial March, just after the end of the small extract I gave above. Check it ! Or you can read Frank M. Lehman thesis

• Frank M. Lehman; “Reading Tonality Through Film: Transformational Hermeneutics and the Music of Hollywood”, Ph.D. dissertation, Harvard University, May 2012

in which he discusses the Imperial March and many other movies. The ark theme continues (at 2:20 in the above video) by a series of chords which follows a path which appears quite structured in the Tonnetz as depicted below

Notice that we find again the $PL$ operation !

Hamiltonian paths in the Tonnetz

There is an interesting mathematical question about the Tonnetz, which has been answered in the following paper

• G. Albini, S. Antonini; “Hamiltonian Cycles in the Topological Dual of the Tonnetz”; Proceedings of the MCM 2009 Conference, CCIS 38, Springer, pp. 1-11

The question is: are there paths (series of operations) which visit each chord exactly once ? This is in fact a famous question in graph theory, and we call such paths “Hamiltonian paths”. It turns out that there are many of these paths in the Tonnetz. Besides, we already know one ! This is the $LR$ operation in Beethoven Ninth symphony which we have seen in a previous post. Here it is in score format

and here is how it looks in the Tonnetz

Pretty regular, isn’t it ?

This is just $(LR)^{12}$ but we can find a hamiltonian path with less repetition. Take for example the path $(LPLPLR)^4$. It looks like this in the Tonnetz (at the lettered points you have to jump to the indicated points. This was easier to visualize, otherwise it wouldn’t fit in my diagram)

and sounds like this

You can even have full length paths such as $LRLPLRLPRLRLPLRPLPRPLRPR$ which looks like this

and sounds like this

In fact, you have 24 full length hamiltonian paths which are obtained by just shifting the operations in the above word. For example, we can have the path $RLPLRLPRLRLPLRPLPRPLRPRL$ which looks like this

and sounds like this

As said before, the transitions between chords obtained with the $P$, $L$ and $R$ operations are quite smooth since they always preserve two tones. The progressions in the above hamiltonian paths are thus acceptable for our ears: you could even use that to write some tunes. In fact, this is just what Moreno Andreatta and Gilles Baroin, from IRCAM, have tried: you can listen to a song they made using, among other things, three hamiltonian paths