Transformational Music Theory (15)

Today, I will come back to the PRL-group that we saw here and here, and illustrate some simple results with particular relationships to the Tonnetz.

To begin, recall that we defined the three operations $P$ , $R$ , and $L$ as transformations of major and minor triads that keep two tones identical and flip the remaining tone (along the symmetry axis which exchange the first two tones). In the Tonnetz (the hexagonal grid representation of tones), these operations occur in the following way.

(Yes, my representation of the Tonnetz has improved since the previous posts. The Latex/TikZ code is available on demand.)

So here, starting from the C major triad (the central triangle connecting C, E, and G), we would get the C minor triad by the $P$ operation, the A minor triad by the $R$ operation, and the E minor triad by the $L$ operation. Just to see if you followed, what would be the result of applying the $L$ operation on the F sharp major triad ?

(Answer: B flat minor)

We have seen that these three operations, acting on the set of the 24 major and minor triads, generate a group isomorphic to the dihedral group $D_{24}$ with 24 elements. In fact, the operation $P$ is not necessary, as we have $P=R(LR)^3$ , and the group can be given by the presentation

$G = \langle R,L \mid (LR)^{12}=R^2=L^2=1 \rangle.$

But the $P$ operation is rather useful, so we’ll keep it around. In fact, what if we removed the $R$ operation and only considered the $P$ and $L$ operations ?

Well, this time you would get a group isomorphic to the dihedral group with six elements $D_6$ , given by the presentation

$G_{PL}= \langle P,L \mid P^2=L^2=1, LPL=PLP \rangle.$

Contrary to the PRL-group, this group does not act simply transitively on the set of major and minor triads: instead, you have four distinct orbits, which are often referred to as hexatonic systems. These hexatonic systems are

{E flat major, E flat minor, B major, B minor, G major, G minor},
{E major, E minor, C major, C minor, A flat major, A flat minor},
{F major, F minor, C sharp major, C sharp minor, A major, A minor}, and
{F sharp major, F sharp minor, D major, D minor, B flat major, B flat minor}

This is seen more clearly on the Tonnetz. Since we removed the $R$ operation, we can only move diagonally in the direction north-east / south-west. So if we start from a C major triad, we will move along the six chords E major, E minor, C major, C minor, A flat major, A flat minor before coming back, along the following strip of the Tonnetz.

(Lighter blue indicates the same six chords repeated due to the toric nature of the Tonnetz)

Had we instead started from F major, we would have visited another strip of the Tonnetz.

You can verify that there are only four such distinct strips (remember that the Tonnetz is toric, hence periodic once unfold), giving the four distinct orbits.

You can read more about hexatonic systems in these papers :

Maximally smooth cycles, hexatonic systems, and the analysis of late-romantic triadic progressions., Cohn, R., Music Analysis, 15(1), pp. 9–40, 1996.
Hexatonic Systems and Dual Groups in Mathematical Music Theory; Berry, C., Fiore, T. M., arXiv:1602.02577

As a cool exercise, you can try to see what happens if we consider only the $P$ and $R$ operations, either by group theory methods, or geometrically from the Tonnetz.

Now, the Tonnetz allows an easy visualization of major and minor chords, but other chords can be represented as well. Consider for example the augmented triads. Here’s one:

And here it is in the Tonnetz:

Clearly, there is only four distincts augmented triads (if we consider enharmonic equivalence). Here they are in the Tonnetz.

The fun fact about these augmented triads is that they connect the hexatonic strips of the Tonnetz. Consider, for example, the C major triad and the F minor triad. They belong to distinct strips, so there is no way to go from the first to the second by using only the $P$ and $L$ operations. However, if we pass through an augmented triad, like this:

we connect the two strips via the C augmented triad, like this:

To know more about the relations between major, minor, and augmented triads, I invite you to read the following references:

Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition, Douthett, J., Steinbach, P., Journal of Music Theory, 42(2), pp. 241-263 (1998)
and the recent book of Richard Cohn: Audacious Euphony: Chromaticism and the Triad’s Second Nature. Oxford University Press, 2012.

There are many other interesting chords in the Tonnetz. I’ll leave it to you to explore these two ones:

And finally, an application (not mine) of all this to the song Take A Bow from Muse:

That’s all for today… There is so much more to talk about Tonnetzes that I’m saving it for later.

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