An introduction to neo-Riemannian theory (6)

The previous post introduced the $D_{24}$ group of transpositions and inversions, often notated as the T/I group, which acts simply transitively on the set of the 24 major and minor triads. It’s time now to introduce another group of transformations acting on the same set, which has a huge number of applications and which forms the core of neo-Riemannian transformational music theory.

But first, some notation. We denote by $[n]$ a single pitch-class, and by $[n_1, n_2, ...]$ a chord comprised of the pitch-classes $[n_1], [n_2]$ etc. We will talk here about chords which have a clearly identifiable root, and we will always order the pitch-classes accordingly. For example, $[0, 9, 5]$ is an F-major chord though ordered in the wrong way: the correct notation would be $[5, 9, 0]$ . A major chord with root $n$ will also be notated as $n_{\text{Maj}}$ , and similarly a minor chord with root $n$ will be notated as $n_{\text{Min}}$ . Hence

$n_{\text{Maj}} = [n, n+4, n+7]$

and

$n_{\text{Min}} = [n, n+3, n+7]$

assuming that the operations are performed modulo 12.

We will graphically represent a major chord (in the example below, a C-major chord) as

A C-major chord

and a minor chord (in this case a C-minor chord) as

A C-minor chord

The T/I group is given by the presentation

$G = < z, I_0 | z^{12}=1, I_0^2=1, I_0.z.I_0 = z^{-1} >$

where $z$ is the transposition operator, which transposes pitch-classes up by one semi-tone (i.e $z.[n] = [n+1]$ ), and $I_0$ is the inversion operator, which inverts pitch-classes around the pitch-class $[0]$ (i.e $I_0.[n] = [-n]$ ).

Let’s now introduce new operations on major and minor triads. We begin with an operation which we will call “ $P$ ” (which stands for “Parallel”). The $P$ operation is the inversion operation which acts on a major triad $n_{\text{Maj}}$ such that the pitch-classes $[n]$ and $[n+7]$ are exchanged. This has the consequence of shifting the pitch-class $[n+4]$ to $[n+3]$ . Graphically, the $P$ operation acts as such :

In this case the $P$ operation acting on the C-major chord is in fact the $I_7$ inversion operation, but beware ! Had we chosen a D-major chord instead, the $P$ operation would in fact have been the $I_{11}$ inversion operation, as shown below

What does the $P$ operation do on minor chords ? Again, it is the inversion operation which exchanges pitch-classes $[n]$ and $[n+7]$ . Graphically, we have

What’s $P^2$ ? From the diagrams above, you can clearly see that this is the identity operation. In other terms, $P$ is an involution. To sum up, we have

$P.n_{\text{Maj}} = n_{\text{Min}}$

and

$P.n_{\text{Min}} = n_{\text{Maj}}$

The second operation is the $L$ operation (which stands for “Leittonwechsel”). When applied on a major chord $n_{\text{Maj}}$ , it is the inversion operator which exchanges the pitch-classes $[n+4]$ and $[n+7]$ . Graphically

When applied on minor chord, it is the inversion operator which exchanges the pitch-classes $[n]$ and $[n+3]$ , such that $L$ is an involution :

So we have

$L.n_{\text{Maj}} = (n+4)_{\text{Min}}$

and

$L.n_{\text{Min}} = (n+8)_{\text{Maj}}$

The last operation is the $R$ operation (which stands for “Relative”). When applied on a major chord $n_{\text{Maj}}$ , it is the inversion operator which exchanges the pitch-classes $[n]$ and $[n+4]$ . Graphically

and the action of $R$ on minor chords is again chosen such that $R$ is an involution:

We thus have

$R.n_{\text{Maj}} = (n+9)_{\text{Min}}$

and

$R.n_{\text{Min}} = (n+3)_{\text{Maj}}$

The $L$ and $R$ operations are said to be contextual. This means that their action, and in particular their action on the root, depends on the type of chord considered, whether it is major or minor. Notice that the $P$ operation is not a contextual operation. Compare also with the $I_0$ inversion operator from the T/I group which switch the chord type, but always send the root $n$ to $(5-n)$ .

It’s quite easy to see that we can form a group with these three $P$ , $L$ and $R$ operations. The identity element is “doing nothing”. The operations can be composed and verifying associativity is straightforward. The inverse elements are well-defined since each operation is an involution. The question is therefore: what is the structure of the group generated by $P$ , $L$ and $R$ .

Before answering that, we can observe that these operations are not independent from each other. Indeed, you can verify that we have

$P = RLRLRLR = R(LR)^3$

We thus need to determine the structure of the group generated by $L$ and $R$ . We have

$LR.n_{\text{Maj}} = (n+5)_{\text{Min}}$

$LR.n_{\text{Min}} = (n+7)_{\text{Maj}}$

therefore $LR$ is of order 12. In addition, it is easy to check that $(LRP)^2=1$ . We therefore have the relations $(LR)^{12}=P^2=((LR)P)^2=1$ which defines group isomorphic to a dihedral group of order 24. The PLR-group

$G = < L , R , P | P=R(LR)^3, (LR)^{12} = P^2 = (LRP)^2 = 1 >$

is therefore isomorphic to the T/I-group. This is no coincidence, and there is in fact a deeper connection between these two groups which we’ll expose in another post.

A very important point is that, similarly to the T/I-group, the PLR-group acts simply transitively on the set of the 24 major and minor chords. It is therefore possible to analyze a sequence of triads by their transformations with elements from the PLR-group, and we’ll provide such an example shortly.

There is also a very nice property of the PLR-group in relation with the T/I-group: all operations of the PLR-group commute with those of the T/I-group. Take for example the $R$ operation, and a transposition $z^p$ from the T/I-group. We have

$(z^pR).n_{\text{Maj}} = z^p.(n+9)_{\text{Min}} = (n+9+p)_{\text{Min}} = (Rz^p).n_{\text{Maj}}$

$(z^pR).n_{\text{Min}} = z^p.(n+3)_{\text{Maj}} = (n+3+p)_{\text{Maj}} = (Rz^p).n_{\text{Min}}$

If you take instead the inversion $I_0$ we have

$(I_0R).n_{\text{Maj}} = I_0.(n+9)_{\text{Min}} = (5-n-9)_{\text{Min}} = ((5-n)+3)_{\text{Min}} =(RI_0).n_{\text{Maj}}$

$(I_0R).n_{\text{Min}} = I_0.(n+3)_{\text{Maj}} = (5-n-3)_{\text{Maj}} = ((5-n)+9)_{\text{Maj}}= (RI_0).n_{\text{Min}}$

and it follows for every inversion $I_p$ since $I_p=z^pI_0$ .

In fact, the proposition is a bit stronger: among all permutations of the 24 triads, those who commute with the PLR-group are exactly those from the T/I-group, and vice-versa. We say that the T/I-group and the PLR-group are dual in the sense of Lewin, meaning that each group, being subgroups of the symmetric group on 24 elements, is the centralizer of the other. Lewin was the first to recognize the notion of duality in his book

“Generalized Musical Intervals and Transformations“, p. 253, David Lewin. Oxford and New York: Oxford University Press, 2007. xxxi, 258 pp. Originally published by Yale University Press, 1987

You can also the check the following references for a more detailed investigation of the notion of dual groups, though the scope of these papers is much larger than this post

“Musical Action of Dihedral Groups”, A. A. Crans, T. M. Fiore and R. Satyendra, The American Mathematical Monthly, vol. 116, no. 6, June 2009, pp. 479-495
“Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality”, T. M. Fiore, T. Noll and R. Satyendra, Proceedings of the MCM 2013 Conference, Springer Lecture Notes in Computer Science, Volume 7937 LNAI, pp. 100-114

I will surely post more about this group and its properties in the future. In the meanwhile, if you’re interested in knowing more, here are a couple of useful references about neo-Riemannian theory :

“Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective”, R. Cohn, Journal of Music Theory, Vol. 42, No. 2, Neo-Riemannian Theory (Autumn, 1998), pp.167-180
“A Formal Theory of Generalized Tonal Functions”, D. Lewin, Journal of Music Theory, Vol. 26, No. 1 (Spring, 1982), pp. 23-60
“Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions”, R. Cohn, Music Analysis, Vol. 15, No. 1 (Mar., 1996), pp. 9-40
“Neo-Riemannian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations”, R. Cohn, Journal of Music Theory, Vol. 41, No. 1 (Spring, 1997), pp. 1-66
“Tonal Intuitions in Tristan and Isolde”, Bryan Hyer, Ph.D. dissertation, Yale University, 1989

To finish this introduction about the PLR-group, here’s a very simple, yet powerful example, which I borrow directly from the paper of Cohn entitled “Neo-Riemannian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations”. We have seen that $LR$ is of order 12, which means that by successively applying $R$ and $L$ , one can go through all 24 triads. Here’s what 19 successive operations gives starting from the C-major triad :

It so happens that this exact cycle is used by Beethoven in its 9th symphony (2nd movement) ! You can listen the above passage starting at 2’29” in this video

There are many other applications of the PLR-group, and more mathematical structure as well, so I’ll keep them all for future posts.

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