An introduction to neo-Riemannian theory (8)

I have left something unfinished in the previous post, namely contextual actions, so it’s time to talk about it.

Last time, we have seen that we can build groups of musical operations using the construction of group extensions. For example, we may have chords which are defined by their root and their type. The roots live in a space, which is usually endowed with a simply transitive group action of a group $Z$ which corresponds to transpositions. This is how we can go from C major to C# major to D major, etc. (or the same for minor chords), and in that case $Z$ is the cyclic group $\mathbb{Z}_{12}$. The types also live in a space, which we may act upon by a simply transitive group action of a group $H$ which corresponds to formal inversions. For example, we have the two elements set $\{\text{Maj},\text{Min}\}$, acted upon by $\mathbb{Z}_2$. Thus the groups of transformations we are looking for are of the form $Z \xrightarrow{i} G \xrightarrow{p} H$

where $i$ and $p$ are respectively injective and surjective homomorphisms, with $Ker(p)=Im(i)$, in other words $G$ is a group extension of $Z$ by $H$.

Once we have found a group we like, we can build a group action, either on the left or on the right, by defining a bijection $\chi : G \to S$ between elements of G and our musical objets of interests, which belong to the set $S$. Then the actions are defined by $g.p = \chi( g.\chi^{-1}(p))$

or $p.g = \chi( \chi^{-1}(p).g)$

with $g \in G, p \in S$. To actually compute the action, we need the group operation, and we have seen that in a group extension, the elements $g \in G$ can be put in the form $(z,h), z \in Z, h \in H$ and that we have $(z_1,h_1) \cdot (z_2,h_2) = (z_1 \cdot \phi_{h_1}(z_2) \cdot \zeta(h_1,h_2),h_1 \cdot h_2)$

where $\phi$ is an action of $H$ on $Z$ by automorphisms, and $\zeta : Z \times Z \to H$ is a 2-cocycle.

Let’s now recall what we mean by contextual actions. Take the $I_0$ operator from the T/I-group. We know that its action on major and minor chords is $I_0.n_{\text{Maj}} = (5-n)_{\text{Min}}$ and $I_0.n_{\text{Min}} = (5-n)_{\text{Maj}}$

The root change here is independent of the chord type, and thus this action is non-contextual. This is the same for transpositions operators. On the other hand, we have seen that the action of the $L$ operator in the PLR-group is $L.n_{\text{Maj}} = (n+4)_{\text{Min}}$ and $L.n_{\text{Min}} = (n+8)_{\text{Maj}}$

and in this case, the action depends on the chord type: it is a contextual action.

Knowing whether we have contextual actions or not in a group extension is actually very simple based on the group operation. We need to distinguish three cases.

Case 1: $G$ is a direct product of $Z$ and $H$

This corresponds to the case where we don’t have an action by automorphism nor a 2-cocycle. In that case, the left action of a group element $(z,h)$ on a musical object $p$ represented by its group element $(z_p,h_p)$ will be given by (we omit the notation for $\chi$ which is implicit): $(z,h)(z_p,h_p) = (z.z_p,h.h_p)$

We see that the root change ( $z.z_p$) does not depend on $h_p$, hence the action will always be non-contextual. Since the group is abelian, the right action is the same, and will be non-contextual as well.

Case 2: $G$ is a semi-direct product of $Z$ and $H$

This is the case where we have an action by automorphisms and no 2-cocycle. In that case the left action is given by $(z,h)(z_p,h_p) = (z.\phi_{h}(z_p),h.h_p)$

which is clearly non-contextual. However, the right action is given by $(z_p,h_p)(z,h) = (z_p.\phi_{h_p}(z),h_p.h)$

and in this case the root change depends on $h_p$, i.e the action is contextual. A direct example of this is given by the group $D_{24}$, a semi-direct product of $\mathbb{Z}_{12}$ by $\mathbb{Z}_2$, whose action on the left corresponds to the action of the non-contextual group T/I, and on the right to the contextual group PLR.

Case3: $G$ is of the most general form

This is the rest of the possible structures, where we can have both an action by automorphisms and a 2-cocycle. The left and right actions are given by $(z,h)(z_p,h_p) = (z \cdot \phi_{h}(z_p) \cdot \zeta(h,h_p),h \cdot h_p)$

and $(z_p,h_p)(z,h) = (z_p \cdot \phi_{h_p}(z) \cdot \zeta(h_p,h),h_p \cdot h)$

Hence we see that the actions can be contextual in both cases.

The semi-direct case is maybe the most interesting, as we have non-contextual actions such as transpositions on one side, and contextual ones on the other side. Finally, there is one last thing we need to talk about, namely duality. Remember that the actions of the T/I group commute with those of the PLR-group: we say that the PLR-group and the T/I-group are dual in the sense of Lewin. It is quite easy to see why from the discussion above: if we have $(z_p,h_p)$ acted upon on the right, then left, it is the same as left-then-right since the group operation is associative. Hence duality always exist between left and right actions.

Contextual actions as “change of basis”

I’d like now to revisit contextual actions from another point of view. Let’s take the $P$ operation of the PLR-group acting on the C major chord : If we want to express it as an operation of the T/I-group, that’s the $I_7$ operation. If we consider now the $P$ operation acting on the D major chord: this is now the $I_{11}$ operation that we need to use (though it is still $P$ in the PLR-group). But if you look more carefully, you will see that, relative to the position of the chord, the position of the inversion has not changed. It is always the inversion operation immediately before the second pitch-class of the chord. In other words, it is as if we always have $I_7$ acting in the basis of the chord.

We can in fact be mathematically more precise about this. Remember that the left action of the T/I-group is given by $g.p = \chi( g.\chi^{-1}(p))$

for any group element $g$. In particular, let’s take a group element $h$ and write $g = \chi^{-1}(p) \cdot h \cdot (\chi^{-1}(p))^{-1}$. Beware that $\chi^{-1}(p)$ is the group element corresponding to the object $p$ and that $(\chi^{-1}(p))^{-1}$ is the inverse of that group element. Now let’s apply $g$ on the left : $g.p = \chi( g.\chi^{-1}(p)) = \chi( \chi^{-1}(p) \cdot h \cdot (\chi^{-1}(p))^{-1} \cdot \chi^{-1}(p))$

and this is equal to the right action $\chi( \chi^{-1}(p).h)$. The formula for the group element $g = \chi^{-1}(p) \cdot h \cdot (\chi^{-1}(p))^{-1}$ is similar to expressions of the type $P.M.P^{-1}$ in linear algebra, and thus the meaning is equivalent: it’s the operation $h$ applied after a change of basis, in that case to be in the basis of the object $p$.

Does it work in the example above ? The D major chord is represented by the group element $T_2$ since it is the transposition of the C major chord (which we have chosen as the identity) by two semi-tones. We have $T_2 \cdot I_7 \cdot T_2^{-1} = I_{11}$

and we recover the proper element of T/I which has the action of the $P$ operator.

What about minor chords ? The $P$ operation applied on the D minor chord should also correspond to $I_{11}$ : The D minor chord is represented by the group element $I_9$ (check it if you are not sure). And we have $I_9 \cdot I_7 \cdot I_9 = I_{11}$

so we recover again the proper operation.

This point of view will allow us later to define contextual actions when applying group elements on the right will not work as well as in the case of groups and group actions.