Transformational Music Theory (16)

The graph below is called the “Cube Dance”, as introduced by Douthett and Steinbach in 1998

• Douthett, J., Steinbach, P. “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition.” Journal of Music Theory, Vol. 42-2 (1998), pp. 241–263. At the nodes of this graph, we find our usual 24 major and minor triads as well as four new chords, the augmented triads. But what do the edges correspond to ? To answer this, let’s examine the neo-Riemannian $P$, $L$, and $R$ operations again.

As we have seen before in this blog, these operations are defined as contextual inversions which permute two notes of a major or minor triad, and change the third. The $P$ operation, for example, is obtained as follows. Similarly, the $L$ operation, is obtained as follows. And this is the $R$ operation. Now, although these operations are all defined as inversions, they are different in terms of voice-leading. The $P$ and $L$ operations move one note by a semitone, whereas the $R$ operation moves a note by a whole tone. In fact, if we take the $C$ major triad as pictured above and move the pitch-class $G$ by a semitone, we get an $A\flat$ augmented triad, as pictured below. In their paper, Douthett and Steinbach introduced the $\mathcal{P}_{m,n}$ relation between chords: two chords are $\mathcal{P}_{m,n}$-related if they have $m$ notes which differ by a semitone, and $n$ notes which differ by a whole tone. Thus, if we apply the $P$ or $L$ operations to a chord, this chord and its image will be $\mathcal{P}_{1,0}$-related, whereas if we apply the $R$ operation to a chord, this chord and its image will be $\mathcal{P}_{0,1}$-related.

With this definition, Douthett and Steinbach define a parsimonious graph for a $\mathcal{P}_{m,n}$ relation on a set $H$ of chords as the graph whose set of vertices is $H$ and whose set of edges is the set $\{(x,y) | x \in H, y \in H, \text{and } x\mathcal{P}_{m,n}y\}$. And the “Cube Dance” is nothing else but the parsimonious graph for the $\mathcal{P}_{1,0}$ relation ! In this graph, any augmented triad is related to six chords by the $\mathcal{P}_{1,0}$ relation, and any major or minor triad is related to three other chords by that relation.

Of course, you can study other parsimonious graphs, and this what Douthett and Steinbach have done. For example, below is the “Weitzmann Waltz”, which is the parsimonious graph for the $\mathcal{P}_{2,0}$ relation. It is disconnected: one can’t reach, say, the $D$ major triad from the $C$ minor triad by a succession of $\mathcal{P}_{2,0}$ relations. Douthett and Steinbach have even studied parsimonious graphs, such as the “Power Towers”, with seventh chords.

Can it be used for music analysis ? Of course ! Take for example the beginning of “The Gunner’s Dream” by Pink Floyd

The path the chords take can be represented as such in the Cube Dance. And here is a more systematic example with “Take a Bow” from Muse. But why does this blog post have “Transformational Music Theory” as a title, when we have just been talking about graphs and relations ? Indeed, the $\mathcal{P}_{1,0}$ relation is not a transformation: you can’t really speak of the image of the $D$ augmented triad by $\mathcal{P}_{1,0}$, since there are six chords which are related to this triad. However, in the context of poly-Klumpenhouwer networks you can, and this is what we’ll see in the next post !