Geometrical Methods in Recent Music Theory

[1] The following remarks are prompted by Joti Rockwell's interesting article, "Birdcage Flights: A Perspective on Inter-Cardinality Voice Leading" (2009). My goal is not to take issue with Rockwell's specific claims but rather to underscore a few details that might escape the casual reader's attention. In particular, I want to stress three basic points.1. Rockwell's discrete "birdcage graphs," which represent efficient voice leadings between chords of different sizes, can be embedded within the infinite-dimensional "OPC space" discussed by Callender, Quinn, and myself.(1) In general, discrete voice-leading graphs can always be embedded within the continuous spaces we describe.2. Because of this, Rockwell's graphs inherit some of the complications of C space, and may distort voice-leading relationships among nonadjacent chords.3. This is actually symptomatic of a more general issue affecting a variety of music-theoretical graphs. Roughly speaking, we have no guarantee that graphs whose edges refer to musical motions of a particular type will give rise to an intuitive or familiar notion of "distance" between nonadjacent chords.[2] This last item is significant because theorists sometimes seem to endorse the following methodology. First, one selects some interesting domain of musical objects and some interesting set of motions among them. (For example, single-voice voice-leading between major and minor triads.) Second, one constructs a graph representing all of these motions between all the objects in question. Third, one interprets the resulting graph as providing a measure of distance. Thus, for example, one might use the graph to analyze music that moves between non-adjacent chords, or claim that larger leaps on the graph are musically disfavored in some way.[3] This last step, however, involves a subtle leap. Consider, for example, the familiar Tonnetz (Figure 1).(2) Two chords are adjacent on this graph if they can be linked by what Cohn calls "parsimonious voice leading": voice leading in which a single voice moves, and it moves by just one or two semitones (Cohn 1996). However, larger distances in the space do not faithfully mirror voice-leading facts. On the Tonnetz, C major is two units away from F major but three units from F minor-even though it takes just two semitones of total motion to move from C major to F minor, and three to move from C major to F major (Figure 2). (This is precisely why F minor so often appears as a passing chord between F major and C major.) It follows that we cannot use the Tonnetz to explain the ubiquitous nineteenth-century IV-iv-I progression, in which the two-semitone motion [arrow right] is broken into the semitonal steps [arrow right][arrow right]. More generally, it shows that Tonnetz-distances do not correspond to voice-leading distances in any straightforward way (Tymoczko 2009).Figure 1. The TonnetzFigure 2(click to enlarge and see the rest)[4] Note that the problem persists even if we try to reinterpret the Tonnetz as representing common tones rather than voice leading: both F minor and E minor are three Tonnetz-steps away from C major, even though C major and F minor have one common tone, while C major and E minor have none. (As before, shorter distances are easier to interpret: two chords are adjacent on the Tonnetz if they have two common tones, and any pair of chords that are two steps away will share exactly one common tone.) Thus, neither voice leading nor common tones allow us to characterize Tonnetz distances precisely. We seem forced to say that Tonnetz-distances represent simply the number of parsimonious moves needed to get from one chord to another-and not some more familiar music-theoretical quality.[5] From this point of view, there is a fundamental difference between the Tonnetz and Douthett and Steinbach's "Cube Dance" (Douthett and Steinbach 1998). (Figure 3) Like the Tonnetz, "Cube Dance" depicts a collection of local moves, in this case the single-semitone voice leadings between major, minor, and augmented triads. …