The Use of Graphs in the Analysis of Post-Tonal Music
The paper linked above uses graphs to analyze post-tonal music; that is, music which does not use the tonal harmonic conventions which dominated the common practice period and thus cannot be analyzed using traditional music theory. The authors present an algorithm to find common streams, which are chains of sequential “events.” An “event” is simply defined as a pitch, a start time, and an end time (in this way, it is essentially a musical note). Streams are significant because they are usually heard as one continuous musical line, just as a melody is in tonal music. Musicologists have been studying streams in post-tonal music for some time, but the process of manually identifying common streams is quite laborious.
The paper uses graphs to facilitate the identification of streams. The nodes of a musical graph are events (or, equivalently, musical notes) and the edges are connections between the events. There are two different kinds of edges: harmonic edges and melodic edges. Harmonic edges are drawn between two events that overlap in time, and melodic edges are drawn between subsequent events. In this manner, the musical graphs are similar to the signed graphs from class, in that not all edges signify the same type of connection. However, the musical graphs are not exactly signed graphs because the melody and harmony are not opposing concepts, they are just different. Similarly, a “balanced” musical graph would not have a clear definition.
The authors use the musical graphs to find which streams are most common in certain pieces of music. The algorithm is quite mathematically complex, and is written out in detail in the paper. Essentially, the algorithm is given a query, which is a set of pitches, and finds the frequency with which these pitches are linked in the graph. At a basic level, the mechanism works by continually reducing the graphs to simpler subgraphs until this can no longer be done. The paper uses an example from Arnold Schoenberg, who developed the twelve-tone technique, which holds that all twelve notes of the chromatic scale should be sounded equally as often in a composition. This does not mean, however, that there cannot be sequences of notes that are more common. In fact, the algorithm in the paper found that in Schoenberg’s Klavierstucke, the pitch-class set {0,1,4} is the most common. When musicologists look for pitch-class sets in this manner, all sets that can be shifted by a constant factor are considered equivalent, so {0,1,4} really represents all sets {n,n+1,n+4}. This set is that of a pitch p, the pitch a semitone above p, and the pitch four semitones above p. The prevalence of the set {0,1,4} matches the findings of manual analysis by musicologists and therefore provides evidence for the effectiveness of the algorithm.