Stability Space and the Below-n Threshold





Stability in tonal and early-atonal music is associated with concepts of “force” as defined by Larson (1994), Lerdahl and Jackendoff (1983), Lerdahl (1994, 2001), Schoenberg (Cherlin 2000), Baroni (1983), and Arnheim (Brower 2000). This paper seeks to frame discussions on stability by constructing a notion of stability space – a nexus of the time-space (T) and pitch-space (Π) in which the forces causing stability and instability occur. Because “force” cannot be delimited by a clear hierarchical structuring mechanism as proposed by Schenker (1935) and Lerdahl and Jackendoff (1983), musical moments can belong to multiple stability spaces of varying temporal and pitch dimensions. The below-n threshold (B) is an algorithm that finds trends between these dimensions by discerning the maximum-sized T containing Π of average cardinality n in various works. Because this divisional scheme necessarily cuts through events and event collections, it does not comport with traditional musical segmentation procedures as defined by Hanninen (2001). Rather, it uses an approach suggested by Mandelbrolt (1964) in analyzing the correlation between measurement (|Π|) and the unit of measurement (T).

A study of Scriabin’s late preludes, for example, find a drastic change in the value of between his extended-diatonic and octatonic pitch languages. The data suggests that, beyond simply adopting a scale with one more pc, Scriabin’s mature style also accessed a more ambitious chromatic palette. Such an assertion is supported by Chang (2006), Callender (1998), Kim (1994), and Meeks (1945).

A second study into Schoenberg’s op. 23 Klavierstücke and op. 25 Suite shows divergence in the below-n threshold after n=9, suggesting that the two collections maintain a similar pitch practice in all but the largest pc sets (Example 3). This claim is supported by Hamao (1988) and Hyde (1985). A musical justification for the divergence between n=9 and n=10 is suggested by Hyde, who identifies serial practices that involve pitch sets of cardinality 8-10 in at least three op. 23 stücke.

Through these case studies, I hope to show how a relatively basic conceptualization of the pitch and time domain can reveal stable and instable uses of pitch that are not suggested by conventional means of musical segmentation.