Divisions without hierarchy: four-dimensional modeling of submeter and its use in empirical analysis of the musics of the new complexity

dissertationCollege of Fine ArtsMusicCentral to virtually all scholarship of meter is a notion of beat hierarchy. However, when beat units and/or tempi are in constant flux, defining a hierarchy becomes nearly impossible. Such impulse structuresâ€"common to music of the New Complexityâ€"resist traditional scholarship of meter/subdivision-based hierarchical layers, yet are also more structured than simple rhythms. This dissertation observes how these structures relate to more traditional definitions of meter and rhythm, and categorizes them as submeter. The number of pulse layers defines each of the three categories. Rhythm is shown to require only a single pulse layer, while meter requires three or more. Submeter, containing the beat unit subdivision of meter without its associated beat hierarchy, falls between rhythm and meter, requiring exactly two pulse layers. After establishing this definition of submeter, this dissertation focuses on a four- dimensional formalization of the defining characteristics of submeter. The formalization models beat unit, subdivision, tempo, and duration as orthogonal dimensions. Through an analysis of Brian Ferneyhough’s Unsichtbare Farben, the empirical modeling and its potential are demonstrated. Comparing the values of each dimension shows that, beneath the notational complexity, there are a highly limited number of subdivisions and beat units, and that tempo has the highest rate of variance. In addition, by analyzing frequency distribution and transitional probabilities of beat units and subdivisions, it is demonstrated that while the majority of cases do not impart statistically significant transitional syntax, beat units can be categorized into three functional groups based on their transitions. The dissertation concludes by comparing the shifts in tempo to pitch transformations. Using the submetric units of the first 22 measures (comprising the first section) of Unsichtbare Farben, the tempo shifts are used to project a series of pitches. These pitches are then processed through Robert Morris’s contour reduction algorithm, with the prime layer projecting the trichord {F, F-sharp, A}. This trichord fits into two possible groups with the trichords presented as the closing pitch material of the opening section, demonstrating the relationship between the submeter and the pitch material in Unsichtbare Farben