In his last works, Stockhausen vigorously explored layering multiple tempos. Whereas many composers who experimented with polytempo or polymeter (such as Carter and Nancarrow) sought to create higher-level unities where pulse layers coincided at various points, Stockhausen vigorously avoided such alignments, practicing a strategy of metric incongruity instead. By pursuing a robust computational analysis of metrical dissonance in Hoch-Zeiten (2001-2), a piece for five choirs and orchestras, this study has uncovered many subtle interactions among pulse streams. My analysis concludes by interpreting the remarkable compositional procedures in a larger metaphysical context, clarifying their meaning and purpose.
Inspired by Viktor Zuckerkandl’s view of meter as wave, I view Stockhausen’s multiple metrical layers as oscillations, finding further justification for this approach in Stockhausen’s essay “...how time passes...” The five metrical layers in Hoch-Zeiten can then generally be represented as superimposed sine waves, and their interactions viewed graphically as sums. This analytical method presents many advantages, especially in its direct visual appeal (a sampling of the results of my graphical analysis can be found on the following page.) As I show, specific musical events in each layer rarely induce any sense of metric regularity; rather, the entire system is designed to efface any hearing that would privilege one layer over the others – though when one takes into account the linguistic content, a layer may temporarily come to the foreground in the chorus version.
My analysis shows two unexpected properties of Stockhausen’s tempo choices in Hoch-Zeiten: first, that partial correspondences among layers generally occur every 16 seconds, surprisingly aligning with page turns in the score. Second, a large-scale composite metric cycle, corresponding to 16 seconds, permeates the metric background with few exceptions. Although this cyclic regularity forms the piece’s metric background structure, there are scant surface markers to confirm it.
To shed light on this question, I turn to Plotinian metaphysics, a branch of thought which has been shown to have strongly influenced Stockhausen’s thinking throughout his life. The metric strategy in Hoch-Zeiten resonates metaphorically with the Plotinian view of the universe as organized by a higher being (“One”) separated from all worldly things, yet present in everything, forming the basis of the Cosmos. Stockhausen’s expression of this, according to Thomas Ulrich, is to create musical systems wherein “each single tone is ... an ensemble of contexts.” As Stockhausen himself wrote, “music is an image of that comprehensive ‘global’ structure in which everything is embraced.” Just as Plotinian order is supra- personal, metric order in Hoch-Zeiten exists far above the level of the individual musical event. Through this lens, the extremely complex and sometimes bizarre ensemble of tempo superimposition begins to make sense as a metaphor for the metaphysical system Stockhausen believed in so strongly.
Paul Miller is a music theorist and a performer specializing in music of the 17th and 18th centuries. Before joining the musicianship department of the Mary Pappert School of Music at Duquesne University in 2015, he served as a Mellon Postdoctoral Fellow at Cornell University and on the faculties of the University of Colorado in Boulder and Temple University.
Paul has presented research at numerous national and regional conferences, and his work has been published in Perspectives of New Music, the American Music Research Center Journal, Twentieth-Century Music and Music and Letters. Forthcoming publications will be in Notes and Opera Quarterly. An expert on the remarkable music of Karlheinz Stockhausen, Paul studied with the composer for six summers and premiered his solo viola work "In Freundschaft" in Europe and the United States. Paul's research has centered on the unusual spatial dimension of Stockhausen's music as well as the phenomenon of metric complexity. In addition, Paul works on ornamentation and improvisation in 18th century music, especially as it relates to the works of Italian composers such as Arcangelo Corelli.