Richard Cohn's Complex Hemiolas, Ski-Hill Graphs and Metric Spaces presents analytical tools for finding and graphing coherence among metric states across the entirety of a work. Cohn maps metric states -- the symmetrical groupings of a composite, non-prime) number of pulses -- onto graphs of "metric space," similar to a graph of abstract pitch space like a Tonnetz. Graphs of metric space allow us to map the various metric states and their distance, or dissonance, from one another. I will address two problems that arise when applying Cohn's theory to certain music. First, Cohn's theory does not account for the asymmetrical partitioning of a time span. Second, while Cohn acknowledges what herald Krebs calls "grouping dissonance," Cohn does not address "displacement dissonances," another important facet of metrical analyses.
Cohn's theory does not account for the asymmetrical partitioning of a time span because his ski-hill graphs utilize only duple or triple relationships. To accommodate music with an asymmetrical partitioning of a time span, either the theory must be modified by altering or omitting the ski-hill graphs, or the music must have underlying symmetrical time spans. This presentation will focus on the later solution. I will show how in the fanfare of Janáček's Sinfonietta, asymmetrical time spans are in fact altered symmetrical time spans. These alterations arise from various types of phrase expansions -- metric phenomenon described by William Rothstein. This presentation will also expand on Cohn's graphs of metric space by accounting for displacement dissonances.