In his study of Robert Schumann's music, Harald Krebs defines two useful types of metrical dissonances: grouping dissonances and displacement dissonances. However, there is an additional kind of metrical dissonance, possible in asymmetrical meters (5/4, 7/8, and the like), which does not fit squarely within the bounds of either category. What I call a shared-cardinality grouping dissonance (SCGD) is the subject of the present paper.
This paper is in three parts. First, I begin with a general discussion of asymmetrical meters, focusing on metrical well-formedness as understood by Fred Lerdahl and Ray Jackendoff, Justin London, and David Temperley. Second, I define the concept of an SCGD and discuss its general characteristics. In this section, which serves to distinguish the differences between an SCGD and Krebs' previously defined dissonance categories, I provide examples of SCGDs through a metrical analysis of the second movement of Tchaikovsky's Sixth Symphony. Finally, I conclude the paper by suggesting some extensions to the SCGD concept.

In recent music theory, graphs--nodes connected by (possibly labeled) arcs or arrows--have become an important tool for modeling the structure of music, musical structures, and composition systems. Transformational networks, compositional spaces, K-nets, Perle cycle-sets, the Tonnetz, voice leading spaces, and canon graphs are examples. The paper examines an important property of such graphs, the number and content of a graph s minimal cycles. This is a useful way to characterize a graph especially when it is complex and its visual presentation is too convoluted to provide much insight into its form and properties. Moreover, minimal cycles can provide ways to compare the musical entities modeled by graphs and assess their similarity.
We begin with the construction of graphs from relations and enumerate the resulting graph's minimal cycles using a computer-implemented algorithm. We then apply the study of minimal cycles to issues in (1) twelve-tone theory, (2) K-nets, (3) modal counterpoint, and (4) tune families extracted from actual compositions or from various modal/tonal theories including Indian music. The paper ends with some thoughts on how minimal cycles might be implicated in models of musical cognition and memory.