Humperdinck and Wagner

Abstract

Building upon recent geometric models of metric states by Richard Cohn and Scott Murphy, this article proposes a complementary model that expands the metric states that can be considered, relates states by symmetries, and creates systems of such states. Application of the model to Engelbert Humperdinck's Hänsel und Gretel and Richard Wagner's Parsifal reveals metric-dramatic strategies common to the two operas: certain metric symmetries parallel certain dramatic ones. recent work by Richard Cohn and Scott Murphy has modeled metric states and their relationships geometrically. Building upon and complementing these models, I introduce a new model that expands the metric states that can be considered, relates states by symmetries, and creates systems of such states. Application of this model to Engelbert Humperdinck's Hänsel und Gretel and Richard Wagner's Parsifal reveals metric-dramatic strategies common to the two operas.The article begins with a brief exposition of Cohn's and Murphy's models. The limitations of these models motivate the development of the metric states, symmetries, and systems of my model, illustrated preliminarily in the scherzo of Beethoven's Ninth Symphony. Fuller application to Hänsel und Gretel and Parsifal demonstrates surprisingly sophisticated links between the two works. The article concludes by suggesting avenues for further exploration. Cohn and MurphyBoth Cohn (2001) and Murphy (2006) base their models on the conception of meter as a hierarchy of pulse levels. Any such hierarchy is here called a metric state, following Cohn (2001, 296). Cohn and Murphy place specific restrictions on the metric states that they model; both writers deal only with states in which Erin Lippard's 2005 paper provided the impetus for this study; I thank her for our subsequent discussions of the topic. I am grateful to Richard Cohn and Scott Murphy for their perceptive comments on an earlier version of this article, Yonatan Malin and Philip Chang for their helpful feedback on an oral version, and Elizabeth Medina-Gray for her useful critique. JMT's anonymous readers also supplied many helpful comments.

Meter Metrics

“…However, both Cohn and Murphy develop only a limited set of metrical relationships, due to their focus on metrical dissonance in tonal music, especially Brahms. (7) (The same is true of Leong's [2007] related work on Humperdinck and Wagner.) They therefore do not engage with mixed meters or the wider set of relations that exist between those structures.…”

“…Complementation is isomorphic in mathematical structure to Leong's (2007) "vertical reflection" relation; however, Leong's relation is designed to address proportional schemas in simple meters, and as such, the similarities reside in the mathematical formalism, and not in the musical comparisons. (76) The more musically relevant notion is the "contour" of a musical rhythm, based on the differences between the lengths of successive beats.…”

“…Return to text 72. The "symmetry" relations discussed here concern beat ordering in mixed meters, an entirely separate consideration from the "symmetry" relations in the literature on simple meters (such as Leong's [2007] "r180" relation and Gotham 2015). In future work that uses these two types of symmetry concurrently, it may be desirable to distinguish an "O[rder]-Sym" (for beat ordering) from a "P[roportional]-Sym" for proportional schemes.…”

“…Return to text 46. The literature on simple meters includesCohn 2001, Leong 2007, and Murphy 2009 as discussed above, as well asLondon's Zeitnetz (2002, 2012 and even the fourteenth-century triangle of Johannes Torkesey (seeGilles and Reaney 1966).Return to text 47. To formalize the relationship, let a and b stand for two meters, expressed by an entry in the Order column of Figure2above.…”

Spectral Meter: Dramatizing Entrainment and Communicating Form in Gérard Grisey’sVortex Temporum I(1994–96)

“…While theories of meter often focus on surface-level rhythms and brief cognitive processes like entrainment, there is also significant precedent for a meter-focused analysis of larger formal units. Authors generally concede that meters cannot create progressions as rich and varied as those generated by tonal harmony, but they nevertheless note similarities between forms suggested by metric changes and those based on tonal structures (Lewin 1981, 261-65;Leong 2007;Murphy 2009). Metric shifts between stability and conflict often outline familiar forms on local and global levels (Malin 2006(Malin , 2008.…”