The article proposes that a construct I call the Dasian space provides an effective framework to interpret harmonic aspects of scale relations in twentieth-century polymodality, particularly in the music of Bartók. Based on Bartók's intuition that the pitch space modeled after his notion of polymodal chromaticism retains integral “diatonic ingredients,” the Dasian space (named after the medieval homonymous scale) establishes a system of relations between all potential diatonic segments, without relying upon traditional constraints, such as complete diatonic collections, harmonic functions, or pitch centricity. The Dasian space is a closed, nonoctave repeating scalar cycle, where each element is identified by a unique coordination of pitch class and modal quality. The dual description of each element enables both the specification of location in a given cycle and the emergence of a group structure, whose generators—named transpositio and transformatio—are also characteristic musical motions and relations. The proposed analytical methodology is probed in a couple of short pieces of Bartók's Mikrokosmos and in the third movement of his Piano Sonata. The article argues that, unlike other tonal and atonal classic approaches, the Dasian framework enables the analyst to reconcile the constructional character of a Bartókian idiomatic feature (the combination of distinct and integral scale strata) with the interpretation of harmonic space in terms of scale-segment interaction and formal processes. The article then contextualizes the structure of the Dasian space within a larger class of constructs, which I call affinity spaces, by generalizing some of its group-theoretical properties that model relations between nondiatonic scalar materials. The analytical pertinence of affinity spaces is probed in Bartók's “Divided Arpeggios,” an intriguing posttonal piece appearing late in the Mikrokosmos set.
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José Oliveira Martins; Bartók's Polymodality: The Dasian and Other Affinity Spaces. Journal of Music Theory 1 October 2015; 59 (2): 273–320. doi: https://doi.org/10.1215/00222909-3136020
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