The key complex system and multiple degree function : a guide to harmonic analysis in the transitional compositions of Arnold Schoenberg
The evolution of diatonic tonality into a more complex, multidimensional order is evident in Arnold Schoenberg's late tonal and early atonal compositions. The development is inclusive of the synthesis of a major key with the related tonic minor key: twenty-four major and minor diatonic keys integrate to form twelve major-minor keys. The effect of the major-minor merger, or, Modal Interchangeability, on the evolution of tonality is that the increased diversity of degree forms results in expanded degree function and metamorphosed harmonic progressions. Schoenberg implements the change in tonal parameters by focusing on degrees with more than one functional interpretation. The concept is codified as 'Multiple Degree Function'. Degrees function simultaneously to implicate two or more keys without explicitly referring to one key. Initially a localized phenomenon, multiplicity of harmonic intent is gradually structural in manifestation. Compositions that use more than one key area through the application of Multiple Degree Function are based on a Key complex. The Key complex is defined as two or more keys interacting to form the harmonic structure of a composition. There are eleven classes of bipartite Key Complexes, based on the interval between two modally interchangeable keys. Potential degree function is systematized according to the particular degree interface between the two key areas. Compound Key Complexes combine more than one class of key relationship. The Key Complex System and the associated concept of Multiple Degree Function comprise the foundation of this study. Analysis of Schoenberg's transitional works demonstrates an evolving harmonic practice in which potential degree function and harmonic progression can be categorized. It is shown through the method of Key Complex analysis that the transitional and 'atonal' compositions of Arnold Schoenberg are structured on a defined set of key areas.