Assertions of musical significance are usually based on unarticulated and often poorly formed assumptions about probability. That is, when we say that a certain note or a certain set is musically/analytically significant, we mean that it would not be likely to occur as a result of random processes, but rather reflects some meaningful compositional design. Assertions of significance are normally made against a background of what might have resulted from a random distribution of tones. But in order to ascertain if some musical event really is significant, we need to have a better sense of what is musically probable or improbable in a particular context. This paper investigates how probabilities can be used to support analyses of post-tonal works. It explains a context-sensitive approach to probability, and presents a simplified method of applying probability formulas. Because finding meaningful applications for probability in analysis can often times be difficult, this paper discusses the various factors involved in choosing a suitable probabilistic model.
Guide for the Perplexed: A Tutorial on Lewinian Boolean Analysis of Babbitt's Composition for Four Instruments
David Lewin’s stature as a theorist derives not only from the sophistication of his musical insights, but the creativity of his methodology. In his 1995 Spectrum article, “Generalizing Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio,” he used Boolean algebra to shed light on Babbitt’s Composition for Four Instruments. This paper provides a guide for the perplexed to explain Boolean concepts and Lewin’s application of this elegant but neglected analytic tool. Additional voice leading applications are proposed.
Boolean algebra is a binary or mod 2 numerical system consisting of two elements or bits, 0 and 1, useful for modeling any musical parameter reducible to an on-off or “light switch” format. Composite timbral states and their transformations can be reduced to binary codes. Lewin’s Boolean analysis reveals that although Babbitt adheres to his aesthetic principle of parametric exhaustion through unique instrumental combinations generating maximum timbral diversity, the music nonetheless encodes a significant degree of structural order through transformational redundancy on hierarchical levels.