Atonality

Theories of Atonal and Serial Music

Influential theories of atonal music analysis were published in the 1960s and 1970s, among them George Perle's Serial Composition and Atonality (first published in 1962), and Allen Forte's The Structure of Atonal Music (1973). In the United States, Forte's system of atonal music analysis was widely adopted in academia and taught in many universities. This system represents the 12 notes of the chromatic scale as integers 0 to 11, assuming octave-related pitches (called “pitch classes”) to be equivalent. Sets are unordered collections of pitch classes, which may be transformed by defined operations. Using set- and group-theoretical principles drawn from mathematics, music theorists established principles by which sets of pitches might be classified as equivalent, such as by transposition or inversion. Together, these principles allowed researchers to generate a list of all possible set classes; the numbers Forte assigned to these were widely used in the United States as analytical labels for atonal melodies and chords. Perle's system also formalized language for discussing symmetries and cycles of repeated intervals.

Serial music differs from other atonal music in its use of ordered series of pitch classes, rather than unordered sets. Although in Schoenberg's original formulation the ordered series contained all 12 tones, other composers wrote serial works with series of differing lengths. Still others serialized elements other than pitch, composing with ordered sets of durations, dynamics, articulations, or timbres to create works of “total serialism.” In classic 12-tone composition, the series or “row” is transformed by four operations: transposition (T), inversion (I), retrograde (R, reversed order), and retrograde inversion (RI, reversed order of the inversion). Through a process known as combinatoriality, rows may be grouped harmonically so that the complete collection of 12 tones circulates more quickly.

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