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The discrete Fourier transform on pitch-class sets, proposed by David Lewin and advanced by Ian Quinn, may provide a new lease on life for Allen Forte's idea of a general theory of harmony for the twentieth century based on the intervallic content of pitch-class collections. This article proposes the use of phase spaces and Quinn's harmonic qualities in analysis of a wide variety of twentieth-century styles. The main focus is on how these ideas relate to scale-theoretic concepts and the repertoires to which they are applied, such as the music of Debussy, Satie, Stravinsky, Ravel, and Shostakovich. Diatonicity, one of the harmonic qualities, is a basic concern for all of these composers. Phase spaces and harmonic qualities also help to explain the "scale-network wormhole" phenomenon in Debussy and Ravel and better pinpoint the role of octatonicism in Stravinsky's and Ravel's music. [ABSTRACT FROM AUTHOR]