Historically, intervals between musical pitches have been understood through two distinct conceptual models: either as distances in an imaginary space or as ratios between frequencies or string lengths. Each model has its own biases: the distance model is well-suited to constructing abstract pitch geometries, while the ratio model offers insight into an interval's sonic quality and stability. In recent music theory scholarship, the popularity of the distance model has led to the neglect of interval ratios: in this dissertation, I argue that a return to the ratio model offers a deeper understanding of many works of contemporary music which are difficult to analyze with distance-based tools.

One subset of the ratio intervals is of particular interest: the just intervals, intervals with frequencies related by simple whole number ratios like 3/2 or 5/4, have historically been associated with musical consonance. The psychoacoustical properties which give simple just intervals their consonant effect persist for intervals with more complex ratios; such "extended just intonation" has been explored by composers including Harry Partch and Ben Johnston. In addition to their consonance, just intervals strongly imply a harmonic root, providing cognitive support for various forms of tonal centricity.

In order to apply just intonation theory to a broader variety of music, I propose a theory of harmonic perception based on Hugo Riemann's Tonvorstellung. I argue that even intervals which are not perfectly just tend to share harmonic properties with the nearest just interval—for example, we hear a piano's tempered major third as essentially equivalent to a just major third, even though the tempered third is slightly wider. I call this process of matching a heard interval to a nearby just interval "tone representation." The theory is developed into three preference rules for practical application—these rules draw on recent research in Gestalt and music psychology.

The theory is used to analyze twentieth-century works by composers including Arnold Schoenberg, György Ligeti, La Monte Young, and Gérard Grisey. As a pragmatic theory based in perception, tone representation can illustrate the common factors between works by composers with very different aesthetics and compositional techniques.