The discrete Fourier transform (DFT) has recently gained traction as a music-theoretic and music analyticaltool, providing a mathematically robust way of modeling various musical phenomena. I examine both local and large-scale harmonic structures by using computational methods to interpret the pitch classes of a digitally encoded musical score through the discrete Fourier transform. On the small scale, my methodology shows relationships between sonorities within Fourier space, and aids in making statements about proximity, similarity, and distance between individual harmonic structures. On the larger scale, my methodology offers a broad view of the backgrounded scales and pitch-class collections of a piece: the "macroharmony. The DFT has the distinct analytical advantage of being stylistically and historically neutral. This allows a single methodology to apply to music from a wide variety of genres, time periods, and styles. In Chapter 1, I present the conversations scholars have had thus far pertaining to the DFT and its applications to harmony and pitch classes. I focus particularly on the contributions of David Lewin, Ian Quinn, and Jason Yust as three of the most influential people in promoting the DFT as a viable music-theoretical tool. Chapter 2 contains an overview of how the DFT applies to pitch classes in 12-tone equal temperament, along with a tutorial on its application. I then discuss some of the challenges of computational approaches to music analysis. In Chapter 3, I focus on simultaneities--the amalgamation of pitch classes sounding in a particular moment in time--and the distances traveled in Fourier space when moving from one to another. My approach is an expansion of Justin Hoffman's cartographies of multisets in Fourier space, which I apply to a chorale by J. S. Bach and a passage from a string quartet by Thomas Adès. In Chapter 4 I expand the span of musical time used to define the multiset. Instead of a single moment in time defined by a discrete harmonic event, I use the technique of overlapping windowing to examine the macroharmony of musical excerpts by W. A. Mozart and Olivier Messiaen. Chapter 5 expands the applications of the DFT even further, now to quarter-tone and other microtonal systems. I apply the techniques from previous chapters to works by Charles Ives and Alois Hába. Finally, Chapter 6 provides a short summary of the project, and includes ideas for future research endeavors.