This article generalizes Ian Quinn's recent harmonic characterization of pitch-class sets in equal tempered spaces to chords drawn from continuous pitch and pitch-class spaces. Using the Fourier transform, chords of any real-valued pitches or pitch classes are represented by their spectra and located in a harmonic space of all possible chord spectra. Euclidean and angular distance metrics defined on chord spectra correlate strongly with common interval-based similarity measures such as IcVSIM and ANGLE. Thus, we can approximate these common measures of harmonic similarity in continuous environments, applying the corresponding harmonic intuitions to all possible chords of pitches and pitch classes in all possible tuning systems. This Fourier-based approach to harmony is used to compare the properties of twelve-note chords in Witold Lutosławski and Elliot Carter, to analyze the opening section of Gérard Grisey's Partiels, and to investigate the structural properties underlying the Z-relation (part of ongoing research with Rachel Hall).