When are two voice leadings similar? This paper proposes an answer in the form of a geometric theory that shows qualitative similarities between non-identical voice leadings (VLs). Consequently, the theory can describe VLs between harmonies of different Tn-types and can recognize similarities between numerically different VLs.

The basis of the theory is a duality between chords and VLs: in cpitchⁿ, each point representing a chord can be identified with a unique VL. This is done by identifying a perfectly even chords with the zero-motion VL of the same cardinality. Every VL away from this origin is identified with the chord that it leads to. This dual meaning for each point in space allows us to adapt the geometries of Sherrill (2025), which describe scale structures, to model VLs. Where Sherrill’s model compares the sizes of intervals within a set, its application to VLs compares the relative motions of voices, identifying for instance which voices converge or diverge. Points in the geometry are described using a “sign vector” whose entries communicate individual facts about a chord or VL’s shape.

The paper’s central analysis uses these concepts to compare several moments from Tristan und Isolde. While these VLs are not identical, this analysis suggests they are all similar: their sign vectors are more alike than different. This analysis makes new connections within the work, especially by connecting set classes other than the “Tristan genus” (Cohn 2012). While Tymockzo (2011) has highlighted that many resolutions of the Tristan chord combine an efficient VL with a voice crossing, this analysis shows how the opera’s opening motivic complex resonates with other progressions that lack a voice crossing. Moreover, it suggests a structural explanation for Wagner’s use of voice crossings: the geometry constrains efficient VLs in a way that magnifies small inflections into large qualitative differences: the four distinct efficient VLs for the Tristan genus result in very different sign vectors. The zones of less efficient VL offer more scope for quantitatively different but qualitatively similar motions. Thus one source of the Tristan chord’s notorious mutability is, surprisingly, the fact that Wagner deploys it unparsimoniously.

The use of the discrete Fourier transform (DFT) in the pitch-class domain has demonstrated considerable analytical and theoretical potential while remaining comparatively underutilized within mainstream music-theoretical practice. Nonetheless, recent scholarship has shown its capacity for nuanced analytical interpretation and continued theoretical refinement. These developments suggest that the DFT can benefit from and help structure further theoretical, analytical, and pedagogical engagement. In addressing the theoretical problem of "splitting and fusing,” this paper facilitates such engagement, offering new visual and conceptual tools for analysts unfamiliar with the DFT.

Using the DFT to quantify harmonic qualia, this paper develops a framework of harmonic parsimony—including intercardinality as a distinct form of parsimony—to characterize and connect set classes. The DFT enables direct comparison between chords of different cardinality, unencumbered by voice-leading imperatives or considerations. In so doing, it becomes possible to measure—and so test—implicit voice-leading assumptions from a harmonic perspective. This new framework—harmonic qualia networks—affords a practical and original conceptualization of harmony through the DFT. It also offers a helpfully variegated view of voice-leading strategies, explored explicitly here through "splitting and fusing.”

Beginning with an exploration of trichord relations and extending observations across set classes from cardinality 2 to 6, this approach reveals a surprisingly restricted but new harmonic landscape defined by harmonic qualia. Finally, returning these insights to Scriabin, where "splitting and fusing" was first deployed, we can see that the implicit harmonic relations of this approach remain just that, implicit. In using the DFT, we are able to map qualitative relationships between these sets, offering not just the flat boundaries of the landscape of Scriabin’s late harmonic world, but peaks and valleys too. Existing voice-leading notions like "splitting and fusing" offer one kind of harmonic insight. This paper shows both that this can be misleading and that the DFT offers us a way to understand and distinguish such difference, musically.

Over the past thirty years, theorists have developed impressive geometrical tools for modeling music. Cardinality equivalence remains a problem, however: (C4, G4, E5) and (C4, G4, E4, E5) are both C major chords, yet it is difficult to model this relationship geometrically, particularly if one cares about voice-leading distance. This is because cardinality changes create “shortcuts” between otherwise distant chords.
My talk makes two new proposals. The first is that cardinality changing voice leadings should be assigned infinite size, since rests are the musical analogues of infinity. Rests are unchanged by transposition and inversion, which correspond to addition and subtraction. Thus the rest R has the properties R = R + c = c – R, for every real number c. Setting c = 0 gives R + 0 = 0 – R = R = –R. There is a mathematical entity with exactly these properties, namely the infinity that appears in the “one-point compactification of the line.” This proposal removes cardinality-changing shortcuts while producing a psychologically plausible model of rests.
The second proposal is that we should reconsider the common belief that concrete voices can “split” and “fuse.” I argue that concrete sound sources never do this; instead, notes change to rests and vice versa. However, abstract voices, or scale degrees, can split and fuse. Abstract and concrete voices have fundamentally different logics: the former are like routes or roads, while the latter are like cars driving along them. Common geometrical assumptions, such as the triangle inequality, are not appropriate for scale degrees.
Together, these suggestions largely solve the conceptual problems surrounding cardinality equivalence. Many of these arise from our failure to appreciate a surprising but productive analogy between rests and infinity.