A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses.

Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. We model this process mathematically, showing that it produces 32 equivalence relations on chords, 243 equivalence relations on chord sequences, and 32 families of geometrical quotient spaces, in which both chords and chord sequences are represented. This model reveals connections between music-theoretical concepts, yields new analytical tools, unifies existing geometrical representations, and suggests a way to understand similarity between chord types.

This article relates two categories of music-theoretical graphs, in which points represent notes and chords, respectively. It unifies previous work by Brower, Callender, Cohn, Douthett, Gollin, O'Connell, Quinn, Steinbach, and myself, while also introducing new models of voice-leading structure-including a three-note octahedral Tonnetz and tetrahedral models of four-note diatonic and chromatic chords. music theorists typically represent voice leading using two different kinds of diagram. In note-based graphs, points represent notes, and chords correspond to extended shapes of some kind; the prototypical example is the Tonnetz, where major and minor triads are triangles, and where parsimonious voice leadings are reflections ("flips") preserving a triangle's edge. In chord-based graphs, by contrast, each point represents an entire sonority, and efficient voice leading corresponds to short-distance motion in the space, typically along an edge of a lattice. This difference is illustrated in Figure 1, which offers two perspectives on the same set of musical possibilities: on the top, we have the traditional note-based Tonnetz, while on the bottom we have Jack Douthett and Peter Steinbach's (1998) chord-based "chicken-wire torus." 1 These figures both represent single-step (or "parsimonious") voice leading among major and minor triads and are "dual" to each other in a sense that will be discussed shortly.In A Geometry of Music (Tymoczko 2011), I provide a general recipe for constructing chord-based graphs, beginning with the continuous geometrical spaces representing all n-note chords and showing how different scales determine different kinds of cubic lattices within them. I also showed that nearly even chords (such as those prevalent in Western tonal music) are represented by three main families of lattices. Two of these are particularly useful in analysis: the first consists of a circle of n-dimensional cubes linked by Thanks to Richard Cohn and Gilles Baroin for helpful comments.

It is frequently observed that over the course of the nineteenth century the chromatic scale gradually supplanted the diatonic. 1 In earlier periods, non-diatonic tones were typically understood to derive from diatonic tones: for example, in C major, the pitch class F≥ might be conceptualized variously as the fifth of B, the leading tone of G, or as an inflection of the more fundamental diatonic pitch class FΩ. By the start of the twentieth century, however, the diatonic scale was increasingly viewed as a selection of seven notes from the more fundamental chromatic collection. No longer dependent on diatonic scale for their function and justification, the chromatic notes had become entities in their own right. Broadly speaking, composers approached this new chromatic context in one of two ways. 2 The first, associated with composers like Wagner, Strauss, and the early Schoenberg, de-emphasized scales other than the chromatic. 3 Chord progressions were no longer constrained to lie within diatonic or other scalar regions. Instead, they occurred directly in chromatic space-often by way of semitonal or stepwise voice leading. Melodic activity also became increasingly chromatic, and conformed less frequently to recognizable scales. Chromaticism thus transformed not only the allowable chord progressions, but also the melodies they accompanied.