# Response to Comment on "The Geometry of Musical Chords"

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Science  19 Jan 2007:
Vol. 315, Issue 5810, pp. 330
DOI: 10.1126/science.1134163

## Abstract

The basic sonorities of traditional Western tonality divide the octave nearly evenly and are found near the center of the orbifolds $Math$ and $Math$. Many common musical patterns exploit this fact, which permits efficient voice leading between structurally similar chords. In actual music, these patterns sometimes appear incompletely or are accompanied by additional notes. Using orbifolds in musical analysis therefore requires interpretive skill.

Theorists agree that when analyzing music, there are two distinct phenomena to consider: the actual notes (the surface) and the common musical patterns those notes may imperfectly embody (the background). Orbifolds can be used to represent either phenomenon, although they are typically most useful when modeling background patterns.

Triads and seventh chords are the conceptual building blocks of Western tonality, the only complete harmonies recognized by traditional theory. These sonorities are found near the center of the orbifolds $Math$ and $Math$. Theorists agree that in actual music these harmonies are sometimes incomplete and are sometimes accompanied by additional notes (doublings and “nonharmonic tones”). Thus, we cannot expect the surface of every tonal piece always to inhabit the center of some orbifold.

Figure 1A, reproduced from (1), is typically accompanied by an additional voice, as in Fig. 1B. The full progression lies on the singular boundary of $Math$. Directly plotting Fig. 1B on this orbifold is not maximally informative, as its lowest voice operates according to what are generally recognized to be distinctive musical principles. My report therefore separates the progression into two parts: the upper voices, which exhibit efficient voice leading between structurally similar chords, and the bass, which adds harmonic support by leaping to the root of each chord (1). The upper voices exploit the geometry of $Math$; the bass plays a different musical role.

Incomplete chords pose a related challenge. Most theorists would understand Fig. 1C to imply a succession of triads, as in Fig. 1B. We can represent the musical surface by plotting the incomplete chords of Fig. 1C on the orbifold $Math$; we can represent the background by plotting Fig. 1A or 1B on the appropriate orbifold. Again, the upper three voices of the background pattern (Fig. 1A) make the most interesting use of orbifold geometry.

Brown and Headlam (2) observe that tonal phrases sometimes cadence on unisons. We can typically model these cadences as incomplete manifestations of a prototypical five-voice background pattern (Fig. 2). The pattern's top four voices use maximally efficient voice leading to connect nearly transpositionally related multisets, interestingly exploiting the geometry of $Math$. If, alternatively, we are interested in the musical surface, we can again use orbifolds. As Headlam and Brown note, when a tonal piece cadences on a unison it moves from the center of an orbifold to its singular boundary. Thus, these cadences do possess a distinctive geometrical signature, even if we restrict our attention to the musical surface.

Given these two analytical possibilities, it is unclear why Headlam and Brown claim that orbifolds cannot effectively model such progressions. Perhaps they have not clearly distinguished the musical surface from the background or have misinterpreted my remarks about the background (that the basic tonal sonorities inhabit the center of their respective orbifolds) as remarks about the surface. It is also possible that they have misinterpreted my Report (1) as an attempt to model all the style-specific norms of 18th- and 19-century composition (3). Instead, it was an effort to model a few general musical principles common to a wider range of Western musical styles.

Orbifolds are tools that can be used for a variety of musical purposes. Like any tools, they have limitations, but these limitations are considerably less constraining when the tools are skillfully deployed.

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