Technical Comments

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


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.

Fig. 1.

(A) A common classical upper-voice pattern that exploits the geometry of Embedded Image. (B) In actual music, the three-voice pattern is often accompanied by an additional bass voice, whose function is to provide harmonic support by sounding the root of each chord. (C) A two-voice passage evoking (B).

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.

Fig. 2.

Many tonal cadences, including all of the examples in (2), can be understood as variants of a prototypical five-voice pattern (A). The pattern's prototypicality is illustrated by the fact that it concludes several of the first pieces in Bach's Well-Tempered Clavier (WTC)(C to F), as well as several of Beethoven's first piano-sonata movements (G to K). By contrast, Headlam and Brown's cadence (B) concludes only one of the first 20 Beethoven piano-sonata movements and none of the first 20 pieces in the WTC. (A) The prototypical five-voice pattern. (B) The cadence from figure 1 in (2). (C) Bach, WTC Prelude 3. (D) Bach, WTC Fugue 3. (E) Bach, WTC Prelude 6. (F) Bach, WTC Prelude 10. (G to H) Beethoven, Piano Sonata No. 1, movements 1 and 2. (I to K) Beethoven, Piano Sonata No. 2, movements 1, 3, and 4.

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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