- Audacious Euphony: Chromaticism and the Triad’s Second Nature by Richard Cohn
Representing an amalgamation of more than two decades of research and development, Richard Cohn’s contribution to the Oxford Studies in Music Theory presents a thorough yet controversial argument for a theory of chromaticism in tonal music during the long nineteenth century. The potential for Audacious Euphony to spark active discussion lies in Cohn’s request for readers “to suspend some overlearned habits” regarding the pedagogy of most North American music theory textbooks (p. x). These habits include the traditional conception—the first nature—of major and minor triads as parts of a diatonic scale, as pitches stacked on top of their roots, and as a consonance in terms of interval content. In other words, these triads constitute a diatonic syntax for analytical and compositional purposes, against which Cohn presents a position based on ideas espoused in nineteenth-century music theory [End Page 113] treatises and in twentieth-century atonal set-theory discourse. The triad’s second nature involves conceiving major and minor triads as independent objects within a closed chromatic space (modulo 12), which opens up an assortment of analytical possibilities, such as the ability to track voice-leading efficiency and create geometric representations of a logical “pan-triadic” syntax (p. xiv).
One of the basic elements of Cohn’s theory, found in chapter 1 (“Mapping the Triadic Universe”), comes from his novel approach to measuring distances between major and minor (hereafter “consonant”) triads by means of “voice-leading work.” In this pan-triadic world, we can quantitatively compare consonant triads based on the number of shared common tones and total amount of voice-leading work through “idealized voice leading,” an abstract concept that measures voice leading in semitones without regard to register (p. 6). Chapter 2 (“Hexatonic Cycles”) introduces a formalized system based on efficient voice leading and shared common tones. For example, both the C major and A–flat major triads are in a “minimal-work relationship” with the C minor triad, which means they share two common tones and the remaining tone moves by semitone (i.e., a single unit of voice-leading work) to produce the other triad. When this system repeats, it creates a hexatonic cycle: a set of six consonant triads—C major, C minor, A–flat major, A–flat minor, E major, E minor, C major (i.e., cyclic closure)—whose three different roots form an augmented triad. There are three other hexatonic cycles (as there are three other augmented triads) that include the remaining eighteen consonant triads, which “produces a preliminary map of the triadic universe” when combined with the initial hexatonic cycle described above. According to Cohn, this “model is sufficient to provide preliminary support for the central claim that the capacity for minimal voice leading between chords of a single type is a special property of consonant triads, resulting from their status as minimal perturbations of perfectly even augmented triads” (p. 17). Because augmented triads divide the octave into three equal parts, separated by major thirds, they accordingly create a “perfectively even” distribution of pitches in chromatic space. Later in the chapter, Cohn emphasizes the dual existence of consonant triads, as both acoustically consonant and nearly even in terms of their distribution of pitches in chromatic space. In this sense, “triads are homophonous diamorphs: one sound, two forms” (p. 40).
Cohn discusses the historical development and use of augmented triads in music theory treatises and compositions during the long nineteenth century in chapter 3 (“Reciprocity”). He gives special attention to Carl Friedrich Weitzmann’s Der übermässige Dreiklang (1853), in which Weitzmann tells various stories about the genesis of the augmented triad. He begins by describing it as subservient to consonant triads, but ends with what Cohn calls a “subtle inversion.” Weitzmann produces a list of the four augmented triads and their six consonant triads (three major and three minor), or “patrons,” that lie within...