What is a Weitzmann Region?

Among my areas of interest in music theory, I am particularly interested in extended tonality, namely the description of chromatically adventurous works of the Romantic period all the way through contemporary artists like Steve Reich. While the term “coloristic” has been used at times to brush past more difficult to analyze chromatic passages, Richard Cohn points out that “coloristic” and “chromatic,” traced backed to their respective Latin and Greek origins, mean the same thing. Needless to say, I won’t make that mistake again.

So what governs these transformations? How can we relate a complex chromatic development to its uniformly diatonic bookends?

Cohn explores these questions in his 2012 book Audacious Euphony. One of the earliest theorists to posit a descriptive language for extended tonality was Carl Friedrich Weitzmann, who explained certain passages in terms of the augmented triad, a black sheep in traditional music theory curriculum.

This comes across at first as somewhat unsettling. Augmented triads are for the most part introduced as passing harmonies, or altered dominants, and their “pivot chord” capabilities had not been discussed in my previous music theory education. In fact, you rarely even see them in the surface level of music, unless briefly or, occasionally, front and center as an “uncanny” harmony.

In practice, however, the augmented triad may not even appear in the surface of a piece. Rather, Weitzmann posits that keys and harmonies pivot around the augmented triad. For example, the augmented triad CEG# is only a half-step shy of six different consonant triads: C+, E+, Ab+, C#-, F-, and A- (Cohn uses + to indicate major and – to indicate minor). The four augmented triads and their six supported chords each make up a Weitzmann region and enable modulation through minimal voice leading. Although we consider C+ and C#- to be remote cousins when examining a circle of fifths, the two are in reality only two half steps away from one another, with C moving to C# and G moving to G#. Uncanny though this may sound, it is a simple transformation from a voice-leading standpoint.

Notably, the augmented triad seems to share the diminished triad’s capacity for enharmonic modulation, enabled by the symmetry with which both chords divide the octave.

Although all triads within a Weitzmann region are equidistant, Weitzmann proposed a prototypical ordering that sequences two transformations: the relative and the nebenverwandt. The relative is movement to the relative major or minor: a- to C+, Ab+ to f-. The nebenverwandt is the iv of a major triad or the V of a minor triad: C+ to f-, g- to D+. These can be chained into the following sequence of the C-E-G# Weitzmann region: E+ a- C+ f- Ab+ db- and Fb+ (enharmonically looping back to E+).

If you’re interested in musical examples, they are prevalent and in Schubert and other Romantic composers, and are central to the exposition of Rimsky-Korsakov’s 2nd Symphony “Antar“. This is only a brief overview of Cohn’s discussion of Weitzmann regions, and I look forward to writing more on it as I explore similar topics in extended tonal harmony.


Generative Theory and Gestalt Psychology

Fred Lerdahl and Ray Jackendoff’s A Generative Theory of Tonal Music (GTTM), written in 1983, is a seminal work of the music cognition literature and has been expanded upon by many more recent works. I recently finished reading the work cover-to-cover, which was difficult to process but overall a rewarding process, most of all in the way it made me reevaluate the way I approach and understand music analysis. I plan to return to the actual analysis in a later post, but here, I will address Chapter 12, which covers the relationship of GTTM to psychology and linguistics.

In a few words, Gestalt psychology posits very theories as to how we organize the chaotic bombardment of stimuli around us. In the visual domain, it seeks to explain how one can distinguish objects from their backgrounds, how we determine depth, and how we perceive interrupted continuity, among other things. Notably, the rules which govern this perception are not absolutes, but rather a series of preferences, which may override one another. When competing preferences cannot be resolved, we experience the optical illusions of well-known “impossible” 3-D figures.

These preference rules, Jackendoff and Lerdahl posit, bear striking similarity to several of the preference rules which they have outlined. In particular, several of their preference rules involve distinguishing a musical object from is surrounding material through parallelism, symmetry, and group boundary definition. These preference rules can override one another at various levels, as seen in the dominance of parallelism in the judgement of the main theme in Mozart’s G Minor Symphony, which is one of a few pieces that GTTM focuses on for its analysis.

As apt as their comparisons may at first seem, they temper their theories with acknowledgement of music’s extreme ambiguity. These preference rules often will not yield absolute interpretations, but rather myriad possible interpretations. When I began reading GTTM, I worried that it would suck the joy from listening to music in the same way a bad teacher can, by demanding only a single correct interpretation for a given work. GTTM reveals the complex mechanisms at work in listening and composition, and encourages explorative listening. Much like multistable constructions such as the Necker cube or the Rubin vase, a Mozart symphony can be examined from slightly different angles to reveal a new work, parallel yet distinct from the performance heard by your fellow audience members.

This is what I value in music theory. Rather than seeking absolutes, let’s explore the vast possibilities which each work offers.