Lerdahl’s Timbral Hierarchies

The real reason, I would argue, why timbre has been regarded as a secondary musical dimension is that, unlike pitch and rhythm, it has lacked any substantial hierarchical organization.

–Fred Lerdahl, 1987

Yesterday I read “Timbral Hierarchies” by Fred Lerdahl, originally published in 1987 in Contemporary Music Review Vol. 2. This article is post-GTTM (A Generative Theory of Tonal Music) and represents an attempt to explain how timbre prolongations, or at the very least timbral hierarchies, might be possible, in much the same vein as L&J-type metrical or tonal hierarchies.

This article is another curious entry in the outpouring of timbre music theory research that occurred in the mid-1980s (see also Cogan 1984, Slawson 1985). Since I wasn’t researching in the 1980s, I’ve wondered myself what the music theory community was like at this time, and what in the culture propelled this sudden interest in timbre. I presumed that this was due to a wider access to 1) spectrograms, a useful visualization tool for timbre, and 2) digital synthesizers, which allow for the level of precise control necessary in many perception studies. Lerdahl identifies out another possible impetus for a sudden rush to theorize timbre: “The issue has sharpened with the recent rise of computer music. There is now such an infinity of timbral possibilities that the need for some kind of selection and organization has become acute” (136).

I’ve found it funny in the past that I study 1980s popular music, and that so many of the existing articles and books on timbre research also date from the 1980s. But this quote in particular helped me realize that the unifying factor in all of this is rapid technological advancement. Advances in computing technology and digital synthesis are what defines the sound of 1980s pop music; these same things also end up defining research trends, including research trends about music. It’s no coincidence, in other words, that timbre research flourished during the same time as the music I’ve chosen to study (although I didn’t choose the music for that reason).

Lerdahl’s article, rather like Slawson’s book, spends a good deal of time establishing what a successful “theory of timbre” would look like. I want my own timbre theory to be a theory that theorists like, so these questions interest me. Lerdahl’s theory is grounded in psychological concepts from the 1980s. Lerdahl relies on association to group things together and organize them—certainly a foundational concept. But of course, simply saying “this is like that” is not a very satisfying analysis to read. Enter hierarchy. Lerdahl correctly notes that cognitive psychology has shown that the mind can learn an organize a great deal more information if it can be organized hierarchically. Therefore, he concludes it would be advantageous to create a timbral hierarchy. In short: “Thus a theory of associations requires a theory of hierarchies” (138).

To get to a timbral hierarchy, Lerdahl has a number of intervening steps. He proposes a system of timbral consonance and dissonance, akin to tonal consonance and dissonance, which can create what he calls stability conditions that will permit the hierarchy. A more stable thing is at a higher level than a less stable thing. A more consonant timbre is more stable than a more dissonant timbre.

But what is a consonant or dissonant timbre? This is where I find Lerdahl begins to tread into the territory of arbitrariness. For practicality’s sake, Lerdahl must focus on only a few timbral attributes. Lerdahl admits that timbre is multidimensional and so creating a continuum from most dissonant to most consonant will not be feasible. So he chooses vibrato and harmonicity (the degree to which partials follow the harmonic series) as his two attributes of focus.

While these steps seem logical enough in isolation, combining these ground rules together to make an analytical theory seems to miss the point.

Lerdahl proposes an “ideal” level of vibrato as the baseline, and says that more or less vibrato is less consonant (i.e., less stable) than this ideal. In other words, a sound that began non-vibrato and then progressed to the ideal level would progress from less stable to more stable. Less arbitrarily, Lerdahl also suggests that a spectrum following the harmonic series is the most consonant and stable, and deviations from this are less stable.

Lerdahl determines his prolongations by saying that unstable things eventually come under the reign of the more stable things. Borrowing terms from counterpoint, Lerdahl articulates the timbral equivalents of arpeggiating, neighboring, and passing functions. He depicts these possibilities using branching graphs, a familiar sight to anyone who knows GTTM. Below are these arpeggiating (Figure 10), neighboring (Figure 11), and passing functions (Figure 12) given certain vowel sounds. The vowel sounds are given in IPA. The increasing subscript numbers indicate an increase in brightness within that vowel sound.

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If I imagine myself saying these vowels (and this is difficult, since Lerdahl does not futher define the qualities of brightness present for each subscript numeral), I start to feel foolish for attempting to think of these as passing, neighboring, or arpeggiating. Perhaps it’s simply the terminological analogy, which as a Schenkerian I find a bit distracting and loaded. But contrapuntal analogies or no, I am skeptical that we can organize timbre hierarchically. Lerdahl writes, “It might be supposed that the pitch-timbre analogues are artifacts of the way the issues have ben posed. But it is more interesting, and I believe more true, to argue that the underlying principles channel musical cognition and that the analogues rely on certain of these principles” (157). I’m no cognition expert but I’m skeptical that these trees necessarily represent our cognition of timbre structures.

The preoccupation with hierarchy is probably typical of music theory in the 1980s, and Lerdahl’s work likely simply reflects that. Instead, reflective of our present day, where music theory continues to borrow more and more from the perspective of musicologists and ethnomusicologists, I prefer to enhance timbral associations through cultural context.  That may be a post for another time.