Tenney’s Effort to Move from Theory to Practice

The ninth chapter of From Scratch: Writings in Music Theory, the University of Illinois Press collection of articles by music theorist and composer James Tenney, seems to be the second-longest in the book: “Hierarchical Temporal Gestalt Perception in Music: A Metric Space Model (with Larry Polansky).” This was written in 1978 and was subsequently published by the Journal of Music Theory in 1980. It basically involves first distilling a hypothesis out of the theoretical speculations that first emerged in Tenney’s “Meta + Hodos” Master’s thesis and then testing the hypothesis by implementing its content in a computer analysis program, which was written by Polansky and executed on input from three scores of twentieth-century music compositions.

Before discussing the model, the hypothesis behind the model, and the testing of the hypothesis, however, I want to call attention to a published review of “Meta + Hodos.” This took place after “Meta + Hodos” itself was published as a monograph by the Inter-American Institute for Musical Research in 1964. The Spring 1966 issue of the Journal of Music Theory published a review of the monograph by A. Wayne Slawson, which was pretty devastating.

Slawson provided the reader with a paragraph explaining the the “birth” of Gestalt psychology as a result of observations made by Max Wertheimer in 1912. However, Slawson then accused Wertheimer and his colleagues of oversimplifying the theory they developed, concluding that “the Gestalt movement failed to go beyond a particularly apt and persuasive presentation of new questions.” This was the stick he used to subject “Meta + Hodos” to a rather merciless beating.

Thus Tenney’s effort to develop a model that could then be realized through software-based testing amounts to a response to Slawson’s review. The fact is that this was a time when painfully little was known about the “wetware” of a brain embedded in the larger complex system of the human body. Thus, even a precept as straightforward as Donald O. Hebb’s famous postulate that “neurons that fire together wire together” could not be tested in the absence of technology for observing the “wiring.” One of the reason’s that I have been citing the work of Gerald Edelman is that he recognized that his own theory of perceptual categorization could not be tested through direct observation; so, as an alternative, his team developed a computer-based simulation model of Hebb’s “wiring” process. Tenney’s model, on the other hand, was based on an attempt to turn the theoretical speculations of “Meta + Hodos” (a product of the theoretical speculations of Gestalt psychology) into practice.

At this point I feel it is important to note that the term “metric space” that appears in the title is never explicitly invoked in the article’s text. Nevertheless, there is an implicit sense of “distance” that Tenney seeks to apply to his “temporal gestalt-units” (TGs), focusing primarily on those constructs captured by the terms “element,” “clang,” and “sequence.” The model requires a quantitative representation of the amount of difference that distinguishes two TGs. A metric space is a topological construct that defines the concept of distance in terms of four criteria:

1. The distance from a point to itself is zero.
2. The distance between two distinct points is a positive number.
3. The distance from point A to point B is the same as the distance from point B to point A.
4. For any point C, the distance from A to B is less than or equal to the sum of the distance from A to C plus the distance from C to B.

(That last criterion is sometimes called the “triangle rule” because the length of the hypotenuse is always shorter than the sum of the lengths of the other two sides of a triangle.)

As we liked to say as freshmen at the Massachusetts Institute of Technology, these properties were “intuitively obvious to the most casual observer.” However, those of us who went on to major in mathematics discovered that the most interesting insights were those involving counterexamples to the constraints of those criteria. Every math major knew about the book Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted and the critical role it played when trying to solve homework problems. So, in the interest of seeking out counterexamples, it is important to determine whether or not the concept of distance can actually be applied to comparing two TGs.

Let’s start with something simple. Imagine music as it is printed on score pages. Imagine, then, that you take a pencil and draw circles around groups of notes that you wish to identify as TGs. You can then take a ruler and measure distances between TGs on the score page. Even if those distances are somewhat rough (since a TG is not a simple point on the page), it is easy to see how the four distance criteria are satisfied.

However, as we all know, the marks on the score pages do not constitute the music. The music only exists through the experience of listening to a performance, even if that performance involves playing a recording. I would now suggest that the time-consciousness required for such listening involves awareness of differences that do not necessarily satisfy the distance criteria.

To make my point, I need to appeal to the reader’s imagination. Think of a flowing stream. Now, imagine that every point along that stream can be established as a fixed position (through a very precise measurement of latitude and longitude, for example). Suppose, now, that “distance” is not measured by difference in latitude and longitude but in the amount of time it takes a reference object, such as a fish, to swim from one point to another. By virtue of the flow of the river, the third property of a distance metric is violated because swimming “downstream” takes less time than swimming “upstream,” even though the beginning and ending points are fixed! I would argue that the passing of time is like the flowing of that stream and that Tenney’s model, while it looks good on paper (such as score pages), does not adequately capture the phenomenology of difference in a situation requiring the dynamic nature of time-consciousness.

After I first performed this exercise, I realized that I had been about as merciless in approaching Tenney’s work as Slawson had been. The reason is that the model that provided Tenney’s point of departure was required a quantitative foundation based on the topological properties of distance. Even the hypothesis being tested in his article, involving relationships among elements, clangs, and sequences, had been undermined in the absence of how those relationships could be represented quantitatively; and, if the hypothesis was no longer sound, then testing it was out of the question.

Once again I find myself thinking about the problems with Tenney being ahead of his time in terms of the tools available for his efforts. This particular paper predates the earliest efforts that would eventually lead to MIDI. For all of its shortcomings, MIDI provided a symbol representation that could capture not only marks on score pages but also the time-dependent factors inherent in any performance involving the interpretation of those marks. As a result of MIDI representations, it has been possible to investigate structural questions that go beyond music notation and enter the realm of time-dependent interpretations of the notation. It is in those interpretations that we need to seek out patterns and make sense of both what they are and why they are, and it is a pity that Tenney did not have such tools at his disposal when he first set off down his phenomenological path of inquiry.