“Meta + Hodos” After a Decade

In working my way through the writings of music theorist and composer James Tenney collected in From Scratch: Writings in Music Theory, some of my more extended efforts went into situating Tenney’s Master’s thesis, “Meta + Hodos: A Phenomenology of Twentieth-Century Musical Materials and an Approach to the Study of Form” in the context of how issues of phenomenology in general and time-consciousness in particular have evolved since that thesis was completed in 1961. Tenney himself took his own retrospective view of his thesis but only after over ten years of experiences and cogitations had ensued. The result was his follow-up essay “META Meta + Hodos,” written in 1975 but not published until 1992, when Frog Peak Music coupled it for publication with the original “Meta + Hodos” document.

The preface to the retrospective document begins as follows:

“META Meta + Hodos” represented an attempt to organize certain ideas first presented in Meta + Hodos in 1961, incorporating insights and revision that have emerged since then. The writing was initially motivated by the desire to provide an outline of my ideas and terminology for use by students in a class in formal perception and analysis at the California Institute of the Arts.

The result is not quite the sort of systematic outline that many of us learned to create in secondary school. However, it amounts to a reasonably well-organized collection of propositions, whose contents are elaborated through systematic definitions. Both propositions and definitions are then further elaborated by comments. For most readers this is hardly a page-turner; but Tenney’s attempt to be systematic facilitates homing in on the ways in which more recent insights reflect on the ideas brewing both in 1961 and subsequently in 1975.

One of my key conclusions about “Meta + Hodos” was that Tenney would probably have altered the path he was following had he been aware of Friedrich Hayek’s book The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology. This book was first published in 1952; and, while it did not cross my path until around 2005, I quickly discovered that it had influenced two of the most perceptive thinkers concerned with the nature of mind that I had been fortunate enough to encounter. One of them was my doctoral thesis advisor Marvin Minsky, a pioneer of artificial intelligence back in the days when understanding the nature of mind was more important than hacking together software that could fool enough people about its inherent “wisdom.” The other was Gerald Edelman, who was inspired to seek out biological processes that might validate what for Hayek was never anything more than mere speculation.

The very first Proposition in “META Meta + Hodos” involves the hierarchical organization of temporal gestalt-units. However, this is quickly qualified with a comment that suggests that the organization does not necessarily follow the strict “tree formation” of a hierarchy. (In abstract mathematics we would call the structure Tenney had in mind as a “partially-ordered set.”) The absence of such strict hierarchies is a significant issue in the biological model that Edelman developed based on a principle he called “perceptual categorization.” Put simply, perceptual categories are not neatly nested, nor, for that matter, are they stable. Indeed, those who have been following these “commentary” articles should know by now that the Necker cube is one of the easiest observed instances of unstable category definition.

Where things start to get a bit dicey, however, is when, in spite of the fluidity of the underlying nature of perceived categories, Tenney succumbs to the influences of his background in working with computers and ventures into a domain he calls “Musical Parameters.” Unfortunately, there is a slippery slope here. The very idea of there being parameters presumes that sound (and, therefore, music) is produced by what in computer-speak we would call a functional module, which has one or more control inputs. “Parameters” are neither more nor less than the quantities passed through those inputs.

Of course, those parameters can be applied to do a variety of things, meaning that they do not all “look alike,” so to speak. Thus, computer science recognizes that parameter values need to be classified, so that they can be distinguished as involving different categories. Those categories are usually called data types. (For the record, in 1975 I was teaching this sort of stuff to both undergraduates and graduate students!)

With such terminology at our disposal, it is easy to appreciate one of Tenney’s Comments:

There is not, in general, a one-to-one correspondence between musical and acoustical parameters.

This can only be appreciated if it is generalized to the above terminology. Given any two data types, does a relationship exist that defines one in terms of the other? From a computational point of view, that relationship needs to be expressed as a function. However, in the language of mathematics, that function may be injective (different inputs mean different outputs), surjective (different outputs mean different inputs), or bijective (both injective and surjective). (A bijective function is also known as an isomorphism.) Until we tease out the mathematics behind it all, any “correspondence between musical and acoustical parameters” can be little more than vague speculation.

Tenney then moves on to a domain that he calls “On Formal Perception and Description.” This is where he probably would have benefitted from a previous encounter with Hayek. As might be expected, The Sensory Order has a good deal to say about “forms” and how they “are formed,” so to speak. Hayek even distinguishes “simple” and “complex” “forms of classification,” the latter having the sort of dynamic qualities we associate with the Necker cube. In that context it seems to be fair to ask whether form, as such, as actually perceived or whether it is essentially a framework for expected perceptions. The latter seems more consistent with Edelman’s model of perceptual categorization, which, in turn, has little to do with Tenney trying to identify state, shape, and structure as “aspects of form.”

To those who may be chafing from my invocation of terminology from mathematics and computer science, I apologize. For better or worse, if we are trying to explain something, it always helps to have a vocabulary that facilitates the process of explanation. Readers may recall my earlier citation of the programming language LISP. An early pioneer of computer science Edsger W. Dijkstra, once praised the language for its ability to allow us to think previously unthinkable thoughts. Decades later a natural language researcher observed, in the middle of a talk he was giving, that it is often easier to explain what you are doing with a LISP program, rather than in text. The bottom line is that you cannot undertake explanation without terminology, and I have tried my best to evoke terminology that is appropriate without being dangerously confusing.

I make this point because, towards the end of the “META” article, Tenney begins to abandon it. There is an entire section of the text based on the mathematical property of ergodicity. However, he does not explain that term; nor does he interpret it the same way that any well-educated mathematician would. (To be fair to Tenney, Milton Babbitt had a tendency to play similar games in applying the terminology of abstract mathematics to his arguments about serial music. Several mathematicians I know have looked at his paragraphs and dismissed them as mere bunk.) The good news is that Tenney enlists such terminology to elaborate on those properties of state, shape, and structure; and the really good news is that those who follow in the footsteps of Hayek and Edelman need not worry about the specifics of those properties. The same can be said about the final section of the “META” article devoted to entropy, which seems to have originated from misconceptions that Tenney probably picked up from Lejaren Hiller at the University of Illinois.