Tenney’s Five-Year Struggle with a Theory of Harmony

If it seems as if it has been a while since I have continued my writing about From Scratch: Writings in Music Theory, the University of Illinois Press collection of articles by music theorist and composer James Tenney, it is because I have been deeply occupied with four consecutive chapters (and one appendix), which collectively account for a five-year period during which Tenney tried to develop a theory of harmony that would apply to contemporary music as effectively as it did to the traditions of the nineteenth and preceding centuries. The texts that occupied my attention for so long are the following:

• Chapter 10: Introduction to “Contributions toward a Quantitative Theory of Harmony” (1979)
• Chapter 11: The Structure of Harmonic Series Aggregates (1979)
• Chapter 12: John Cage and the Theory of Harmony (1983)
• Chapter 13: Reflections after Bridge (1984)
• Appendix 3: Excerpt from A History of ‘Consonance’ and ‘Dissonance’ (1988)

As can be seen from the first of these titles, Tenney’s orientation throughout these chapters is primarily quantitative. His methods are grounded in mathematics, but it is a mathematical orientation that reflects his experiences in using computers to manipulate properties that are fundamentally numerical in nature. It is from that foundation that he believed that there can be “a quantitative theory of harmony.” So it is that we encounter this sentence in Chapter 10:

Unless the propositions, deductions, and predictions of the theory are formulated quantitatively, there is no way to verify the theory and thus no basis for comparison with other theoretical propositions.

This is a rather unfortunate attempt to apply the methods of formal logic to quantitative properties. Sadly, it leads to a misunderstanding of what formal logic can and cannot do. While those who work in formal logic will use terms such as “truth value” casually, Tenney seems to have overlooked that such logicians are not constrained by dictionary definitions of the noun “truth.” The “mission” of formal logic is never anything more than a means to establish whether a collection of propositions is consistent; and deductions involving the determination of truth values is the tool for seeking out an inconsistency. (It takes only one inconsistency for the whole collection to dissolve into uselessness.)

In pursuing that mission, whether or not a proposition is “formulated quantitatively” is not relevant. Identifying an inconsistency is a matter of symbol manipulation, the manipulations being the workings of deduction. As I have previously observed, had Tenney enjoyed the benefit of an intellectual community in which “computing” was more concerned with manipulating symbolic structures, rather than evaluating complex numerical forms, he would have realized that the role of numbers in a “theory of harmony” is only part of the story, a story that is more concerned with finding useful symbolic constructs to represent the nature of signals that must be processed when either making or listening to music.

As a result, even after Cage had radically broadened Tenney’s view to accept that a “harmonic structure” may involve any simultaneity of sounds, Tenney continues to be obsessed with the integers that represent the overtone series. This is understandable, but it also distracts from where the real questions reside. Consider, for example, some of the ways that Cage worked with a piano (both with and without “preparation”). Clearly, he understood that composition and performance needed to deal with simultaneities of sounds. However, Cage was willing to deal with a sequence of such simultaneities as if it were a progression no different from the progression of chords in a four-voice hymn setting. In Cage’s case, however, one could not reduce that progression to a sequence of Roman numerals or figured bass integers.

Tenney clearly appreciated this quality of Cage’s music; but, by the end of Chapter 12, one gets the impression that he had not quite figured out what, in his capacity of theorist, he should be doing about it. The good news is that he was aware that there are actually (at least) two different kinds of simultaneity. In one case, such as in those hymn settings, one is aware of both the individual notes and the chords that they form. In another case, such as one of Henry Cowell’s tone clusters, the individual notes “fuse” into a single “sonorous object,” whose “signal” is an integrated whole, rather than a superposition of recognizable parts.

The good news is that, by the time Tenney wraps up Chapter 12, he is beginning to appreciate that time-consciousness is more relevant to perception than “score reading,” associating correlations between auditory constructs and symbols on staff paper. Once again, however, I need to be fair to Tenney. He was writing at a time when few were focusing on issues of time-consciousness raised by Edmund Husserl and Martin Heidegger, and even fewer were trying to relate their focus to the cognitive foundations behind listening to music. Those who read From Scratch today are better equipped to consider going down roads not taken by Tenney; and, if Tenney’s efforts did not lead very far, they may yet provoke a new generation of readers to seek out new paths.