The Structure and Function of Musical Theory: I
I like to believe that a not insignificant consequence of the proper understanding of a proper theory of music is to assure that a composer who asserts something such as: "I don't compose by system, but by ear" thereby convicts himself of, at least, an argumentum ad populum by equating ignorance with freedom, that is, by equating ignorance of the constraints under which he creates with freedom from constraints. In other words, musical theory must provide not only the examination of the structure of musical systems—familiar and unfamiliar by informal conditioning—as a connected theory derived from statements of significant properties of individual works, a formulation of the constraints of such systems in a "creative" form (in that, as a language grammar does for sentences, it can supply the basis for unprecedented musical utterances which, nevertheless, are coherent and comprehensible), but—necessarily prior to these—an adequately reconstructed terminology to make possible and to provide a model for determinate and testable statements about musical compositions.
Whether one prefers to declare that a theory must be, should be, or is a mere symbolic description, or a structured formulation of statements of relations among observed phenomena, or a collection of rules for the representation of observables, or an interpreted model of a formal system, or still none of these, presumably it can be agreed that questions of musical theory construction attend and include all matters of the form, the manner of formulation, and the signification of statements about individual musical compositions, and the subsumption of such statements into a higher level theory, constructed purely logically from the empirical acts of examination of the individual compositions. Surely there is no more crucial and critical issue in music today, no more central determinant of the climate of music today, than that of the admittedly complex and intricate problems associated with assertions about music. Perhaps there have been eras in the musical past when discourse about music was not a primary factor in determining what was performed, published, therefore disseminated, and—therefore—composed, and when the criteria of verbal rigor could not be inferred from either discourse in other areas or from the study of the methodology of discourse, when—indeed—the compositional situation was such as not to require that knowing composers make fundamental choices and decisions that require eventual verbal formulation, clarification, and—to an important extent—resolution. But the problems of our time certainly cannot be expressed in or discussed in what has passed generally for the language of musical discourse, that language in which the incorrigible personal statement is granted the grammatical form of an attributative proposition, and in which negation—therefore—does not produce a contradiction; that wonderful language which permits anything to be said and virtually nothing to be communicated. The composer who insists that he is concerned only with writing music and not with talking about it may once have been, may still be, a commendable—even enviable—figure, but once he presumes to speak or take pen in hand in order to describe, inform, evaluate, reward, or teach, he cannot presume to claim exemption—on medical or vocational grounds—from the requirements of cognitive communication. Nor can the performer, that traditionally most pristine of non-intellectuals, be permitted his easy evaluatives which determine in turn what music is permitted to be heard, on the plea of ignorance of the requirements of responsible normative discourse. Nor can the historian, in the sanctified name of scholarship, be allowed such verbal act as the following: "There can be no question that in many of Mendelssohn's works there is missing that real depth that opens wide perspectives, the mysticism of the unutterable." Can one conceive of a possible interpretation and application of those mild Humean criteria so liberal as to save a book containing a sentence such as this from the flames? And what of the more apparently factual scholarly statement that the c-flat of measure 53 of the second movement of the Mozart G Minor Symphony (K. 550) is "an unexpected c-flat"; overlooking for the moment the dubious status of such expressive descriptives, what can the term "unexpected" be inferred to designate when applied to the succession b-flat, c-flat, which had been stated in the movement in question at the outset in measure 2? If nothing else, such verbal phenomena would appear to be instances of the situation characterized by Quine's conservative observation: "The less a science is advanced, the more its terminology tends to rest on an uncritical assumption of mutual understanding." In attempting to preserve or defend this self-indulgent, unwarranted state, it may be asserted that music is not a science. This, naturally, is not the point, not even a point. Parenthetically, it may be said that neither the proclamations of those who work in what are traditionally termed sciences as to the essentially artistic nature of their activity, or the proclamations of those far fewer who work in what are traditionally termed the arts and humanities as to the appropriateness or necessity of scientific method in their activity, need be cited to discredit an already indefensible dichotomy or multichotomy, the perpetuation of a linguistic fortuity as if it embodied a fundamental and persistent truth. Whatever such categorizations, if any, of fields of intellectual creation are justified or fruitful must await an investigation which is not even yet begun, but even were it now completed, there would remain the still more germane question for us as to whether the term "musical composition" is accurately applicable to the many apparently diverse activities to which it is now applied, or—perhaps more importantly—to all of those or any of those activities and to any activity of, say, a half century ago. It is at least worth asking whether so generic a description has survived that revolution in musical thought which has been and is still in progress.
Certainly it is tantalizing to conjecture why, far more violent than the responses to the music of this period, responses which have ranged usually from patronizing tolerance to amused tolerance, have been those to the verbal activities. But, since the subject here is musical theory rather than clinical psychology, only one such conjecture will be pursued and this only to return to a temporarily abandoned line of discussion. The issue of "science" does not intrude itself directly upon the occasion of the performance of a musical work, at least a non-electronically produced work, since—as has been said—there is at least a question as to whether the question as to whether musical composition is to be regarded as a science or not is indeed really a question; but there is no doubt that the question as to whether musical discourse or—more precisely—the theory of music should be subject to the methodological criteria of scientific method and the attendant scientific language is a question, except that the question is really not the normative one of whether it "should be" or "must be," but the factual one that it is, not because of the nature of musical theory, but because of the nature and scope of scientific method and language, whose domain of application is such that if it is not extensible to musical theory, then musical theory is not a theory in any sense in which the term ever has been employed. This should sound neither contentious nor portentous, rather it should be obvious to the point of virtual tautology. Assuredly, I am not stating that all of the problems of musical theory can be resolved automatically and easily by our merely embracing the latest formulations of the philosophy of science, for neither music nor the philosophy of science is that simple and static; and the problems of musical theory are, in many ways, so complex as to carry one unavoidably and quickly to still highly controversial, still unresolved questions in philosophy and related fields. For instance, functionality in the traditional tonal sense probably can be formulated only as a disposition concept, which may account for the unsatisfactory character of less formal attempts to "define" tonality; musical analysis involves many of the contested problems of explanation, postdiction, and prediction, which are regarded by many as the most crucial components in the construction of a possible musical theory; concept formation in music involves those problems of intersubjectivity and of verbal utterances as empirical data with which psychological theory has been grappling, and for the formulation of which it has been obliged to employ advanced and novel notions and techniques. One need but recall the forbidding appearance, to musicians, of Suppes' partial formalization of the notion of "finite equal difference structure," of which the musical concept of interval is a familiar instance.
But if there are obstacles to the construction of a satisfactory musical theory in the form of such systematic difficulties, there are obstacles also in the form of the task being too easy, for if a composition be regarded—as manifestly it can be, completely and accurately—as events occurring at time-points, then there are an infinity of analytical expressions which will generate any given composition, and one moral of this casual, but undeniable, realization is that the relation between a formal theory and its empirical interpretation is not merely that of the relation of validity to truth (in some sense of verifiability), or of the analytic to the synthetic (be this or not an untenable dualism or a dogma of empiricism), but of the whole area of the criteria of useful, useable, relevant, or significant characterizations. Another facet of the same question, a facet which had made possible a great many of the analyses by which we find ourselves confronted at the moment and which perhaps underlies our dissatisfaction with a great deal of traditional theory, is that there are an infinite number of true (or false) statements that can be made about any composition, and—therefore—any collection of compositions. Putting aside formalities, only because the result of not putting them aside is known and herein represented, any theory is a choice from an infinite number of possible theories, and the choice is determined by what can be termed a criterion of significance in the selection, first, of primitives, whatever the linguistic form of these primitives. Whether this significance be expressed in terms of predictive power, explanatory scope, simplicity, or some other criterion, the decision is not easily made or ever surely made, which is only to state that an empirical theory is subject always to revision and reformulation. The question of significance arises as soon as one seeks to formulate concepts for the analysis of an individual piece founded on the notion of, say, pitch. One cannot be at all certain that any concept is necessarily a fruitless or an absurd one; one simply does not know if it is or not until one has tested the results of the application of this concept for correlation with other independent concepts, for invariance under non-vacuous conditions. The musical naive realist impatiently may dismiss concepts involving the conjunction of primitive and logical terms as normally fruitless from his point of view of aural immediacy, so I shall begin with a concept to which he could not possibly take exception, that of "interval" in the usual sense of measure of distance between two pitches or pitch-classes. We can explicate it as the result of applying to two pitch terms (represented in conventional numerical pitch-class notation) the operation of subtraction mod. 12, or by constructing its geometrical isomorph and regarding it as directed distance. Either of these representations should satisfy the realist, for they both entail that characteristic of transposition which must be regarded as primary: the invariance of the interval value under transposition. In terms of its geometrical analog, this is a statement that translation is distance preserving. Still, however immediate the experiential fact of interval may be, it is here not only a concept, but a theoretical construct, a two-place predicate, and if on no other basis than the role it has played in all component statements about musical composition, it is probably generally acceptable as a significant concept. In order to compare it with another, less familiar concept, I shall examine a statement including the concept "interval," that concerning the hierarchical implications of the intervallic characterization of a pitch-class 2 collection. Given n pitches or pitch-classes, the non-zero intervals or interval classes they define can be collected in equal interval categories, and the multiplicity of occurrence number associated with each of the categories determines the number of pitches in common between the original collection and the collection transposed by the interval associated with a particular category. That is, in the equivalent geometrical language, a fairly obvious statement of the relation between the distances defined by a point collection, and the distance defined by these points and their images under a given translation. An immediate and simple consequence of this property is the "circle of fifths," a symmetric (because of the equivalence of complements in an unordered collection) hierarchization of major-minor scale content in terms of pitch-class intersection. This property is inferrable therefore from the interval structure of the scale, since each interval occurs with unique multiplicity. It is easily shown that the major scale is a maximal structure possessing this property in the usual equal-tempered division of the octave. The compositional consequences in the tonal system of this property are so fundamental as to require no statement of specifics, but one of the importantly therapeutic values of such a generalized formulation is that of, at least, restraining that pedagogical liberalism which would urge students to "experiment" with other, less familiar, more exotic scales, on the basis that they are just as fruitful as our traditional scales. But in general they are not from a structural point of view, since they do not admit comparable properties of hierarchization.
It may be less obvious that the same concept is equally consequential in twelve-tone composition, since all sets have the same total interval content. But the utilization of subsets as combinational units within the sets—for example, the hexachord in Schoenberg's and Stravinsky's twelve-tone music—leads to the contextually hierarchical function of this property. Assuming that this is either familiar or can become so by the consulting of the available literature, I shall draw musical examples of this property in those works of Stravinsky which are most efficiently described as serial, but not twelve-tone, where the serial unit is—therefore—in general uniquely characterized, to within transposition—and possibly inversion—by its intervallic content. Perhaps the most general and inclusive basis of relationship in such music resides in pitch-class identification between and among the compositionally defined serial units, since they are not embedded in larger serial units as in twelve-tone works, or associated with concepts of functionality as in tonal music. The pitch collection from which the serial unit of the Gigue of the Septet is formed is so constructed that maximum identification of pitch content is achieved by transposition by an interval of 5 or 7; this reflects the compositional design of the movement, with the serial unit employed as a thematic entity in what may be described as a succession of fugal entries. In In Memoriam the transpositions effecting maximum intersection are 1 and 11, reflecting the fact that the succession of serial units is in the vocal part, as a linear succession. The concept of interval, and at least this one of its applications would appear to belong to a theory that subsumes traditional tone, twelve-tone, and non-twelve-tone serial theory, but what of another concept closely similar in arithmetical form, represented by, instead of the difference of pitch-class numbers, the sum of pitch-class numbers: a + b (mod. 12), where a and b are pitch class numbers. This may appear to smack of that familiar pedagogical procedure of demonstrating that the analytically valid is not necessarily empirically true or even meaningful. But we are not dealing here with analytical validity, only with a formal operation which can be performed with great ease, but appears to lack any interpretation and application in the musical domain. What could be meant or designated by a sum of pitch numbers? The difference between such numbers defines what has always been termed an interval, but the sum represents no traditional property, and there is no term, no abbreviative definition, for it. Surely the sum, like the difference of pitch numbers is not a pitch number, and there appears to be no observation concept associated with this arithmetical expression. But the sums of set numbers of a collection characterizes such a collection in terms of its inverted forms in precisely the same way as the differences characterize a collection in terms of its transposition. Why this is so becomes clear when it is recalled that the sums defined by order corresponding elements of inversionally related collections are equal, and that inversion is definable in terms of complementation. The applicability of this concept to twelve-tone set construction is generally understood, but—again—I choose examples from non-twelve-tone serial works of Stravinsky. The Ricercar II of the Cantata employs a six note collection which permits maximum intersection of pitch content by inversion at the interval 6, which is employed by Stravinsky in the initial statement of forms of the collection, while the collection of the In Memoriam is inversionally symmetrical, thus permitting total pitch intersection at the interval of 4.
There is, then, this close analogy between interval and whatever we wish to call the concept represented by the sum of pitch-class numbers. And yet, in some musically important sense, these two concepts would seem to require differentiation at some level. Surely interval is an "observation concept"; does the other concept require categorization as "theoretical," in the usual sense of the term, since it is not apparently translatable into perceptual terms? Until, if ever, an ultimate disposition is made of this terminological differentiation, this latter concept, for all of its hierarchical implications, will be formulable only in theoretical terms.
The whole question of the status of the notion of the overtone "system" (surely not a system, but a phenomenon), the checkered history of this status for two centuries, and that of its predecessor—the divisions of the string, must occupy a central place in any discussion of musical theory. Naturally, since I am not concerned with normative allegations, I cannot be concerned here with the invocation of the overtone series as a "natural" phenomenon, and that application of equivocation which then would label as "unnatural" (in the sense, it would appear, of morally perverse) music which is not "founded" on it. Now, what music, in what sense, ever has been founded on it? Experimentally, the intensity of harmonic, and non-harmonic, partials in a spectrum associated with a given sound source would appear to be an important determinant, but by no means the sole determinant, of what is ordinarily termed tone color. But what is, what can be, the status of the overtone series in a theory of the triadic, tonal pitch system? For it to furnish the criterion of the structure of the major triad it is necessary—first—to append the independent assumption of octave equivalents, for to assert that the overtone series itself supplies this criterion because the octave is the first interval above the fundamental, or the interval determined by frequencies whose ratio is a power of 2, or etc., obviously is to adjoin independent assumptions of the equivalence priority of the first interval, or of the intervals determined by powers of 2, etc. Then, the independent assumption of the significance of the number 6, or 5 as that which determines the highest partial to be included as a specifically realized pitch, has required "justifications" which have ranged numerologically from the number of planets to the number of fingers on the human hand. And again, the principle which permits one to proceed from the assertion that "associated with a given frequency produced at a given intensity by a given instrument are other frequencies" to the assertion that "such other frequencies always may be explicitly presented on any instrument simultaneously with a given frequency" must be combined with another rule which prohibits this process from continuing, this principle then being further applied to the frequencies so explicitly presented. And still, the structure of the harmonic series does not supply a basis for the status of the minor triad in tonal music. It either dissonantly "contradicts" it or requires the invocation of still further assumptions of intervallic permutability or numerology. And yet, the succession of intervals in the overtone series does not correspond to the categorizations of "consonant" and "dissonant," even in relative terms, whether one asserts the independent assumption of adjacency or of relation to the first partial. Under the former criterion, the fourth would be termed more consonant than the major third; under the latter, the minor seventh and major second would be termed more consonant than the major or minor sixth, or the minor third. The concepts of consonance and dissonance have induced centuries of a comedy of methodological errors, from the rationalistic stage, through the so-called "experimental stage," without its having been clear or inquired at any time as to the object of the rationalizing or the experimentation. Clearly, this is because consonance and dissonance are context dependent tonal concepts; it is impossible to assert that an interval is consonant aurally, since it always can be notated as dissonant, and this notation reflects a possible context.
One can continue with the overtone follies, with what having the overtone series commits one to eat, but perhaps it is necessary only to point out that a theory compounded from statements descriptive of a body of representative works of the 18th and 19th centuries undoubtedly would include the concepts of the major and minor triad as definitional, and as instancing the property of consonance, which, with the property of non-consonance, describe the two basic states of a composition which determine the modes of succession to the next state, octave equivalence classes (identical, in this body of literature, with function equivalence classes), the major scale (as completing independently the concept of consonance and providing the criterion for proceeding from state to state). These concepts hardly suggest the postulation of an overtone series as a master concept entailing them.
But from this body of works one probably would formulate, for example, a law regarding the "prohibition" of motion in parallel fifths (not, however, of unisons or octaves, since these scores are packed with such parallelisms; it does not do to say that these are the "same notes," but rather it is this parallelism which suggests the formulation of function class equivalence, and thus octave equivalence, and thus this degree of "sameness"). The formulation surely would not take the form of: "Parallel intervals of the unison, octave, and perfect fifth have been systematically avoided by composers of the 18th and 19th centuries, whenever it has been their intention to write a basic four-part texture," for all that this is the most popular of formulations. Nor would it or could it be explained by some statement such as: "Fifths are too closely related." What does it mean to be too closely related? To be fifths. It is difficult to see how the law can be derived from other, non-tautologically related, more general laws of "tonality," nor should it, if—for example—Debussy's music is to qualify as tonal.
Empirical theory construction to the end of either discovering a known formal theory of which the empirical theory is an interpretation or constructing such a formal theory, serves not only the goal of clarity, precise communication, and efficiency, but of providing knowledge of general and necessary characteristics of the empirical system through the structure of the formal model. It is well known that it can be shown easily that the rules of formation and transformation of the twelve-tone system are interpretable as defining a group element (a permutation of order or set numbers) and a group operation (composition of permutations). There then follows from a deduced property of inverse permutations the following property of twelve-tone sets, a now familiar property whose discovery in all its generality scarcely could have been accomplished—perhaps not even suspected—without such a formal model. The theorem states, in terms of the twelve-tone system, that two transposed set forms which are complementary with regard to a third have the same number of order inversions (and, therefore, permanences) with relation to this third set. This property is explanatory in explaining the compositional use of such related sets, and—by extension in suggesting more general applications of complementation, as in the case of the operation of inversion. This property also functions as predictive in determining possible attributes of future works concerned with exposing this property. So too, for example, the even less intuitively manifest property of the systems of common representatives shared by any two similar partitions of the same collection of, say, pitch-classes reveals what is, in some reasonable sense, a new facet of the possible relation between analytical explanation and creative prediction.
The Schenkerian theory of tonal music, in its structure of nested transformations so strikingly similar to transformational grammars in linguistics, provides rules of transformation in proceeding synthetically through the levels of a composition from "kernel" to the foreground of the composition, or analytically, in reverse. Since many of the transformational rules are level invariant, parallelism of transformation often plays an explanatory role in the context of the theory (and, apparently, an implicitly normative one in Schenker's own writing). The formulation of this theory in relatively uninterpreted terms (as Kassler is doing), as a partially formalized theory, serves to reveal not only its essential structure but its points of incompleteness, vagueness, and redundancy, and the means for correcting such flaws. The laying bare of the structure of an interpreted theory in a manner such as this is an efficient and powerful way also of detecting false analogies, be they between systems (for example, the "tonal" and the "twelve-tone"), between compositional dimensions (for example, that of pitch and that of timbre), or between compositions (with a composition regarded as an interpreted theory).
Such concerns with and, hopefully, contributions to verbal and methodological responsibility (far more than whether theoretical instruction begins with "tonal," "atonal," or "all" music, with species counterpoint or Webern counterpoint) must be central to the instruction of the student of music theory in the liberal arts college, only a rare one of whom will employ such theory creatively as a composer or professionally as a theorist, if he—as a student of contemporary philosophy and science—is not to dismiss the theory and—therefore, probably—the music as immature and irresponsible, or if he—as a student of predominantly literary orientation—is not to transplant mistakenly the prevalent verbiage of that domain to our, at least, more modest area of activity, and if he is to attain that rarest of all states: that of the concerned and thoughtful musical citizen.