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COPYRIGHT 2003 University of Washington
Art, and above all, music has a fundamental function, which is to catalyze the sublimation that it can bring about through all means of expression. It must aim through fixations which are landmarks to draw towards a total exaltation in which the individual mingles, losing his consciousness in a truth immediate, rare, enormous, and perfect. If a work of art succeeds in this undertaking even for a single moment, it attains its goal.
--Iannis Xenakis
0. INTRODUCTION
0.1 NOMOS ALPHA
FOR MANY OF US, Iannis Xenakis's 1966 solo cello composition Nomos alpha is one of these works of art. It gathers us as listeners into a sonic world which speaks to, indeed argues with, our notions of logic, philosophy, affection, and beauty. One of the piece's most engaging aspects is its difficulty, presenting substantial challenges to us as performers, listeners, and analysts. In the artist preparing a performance of Nomos alpha, all three of these personae must work together to overcome its obstacles: its formidable technical difficulties and, perhaps greater, the problem of forming an interpretation. The piece is extremely rich in detail and design, yet how do these aspects translate into the field of interpretation? George Fisher and Judy Lochhead (1993, 5) state the general problem most succinctly: "The basic question thus becomes not what bearing analysis should have on performance, but what bearing it can have."
The opinion we explore here is that, overall, Nomos alpha is a piece about time. As we will see later, the work's primary dialogue does not takes place in aspects of its pitch structure, but rather in the categories of time (linear, non-linear, etc.) which it incorporates. Consequently, to arrive at an appropriate interpretive model of the work, the performer must first have a clear understanding of the form of the piece, and how it works with, or against, notions of time in music. In the most general sense, such interpretation is a question of rhythm. As Edward T. Cone (1968, 38-9) states, "We must first discover the rhythmic shape of a piece--which is what is meant by its form--and then try to make it as clear as possible to our listeners." Like performers, we as analysts and, even more generally, listeners of Nomos alpha, must first discover the work's form to come to an understanding of the piece.
0.2 THE ROLE OF LISTENING IN FORMING AN INTERPRETATION
Let us assume, for the moment, that our analysis will lead to a discernible structure in Nomos alpha that may be communicated successfully to a listener. Our first step in forming an interpretive model of this structure might be from the listener's perspective; we must understand something of one's capacity for comprehending musical structures, however complex. We will begin with the ideas of Aristoxenus, (1) one of the dedicatees of the piece, on the role of listening in the context of music theory, since the consequences of his thoughts appear throughout the modern music theoretic literature.
Aristoxenus differentiates between his own approach to music theory, which seeks to embrace all the "phenomenal principles," and that of his predecessors, who "speak irrelevantly, ignoring the senses as not being exact, build contrived causes, pretend that there are certain ratios of numbers and reciprocal velocities in which the high and the low arise, and propose considerations totally alien to all things and completely opposite to the phenomena." (2) Aristoxenus proposes that a combination of hearing and intellect is necessary to an understanding of music. Hearing, through a discerning ear, is of tantamount importance to the intellect, as it determines first the characteristics of musical phenomena. The intellect, then, ascertains the functions of these phenomena, their intervallic and rhythmic magnitudes. The failure of Aristoxenus's predecessors' method would appear to be its disregard for the senses.
Particularly with regard to rhythm, Aristoxenus's ideas hold further relevance for our discussion. For him, the context of rhythm is time, or chronos. Chronos, and our sensing of it, form the initial principles of the science of rhythm. Chronos may be divided into rhythmized patterns, and we must first be able to sense these patterns in order to appreciate them. Aristoxenus differentiates between rhythmization (e.g., of melodic or verbal patterns) and rhythm itself. By way of analogy, a body may be formed into a triangle, but it is not the form itself. It is not a triangle; rather, the triangle is an abstract mental construct. Thus, for the intellect to apprehend a wholly abstract form or rhythm--ontologically, for it to have being--some perceptible entity must first exist to be formed or rhythmized, and we must have the faculties to discern it.
These ideas of Aristoxenus are relevant to our endeavor of forming an interpretation of Nomos alpha. In particular, it is in the domain of hearing that performer and analyst meet. Certainly, the performer must have a preconceived aural image of the piece. Moreover, the analyst may incorporate many techniques, but perceptually based relations have the greatest impact on performance. Specifically, such an analysis should incorporate techniques which draw explicitly on the music's temporal unfolding. These methods may involve a dramatic or narrative conception of the piece's chronological organization, and perhaps also more complex conceptions in which events are perceived, retained, and projected in various ways.
Thus, our task will be to create an interpretive model of Nomos alpha. This model should take into account the piece's unfolding in time--its form--not by "building contrived causes," "pretending certain numerical relations correspond to musical events," or "proposing considerations alien to the phenomena," but through discerning use of our senses. The "rhythmized patterns" we perceive will suggest the appropriate forms and functions to our intellects.
0.3 THE LITERATURE ON NOMOS ALPHA
We turn next to an examination of the literature on Nomos alpha. Xenakis's own words on the piece are few, yet are imbued with significance. They derive mainly from Chapter VIII of Formalized Music, "Towards a Philosophy of Music" (Xenakis 1992, 201-41), and the preface to the Boosey & Hawkes score (Xenakis 1967). The latter states:
Symbolic music for solo cello, possessing an extra-temporal architecture based on the theory of groups of transformations. In it is made use of the theory of "sieves," a theory which annexes the congruences modulo z and which is the result of an axiomatic theory of the universal structure of music. This work is an act of homage to the imperishable work of Aristoxenus of Tarentum, musician, philosopher and mathematician and founder of the Theory of Music; of Evariste Galois, mathematician and founder of the Theory of Groups, and of Felix Klein, his worthy successor. Written for Siegfried Palm, it was commissioned by Hans Otte of Bremen Radio.
In Formalized Music, Xenakis provides both a definition of his concept of "symbolic music," (3) and an exposition of the philosophical and mathematical issues present in the piece. Chapter VIII, "Towards a Philosophy of Music," begins as follows: "We are going to attempt briefly: 1. an 'unveiling of the historical tradition' of music, and 2. to construct a music." The primary treatment of Nomos alpha in the book is contained in this chapter, particularly as an example of musical composition ex nihilo (Xenakis's term). (4) Consisting mainly of his precompositional plans for the work, it demonstrates the various structures he used, their mathematical bases and their realization as musical events, and places them ultimately within a broader philosophical context. We will return to these concepts later in [section] 0.4.
Soon after the piece's publication, an article by Fernand Vandenbogaerde (1968) appeared which puts forth the precompositional materials of Nomos alpha. Next, two significant studies of the piece appeared in print in the early 1980s: one by Thomas DeLio (1980), and the other by Jan Vriend (1981). DeLio presents an analysis of the piece's "extra-temporal architecture," augmenting Xenakis's exposition. He comes to two significant conclusions. First, several aspects of the piece suggest its formal division into halves. Second, the piece incorporates two processes, which he labels levels I and II. These processes are distinct and in a dialectical relation to one another, I being an example of a discrete structure, while II is essentially continuous. Whereas I and II are not levels in a concurrent or hierarchical sense, for consistency we will preserve DeLio's labeling.
On level I eight distinct elements are defined for each parameter and then used to articulate the permutation schemes. In contrast, on level II the sonic elements seem tied to one another within an unbroken temporal/spatial continuum. The form of Nomos Alpha, then, would appear to have as its source the juxtaposition of two radically different methodologies. From their interaction a magnificent dialogue evolves and over the course of this dialogue some of the deepest issues concerning the evolution of structure are illuminated. (DeLio 1980, 91)
In the 1981 article, Vriend presents an exhaustive analysis of the structure of the piece, and raises several problems which confront listeners as well as analysts.
The "logic" supposedly present in the chains of group transformations is not simply transplantable to questions of logic in the domain of a listening strategy: for example, if we would like a listener to be able to follow the string of transformations and, when completed, to realize that a "loop" is closed at the end ... that would not be a fair, or adequate problem: it is not sufficiently stated in musical terms, it is not sufficiently a musical problem. (Vriend 1981, 74)
In addition, Vriend points out several inconsistencies and ambiguities between Xenakis's precompositional exposition in Formalized Music and its realization in the score; these, he relates from personal correspondence with the composer, are the result of either "slips of the pen," changes made for musical reasons, or theoretical mistakes. (5)
In the final analysis, Vriend considers the piece to be a type of variation form. Furthermore, he writes that Xenakis deals only intuitively with "purposeful variation, or transformation with a target of particular (i.e., well-defined and specific) interest from the point of view of perception," unless Xenakis considers "variation in itself a worthwhile goal" (74). In either case, Vriend estimates the piece contains too many difficulties for the listener to be perceived as a continuous form. It is, therefore, discontinuous. We note that this attitude contrasts with that of DeLio, who believes that the musical materials are chosen to articulate, presumably successfully, the piece's group-theoretical structure.
0.4 XENAKIS'S CONCEPTIONS OF SYMBOLIC MUSIC AND THE PHILOSOPHY OF MUSIC
But everything in pure determinism or in less pure determinism is subjected to the fundamental operational laws of logic, which were disentangled by mathematical thought under the title of general algebra. These laws operate on isolated states or on sets of elements with the aid of operations, the most primitive of which are the union, notated [union], the intersection, notated [intersection], and the negation. Equivalence, implication, and quantifications are elementary relations from which all current science can be constructed. Music, then, may be defined as an organization of these elementary operations and relations between sonic entities or between functions of sonic entities. (Xenakis 1992, 4)
For Xenakis, the phenomena of music may be explained or constructed axiomatically according to the rules of logic. This concept, which owes a particular debt to symbolic logic, forms the basis of Xenakis's theory of "symbolic music." (6)
In general, in symbolic music, we begin with the postulate of a non-qualitative sonic event, which possesses some frequency, intensity, and duration. We may represent this event graphically with some variable, for instance A. Given any two such sonic events A and B, where A is recognizably distinct from B, and considered independently of the temporal continuum--Xenakis's "outside-time"--we observe that AB is no different than BA. In other words, events "outside-time" may be considered elements of an unordered set. However, when the events are considered in the temporal continuum, "in-time," their ordering is significant. Symbolic music also admits the associative property. For any three distinct sonic events A, B and C, we observe that (AB)C = A(BC).
Next, symbolic music defines a conception of distance. Xenakis argues that the distance between any two generic sound events is not sufficient to create a notion of interval. Rather, the listener requires a third element to form a concept of relative size. When listening to music, we may rank the set of all such distances, using the equivalence relation "is greater than or equal to," and this set of intervals is isomorphic to the equivalence classes of the N x N product set of natural numbers. That is, it possesses the qualities of an infinite additive group structure. (8)
By taking the product of this interval group with a field structure, such as the set of real numbers R, or the set of points on a straight line to which R is isomorphic, we form a vector space (Xenakis 1992, 210-1). In the Pythagorean sense, such a space permits us to describe its elements numerically--to represent music in terms of numbers. However, this ability is neither the sole nor primary function of symbolic music. Rather, as Xenakis (155) states: "We ... shall simply try to understand more clearly the phenomenon of hearing and the thought-processes involved when listening to music. In this way we hope to forge a tool for the better comprehension of the works of the past and for the construction of new music" (emphasis added).
For Xenakis, the Pythagorean concept of numbers and the Parmenidean dialectics--the two schools which formed the basis of the idealism of later Greek philosophy--also form the basis of a philosophy of music. The Pythagoreans taught to emphasize form over matter; all things are either numbers themselves, furnished with numbers, or similar to numbers. Xenakis argues that all music theorists, from Aristoxenus to Rameau, are indebted to this branch of philosophy: "We are all Pythagoreans" (202). The second primary school of philosophy to which Xenakis alludes is the Eleatic school, specifically Parmenides's concept of Being versus not-Being. The appearance of movement, and the existence of separate objects in the world, are mere illusions; they only seem to exist. True Being, the infinite and unchanging universe, is beyond human sensory comprehension, but may be known through philosophical reflection, through reason. Xenakis considers that perhaps the most important aspect of the Parmenidean dialectics is determinism: "If logic indeed implies the absence of chance, then one can know all and even construct everything with logic. The problem of choice, of decision, and of the future is resolved" (204).
Finally, Xenakis asks:
1. What consequence does the awareness of the Pythagorean-Parmenidean field have for musical composition? 2. In what ways? To which the answers are: 1. Reflection on that which is leads us directly to the reconstruction, as much as possible ex nihilo, of the ideas basic to musical composition, and above all to the rejection of every idea that does not undergo the inquiry. 2. This reconstruction will be prompted by modern axiomatic methods. (207)
For Xenakis, these axiomatic methods are the construction of scale types, or "sieves," (9) which is outside the scope of the present study, and the formation of a vector space. Xenakis maintains that all sound, hence all music, is a consequence of a correspondence of these two principles, and this fact leads us to the ability to construct ex nihilo a music: Nomos alpha.
1. TOWARD AN APPROPRIATE CONCEPT OF FORM
Before beginning our discussion of form, we must first make a few observations about Nomos alpha. We will explore the consequences of these factors on our model of form and, by extension, interpretation. Whereas pitch relations, particularly with regard to voice-leading connections and tonal or quasi-tonal centers, are a primary determinant of form in many repertories, we must look elsewhere in Nomos alpha for the progenitors of form. Except for the particular registers and...
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