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Toward an interpretation of Xenakis's Nomos alpha.

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 sieves used at any one time, the specific pitches and their orderings are chosen freely.

In personal correspondence with Vriend, Xenakis states that the specific choices of notes used in the piece are arbitrary--inasmuch as this arbitrariness derives from his "experience in stochastics, and therefore an arbitrariness steeped in experience." Whereas Nomos alpha is not a stochastic composition per se, Xenakis seeks to reflect in it the asymmetrical, non-periodic, non-repetitious pitch environment of his stochastic pieces (Vriend 1981, 68). Thus, we must determine form in Nomos alpha much in the same way we would approach an indefinite-pitched percussion piece: we rely on rhythm, register, contour, etc. This gives us some reason to focus on time as a primary determinant of form.

1.1 ARISTOXENUS'S CONCEPT OF TIME

Again, we turn to Aristoxenus's concept of time from the Elementa Rhythmica. For Aristoxenus, chronos is divided by rhythmized patterns. However, not all patterns are comprehensible. "Many symmetries and orders of chronoi appear to be quite alien to the sense, while certain few are suitable and can be applied to the nature of rhythm."

The nature of chronos allows it to be divided. Therefore, we may discover its basic, smallest unit, which Aristoxenus designates "protos chronos." Depending on whether the chronos is divided by diction, melos, or bodily motion, a protos chronos may hold a single syllable, note, or point. Furthermore, we may perceive that a division of chronos may be simple, complex, or both. It is simple if only the protos chronos divides it; complex if all rhythmized patterns divide it; and both if some, but not all, rhythmized patterns divide it. According to this view, the protos chronos corresponds to an integer, whereas a complex rhythm corresponds to a ratio (or fraction).

Aristoxenus holds that our senses play a role in the study of rhythm; indeed, our perception of chronos is elemental to the science of rhythm. Moreover, the various divisions of chronos must be articulated in some way to be sensed. For Aristoxenus, this method of articulation was the poetic foot, and this concept brings us to the twentieth-century theories of rhythm and meter elucidated by Meyer et al.

1.2 THE PROBLEM OF ACCENTUATION

Let us continue with a few more observations concerning the analysis of Nomos alpha, particularly regarding the problem of accentuation. If we accept an accent as a salient musical event, we may observe it ontologically as a result of various acoustical phenomena: amplitude, frequency, and duration. In general, events requiring more energy in these acoustical domains are considered accented, and such phenomena correspond respectively to the traditional dynamic, tonic, and agogic forms of accentuation. Metric accent, on the other hand, has no such acoustical basis.

Now, Nomos alpha is largely metered (in quadruple simple time), but it is arguably not a metric piece. Xenakis has also subjected dynamics and subsectional durations to a permutational scheme which places an equal emphasis on all its members (sec [section] 2.1.3 below). Like the successions of pitches, the piece's patterns of different kinds of microrhythms are not determined by a particular precompositional scheme. However, unlike Xenakis's use of particular sieves for pitches, the microrhythm does not even correspond broadly to the division of the work into sections. Therefore, we are left with none of the above means of accentuation. We must look elsewhere for events which are salient, but where?

The answer, of course, is that salience in Nomos alpha is defined contextually by processes, formal relations, or some combination thereof, and this salience projects the form of the piece. Therefore, the rhythmized patterns we describe will be in terms of these contextually-defined processes and relations. For example, we may think of a piece's beginning and ending as salient events; they are associated with the significant processes of beginning and ending, respectively, as in Cone's concept of "frame" (Cone 1968, 11-31). Furthermore, the elements of a frame are also formally related, regardless of their positions and associated processes. They serve to demarcate a piece of music from the external environment.

Here, we come to an important point which will be of considerable importance later: the ordering of elements in a process is important, but that of elements in a formal relation is not. For example, let E be the external environment, F be an element of the frame, and ~ be the equivalence relation "is the same process as." The composition EF (E followed by F), as a process, implies a beginning, whereas FE implies an ending. Clearly, EF [??] FE; a beginning is not the same process as an ending, and vice versa. Now, however, let ~ be the equivalence relation "contains the same elements as." Therefore, as a formal relation, EF ~ FE. (10) Aside from their roles as processes, they serve the same function: to describe the limits of the work, and in this regard they are equivalent.

In certain pieces of music, including Nomos alpha, specific components of the flame, or even an image of the frame, may appear in the context of the music, as a part of the "internal environment," if one may be said to exist. One particularly apt example may be found in Modest Mussorgsky's Pictures at an Exhibition. The "Promenade" serves, at the very least, as a frame at the beginning of the piece. Then, it recurs sporadically between movements or sets of movements, framing them. Indeed, the final movement, and ending flame, "The Great Gate of Kiev," is highly related to the "Promenade." Another example may be found in the first movement of Lutoslawski's String Quartet. Here, the articulator of the various sections of aleatoric counterpoint is a repeated-note motive on octave Cs in the cello. Whereas this motive is not associated with the processes of beginning or ending, it clearly has the function of a flame, much like the leading in a stained-glass window. However, it also has a much more significant role; Lutoslawski builds the climax of the movement out of the articulator, out of the frame. Similarly, in Nomos alpha, we find certain sections which function as framing material. These sections are the six which correspond to DeLio's level II. Five of these six sections, or "Intermezzi" (Vriend, 1981, 22), have an internal framing function. The sixth, then, becomes the piece's final, cumulative section and ending frame.

Cone makes another observation regarding the concept of frame which has particular relevance for our purposes:
   Music that is intrinsically formless, in the sense of having no
   apparent musical reason for beginning and ending when and as it
   does, may sound like an arbitrarily framed segment of an
   indefinitely extending sound-continuum. This is indeed the effect of
   much "totally organized" serial music, and equally of much music
   composed by methods of pure chance. (Cone 1968, 15-6)


This statement calls to mind Stockhausen's distinction between, on one hand, "beginning" (Anfang) and "ending" (Ende), and on the other, "starting" (Beginn) and "stopping" (Schluss). He associates the latter two with caesurae, which delineate the perimeter of a section within an indefinitely extending sound-continuum. Thus, he may speak of infinite forms, even though, for practical reasons, they must start and stop at some point (Stockhausen 1963, 207).

We find an example of such an indefinitely extending sound-continuum in Nomos alpha. The eighteen sections of DeLio's level I describe a cycle (see [section] 2.1.2 below), which essentially has no beginning or ending. It has a start and a stop, but these points are inherently arbitrary. Thus, the interaction between the non-processive level I and the processive level II (see Example 1) will be of considerable interest in our analysis.

1.3 TIME IN NOMOS ALPHA

Now we come to a central question in the interpretation of Nomos alpha. As we stated at the outset, Nomos alpha is a piece about time. However, is this time linear or nonlinear (discontinuous, as Vriend defines it); or is it some combination of the two (as in DeLio's dialectical conception)? If it is linear, is it goal-directed or nondirected? If nonlinear, is it an example of moment time or vertical time? Is it even possible for a single piece of music to be both linear and nonlinear? Let us turn briefly to Kramer's treatment of these topics for some answers to these questions.

Kramer (1988, 20) makes a fundamental distinction between linear time, "the temporal continuum created by a succession of events in which the earlier events imply the later ones and later ones are consequents of earlier ones," and nonlinear time, "the temporal continuum that results from principles permanently governing a section or piece." Immediately, we notice a distinction here between events which follow a certain succession, and principles which do not.

In the repertoire of post-tonal music, linear music must exhibit continuity, described by one or more directional processes. If it has goals--if it is directed--they may be described by the processes themselves, contextually; or they may be deduced, a priori, from general stylistic considerations. In either case, their arrival must be supported by some means of accentuation. Nondirected linearity, on the other hand, possesses the same qualities of continuity as defined by processes; however, the goals of these processes are not clear from the outset. "Such music carries us along in its continuum, but we do not really know where we are going in each phrase or section until we get there" (40).

In contrast, nonlinear music is either discontinuous and seemingly random, or wholly continuous and static. In both cases, we are unable to perceive continuity based on process; in the second, the process is trivial, and contains no information. Accordingly, Kramer describes the former category of music as existing in moment time, and the latter in vertical time. (11) Moment time is constructed of self-contained segments which are capable of existing autonomously, related to each other only according to globally defined principles. Each may be governed internally either by stasis or by some process, but we find no linearity connecting moments. They must appear to follow in arbitrary succession. Return of moments is possible, (12) but it must appear arbitrary; no process of implication may be present (207-8).

Nonlinear time may also be achieved through stasis, what Kramer labels "vertical time." Here, however, rather than discontinuity, nonlinearity is realized through extreme consistency. (13) At first, we may expect significant change; but ultimately, we are resigned to accept the nature of the music as it is. "The result is a single present stretched out into an enormous duration, a potentially infinite 'now' that nonetheless feels like an instant" (55). In other words, vertical time consists of a single moment.

Before considering the time of Nomos alpha itself, let us make a few comparisons between Kramer's and Aristoxenus's concepts. First, from the perspective of perception, we might say that music which belongs to linear time corresponds to Aristoxenus's concept of rhythm: first, it is a division of time, an ordering of chronoi, and a rhythmized pattern. Furthermore, it appeals to our intellects. We are able to apprehend and associate it--either initially, directed; or retroactively, nondirected--with a specific abstract mental form: the process. In other words, all linear rhythmized patterns may be said to correspond to rhythmic patterns as perceived in the music. In contrast, music which belongs to nonlinear time is arrhythmic. If it is in moment time, its rhythmized patterns, the order of its chronoi, seem arbitrary. If it is in vertical time, it consists of a single moment, a trivial division of time, or the protos chronos. It is what Aristoxenus labels a simple division of the chronos.

Thus, we come to the question: what is the category (or, are the categories) of time of Nomos alpha? Our answer to this question will help us arrive at an interpretive model in [section] 3. Even though we consider one possibility here, we must emphasize that no single correct answer exists to this rather complex question. We accept initially DeLio's assertion that the piece exists on two levels: one discrete, and the other continuous. However, for instance, is level I in moment or vertical time, and is level II goal-directed or nondirected? Moreover, what may we say about the time of the piece as a whole?

Regarding level I, we also accept, for now, Vriend's notion of discontinuity (noting, however, that this discontinuity does not extend to the six Intermezzi of level II). Level I's eighteen sections each divide into eight subsections, the order of which is determined by a permutational scheme (see [section] 2.1.2 below). The eighteen sections, then, are ordered by a series of group transformations. Whereas this scheme is highly "logical," we are not concerned here with that logic; rather, we will focus on the seemingly arbitrary succession of 144 subsections. In [section] 2.3, we will attempt to determine if this logic, or some consequence of it, may be used to inform our interpretive model of the piece--whether this division of the chronos may be said to be rhythmic or arrhythmic.

The division of level I into subsections is articulated by Xenakis's use of sound complexes, densities, intensities, and durations, and further articulated by method of playing, register, and choice of sieve (see [section] 2.1.3 below). Accordingly, each of the 144 subsections in level I is self-contained; some are processive and some are static (see Example 2; Example 3 contains an explanation of the notational conventions in the score). As the order of subsections, including recurrences, seems arbitrary, we may say level I is in moment time. (14)

[ILLUSTRATIONS OMITTED]

Level II, in contrast, is continuous. Its six sections describe a process of registral evolution which is discussed by both DeLio and Vriend. The first Intermezzo explores exclusively the extreme high range of the cello. The next provides a transition to the third Intermezzo, which then explores exclusively the extreme low register. The fourth and fifth Intermezzi develop ideas from the highest and lowest registers, respectively, and the sixth synthesizes these ideas. This process is not implicit: we are not aware of it at its outset, and we do not fully apprehend its shape and logic until the final, cumulative section. Therefore, we will interpret level II to be in nondirected linear time. It is marked by a process, but the goal is not unequivocal; particularly on first hearing, we realize its significance only upon hearing it.

Now we direct our attention to the interaction of these two distinct levels. Whereas the piece contains linear and nonlinear elements, the whole must be one or the other. As neither level I nor level II exists in isolation, what may we say about the time of the piece in its entirety? We turn to logic for the answer: we have established that the events of nonlinear time, those which are governed only by permanent principles, hence formal relations, do not require a particular ordering. In contrast, the events of linear time, those which appear in implicative successions, hence processes, do rely on a specific ordering. Therefore, since ordering is important in at least portions of Nomos alpha, we will make the assumption here that Nomos alpha, as a whole, is indeed linear.

For instance, we are given a musical example of events ABCD. In this example, events A and B are in nonlinear time; neither implies any subsequent event. Moreover, C and D are in linear time; specifically, C implies D. Therefore, we may view our set as partially ordered. As such, AB(CD) ~ BA(CD). This situation is true only if AB ~ BA; and, as A and B are only formally related, their presentation in either order has the same result. On the other hand, AB(CD) [??] AB(DC). The events in linear time must remain in some specific order, even if they are interrupted by nonlinear events.

Thus, we have established: (1) a unique ordering of the particular events in linear time, and (2) the need for the events in nonlinear time not to disrupt that continuity. However, we still cannot claim an overall context of truly linear time until we can demonstrate further that the composition (AB) ~ (BA) implies C, which, in turn, implies D. Again, if this is not the case, then the example is in nonlinear time. In other words, if (AB) ~ (BA) does not imply CD, the composition (CD) is perceived only as a processive moment in an otherwise discontinuous context.

This process of implication is defined in Nomos alpha, not at the outset, but gradually throughout the course of the piece. We hear a passage of discontinuous music (sections 1-3 in the score), then a continuous one (section 4), another discontinuous passage (sections 5-7), then another continuous one, etc., until we exhaust the score's twenty-four sections (see Example 1). Whereas the implication of a subsequent continuous section, 4, is not at all present in sections 1-3, by sections 21-3 it is well established: at this point, we expect a continuous section 24. In Kramer's words, "we do not really know where we are going until we get there"; and by the end of the piece, the process is revealed to us. Thus, we conclude that Nomos alpha, as a whole, is in nondirected linear time. We must caution, however, that this result does not establish a priori some hierarchy among its linear and non-linear events, in which the processive elements are given greater structural importance. If this is indeed the case, it will have to arise from musical issues.

Let us summarize here briefly before beginning the analysis. We concluded that, even though it seems to contain certain aspects of moment time, Nomos alpha is, as a whole, in nondirected linear time. Its processive elements are the Intermezzi, which also serve as the internal and closing frames. The remaining eighteen sections have the initial impression of being arbitrarily ordered. However, our awareness of the Pythagorean-Parmenidean field permits us to reconstruct logically the basic ideas of Nomos alpha, even its seemingly arbitrary sections. Our task in [section] 2 will be to conclude whether or not this determinism may serve the piece's overall structure in nondirected linear time. Ultimately, in [section] 3, we will examine interpretive strategies for performance of the piece based on this analysis and our conclusions.

2. MATHEMATICAL AND MUSICAL ANALYSIS

In [section] 2.1, we will examine the structure of Nomos alpha as Xenakis describes it in Formalized Music, focusing on its logical and systematic construction. Whereas this section is rather abstract, the musical consequences of the issues it raises will later inform our interpretations in [section] 3. In [section] 2.2, we will examine some issues which Xenakis uses deliberately, or unintentionally, to disrupt the symmetry of his pre-compositional design. Finally, in [section] 2.3, we will endeavor to reconcile these differences in an analysis which may ultimately inform our interpretation.

2.1 THESIS

2.1.1 The group-theoretical structure of Nomos alpha

The organization of Nomos alpha has already been discussed in several sources, including Xenakis (1992), Vandenbogaerde (1968), DeLio (1980), and Vriend (1981). Therefore, we will not give a full account of it here, but will deal more particularly with its aspects which will be of consequence later in our discussion of interpretation.

In composing Nomos alpha, Xenakis chooses as a point of departure the octahedral group of rigid motions, or rotational symmetry group, which maps the eight vertices of a cube (Example 4) onto themselves. (16) We will refer henceforth to this group as CUBE.

[ILLUSTRATION OMITTED]

DEFINITION 2.1.1.1. CUBE: The group of rigid motions which map the vertices of a cube onto themselves.

CUBE is isomorphic to the symmetric group [S.sub.4]; accordingly, CUBE consists of twenty-four (4!) group elements. (See Example 5.) The vertices of the cube in Example 4 may be partitioned into two sets. The first consists of vertices 1-4, and the second of 5-8; each belongs to a separate tetrahedron.

2.1.1.2. The cube is partitioned into tetrahedra (1234) and (5678).

We may observe the isomorphism of CUBE to [S.sub.4] in the permutations of either tetrahedron's vertices, such as (1234), (2143), etc. Furthermore, as we may describe the elements of CUBE as mappings, or permutations, of the cube's eight vertices, CUBE (and, by extension, [S.sub.4]) is also isomorphic to the order 24 subgroup T of the symmetric group [S.sub.8].

2.1.1.3. CUBE [congruent to] [S.sub.4] [congruent to] T [subset] [S.sub.8].

This point will become increasingly relevant in [section] 2.3.

Since CUBE consists of rigid motions of a cube, it preserves the orientations of vertices relative to the cube's four axes. Using Xenakis's labeling of vertices, these axes consist of vertices 1-5, 2-6, 3-7 and 4-8 (Example 4). Therefore, the permutations of 5-8 always parallel those of 1-4.

We define a relation X ~ Y between any elements X and Y of CUBE.

DEFINITION 2.1.1.4. The relation X ~ Y: for any X,Y [member of] CUBE, X ~ Y if the factorization of both elements' images in [S.sub.4] as products of (mathematical) transpositions contain the same parity, even or odd, of factors. (17)

Clearly, ~ is an equivalence relation: it is reflexive, as X ~ X; it is symmetric, since X ~ Y implies Y ~ X; and it is transitive, because if X ~ Y and Y ~ Z, then X ~ Z. Thus, ~ effects a partition of CUBE into two equivalence classes, those corresponding to the even and to the odd permutations of [S.sub.4].

The twelve elements of CUBE which Xenakis labels I-[L.sup.2] map each tetrahedron (1234) and (5678) onto itself. This partition forms a subgroup of CUBE, Xenakis's subgroup A. A is isomorphic to the alternating group [A.sub.4], the subgroup of even permutations in [S.sub.4]. The twelve elements of CUBE which map the tetrahedra onto each other form a coset of A. Xenakis gives these elements labels in the form [Q.sub.i]. We will refer to this coset using his label Q.

DEFINITION 2.1.1.5. A: The subgroup of CUBE whose elements have images in the even permutations of [S.sub.4]; therefore, A [congruent to] [A.sub.4].

DEFINITION 2.1.1.6. Q: The coset of A whose elements have images in the odd permutations or [S.sub.4].

2.1.2 The cosets [V.sub.i], and the paths B and [DELTA] (18)

We begin with the process Xenakis uses to determine the materials of level I: the path B. The subset of elements [V.sub.1] = {I,A,B,C} forms a subgroup of CUBE.

DEFINITION 2.1.2.1. [V.sub.1] = {I,A,B,C}.

We observe further that [V.sub.1] is normal in CUBE; that is, for any X [member of] CUBE, X[V.sub.1] = [V.sub.1]X. (19)

2.1.2.2. [V.sub.1] is the normal subgroup of CUBE of order 4.

Thus, [V.sub.1] induces a regular partition (20) of CUBE into 24/4 = 6 cosets. Xenakis gives these cosets labels in the form [V.sub.i], where i = 1,...,6; each is a distinct collection of transformations in CUBE. (See Example 6.) Since this partition is regular, it defines the quotient group CUBE/[V.sub.1] of CUBE by [V.sub.1].

DEFINITION 2.1.2.3 CUBE/[V.sub.1] = {[V.sub.i]:i = 1,...,6}.

The elements of CUBE/[V.sub.1] are the six cosets in CUBE defined by X[V.sub.1] (or [V.sub.1]X) for all X [member of] CUBE. The identity element is [V.sub.1].

With regard to CUBE/[V.sub.1], four elements, [V.sub.1], [V.sub.4], [V.sub.5], and [V.sub.6], generate involutions; that is, each of these elements is its own inverse.

2.1.2.4. [V.sup.2.sub.1] = [V.sup.2.sub.4] = [V.sup.2.sub.5] = [V.sup.2.sub.6.] = [V.sub.1].

[V.sub.2] and [V.sub.3] are inverses of each other; moreover, [V.sup.2.sub.2] = [V.sub.3], and [V.sup.2.sub.3] = [V.sub.2].

2.1.2.5. [V.sub.2][V.sup.3] = [V.sub.3][V.sub.2] = [V.sub.1]; [V.sup.2.sub.2] = [V.sub.3], and [V.sup.2.sub.3] = [V.sub.2].

Using the structure of CUBE/[V.sub.1] and a Fibonacci-series-like process, Xenakis identifies eight distinct paths through the various elements of the quotient group. This process may be defined as follows:

DEFINITION 2.1.2.6. FIB: The path induced by ab: (a, b, ab = c, bc = d, cd = e, ...). (21)

For example, using members of the quotient group, the consecutive, concatenated compositions [V.sub.1][V.sub.4] = [V.sub.4], [V.sub.4][V.sub.4] = [V.sub.1], [V.sub.4][V.sub.1] = [V.sub.4], etc., generate the 3 cycle ([V.sub.1] [V.sub.4] [V.sub.4]). Any such FIB cycle which is induced by [V.sub.1], the identity element, and another involution (not [V.sub.1]) will be a two-valued 3 cycle. We may express these cycles abstractly using the structure (a b b). [V.sub.1] [V.sub.1] induces the trivial FIB cycle (a). The FIB cycle induced by [V.sub.1] and either [V.sub.2] or [V.sub.3] is a three-valued 8 cycle, (a b b c a c c b). (22)

For an uncited reason, but presumably to achieve a maximum variety, (23) Xenakis chooses to use the five-valued 6 cycles. Each of these FIB cycles excludes the identity element, and involves the remaining five values, [V.sub.2]-[V.sub.6]. Moreover, each of these cycles may be expressed abstractly as using the following structure: (a b c d e c).

2.1.2.7. The FIB cycles induced by [V.sub.2][V.sub.4], [V.sub.2][V.sub.5], and [V.sub.2][V.sub.6] all have the structure (a b c d e c), where a,...,e represent distinct elements of CUBE/[V.sub.1].

We will return to this point shortly.

We may describe further FIB cycles, similar to those above, now using the specific members of CUBE instead of those of CUBE/[V.sub.1]. Furthermore, since each member [V.sub.i] of CUBE/[V.sub.1] represents four different permutations, each of the eight paths above represents potentially [4.sup.2] = 16 distinct cycles of permutations, for a total of ( 16 * 8) = 128 potential cycles. (24)

In Nomos alpha, Xenakis chooses to use two of these cycles of permutations, one each from two 6 cycles on CUBE/[V.sub.1]: the first induced by D[Q.sub.12] (from [V.sub.2][V.sub.4]) and the second induced by D[Q.sub.3] (from [V.sub.2][V.sub.5]). As we noted in (2.1.2.7) above, the cycles of [V.sub.2][V.sub.4] and [V.sub.2][V.sub.5] are structurally identical. This property extends to the cycles generated respectively by the elements of [V.sub.2] and [V.sub.4], and [V.sub.2] and [V.sub.5], with only slightly more complexity.

D[Q.sub.12] induces a thirteen-valued 18 cycle:

DEFINITION 2.1.2.8. [F.sub.1]: The cycle induced by D[Q.sub.12],

(D [Q.sub.12] [Q.sub.4] E [Q.sub.8] [Q.sub.2] [E.sup.2] [Q.sub.7] [Q.sub.4] [D.sup.2] [Q.sub.3] [Q.sub.11] [L.sup.2] [Q.sub.7] [Q.sub.2] L [Q.sub.8] [Q.sub.11]).

We note that five elements, [Q.sub.4], [Q.sub.8], [Q.sub.2], [Q.sub.7], and [Q.sub.11] recur in order positions (3,9), (5,17), (6,15), (8,14), and (12,18), respectively. D[Q.sub.3] also generates a thirteen-valued 18 cycle:

DEFINITION 2.1.2.9. [F.sub.2]: The cycle induced by D[Q.sub.3],

(D [Q.sub.3] [Q.sub.7] L [Q.sub.11] [Q.sub.6] [L.sup.2] [Q.sub.5] [Q.sub.7] [D.sup.2] [Q.sub.9] [Q.sub.1] G [Q.sub.5] [Q.sub.6] [G.sup.2] [Q.sub.11] [Q.sub.1]).

It also has five recurring elements, [Q.sub.7], [Q.sub.11], [Q.sub.6], [Q.sub.5], and [Q.sub.1], in respective order positions (3,9), (5,17), (6,15), (8,14), and (12,18). Thus, the cycles induced by D[Q.sub.12] and D[Q.sub.3] both have the same structure:

DEFINITION 2.1.2.10. F: The abstract cyclic structure of [F.sub.1] and [F.sub.2],

([a.sub.1] [b.sub.1] [c.sub.1] [d.sub.1] [e.sub.1] [c.sub.2] [a.sub.2] [b.sub.2] [c.sub.1] [d.sub.2] [e.sub.2] [c.sub.3] [a.sub.3] [b.sub.2] [c.sub.2] [d.sub.3] [e.sub.1] [c.sub.3]); where [a.sub.i] [member of] a, [b.sub.i] [member of] b, [c.sub.i] [member of] c, [d.sub.i] [member of] d, and [e.sub.i] [member of] e, from (2.1.2.7).

We note further that the various elements of [F.sub.1] and [F.sub.2] belong to both the A and Q subgroups of CUBE (2.1.1.5-6). These subgroups are represented in the FIB cycles in a two-valued 3 cycle, given by the following structure:

2.1.2.11. The subgroups A and Q of CUBE are represented by the successive elements of [F.sub.1] and [F.sub.2] in the following cycle: (A Q Q).

This point will be of consequence later in our discussion of the cycle of transformations [lambda] in [subsection] 2.2-3.

Since Xenakis unfolds [F.sub.1] and [F.sub.2] in a one-to-one correspondence (i.e., D [member of] [F.sub.1] is presented together with D [member of] [F.sub.2], [Q.sub.12] [member of] [F.sub.1] with [Q.sub.3] [member of] [F.sub.2], [Q.sub.4] [member of] [F.sub.1] with [Q.sub.7] [member of] [F.sub.2], etc.), it suffices to refer to their mutual abstract structure as given above in (2.1.2.10) when knowledge of their specific permutational elements is not necessary. This point will be of considerable importance in [section] 2.3.

Before moving on, we must point out that Formalized Music presents the beginnings of a similar approach to the materials of level II, the path [DELTA]. For an unstated reason, Xenakis abandons this scheme. Nonetheless, he preserves some of its aspects: each Intermezzo uses a particular sieve; the Intermezzi use only the extreme high and low registers of the instrument; and they use only extreme dynamic levels, pppp, pp, ff, fff. Otherwise, they are apparently freely composed.

2.1.3 The determination of musical materials

In level I of Nomos alpha, Xenakis uses the elements of the FIB cycles [F.sub.1] and [F.sub.2], and their respective members of CUBE, directly to determine the order of sound complexes, and the values of densities, intensities, and subsectional durations. He also incorporates them with several kinematic diagrams to establish registers and methods of playing. Finally, they decide, indirectly, the placement of Intermezzi and choices of sieve. We will proceed through these aspects in order.

Xenakis uses eight sound complexes in Nomos alpha. Initially, (25) these ideas are associated with the eight vertices of the cube as per Example 7. (26) Their order is determined by the members of [F.sub,1]; thus, in succession, we find D = (23146758), [Q.sub.12] = (56871243), [Q.sub.4] = (67852341), etc. (See Example 8.)

[ILLUSTRATION OMITTED]

Also in level I, the densities, intensities, and subsectional durations are determined by a FIB cycle, in this case by the members of [F.sub.2]. This FIB cycle unfolds in a one-to-one correspondence with the one above; its first few members are, in order: D = (23146758), Q.sub.3] = (86754231), [Q.sub.7] = (87564312). Using the techniques of symbolic music described above in [section] 0.4, Xenakis defines three sets, D, G, and U, one each for the respective parameters above, and these sets are "mapped onto three vector spaces or onto a single three-dimensional vector space" (Xenakis 1992, 218). The vector [k.sup.r.sub.i] consists of the coordinates [d.sub.i][g.sub.i][u.sub.i]: the first for density (events per second), [d.sub.i]; the second for average intensity, [g.sub.i]; and the third for subsectional duration in seconds, [u.sub.i]. (27)

2.1.3.1. The sets D (densities), G (intensities), and U (durations)are mapped onto the vector space [k.sup.r.sub.i] = [d.sub.i][g.sub.i][u.sub.i].

Initially, Xenakis associates the values of D, G, and U with the eight vertices of the cube in the scheme shown in Example 9. (Compare with Example 8.)

[F.sub.1] also establishes particular registers and methods of playing; they are determined further by several kinematic diagrams which Xenakis designs to achieve maximum expansion (minimum repetition) and maximum contrast (minimum resemblance) of materials (Xenakis 1992, 228-9). Initially, in level I, set H consists of four distinct registers (see Example 10), and set X contains eight methods of playing. The methods of playing are associated with the eight vertices of the cube as shown in Example 11. The kinematic diagrams decide the particular choices from among these possibilities, as well as among registers.

The product set H x X defines the set of vector spaces [h.sub.i][x.sub.i] which the kinematic diagrams and [F.sub.1] traverse. For example, the first member of [F.sub.1], D, begins with vertices 2, 3, and 1. Using these vertices and the kinematic diagram in Example 12, (28) we should find, in order, pizzicato glissando in register IV, col legno battuto in register II, and pizzicato in level IV. (The actual registers in the music are III, I, and III, respectively; see Example 8.) (29) At this time, we switch to another diagram to accommodate the playing methods of vertices 4-6, etc. In short, Xenakis uses the registers and methods of playing as further means of distinguishing the subsections from each other. (30)

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Finally, we come to the placement of the Intermezzi and the choices of sieve, both of which are also determined indirectly by the FIB cycles. In particular, these aspects follow the (A QQ) cycles (2.1.2.11) of [F.sub.1] and [F.sub.2] above. We begin with the sieve ^(11,13). After the first complete (A Q Q) cycle (D/D [Q.sub.12]/[Q.sub.3] [Q.sub.4]/[Q.sub.7]), Xenakis inserts the first Intermezzo (from level II) with its own associated sieve, ^(11,11). Then, we find the members of the next (A Q Q) cycle using sieve ^(13,11), followed by an Intermezzo using ^(13,17), etc.

The six sections of level II are readily perceived as cohesive units. Therefore, assuming that one may perceive the eighteen sections of level I as discrete entities of eight subsections each, a diagram of the form of Nomos alpha, as we have defined it so far, is given in Example 13. We find five sections which recur in level I: specifically, C, E, F, H, and K. Examples 14 and 15 show the two respective occurrences of section F.

[ILLUSTRATIONS OMITTED]

Thus, as DeLio maintains, the elements of Nomos alpha were chosen to articulate its group structure. Certainly, Xenakis has taken great efforts to assign distinct sonic properties to the various group elements represented in the scheme of the piece. However, as we will explore in the next section, and particularly as regards the [lambda] transformations, is it likely, or even possible, that we can apprehend this structure aurally? (31) Should we, therefore, use it as the basis for our interpretive model of the piece? Or, might some aspect or consequence of it inform our interpretation of the work?

2.2 ANTITHESIS

2.2.1 The effects of the transformational scheme [lambda] = ([beta][gamma][alpha]) on level I

Another significant aspect of level I in Nomos alpha is determined by (A Q Q) cycles: the scheme [lambda]. As we will see below, this scheme disrupts much of the continuity we have defined so far. Ultimately, in [section] 2.3, we hope to show its logic and its consequences for the form of the piece. [lambda] contains three elements which appear cyclically in the order ([beta] [gamma][alpha]). (32)

In terms of [F.sub.1], [lambda] is manifest by a series of permutational transformations which affect the ordering of sound complexes. (N.B.: it does not impact the orders of methods of playing, which are still determined by the original members of [F.sub.1].) These transformations are given in Example 16. [lambda] transforms the first three elements of [F.sub.i], those belonging to the first (A Q Q) cycle, by [beta]. As it is the identity element, [beta] leaves them unchanged. After the first Intermezzo, the next three elements, those belonging to the next (A Q Q) cycle, are transformed by [gamma]. After the next Intermezzo, we find the next three elements transformed by [alpha]. Then, the cycle begins again, etc. Therefore, for example, the fifth section of the piece uses the ordering E = (24316875) from [F.sub.1]. E is now transformed by [gamma] = (25)(36)(47), with the resulting permutation [gamma](E) = (57613842). This reordering, then, determines the succession of sound complexes.

[lambda] also impacts the elements of [F.sub.2], in particular, with regard to the vector coordinate [d.sub.i] (2.1.3.1) for number of events per second. These values vary with each change of [lambda]. The initial values, those used by [beta], are 0.5, 1.08, 2.32, and 5.00 events per second. Under [gamma], these values become 1, 2, 3, and 4; [alpha] yields 1.0, 1.5.2.0, and 2.5.

We will address the significance of [lambda]'s impact on [F.sub.2] first, as it is less disruptive on the logic of level I. As we noted above, [lambda] affects the number of events per second within subsections. Whereas each transformation [beta], [gamma], and [alpha] maintains a relative less-to-more ranking of densities, without our specific knowledge of a change in scale, a potential risk of confusion exists. For example, one subsection may have a density of 5 events/sec., the highest ranking under [beta]; later, under [alpha], a parallel subsection may have a density of 2.5 events/sec., again, its respectively highest ranking. However, without something to signal a change of scale to us, how are we to know that the latter's density is relatively higher than a third subsection's 3 events/sec, under [gamma]?

The [lambda]. transformations also impact the elements of [F.sub.1], with more serious consequences. Namely, its members [gamma] and [alpha] disrupt the ordering of the sound complexes. Let us consider once more the preliminary form diagram of Example 13, now taking into account the elements of [lambda] (33) (see Example 17). In terms of the original organization of sound complexes, we observe that four of the five recurring sections are now reordered. For example, section H uses the series [Q.sub.7] of sound complexes. We observe that [alpha]([Q.sub.7]) = (82564317) [not equal to] [gamma]([Q.sub.7]) = (84237615). Only one section, F, recurs under the same element, [gamma], of [lambda]. F uses [Q.sub.2]; and, clearly, [gamma]([Q.sub.2]) = [gamma]([Q.sub.2]) (see Examples 14 and 15).

[ILLUSTRATIONS OMITTED]

Since it is very unlikely that, without some prior knowledge of the [lambda] transformations, a listener would perceive a connection among the other four formerly recurring sections, we must propose a new form diagram. Still assuming that one may perceive level I as eighteen discrete entities, this new diagram takes the configuration of Example 18. With the exception of section F's recurrence, this scheme suggests Kramer's process of "constant newness" (see Note 12).

[ILLUSTRATION OMITTED]

The [lambda] transformations on [F.sub.1] also impact the continuity of sound complexes within a section, based on DeLio's "staccato" and "legato" articulations. The original orderings of sound complexes derive from CUBE, and the elements of CUBE divide the eight vertices into parallel permutations of 1-4 and 5-8 (2.1.1.2). Accordingly, the sound complexes divide each section into four adjacent staccato, and four adjacent legato subsections, respectively. After [alpha], sound complex 2 is transposed with 7; hence, one legato subsection is introduced into the set of staccato subsections, and vice versa, [gamma], which transposes 2 with 5, 3 with 6, and 4 with 7, preserves only one subsection of the original ordering; therefore, it replaces three of the four staccato subsections with legato counterparts, and vice versa. It has the same effect as [alpha], but also reverses the order of articulation types, staccato or legato, within the section.

Because of its impact on the structure we defined in [section] 2.1, [lambda] has the effect of nearly abolishing the patterns of recurrence in level I's precompositional scheme. In fact, Xenakis states that he incorporates it to make things less predictable (Vriend 1981, 32), and in this regard, it may be argued, he succeeds.

2.2.2 Xenakis's apparent deviations and/or mistakes in level I

Two other categories of complications are potentially disruptive to the structure of level I: mistakes, and alterations in the realization of the precompositional plan. Vriend (1981, 44) quotes Xenakis from his personal correspondence:

a) [I]n the heat of the action I made slips of the pen which I discovered only too late, after publication; b) I sometimes changed details because they appear to me more interesting for the ear and c) I make theoretic errors which entail errors in the details.

Accordingly, the structure of level I which we defined above may not actually be expressed in the music.

Vriend gives two examples in the first section of the piece. Using the values for D in [F.sub.2], the duration of the third subsection is given by [k.sup.[beta].sub.4]] as two seconds, hence a total duration of twenty sixteenth notes. Checking the score (see Example 8), we see that the third subsection, including the rests, contains only ten sixteenth notes. (34) If we assume a dot on the quarter note, it still brings the total subsectional duration only to 1.2 seconds. The next subsection also has a different duration than its precompositional counterpart: the score gives a duration of eighteen instead of twenty sixteenth notes. Further inconsistencies are left for the reader.

These mistakes or intentional deviations obfuscate further the structure of level I, again raising the question of whether it can be communicated aurally to a listener. At this point of our analysis, we must conclude that this structure is not sufficient for our interpretive model of the piece, unless we wish to emphasize its seeming arbitrariness. However, we echo Vriend's sentiment that "to conceive of a machinery as sophisticated as the one we have been reconstructing so far, with all its philosophical pretensions and historical ambitions and then to destroy it with purposefully planned deviations and confusions, would likewise destroy the value of the whole conception" (Vriend 1981, 54). Yet, perhaps we may find some aspect of the logic of this design which we may project in performance.

2.3 SYNTHESIS

Of all the aspects we discussed above which are potentially disruptive to the logic of level I, the [lambda] transformations on FI seem to be the most profound. The modifications to the ranking of densities in [F.sub.2] by [lambda], as well as errors or adjustments in the realization of the piece's precompositional scheme, do not have the same effect of "destroying the machinery" to the extent of the former. If we can perceive the [lambda] transformations on [F.sub.1]--if we can observe some logic for their choice and application, and corroborate this logic with tangible musical results--the latter aspects should follow. So, we begin: why [lambda]?

In seeking an answer to this question, we turn to Xenakis's thesis defense for the degree Doctorat d'Etat. (35) In one of his questions, Olivier Messiaen asks Xenakis:
   You know as well as I do that a certain number of objects gives a
   certain number of permutations, and the more the number of objects
   increases, the more the number of permutations increases and with a
   speed and in quantities which can seem disproportioned. So, three
   objects give six permutations, six objects give seven hundred and
   twenty, and twelve objects give (if I'm not mistaken) four hundred
   ninety seven million, one thousand six hundred permutations. (36)
   Suppose these objects correspond to durations: I would have to write
   out these durations in order to know what gesture or what movement
   they could create in time. There has been a lot of talk about
   retrograde movement these days: this is but one movement, one single
   movement among thousands of others, and its permutation follows the
   original trajectory. And all the other permutations? I can't write
   out the millions and millions of permutations ... and yet
   I must write them out in order to know them and to love them (I
   insist on the verb to love!). In your case, a machine will give you
   the millions of permutations within a few minutes: it's a cold and
   unexplicit list. How can and do you choose directly from within this
   immense world of possibilities without intimate knowledge or love?
   (Xenakis 1985, 31)


In the case of Nomos alpha, by introducing the permutations ct and [gamma], Xenakis is no longer working only with the 4! = 24 elements of CUBE. Rather, he is choosing from among the 8! = 40,320 elements of the symmetric group Sa. How does he choose these particular permutations from this comparatively large list? Xenakis replies thus to Messiaen's question: (37)
   I can imagine--I don't need a machine for that--I can imagine and
   intellectually make a choice. There are several ways of making this
   choice. It's true that when there are a few sounds, or more
   precisely, a few pitches to control, it is easy to proceed in an
   arbitrary or intuitive manner, directly. But, when it's a question
   of a great quantity of sounds, well, there it would be handy to
   borrow from other domains. When I look at a small number of
   individuals, I see them as individuals; I see their relationships,
   their characteristics, and their relations to space and time, their
   own physiognomies, etc. But if there is a crowd, I can no longer
   distinguish the individuals, because they are too numerous. On the
   contrary, what I can see are the aspects, the characteristics of
   the crowd. (Xenakis 1985, 33-4)


Therefore, even though we do not know Xenakis's actual reason for using the [lambda] transformations on [F.sub.1], we can make some conjectures by "seeing the characteristics of the crowd," by borrowing from the domain of mathematics, and examining some of the properties of [S.sub.8].

2.3.1 The [alpha] transformation

Our logic here may not be the same as that of Xenakis, but it does lead to an interesting and satisfactory result. Let us decide first that we wish to introduce a transformation [alpha] which exchanges only two vertices; this transformation leaves the other six vertices fixed. We may already observe a few aspects of [alpha]:
   2.3.1.1. [alpha] is of order 2. [alpha] contains a single
   transposition, [alpha] is therefore not a member of CUBE, as all
   permutations in CUBE are even. For the same reason, [alpha] is also
   not in the alternating group  [A.sub.8]. [alpha] fixes at least two
   of eight vertices; therefore, its transforms act on [S.sub.6].


The centralizer of [alpha] in any permutation group G--in other words, the set of elements in G with which [alpha] commutes--is given by the following definition:

DEFINITION 2.3.1.2. For any [alpha], [C.sub.G]([alpha]) = {[sigma] : [sigma]([alpha])[[sigma].sup.-1]=[alpha];[sigma][member of] G}.

Therefore, the centralizer of [alpha] in [S.sub.8] is the set of elements in [S.sub.8] with which [alpha] commutes.

2.3.1.3. [C.sub.S8] ([alpha]) forms a subgroup [S.sub.8].

The size of [C.sub.S8] ([alpha]) is given by the following formula:

DEFINITION 2.3.1.4. |[C.sub.S8]([alpha])| = |<[alpha]> x [S.sub.6]|.

Thus, [C.sub.S8]([alpha]) contains (2 x 720) = 1,440 elements.

Furthermore, all elements [mu] [member of] [S.sub.8] which are formed by [sigma]([alpha])[[sigma].sup.-1] are conjugates of [alpha]. [[[alpha].sup.S8]], then, is the set of all conjugates of [alpha], or the conjugacy class of [alpha].

DEFINITION 2.3.1.5. [[[alpha].sup.S8] = {[micro] : [micro] = [sigma]([alpha])[[sigma].sup.-1]}.

The size of this conjugacy class is given by the index of the centralizer.

DEFINITION 2.3.1.6. |/[[alpha].sub.s8]| = | [S.sub.8]/[C.sub.S8]([alpha]) |.

In other words, [[alpha].sup.S8] contains 40,320/1,440 = 28 permutations. These conjugates are given by all the single transpositions in [S.sub.8]; that is, [[alpha].sup.S8] contains (12), (13), (14), ..., (18), (23), (24), (25), ..., (28), ..., (78), for a total of(7 + 6 + 5 + 4 + 3 + 2 + 1) = 28 elements. Therefore, if Xenakis wishes to choose some transformation which leaves six of the eight vertices fixed, he has to choose from among only twenty-eight possibilities.

Now, the centralizer of [alpha] in CUBE, [C.sub.CUBE]([alpha]), consists of the following set:

DEFINITION 2.3.1.7. [C.sub.CUBE]([alpha]) = {[sigma] : [sigma]([alpha][[sigma].sup.-1] = [alpha]; [sigma][member] CUBE}.

Accordingly, by (2.3.1.3), [C.sub.CUBE]([alpha]) is a subgroup of CUBE. We may then calculate the size of the conjugacy class of [alpha] in CUBE as follows:

DEFINITION 2.3.1.8. |[[alpha].sup.CUBE]| = |CUBE/[C.sub.CUBE] ([alpha])|.

For any [alpha], the size of [C.sub.CUBE]([alpha]) is two: the identity element, I, and one other member of CUBE. So, [[[alpha].sup.CUBE]] contains 24/2 = 12 elements. Therefore, let us assume Xenakis intends a transformation which leaves six of the eight vertices fixed; furthermore, for a more interesting result, he selects one which is not in the conjugacy class of such elements in CUBE. He need now choose from among only (28 - 12) = 16 possibilities. He limits his choice further by eliminating the permutations which do not leave vertices 1 and 8 fixed.

2.3.1.9. [alpha] leaves vertices 1 and 8 fixed.

He further limits his choice by deciding on a transformation which transposes one vertex from 1-4 with another from 5-8.

2.3.1.10. [alpha] transposes one vertex from 1-4 with another from 5-8.

Therefore, with this number, it is possible to "see the characteristics of the individuals." He ultimately chooses [alpha] = (27).

2.3.2 The [gamma] transformation

Following our train of logic from the previous section, we ask how Xenakis determines the other members of the [lambda] cycle, [beta] and [gamma]. The rationale for [beta] is fairly obvious: it is the identity element; accordingly, [beta] leaves the elements of [F.sub.1] in their original states.

Now, from [alpha], it takes only a little more effort to arrive at [gamma]. We noted above that [alpha] fixes six vertices and permutes two (2.3.1.1). Now, let us assume that Xenakis chooses another transformation, [lambda], which is structurally opposite of [alpha]; it fixes two vertices and permutes six.

2.3.2.1. [gamma] fixes two vertices and permutes six.

We noted in (2.3.1.1) that [alpha] is of order 2, and further that [alpha] is not in [A.sub.8]. Only one other conjugacy class in [S.sub.8] has these same properties: the set of transformations consisting of three transpositions. The centralizer of such transformations in [S.sub.8] contains ([2.sup.5] * 3) = 96 elements; therefore, it has 420 conjugates. This number seems too large for Xenakis's purposes. However, because [alpha] leaves at least two vertices fixed, we also observed in (2.3.1.1) that it acts on the symmetric group [S.sub.6]. Specifically, by (2.3.1.9), we noted that a fixes vertices 1 and 8. Therefore, we may regard [alpha] in [S.sub.6] as permuting vertices 2 through 7. If Xenakis wishes to pick another transformation [gamma], in that [S.sub.6] which leaves vertices 1 and 8 unchanged, and which has the properties described in the previous paragraph, he need now choose from among only fifteen possibilities.

Finally, in (2.3.1.10), we noted that [alpha] transposes one vertex from 1-4 with another from 5-8. Xenakis decides that each of the three transpositions of [gamma] will have the same property. This quality limits his choices further to four; ultimately, he chooses [gamma] = (25)(36)(47). In other words, having selected [alpha], he is able to obtain [gamma] through a conjugation by an outer automorphism of [S.sub.6] of order 4.

2.3.3 Invariant pairings of order positions in level I under [lambda]

We now address the question of Xenakis's reasons for implementing such a transformational scheme. In answer to Vriend's inquiry, as we noted at the end of [section] 2.2.1, Xenakis said it was to make things less predictable; not, however, to make things unpredictable. Still, as we observed earlier in [section] 2.2.1, the form of level I under [lambda] suggests Kramer's process of "constant newness," making it either wholly unpredictable or wholly predictable, depending on one's perspective (see Example 18). That does not seem to be Xenakis's intention. Rather, there is an attribute of the particular transformational scheme he chooses, a result of the characteristics we defined above, suggesting a form in level I which is slightly more complex than Example 18, but is also potentially more musically interesting.

For any X [member of ] CUBE, all three transformations in [lambda] fix the positions of vertices 1 and 8. For instance, using [Q.sub.8] = (75863142), we note the following transformations: [alpha]([Q.sub.8]) = (25863147), [beta]([Q.sub.8]) = (75863142), and [gamma]([Q.sub.8]) = (42836175). In each case, vertices 8 and 1 appear in order positions 3 and 6, respectively. Now, since the elements of [F.sub.1] under [lambda] determine the order of sound complexes in Nomos alpha, and the elements of [F.sub.2]--which do not undergo such a reordering--determine the densities, intensities, and subsectional durations, we may say that whenever [Q.sub.8] appears in [F.sub.1], its third and sixth subsections will have the same characteristics: that is, they will use the same sound complex, and have relative densities, intensities, and durations, regardless of member of [lambda]. However, to say that two sections are perceptibly related if they each contain only two similar subsections of eight is pushing the threshold of our abilities. This correlation is not altogether a satisfying progenitor of form from a perceptual standpoint, but it does point us in the right direction.

For us to claim musical connections between sections R and S, we should base these correlations on some perceptual criteria. First, we recall that R contains eight subsections, [r.sub.1], ..., [r.sub.8], which are determined by some W = ([w.sub.1], ..., [w.sub.8]) [member of] [F.sub.1] and X = ([x.sub.1], ... [x.sub.8] [member of] [F.sub.2]. Similarly, subsections [s.sub.1], ..., [s.sub.8] of S are determined by some Y = ([y.sub.1], ..., [Y.sub.8]) [member of] [F.sub.1] and Z = ([z.sub.1], ..., [z.sub.8]) [member of] [F.sub.2]. To assert a perceptual correlation, we will say that at least half of the eight subsections of R need to have a close resemblance to counterparts in S, with certain aspects of order preserved as well. We label this relation [R.sub.SIM] and define it formally as follows:

DEFINITION 2.3.3.1. For any two sections R and S, we may say R [R.sub.SIM]S if the following minimal conditions hold:

1) four ordered pairs ([W.sub.i], [X.sub.i]) in R are equal to four ordered pairs ([y.sub.i],[z.sub.i]) in S;

2) these ordered pairs must appear in two adjacent order positions [r.sub.j],[r.sub.j+1] and [r.sub.j+4], [r.sub.j+5] in R, and two adjacent order positions [s.sub.k], [s.sub.k+1] and [s.sub.k+4], [s.sub.k+5] in S; and

3) R and S must be transformed either by the same element of [lambda], or by some combination of [alpha] and [beta] (but not [gamma]).

Certainly, we may still say that R and S are related if these criteria are exceeded, as in the case of section F and its reprise (see Example 18).

Xenakis's choices of [alpha], [beta], and [gamma] guarantee this minimal correspondence among certain members of CUBE under [lambda]. We recall that the a transformation leaves six positions fixed. It transposes one vertex from 1-4 with another from 5-8, leaving at least two pairs of the original vertices adjacent: one in each partition. Because CUBE is isomorphic to [S.sub.4], at least one other member of CUBE exists in which these pairs of vertices are adjacent. Furthermore, the [gamma] transformation transposes 2 with 5, 3 with 6, and 4 with 7; but it also preserves the orders of (2, 3, 4) and (5, 6, 7), respectively. So, the same argument works here as well. These facts ensure conditions (1) and (2) of (2.3.3.1) above for certain pairs in CUBE, provided that its members are transformed by the same element of [lambda], or that one is transformed by [alpha] and the other by [beta], as described by Condition (3).

Regarding these correlations in level I, we observe that sixteen of its eighteen sections display this minimal correspondence with at least one other section. Based on these observations, we can propose a new form diagram which accounts for these similarities among sections (see Example 19). Only two sections do not relate to any others: D and I. Sections A, G, and H relate to A', G', and H'; and sections B and C relate to B' and B", and C' and C", respectively. As we noted before, section F recurs nearly intact. Taken in this light, level I indicates some sort of bipartite division. The first half consists of A, B, C, D, E, F, G, H, and C'. The second part restates this structure, but reverses the orders of B and C, and recalls B in its closing (see Example 20).

[ILLUSTRATION OMITTED]

As an illustration of these sectional relations, Example 21 gives the parallel section to Example 8. Its first and second, and fifth and sixth subsections correspond to the second and third, and sixth and seventh of the earlier example. We recall that, due to the particular system Xenakis uses, we cannot claim any correlations between pitches or surface rhythms. Rather, we are interested primarily in the elements determined by [F.sub.1] and [F.sub.2]: sound complexes, and densities, intensities, and subsectional durations.

[ILLUSTRATION OMITTED]

So, our analysis leads us to an interesting result: whereas level I of Nomos alpha initially seems to indicate an arbitrary succession of events, after careful study, we find that it has a perceptible, albeit nonlinear, directed motion. The task of the performer, then, is to decide whether to show or bide the directed motion. Since either interpretation is feasible, the performer should then take certain steps in communicating his or her conception of the work to a listener.

3. TWO INTERPRETIVE MODELS OF NOMOS ALPHA

In this section, we will focus on techniques performers might use to project these two interpretations of the piece. Many of our ideas here on the relation of analysis to performance derive from Wallace Berry's 1989 monograph, Musical Structure and Performance. In this work, certain of Berry's ideas derive from his earlier, 1976 study, Structural Functions, in which he describes five "consistently relevant processes": (1) preparation or introduction, (2) expository statement (or expository restatement), (3) transitional bridging, (4) development, and (5) closure. We will propose two approaches to Nomos alpha--one in which level I is in moment time, and the other in which level I is in nondirected linear time--based on these processes.

3. 1 CASE 1: LEVEL I IN MOMENT TIME

We begin with the assumption that the logic of event succession in level I is not readily perceptible; rather, it is seemingly arbitrary. Therefore, level I, at least taken in isolation, is in moment time. What, then, may we say about its interaction with level II, and how do we communicate this structure to a listener?

As we noted in [section] 1.3, the time of Nomos alpha is revealed throughout the course of the piece. At first, we hear an arbitrary succession of isolated moments--or subsections, but at this point we have no way of hearing them as such. Then, in measure 46, we hear a longer, continuous section which explores the extreme high range. These sets of arbitrary events continue to alternate with longer continuous sections which explore the extreme ranges of the instrument, and we begin to anticipate the occurrence of these longer sections in the sets of isolated events. This anticipation is not the result of any specific aspect of the materials in the arbitrary successions, it is rather the result of our conditioning in some way such as Meyer's (1956) implication-realization model. Finally, we hear a continuous section which synthesizes the previous Intermezzi's explorations in a stunning way: a contrary motion passage, which, seeming to be impossible on cello, we realize retrospectively is the goal of a process. Here, the piece "ends," not "stops."

According to Berry (1989, 2), "the central issue of interpretation can be summarized in two questions: In a particular unit of musical structure, to and from what points (and states) can directed motions be said to lead? And what is the performer's role in projecting and illuminating essential elements of direction and continuity?" In a work such as Nomos alpha, we must amend Berry's first question somewhat: "In a particular unit of musical structure, to and from what points (and states) can linear motions be said to lead, and what are the boundaries of nonlinear material?" The second question follows accordingly: "And what is the performer's role in projecting and illuminating essential elements of linearity and continuity, or nonlinearity and discontinuity?"

We have already dealt with the first question, so let us now consider the performer's role in communicating such an interpretation to a listener. Again, we turn to Berry (1989, 10-44), and address seven of his twelve "questions arising in the relations of analysis to performance," (38) inasmuch as they can apply to these further concepts of form and time. We will begin with two questions which impact our interpretation in a related tangential manner.

1. Is a given motive self-evident in the music, or does it require the performer's intervention?

2. By what means and to what extent should relatively disguised imitations be brought to the fore?

The concepts of motive and imitation are not particularly relevant to Nomos alpha, but they suggest the further question of what the performer should do in circumstances where conformance is inconspicuous, such as in level I. (39) Should such recurrences be made explicit, and if so, how would this interpretation impact the seeming arbitrariness of events? Let us begin with the latter half of the question. Since the conformances in level I are tenuous, they will most likely not be perceived as recurrences unless the performer makes them obvious. Moreover, without recurrence, we are left with Kramer's process of "constant newness," and the succession of materials will not seem truly arbitrary.

Hence, to project a true sense of arbitrariness, these conformances should be made explicit, but not set up by some process of suggestion or structural upbeat, resulting in a sense of "relative newness." In Berry's (1989, 218) words: "The performer must be conscious of unifying recurrences apart from those of a motive in its usual sense ... Their prevalence often suggests ... some marked attention to formulations of surface rhythm, harmony, or figural pattern ... that function critically in a discerned, vital structural tendency." In the case of level I, this structural tendency is arbitrariness.

The next two questions we consider are similarly related, and have a potentially major impact on interpretation. In fact, Berry (1989, 3) names articulation and tempo as the two essential categories of interpretive intervention.

3. What dynamic inflections should be added where none are indicated?

4. What factors bear upon choosing a tempo?

First, the tempo is determined at the beginning of the piece, but tempo is also potentially subject to expressive fluctuations. In either case, that of dynamics or tempo, we should not allow our interventions in level I to interfere with the scheme of dynamics and subsectional durations as defined by [F.sub.2], which already ensures a seeming arbitrariness. In particular, we should not use inflections to suggest implicative processes; for example, an expressive crescendo or accelerando might suggest a structural upbeat. Regarding the sections of level II, we might incorporate a minor caesura before and after each to signal its separation from the surrounding material; similarly, a slightly different tempo will help set these sections off from level I. In terms of dynamic inflection, level II should be contrasting to level I. Gradual dynamic shaping processes will serve to emphasize these sections' continuous nature.

The continuity of Level II also extends to the connections between its sections.

5. What kind of intervention is required to bring out a significant but not necessarily explicit voice-leading connection?

Whereas this question does not address the problem directly, level II presents a difficulty in terms of projecting voice-leading (i.e., registral) connections over relatively large expanses of time, with intervening material. The performer needs to convey a sense of its sections' participation in a large-scale process which is ultimately realized in the final section. In addition to Xenakis's use of dynamics and register to articulate these sections, the performer might implement some sense of timbral continuity from one section of level II to another, particularly by incorporating some timbre which is not used in level I. Similarly, a slight dwelling on pitches in the same register at the end of one section and beginning of the next might draw attention to their connection.

Finally, we turn to Berry's questions regarding the performer's awareness of form.

6. How important is it for one to know where one is within a formal process?

7. How should the groupings that arise from an awareness of form and structure be projected?

These questions are difficult to answer in pieces which incorporate aspects of nondirected and nonlinear time. Certainly, we agree with Berry's conclusion that it is important for a performer to know where he or she is in a formal process. But what should the performer do in circumstances in which no process exists, or where a process is not implicit?

In general, we may say that in level I, the performer should take care not to project any sense of groupings. He or she should, as stated above, avoid processive interventions which adhere moments. In level II, the awareness of the process should not be communicated as a directed motion to an unequivocal goal. Rather, the performer should merely project a sense that a process is present. Then, on reaching the final section, he or she should indicate--perhaps by way of a dramatic Luftpause and/or a slight initial broadening of tempo--the arrival of the culmination of this process.

Thus, in terms of Berry's five "consistently relevant processes," we make the following observations. In moment time, all material is either expository statement or expository restatement. Since no processes of beginning or ending exist, it has no need for introduction or closure. Similarly, as moment time utilizes essentially no connection between events, it has no need for transitional bridging. Any sense of development would be limited to variation among recurring events, but it should not imply a process of development. In contrast, level II does imply a process. Its six sections may be divided as follows. The expository statements are Intermezzi I and 3, which explore the extreme high and low registers, respectively. Intermezzo 2 is transitional bridging between these two. The materials of Intermezzi 1 and 3 develop in 4 and 5, respectively. Finally, Intermezzo 6 develops these two materials simultaneously.

3.2 CASE 2: LEVEL I IN NONDIRECTED LINEAR TIME

Now we explore the other of our two interpretative models: that level I could be said to be in nondirected linear time. This perspective requires a different interpretive approach; and, while some of our interpretive decisions from the previous section will hold here, we will come to them from a new rationale.

First, let us explore the general ramifications of a linear level I. Does this process operate with, or separately from, that of level II? If the two levels work together, what is the goal of their mutual process? If they act independently, where are their respective goals? How do level I's elements relate to Berry's five "consistently relevant processes"? We must answer these questions before we are able to model our interpretation.

First, do the processes of levels I and II work together or individually? We recall that level II contains six sections, divided into two parts: the first, Intermezzi 1-3, primarily expository; and the second, Intermezzi 4-6, primarily developmental. The goal of the entire process is the sixth Intermezzo, which synthesizes the two contrasting ideas of extreme high and extreme low. Level I, then, as we deduced in our analysis from [section] 2.3, consists of a bipartite AA' design of nine sections each (Example 19). Moreover, this division was based on the recurrence of material: sections A-C' have certain adjacent subsections whose sound complexes, densities, intensities, and durations recur in adjacent subsections of corresponding sections in A'-B". (These subsections appear bold or underlined in Example 20.) Whereas both levels incorporate a division into halves, their processes are not the same. The goal of the process of level I is the modified restatement in the second half of the materials from the first. In level II, the goal of the process is the final, developmental Intermezzo. So, the processes work together only inasmuch as they serve to divide the work into halves.

Next, how do level I's elements relate to Berry's five "consistently relevant processes"? First, we must define a structural hierarchy. To perceive this form, it is necessary for a listener to apprehend a division into sections which is further divided into subsections. For the appropriate recurring subsections to be perceived, and ultimately grasped as the progenitors of overall form, they need to be treated as either expository statements or restatements. The subsections surrounding them, then, are introductory, transitional, or closing materials in each section. On a higher architectonic level, the eighteen sections also function in terms of these processes. In general, we may say that sections A, B, C, E, F, G, and H are expository statements; D is transitional; and C' is closing. The counterparts of these sections generally have the same functions, only as varied restatements instead of statements. (40)

Now we are ready to consider interpretive techniques. First, we wish to partition levels I and II, using ideas similar to those in the previous section. However, both levels should also mark the midpoint of the piece: measure 187. (41) Here, the performer might incorporate a slight rallentando at the end of the preceding level II material, and a purposeful quality to the ensuing level I material. In our previous interpretive model, the listener ultimately began to anticipate level II material during that of level I. Here, the performer might think more in terms of level II's material as being an interruption to that of level I (and vice versa): each level should maintain its own character.

The procedures for maintaining continuity in level II hold here as well. However, we should also note certain approaches to the articulation of a linear level I. In general, introductory material should have a quality of leading somewhere--either deliberately, by being some type of obvious structural upbeat, or more subtly, through stasis or some other means of expectation. Aspects of tempo, including rubato, could help to communicate such an interpretation. The performance of expository material should be deliberate and obvious, and possess a certain stability. In contrast, transitional bridging should have a sense of motion, and be unstable. Instability also occurs in developmental material, which, however, should not be fleeting; it should call attention to the processes which it explores. Last, one should present closing material with a sense of finality, inclined toward relatively stable and resoluble conditions.

In terms of Berry's questions which we examined above, we observe the following points with regard to this interpretation. The hidden recurrences in level I should be brought to the fore, and made as clear to the listener as possible, as they are ultimately the determinants of form. Fluctuations of dynamics and tempo may be used to help achieve a sense of direction in the non-expository sections, as long as they do not become obtrusive. Finally, it is of paramount importance that the performer consider his or her location in the formal process; the interpretive methods discussed above will help to articulate this cognizance to the listener.

3.3 CONCLUSIONS

Xenakis (1992, 178-9) observes that making music is a sonic expression of intelligence, incorporating not only pure logic, but also the "logic" of emotions and intuition. He states further that the technics of Formalized Music, including those used to compose Nomos alpha, possess a rigorous internal structure. This precision allows us to consider music universally, as did the ancients, equal to astronomy or mathematics. Nonetheless, this structure also contains "openings through which the most complex and mysterious factors of the intelligence may penetrate." Accordingly, in music, as in the sciences, our intellects and imaginations must interpret together to achieve a truly innovative result.

Such rigor has a basis in either determinism or chance, extremes united by philosophy and modern scientific thought. Interpretation, then, takes place in the continuum between these poles: while being neither wholly, it has properties of both. Our analysis of Nomos alpha demonstrates this aspect, as it leads to different interpretations, incorporating different degrees of determinism. Similarly, one might formulate other effective interpretations, following new lines of reason. Therefore, the task of the artist, as simultaneously a performer, analyst, and listener, is threefold: to establish a conception of the work, to execute this conception, and to test its validity.

In a piece of music such as Nomos alpha, one encounters several difficulties in this process. Berry's ideas, being concerned more with traditional Western art music, do not wholly apply to the issues arising in this piece. Similarly, Nomos alpha falls between categories in Kramer's study of time. Therefore, we must answer for ourselves questions such as those that follow, which we have answered provisionally in this study: What defines salience or accentuation where traditional means are not available? Which conformance (or non-conformance) is relevant to the design of the work? What is the time of Nomos alpha? Docs the [lambda] scheme of transformations destroy the continuity of level I; and if not, how do we project its structure? Of course, ours are not the only valid answers to these questions, but any artist preparing a performance of the piece should address them in some way, and, in doing so, begin to fulfill Xenakis's charge:
   All of a sudden it is unthinkable that the human mind forges its
   conception of time and space in childhood and never alters it. (42)
   Thus the bottom of the cave would not reflect the beings who are
   behind us, but would be a filtering glass that would allow us to
   guess at what is at the very heart of the universe. It is this
   bottom that must be broken up. Consequences: 1. It would be
   necessary to change the ordered structures of time and space, those
   of logic, ... 2. Art, and sciences annexed to it, should realize
   this mutation.

      The space ships that ambitious technology have produced may not
   carry us as far as liberation from our mental shackles could. This
   is the fantastic perspective that art-science opens to us in the
   Pythagorean-Parmenidean field. (Xenakis 1992, 241)

EXAMPLE 1: THE SECTIONAL DESIGN OF NOMOS ALPHA

Section:    1    2    3    4    5    6    7    8
Level:      I    I    I   II    I    I    I   II
Section:    9   10   11   12   13   14   15   16
Level:      I    I    I   II    I    I    I   II
Section:   17   18   19   20   21   22   23   24
Level:      I    I    I   II    I    I    I   II

EXAMPLE 5: THE ELEMENTS OF CUBE

Label        Order of Vertices

I                12345678
A                21436587
B                34127856
C                43218765
D                23146758
[D.sup.2]        31247568
E                24316875
[E.sup.2]        41328576
G                32417685
[G.sup.2]        42138657
L                13425786
[L.sup.2]        14235867
[Q.sub.1]        78653421
[Q.sub.2]        76583214
[Q.sub.3]        86754231
[Q.sub.4]        67852341
[Q.sub.5]        68572413
[Q.sub.6]        65782134
[Q.sub.7]        87564312
[Q.sub.8]        75863142
[Q.sub.9]        58761432
[Q.sub.10]       57681324
[Q.sub.ll]       85674123
[Q.sub.12]       56871243

EXAMPLE 6: THE ELEMENTS OF CUBE/[V.sub.1]

Elements of
CUBE/[V.sub.1]   Members of CUBE

[V.sub.1]        I, A, B, C
[V.sub.2]        D, [E.sup.2], G, [L.sup.2]
[V.sub.3]        [D.sup.2], E, [G.sup.2], L
[V.sub.4]        [Q.sub.1], [Q.sub.6], [Q.sub.7], [Q.sub.12]
[V.sub.5]        [Q.sub.3], [Q.sub.5], [Q.sub.8], [Q.sub.10]
[V.sub.6]        [Q.sub.2], [Q.sub.4], [Q.sub.9], [Q.sub.11]

EXAMPLE 7: INITIAL ASSOCIATION OF VERTICES AND SOUND COMPLEXES

Vertex                           Sound Complex

1                        Ataxic cloud of sound points
2              Relatively ordered ascending or descending cloud
                                of sound-points
3             Relatively ordered cloud of sound-points, neither
                           ascending nor descending
4           Ionized atom represented on a cello by interferences,
                           accompanied by pizzicati
5                       Ataxic field of sliding sounds
6              Relatively ordered ascending or descending field
                               of sliding sounds
7            Relatively ordered cloud of sliding sounds, neither
                           ascending nor descending
8        Atom represented on a cello by interferences of a quasi union

EXAMPLE 9: THE INITIAL VALUES OF D, G, AND U

                                             U = subsectional
Vertex   D = events/sec.   G = intensities      durations

1             0.5                mf                 2
2             0.5                fff               4.5
3             5.0                fff               4.5
4             5.0                mf                 2
5             1.08                f                2.62
6             1.08               ff                3.44
7             2.32               ff                3.44
8             2.32                f                2.62

EXAMPLE 10: THE SET H OF REGISTERS IN NOMOS ALPHA

               Normale   Harmonic
Register I     C3-F#3     A4-D5
Register II    F#3-C4     D5-G5
Register III   C4-F#4     G5-C6
Register IV    F#4-C5     C6-F6

EXAMPLE 11: THE SET X OF METHODS OF PLAYING

Vertices   Methods of Playing

1-3        pizzicato, col legno battuto, arco normale,
           pizzicato glissando

4-6        arco normale, arco tremolo, harmonic tremolo,
           arco sul ponticello, arco sul ponticello tremolo

7-8        arco normale, arco with interferences

EXAMPLE 16: THE ELEMENTS OF [lambda]

Label         Elements of [lambda]   Cyclic notation

[beta] (=I)        (12345678)              (1)
[gamma]            (15672348)         (25)(36)(47)
[alpha]            (17345628)             (27)

EXAMPLE 20: THE EIGHTEEN SECTIONS OF LEVEL I (BOLD NUMBERS
CORRESPOND HORIZONTALLY--OR DIAGONALLY, FOR SECTIONS B AND
B', AND C AND C"; UNDERLINED NUMBERS CORRESPOND VERTICALLY)

                                            Sections A-C'
             Section                               A
Member of [F.sub.1] under [Beta]    [Beta](D) = (23146758)
Member of [F.sub.2] under [Beta]    [Beta](D) = (23146758)
             Section                               B
Member of [F.sub.1] under [Beta]    [Beta]([Q.sub.12]) = (56871243)
Member of [F.sub.2] under [Beta]    [Beta]([Q.sub.3]) = (86754231)
             Section                               C
Member of [F.sub.1] under [Beta]    [Beta]([Q.sub.4]) = (67852341)
Member of [F.sub.2] under [Beta]    [Beta]([Q.sub.7]) = (87564312)
             Section                               D
Member of [F.sub.1] under [gamma]   [gamma](E) = (57613842)
Member of [F.sub.2] under [gamma]   [gamma](L) = (13425786)
             Section                               E
Member of [F.sub.1] under [gamma]   [gamma]([Q.sub.8]) = (42836175)
Member of [F.sub.2] under [gamma]   [gamma]([Q.sub.11]) = (85674123)
             Section                               F
Member of [F.sub.1] under [gamma]   [gamma]([Q.sub.2]) = (43286517)
Member of [F.sub.2] under [gamma]   [gamma]([Q.sub.6]) = (65782134)
             Section                               G
Member of [F.sub.1] under [alpha]   [alpha]([E.sup.2]) = (41378526)
Member of [F.sub.2] under [alpha]   [alpha]([L.sup.2]) = (14235867)
             Section                               H
Member of [F.sub.1] under [alpha]   [alpha]([Q.sub.7]) = (82564317)
Member of [F.sub.2] under [alpha]   [alpha]([Q.sub.5]) = (68572413)
             Section                               C'
Member of [F.sub.1] under [alpha]   [alpha]([Q.sub.4]) = (62857341)
Member of [F.sub.2] under [alpha]   [alpha]([Q.sub.7]) = (87564312)

             Section                        Sections A'-B"
Member of [F.sub.1] under [Beta]    [Beta]([D.sup.2]) = (31247568)
Member of [F.sub.2] under [Beta]    [Beta]([D.sup.2]) = (31247568)
             Section                               C"
Member of [F.sub.1] under [Beta]    [Beta]([Q.sub.3]) = (86754231)
Member of [F.sub.2] under [Beta]    [Beta]([Q.sub.9]) = (58761432)
             Section                               B'
Member of [F.sub.1] under [Beta]    [Beta]([Q.sub.11]) = (85674123)
Member of [F.sub.2] under [Beta]    [Beta]([Q.sub.1) = (78653421)
             Section                               I
Member of [F.sub.1] under [gamma]   [gamma]([L.sup.2]) = (17562834)
Member of [F.sub.2] under [gamma]   [gamma](G) = (32417685)
             Section                               E'
Member of [F.sub.1] under [gamma]   [gamma]([Q.sub.7]) = (84237615)
Member of [F.sub.2] under [gamma]   [gamma]([Q.sub.5]) = (68572413)
             Section                               F
Member of [F.sub.1] under [gamma]   [gamma]([Q.sub.2]) = (43286517)
Member of [F.sub.2] under [gamma]   [gamma]([Q.sub.6]) = (65782134)
             Section                               G'
Member of [F.sub.1] under [alpha]   [alpha](L) = (13475286)
Member of [F.sub.2] under [alpha]   [alpha]([G.sup.2]) = (42138657)
             Section                               H'
Member of [F.sub.1] under [alpha]   [alpha]([Q.sub.8]) = (25863147)
Member of [F.sub.2] under [alpha]   [alpha]([Q.sub.11]) = (85674123)
             Section                               B"
Member of [F.sub.1] under [alpha]   [alpha]([Q.sub.11]) = (85624173)
Member of [F.sub.2] under [alpha]   [alpha]([Q.sub.1]) = (78653421)


NOTES

(1.) Our account of Aristoxenus's writings is taken from Mathiesen (1999, 294-344). All translations from the Greek are by Mathiesen.

(2.) Presumably, Aristoxenus is referring here to the Eleatic school, which holds that the senses cannot conceive of the nature of True Being; and the Pythagorean school, which maintains that all things may be explained in numerical terms.

(3.) Xenakis (1992, 155-77). Chapter VI, "Symbolic Music," deals mainly with manifestations of set-theoretical principles in music. The application of group-theoretical concepts in Nomos alpha is an extension of these ideas.

(4.) This chapter also contains a similar analysis of a companion orchestral piece, Nomos gamma (1967-8).

(5.) Vriend (1981, 44). Vriend holds that, whereas they are not the primary cause, these inconsistencies and ambiguities contribute to the piece's discontinuity, hence its inability to be perceived as an expression of the group transformations.

(6.) As Griffiths (1975, 330) points out, "'sound logic' might have been a more logical analogical title."

(7.) We note, however, that in traditional accounts of musical form, the associative property may not hold. For example, ||: AB :|| C is different from A [parallel]: BC :[parallel].

(8.) Within the linear context of time, the distances between musical events map onto the equivalence classes of ordered-pair elements <a,b> of N x N, using the equivalence relation "has the same difference b-a." This contrasts with the more standard definition of distance, |b-a|.

(9.) The topic of differentiating among scale types is problematic. Vriend (1981, 60-5) points out several difficulties which face the listener in trying to perceive aurally a distinction among the twelve sieves used in the piece. In addition to incorporating notes outside the system, Xenakis does not provide us with a large enough sample of the pitches in each sieve to apprehend its highly complex structure (e.g., the first division omits thirteen of thirty-two pitches). This situation is especially problematic as the sieves' respective consecutive intervallic patterns change throughout their gamuts, and we are given pitches only in select registers. Thus, as we are unable to distinguish the sieves, we are clearly unable to perceive changes from one to another.

(10.) We might also say the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], sent the upper and lower boundaries of the frame.

(11.) Kramer (1988, 46-9) also gives a third category of nonlinear time: multiply-directed time. It achieves discontinuity by segmenting and reordering linear time. Subsumed within it is gestural time, in which gestures are out of alignment with absolute time. Since neither of these concepts is appropriate to Nomos alpha, we mention them here only in passing.

(12.) In fact, Kramer (1988, 207-8) demonstrates that, contrary to Stockhausen's original conception of moment form, return is in fact necessary; without it, the listener may assume a process of "constant newness."

(13.) Kramer (1988, 55) lists Xenakis's Bohor I (1962) as an example of vertical time.

(14.) The overall effect of level I is, on the other hand, one of stasis. We apprehend quickly a finite number of sound-states, and relinquish expecting there to be change from the system. However, primarily because it has more aspects of a complex division of the chronos (all rhythmized patterns in the permutational scheme divide it), rather than a simple division of the chronos (divided only by the protos chronos), we consider it to be in moment time, not vertical time.

(15.) Readers not familiar with the mathematical concepts presented in this section may wish to consult an abstract algebra text, such as Dean (1966), and a text which deals with crystallographic point groups, such as Coxeter and Moser (1965).

(16.) Xenakis's (1992, 220) Figure VIII-6 shows all twenty-four rotations. We favor the more mathematically standard term "octahedral" to the less standard "hexahedral" used by Xenakis et al. For more information on this group and its designation, see the references in Coxeter and Moser (1965).

(17.) A transposition, in the mathematical sense, not in the usual music-theoretical sense, is an exchange of two elements. A standard theorem of abstract algebra demonstrates that all permutations may be conceived of as the product of some number of translations. If this number is even, the permutation is said to be even, odd if it is odd. The subgroup of all even permutations in any symmetric group [S.sub.n] is the alternating group [A.sub.n].

(18.) Following Vandenbogaerde and Vriend, we use Xenakis's labeling of the paths B and [DELTA]. Elsewhere, Xenakis uses V1 and V2 for the same paths.

(19.) We note, however, that the individual elements of a normal subgroup do not necessarily commute with every member of the larger group.

(20.) A partition of a group is regular if, for any elements [x.sub.1] and [x.sub.2] of one class, and any elements [y.sub.1] and [y.sub.2] from another class, the elements [x.sub.1][y.sub.1] and [x.sub.2][y.sub.2] belong to the same class.

(21.) In right autography, the FIB series of transformations is as follows: a, ab, bah, abbab, bababbab, etc. I would like to thank Robert Morris for pointing out that if a is the identity, the FIB series becomes a cyclic group, and if the group is commutative, then it is an automorphism of a cyclic group.

(22.) For graphic representations of these cycles, see the figures in Xenakis (1992, 223) and Vandenbogaerde (1968, 44-5).

(23.) See Vriend (1981, 27-8) for further discussion on the rationale for the choice of these particular paths.

(24.) However, we would find some degenerate cycles, so the actual number of cycles is less than 128. For example, if c and d lie as adjacencies on a cycle generated by a and b, both ab and cd will induce the same cycle. Similarly, if two permutations a and b commute, ab will generate the same cycle as ba. The interested reader may wish to discover the number of distinct cycles, and perhaps uncover Xenakis's rationale for choosing the ones he uses in the piece.

(25.) We say "initially" because Xenakis changes certain of these associations during the course of the piece. See [sections] 2.2.1.

(26.) Xenakis discusses the terminology of Example 7 in Chapter II of Formalized Music, "Markovian Stochastic Music Theory." Essentially, "clouds" are systems of "grains," which, in our case, are "sound points," or musical events. The degree of probability used in predicting the emissions of grains determines the "ataxy," or entropy, of the cloud. A wholly predictable system has minimum ataxy, whereas a chaotic one has maximum. Between the two states are infinite gradations, in varying degrees of "relative ordering." (See Xenakis 1992, 63-78.)

(27.) The metronome marking of a [crotchet] = 75MM describes exactly ten sixteenth notes per second.

(28.) The diagram appears in Formalized Music on page 228, third row, second column.

(29.) For more discussion of the deviations in this scheme, see Vriend (1981, 80, n. 20).

(30.) Vandenbogaerde (1968, 47) states that kinematic diagrams also determine the rest distributions and glissando slopes in Nomos alpha, according to the Golden Section. However, he gives no examples, nor does Xenakis indicate this application in his writings.

(31.) Vriend (1981, 30-1) makes the case that there is no logical association between the eight vertices of a cube and the eight sound-states. A cube is a solid object; its vertices are in a fixed orientation to one another. However, no such device may be said to fix sound-states to one another. Thus, even if we were to perceive the logic of the succession of cube rotations, it has no particular relevance for musical ordering.

(32.) We note, however, that the cycle ([beta] [gamma] [alpha]) is not a cyclic group.

(33.) We recall that [lambda] does not affect the Intermezzi.

(34.) The insertion of rests adds yet another level of difficulty to the comprehension of the structure.

(35.) The thesis defense took place on 18 May 1976 at the University of Paris-Sorbonne. The members of the jury were Bernard Teyssedre (presiding), Olivier Messiaen, Michel Ragon, Olivier Revault d'Allones (thesis director and advisor), and Michel Serres. Xenakis was awarded the degree with Very Honorable mention.

(36.) The actual number of permutations given by 12! is 479,001,600.

(37.) Before Xenakis answers the question, he and the committee discuss what it means to know something intimately, and the necessity of such knowledge in using it.

(38.) The questions we address are, in Berry's order, 2, 7, 3, 8, 4, 5 and 6. The remaining questions are not of particular relevance to Nomos alpha.

(39.) Level II does not contain any significant recurrences, so we omit it from this discussion.

(40.) Since level I ultimately forms a loop, we might say abstractly that its first nine sections are also a varied restatement of the second nine. However, it is unlikely that this logic is perceptible in audible terms.

(41.) This interpretive strategy is further substantiated be DeLio (1980), who gives several reasons for considering the piece to be divided into halves.

(42.) Here, Xenakis is referring to the ideas of Jean Piaget on the development of notions of time and space in children.

REFERENCES

Berry, Wallace. 1976. Structural Functions in Music. Englewood Cliffs, New Jersey: Prentice-Hall.

--. 1989. Musical Structure and Performance. New Haven: Yale University Press.

Cone, Edward T. 1968. Musical Form and Musical Performance. New York: W. W. Norton.

Coxeter, Harold Scott McDonald, and William O.J. Moser. 1965. Generators and Relations for Discrete Groups. 2d ed. Berlin: Springer.

Dean, Richard A. 1966. Elements of Abstract Algebra. New York: John Wiley and Sons.

DeLio, Thomas. 1980. "Iannis Xenakis's Nomos Alpha: The Dialectics of Structure and Materials." Journal of Music Theory 24, no. 1 (Spring): 63-95.

Fisher, George, and Judy Lochhead. 1993. "Analysis, Hearing, and Performance." Indiana Theory Review 14, no. 1 (Spring): 1-36.

Griffiths, Paul. 1975. "Xenakis: Logic and Disorder." Musical Times CXVI/1586 (April): 329-31.

Kramer, Jonathan D. 1988. The Time of Music: New Meanings, New Temporalities, New Listening Strategies. New York: Schirmer.

Lewin, David. 1986. "Music Theory, Phenomenology, and Modes of Perception." Music Perception 3, no. 4 (Summer): 327-92.

Mathiesen, Thomas J. 1999. Apollo's Lyre: Greek Music and Music Theory in Antiquity and the Middle Ages. Lincoln: University of Nebraska Press.

Meyer, Leonard B. 1956. Emotion and Meaning in Music. Chicago: University of Chicago Press.

--. 1973. Explaining Music: Essays and Explorations. Berkeley: University of California Press.

Piaget, Jean. 1946. Le developpement de la notion de temps chez l'enfant. Paris: Presses Universitaires de France.

--. 1948. La representation de l'espace chez l'enfant. Paris: Presses Universitaires de France.

Stockhausen, Karlheinz. 1963. "Momentform." In Texte zur elektronischen und instrumentalen Musik 1. Cologne: DuMont.

Vandenbogaerde, Fernand. 1968. "Analyse de Nomos Alpha de I. Xenakis." Mathematiques et sciences humaines 7, no. 24: 35-50.

Vriend, Jan. 1981. "Nomos Alpha for Violoncello Solo (Xenakis 1966): Analysis and Comments." Interface 10: 15-82.

Xenakis, Iannis. 1967. Nomos alpha. London: Boosey & Hawkes.

--. 1985. Arts/Sciences: Alloys: The Thesis Defense of Iannis Xenakis before Olivier Messiaen, Olivier Revault d' Allones, Michel Serres, and Bernard Teyssedre. Trans. Sharon E. Kanach. New York: Pendragon.

--. 1992. Formalized Music. Revised edition. Stuyvesant, New York: Pendragon.

ROBERT W. PECK is Assistant Professor of Music Theory at Louisiana State University. He is also an active composer and cellist.
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