Introduction and general view The complex mathematical procedures under the strict musical layers of Nomos Alpha

are an essential feature of this piece. To fully comprehend the compositional process behind the piece one should grasp a few basic rules pertaining Set Theory and the use of statistics; both, mathematical concepts. In what follows we will attempt to lay down those rules in a clear and concise manner. This assignment follows closely the thorough analysis of Xenakis work and life by Nouritza Matossian, whose book Xenakis is the most comprehensive and detailed account of the composer available to me. Traditionally the discipline of mathematics has been only vaguely associated with art music. If we disregard the physical properties of sound (to our own peril) we still can achieve a musically satisfying composition following rational procedures other than algebra or calculus; that goes almost without saying. It was not until serialism had established itself that, for the first time, a precise mathematical rationale could be deduced from the musical discourse; combinatory. Most of the common manipulations of the serial row belong to that chapter of calculus. And yet, the concern of some contemporary composers (think of Varse and crystallography, or the practical influence that the divinatory Chinese book of the I Ching had on Cage) with areas of knowledge traditionally alien to music, seems to have opened a radical new chapter in music history. And Xenakis belongs, without a doubt, to that chapter. Born in Braila, Romania, and reared in Greece, he first studied engineering and fought in the Greek resistance movement during World War II, (due to an injury he lost one eye). After the war he had to flee from Greece because of his political activities (he opposed British rule in Cyprus and, later on, the logistic support by the USA to the right wing government in Grecce). He went to France, where he lived for the rest of his life (he became a French citizen in 1965). He studied in Paris with the French composers Arthur Honegger, Darius Milhaud, and Olivier Messiaen. He became assistant to the French architect Le Corbusier in 1948-1960. As a member of

Le Corbusier's office he designed the Philips pavilion at the 1958 Brussels International Exhibitiona design derived in part from analogies with his orchestral work Metastaseis, which will be approached succinctly later on.

Music and architecture: the concept of Field According to Nouritza Matossian an identity of approach to architecture and music is the key to Xenakis creative awakening (philosopher and biographer of who extensive use will used in this assignment). That creative awakening can be seen, a posteriori, like a product of the intellectual and emotional tensions that had, since his early youth, keep occupied the mind of the French composer. His first major work, Metastaseis, characteristically shows Xenakis concern with events taking a time (rhythm) dimension in space. Drawing back on a dramatic anti-Nazi demonstration in Athens, Xenakis recalls the chanting of slogans like a gigantic rhythm, bursting into a chaos of sharp sounds when combat with the enemy broke; and then the silence. To capture the shape of this events musically, in other words, to give them a fourth time dimension, the composer could not avail himself of any traditional compositional method. Xenakis, self-taught but undaunted by his own limitations, sought to solve this problem by applying to the musical discourse the solutions and procedures that architecture (through mathematics) provided in dealing with the distribution of masses in space. He analysed complex sounds (people chanting, machine guns and screams, and so on) in order to extricate the particular characteristics (physical qualities) that identify each unique event; he concluded that, not the intrinsic qualities of sounds (pitch, timbre), but the characteristic distribution of vast numbers of events is what produced a unique, composite, living sound. What it is essential, then and this is our very first step in understanding Nomos Alpha, is that for Xenakis:

1. The polarity between architecture and music is no other than that of space and time; 2. That the individual characteristics of sound are not separately perceptible in dealing with mass events 3. That, whereas space is reversible (we can re-organize masses-events in space) time is irreversible. It passes. To characterize the experience of sounds, thus perceived, Xenakis drew on the notion, borrowed from Physics, of field: a region of space subject to force (electromagnetic, gravitational). This way, fields of sound might be created by varying the quantities and direction of the forces (dynamics, frequency, intensity, duration). This ideas were partially realized in Metastaseis, what is now considered to be his first step in initiating a new (according to Matossian) brutalism in music, which relied not on emotional values but on the display of technical resources in a totally new way. Metastaseis (transformations) is a continuous conversion of one state (or sound field) into another, where clear musical ideas occur, merge into unidentifiable states, and result in new phenomena. It involved the transformation of mathematical rules and proportions into compositional procedures.

Serialism and stochastic music Xenakis, still then a serialist composer, agreed with Boulez and Stockhausen that the Second Viennese School had overestimated the predominance of pitch. But he broadened his criticism to what he saw as serialism main weaknesses: 1. The series; proceeding from a linear category of thought that strings a finite number (traditionally the 12 tones) of objects together; ignoring other elements and their possible combinations;

2. Poliphonic structure; were the enormous complexity of the various tones in different registers prevents the hearer from following the crisscrossing of lines; this has, according to Xenakis, an effect of an irrational and fortuitous dispersal of notes across the whole range of the sound spectrum. Xenakis asked why in large-scale forms one permutation (or row) should be placed to the next, in the absence of a general logic emanating from the system itself (Boulez had said La Revue Musicale, 1952- that linking serial permutations one to another is the most nave but apparently most delicate of problems). While Stockhausen and Boulez developed new ways of solving the serialists problems Xenakis finally distanced himself. He introduced in his new work, Pithoprakta, the theory of probability (developed in the 50s to cope with some results of quantum physics and Einsteins theory of relativity). It is, in fact, an extension of sound fields in which a specific mean or average is calculated; the experience of this music goes hand in hand with the conception that in vast configurations the individual sound itself is not as important as the position in relations to other sounds in the field. Xenakis use the word stochastic to express the ideas of masses tending towards a mean or a goal such as a stable state. This amounts to construct the music from the principle of indeterminism, but, at the same time, provide it with a sense or development which is, largely, what the listener is expected to capture: the overall spatial form as it evolves.

A New Music History After the two major works already mentioned, Metastaseis and Pithoprakta, Xenakis continued working on a remarkable journey of self discovery that would take him from one theoretical landmark to another; to mention some, Analogiques (1959), for 9 instruments and electronic tape, Herma (1960-61), for solo piano, Eonta (196364) for piano and five brass instruments! Among many others. The chosen path

seemed to keep up to date with the most important developments in science; as if, through the genius of the composer, these developments could be translated into architectural sounds. It should be noted that, regardless of his dedication and seriousness, only this reason alone, the attempt to convey musically the empirical and theoretical advances in human knowledge, would be sufficient to grant Xenakis a place apart in music history. And yet, Xenakis was determined that this would not remain a mere curiosity. His concerned with technique, his effort to find ways to translate his spatial vision into sounds (through mathematics), made him, time and again, push the technical limits of the instruments with exacting demands on the players. Around 1960 Xenakis broke with Le Corbusier, the exceptional talent that had inspired Xenakis flights of architectural imagination. Dramatic as it was, the uncompromising Xenakis had, at last, made the difficult choice for him- between both disciplines; in his own words: it was too late to go back. I was sure of one thing; all I wanted to do was to compose, to think about the problems of music and to write about them but above all to compose. Not happy with the modern accounts of the origin of scales and pitch systems Xenakis turned to the earliest direct sources, the writings of Pythagoras and Aristoxenus. As Nouritza Matossian tells us, one was the account of a mathematician giving the additive method of calculating a scale, and the other a musician concerned with the physical subdivision of a string in ratio, the geometric (or exponential) method. In view of Xenakis this two systems these two systems had been confused giving rise to a muddle account of the scale system (!). as he himself said, they are both expressions of group structure with different operations (addition in one case, multiplication on the other), and thus, only formally compatible (not exchangeable). Later developments which limited music to a diatonic or chromatic level restrict the possibilities of the systems. Xenakis, in keeping with modern philosophy, attempted to go beyond the ambiguities of language to a more abstract, formal level.1

Accordingly, he divided music of the past in: 1. Outside-time: ancient music, relying on pitch scales and tetrachords (Byzantine and Gregorian); 2. In-time structures: Western music, subject to functional harmony and to a system of durations, intervals and successions (namely, polyphony that highly original invention of the barbarous and uncultivated Occident following the schism of the churches; Xenakis)2.

Nomos Alpha: Set Theory and Theory of Sieves In 1963 Xenakis visited the USA, guest to no other than Aaron Copland himself (who had personally invited him to the Summer Course in the Berkshire Music Centre, at Tanglewood). In the chose topics that summer we see the French composer treading with firm foot on what was to become (if it had not already done so) the formal backbone of his musical premises. It is worthwhile looking at it in a little detail: 1. Formal and axiomatic tendency in musical composition 2. Deterministic and probabilistic thought used in composition 3. Mathematical model for a stochastic (goal orientated) music 4. Mechanisation and computation of stochastic music 5. Theory of games and musical composition 6. Introduction of Set Theory and symbolic logic in musical composition 7. Some problems of electromagnetic music A) Theory of Sieves We reached that stage, in the formal development of Xenakis music, where all the previous theoretical ideas take a final, coherent form. What has been said so far, is a kind of introductory preamble to fully understand the enormous weight and articulate set of principles behind Nomos Alpha; the first major synthesis of these principles he called Theory of Sieves; a theoretical axiom which is a resultant of his aesthetic and formal views about music and the history of music in general.

In the Theory of Sieves, pitch and time are continuums which can be ordered in a rational manner as scales; if the spectrum of pitch is pictured as a sieve through which only the intervals of a semi-tone may pass, we obtain the traditional chromatic scale. Further steps (fifths, fourths, quarter tones) will provide us with all the known scales plus new ones (1,1,1/2, would give us a major tetrachord, and so on). Extending this principle to the time continuum we will obtain, literary, a time scale that functions as a formal structuring principle (with equal structuring value as, say, C major diatonic scale).

B) Set Theory The second synthetic axis, following the same rational ordering principle is the combination of other sound characteristics (excluding timbre) such as intensity, density (number of events per unit of time), duration as sets of classes whose member are particular values (in fact, articulate scales), as in a scale of intensity with 5 members ordered as: ff, f, mf, p, pp. A sound is the result of combining different members of these sets. The ways of combining fall, in Nomos Alpha, beyond the stochastic into the more comprehensive Group (set) Theory. Instead of collection of different strands (qualitative scales) we possess a framework in which all those different strands are comprehended and articulated around a few basic mathematical rules. For claritys sake, we will say that these rules order in which way the different elements are to be distributed and associated to each other (strictly speaking, addition, multiplication and the associative law that says that A . (B+C)=A . B+A . C). In Nomos Alpha Xenakis found a way of conveying the hierarchy that lies beyond large-scale massed textures of orchestral works (subsuming the probability and the stochastic in the process) . Through a historical coincidence, Xenakis hit upon the structure of crystals (the same principle Varse himself, personal acquaintance of the composer, had applied to the movement of sound masses). By mapping the different vertices of a cube (in this case) onto one another following the rules of Group Theory, Xenakis was able to develop a musical discourse that resulted in periodic repetition. To each vertices of the cube correspond a musical value, we combine a number of cubes following the strict rules of set theory (in other words, by associating different sets of musical values until we accomplish the 8 rotations that complete the cycle, all possible associative points).

Mapping a given macroscopic (events) cube, with its set values, onto another in-time cubes (whose vertices are dynamics, techniques col legno, arco, and so on).

In the last instance we provide those sets of values with a pitch. In Nomos Alpha Xenakis organized the pitches onto two sieves of a quarter-tone and three quarter-tone interval size.

We effect the rotations until all the elements are successfully combined with the different set of pitches.

The form of Nomos Alpha is perfectly symmetrical (a step beyond the stochastic); 24 sections played in consecutive order. Of these sections, 18 are constructed following what we have just said. In order to maintain continuity between these sections of such disparate character, another section, freely composed, is inserted every 3 sections:

These six free sections (shoes pitches are take from the three quarter-tone sieve) become pivots for the composition as a whole. The relentless structure of the process prevents, for instance, events that share a common structure from being grouped together, and only when sufficient time is allotted (through a rotation) to a specific event, say a rhythmic cell, a recognizable (or catchy) rhythm is heard; as always, density, speed, duration (vertices values) will shape these events in one form or another. Following Messiaens criticism of the Second Viennes School, in the sense that it confined serialism to pitch alone, Xenakis Nomos Alpha can be seen, according to Nouritza Matossian, as the unrecognised heir to a line of succession starting with the 12-tone composition.

Notes
1The first order or layer consists of the whole tone, defined as the amount by which the interval of the fifth exceeds the fourth, while the semi-tone and quarter-tone are simple subdivisions of it. The secondary order is derived by the rearrangements within an octave of the small intervals (whole tone and semi-tone) in tetrachords and other intervals, keys and modes which are obtained from the first order, which is basic. 2 The tonal organization that has resulted from venturing into polyphony and neglecting the ancients has leaned strongly, by virtue of its very nature, on the temporal category, and defined the hierarchies of its harmonic functions as the In-time categories. Outside-time is appreciably poorer, its harmonics being reduced to a single scale octaveThis degradation of the outside-time structures of music since late medieval times is perhaps the most characteristic fact about the evolution of Western European music and it has led to an unparallel excrescence of temporal and in-time structures. Xenakis. He saw Debussy and Messiaen (modes of limited transposition, non-retro gradable rhythms) as outside-time structures, while atonality as the reinstatement of the outside-time neutrality of the half-tone scale.

Understanding Nomos Alpha: Xenakis compositional practice in context Jorge Rodriguez