Xenakis: Logic and DisorderAuthor(s): Paul GriffithsSource: The Musical Times, Vol. 116, No. 1586 (Apr., 1975), pp. 329-331Published by: Musical Times Publications Ltd.Stable URL: http://www.jstor.org/stable/960327Accessed: 11/10/2009 10:27

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Like all the operas after 1749, however, Les Boreades lacks a prologue, and its overture is thus linked directly to the main action. It is in the Italianate three-movement form in which the majority of his later overtures are written. As there was no curtain at the Opera, the opening forest scene would be visible throughout, which, with the horn-calls and fanfares of the outer movements, prepares us for the entry of a hunting party led by Alphise. Rameau actually dovetails the end of the overture with the opening recitative: the last chord is marked with a slow diminuendo, and an autograph note suggests that Alphise may begin singing before the chord ends. The link with the overture does not end there; horn-calls interrupt the recitative at intervals throughout the first scene, leading to a clear allusion to the overture's final bars in the introduction to scene ii. Though Gluck was certainly aware of earlier attempts to link overture and opening scene,

Like all the operas after 1749, however, Les Boreades lacks a prologue, and its overture is thus linked directly to the main action. It is in the Italianate three-movement form in which the majority of his later overtures are written. As there was no curtain at the Opera, the opening forest scene would be visible throughout, which, with the horn-calls and fanfares of the outer movements, prepares us for the entry of a hunting party led by Alphise. Rameau actually dovetails the end of the overture with the opening recitative: the last chord is marked with a slow diminuendo, and an autograph note suggests that Alphise may begin singing before the chord ends. The link with the overture does not end there; horn-calls interrupt the recitative at intervals throughout the first scene, leading to a clear allusion to the overture's final bars in the introduction to scene ii. Though Gluck was certainly aware of earlier attempts to link overture and opening scene,

some of his supporters were not, for Castillon had to remind them, after the premiere of Iphigenie en Aulide (1774), that 'an injustice had been done [to Rameau] by bringing forward this idea as entirely new'.5

If it had appeared in the so-called Oeuvres completes (Durand), Les Boreades would never have been so unknown today. It is without doubt a work of very considerable stature, a worthy successor to the earlier tragedies and in no sense a rehash of stale ideas. It is sad that so important an opera should have had to wait 211 years for its first performance, and only slightly less for its first scholarly edition.6

5quoted by P.-M. Masson: 'Rameau and Wagner', MQ, xxv (1939), 475 6M. Terey-Smith: Jean-Philippe Rameau: Abaris ou les Boreades: a critical edition (Ph.D., University of Rochester, 1972); for this month's performance, John Eliot Gardiner has prepared his own edition.

some of his supporters were not, for Castillon had to remind them, after the premiere of Iphigenie en Aulide (1774), that 'an injustice had been done [to Rameau] by bringing forward this idea as entirely new'.5

If it had appeared in the so-called Oeuvres completes (Durand), Les Boreades would never have been so unknown today. It is without doubt a work of very considerable stature, a worthy successor to the earlier tragedies and in no sense a rehash of stale ideas. It is sad that so important an opera should have had to wait 211 years for its first performance, and only slightly less for its first scholarly edition.6

5quoted by P.-M. Masson: 'Rameau and Wagner', MQ, xxv (1939), 475 6M. Terey-Smith: Jean-Philippe Rameau: Abaris ou les Boreades: a critical edition (Ph.D., University of Rochester, 1972); for this month's performance, John Eliot Gardiner has prepared his own edition.

Xenakis: Logic and Disorder Paul Griffiths

Xenakis: Logic and Disorder Paul Griffiths

Most of the compositions by Xenakis mentioned below are to be performed in two English Bach Festival concerts: at the Purcell Room on April 20 ('ST/4-1, 080262' and 'Nomos alpha') and at the Festival Hall on April 25 ('Metastaseis', 'Nuits', 'Antikhthon', 'Linaia-agon' and 'Evryali').

'Linear polyphony destroys itself by its very complexity; what one hears is in reality nothing but a mass of notes in various registers.' The quotation, from Xenakis's criticism of post-1950 serialism,1 has relevance to a consideration of order (i.e. con- sistency with an arbitrary system) in his own music, for it raises a central problem: the composer's order may be different from that perceived by the listener, and/or, one might add, from that which the score- reader discerns or the performer attempts to elucidate. This confusion may be expected to be particularly great when the compositional system is unfamiliar to the observer (Schoenberg's avoidance of octaves in early serial works may be seen as an attempt to circumvent misunderstandings of this type) or when a high degree of disorder is composed into the music. In that Xenakis has been active in both devising new systems and composing with probabilities rather than certainties, his work may demonstrate some of the difficulties in these areas.

His first acknowledged composition, Metastaseis for orchestra (1953-4), at once shows a broaching of chaos; the new systems were to come later. Here there is some use of serial principles, though that, Xenakis later decided, was the work's 'fault'2 and the composition can be associated with the attack from which the above quotation is taken. His

11. Xenakis: Formalized Music (Bloomington and London, 1971), 8 2Iannis Xenakis: the Man and his Music (London, 1967), 16

Most of the compositions by Xenakis mentioned below are to be performed in two English Bach Festival concerts: at the Purcell Room on April 20 ('ST/4-1, 080262' and 'Nomos alpha') and at the Festival Hall on April 25 ('Metastaseis', 'Nuits', 'Antikhthon', 'Linaia-agon' and 'Evryali').

'Linear polyphony destroys itself by its very complexity; what one hears is in reality nothing but a mass of notes in various registers.' The quotation, from Xenakis's criticism of post-1950 serialism,1 has relevance to a consideration of order (i.e. con- sistency with an arbitrary system) in his own music, for it raises a central problem: the composer's order may be different from that perceived by the listener, and/or, one might add, from that which the score- reader discerns or the performer attempts to elucidate. This confusion may be expected to be particularly great when the compositional system is unfamiliar to the observer (Schoenberg's avoidance of octaves in early serial works may be seen as an attempt to circumvent misunderstandings of this type) or when a high degree of disorder is composed into the music. In that Xenakis has been active in both devising new systems and composing with probabilities rather than certainties, his work may demonstrate some of the difficulties in these areas.

His first acknowledged composition, Metastaseis for orchestra (1953-4), at once shows a broaching of chaos; the new systems were to come later. Here there is some use of serial principles, though that, Xenakis later decided, was the work's 'fault'2 and the composition can be associated with the attack from which the above quotation is taken. His

11. Xenakis: Formalized Music (Bloomington and London, 1971), 8 2Iannis Xenakis: the Man and his Music (London, 1967), 16

reaction to the contradiction he saw in serial music between method and result was a search for com- positional means that might define 'sound masses'- 'galaxies', 'clouds', 'nebulae' as he has often termed them-in a global manner, directing attention away from individual events. In Metastaseis he found an answer by forming sound analogues of 'ruled sur- faces' (surfaces in three dimensions generated by the continuous motion of a straight line): complexes of glissandos in which each instrument, from a body of 46 strings, describes a straight line inclined to those of the others, so that the resultant is a sweeping curve. This concentration on simple macroscopic shapes presented few problems of comprehension (if not, at first, of acceptance), but the technique was obviously a limited one.

Xenakis's next step was the application of the mathematics of probability, and in particular of the law of large numbers. In the form used by Xenakis this states that, as the number of observations of a process increases, the distribution of outcomes tends toward the distribution of their respective probabilities. (For example, as the number of times an unbiased coin is tossed increases, the results tend toward an equal distribution between heads and tails; note that the result of any particular toss is unpredictable.) Xenakis borrowed a term from this field of mathematics in naming his 'stochastic music', of which ST/4-1,080262 for string quartet (1956-62) is an example (the title indicates that this is STochastic music for four instruments, the first of its type, and developed from electronic computa- tions of 8 February 1962). Essentially the composi- tion of stochastic music entails the provision of limits for sound aspects (for example, a range of equal-tempered semitones) and of a formula to determine the probabilities of choices within those

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reaction to the contradiction he saw in serial music between method and result was a search for com- positional means that might define 'sound masses'- 'galaxies', 'clouds', 'nebulae' as he has often termed them-in a global manner, directing attention away from individual events. In Metastaseis he found an answer by forming sound analogues of 'ruled sur- faces' (surfaces in three dimensions generated by the continuous motion of a straight line): complexes of glissandos in which each instrument, from a body of 46 strings, describes a straight line inclined to those of the others, so that the resultant is a sweeping curve. This concentration on simple macroscopic shapes presented few problems of comprehension (if not, at first, of acceptance), but the technique was obviously a limited one.

Xenakis's next step was the application of the mathematics of probability, and in particular of the law of large numbers. In the form used by Xenakis this states that, as the number of observations of a process increases, the distribution of outcomes tends toward the distribution of their respective probabilities. (For example, as the number of times an unbiased coin is tossed increases, the results tend toward an equal distribution between heads and tails; note that the result of any particular toss is unpredictable.) Xenakis borrowed a term from this field of mathematics in naming his 'stochastic music', of which ST/4-1,080262 for string quartet (1956-62) is an example (the title indicates that this is STochastic music for four instruments, the first of its type, and developed from electronic computa- tions of 8 February 1962). Essentially the composi- tion of stochastic music entails the provision of limits for sound aspects (for example, a range of equal-tempered semitones) and of a formula to determine the probabilities of choices within those

329

limits; the calculations may be carried out manually or automatically. Xenakis's aim in developing stochastic music was 'to attain the greatest possible asymmetry (in the etymological sense) and the minimum of constraints, causalities and rules'. 3

Since it is precisely the presence of 'constraints, causalities and rules' that is responsible for order, the composer's ordering is restricted to the deter- mination of limits and formulae, which in Xenakis's stochastic music are highly accommodating. How- ever, Xenakis does retain the option that he may 'instil his own personality in the sonic result'4 after computation. There is another problem, which may be made apparent by considering the simple situa- tion of the observer of a sequence of coin tosses. Such an observer may well take more notice of small-scale indeterminate occurrences-for example, a succession of four heads-than of the determinate goal, the perception of which requires a feat of memory. And if he repeatedly observes the same sequence, on film as might be, he has no way of telling the indeterminate from the determinate: the sequence might have been put together in such a way that particular successions occur and not others. In the very much more complicated stochastic processes of a work such as ST/4-1,080262 the goals are a great deal more difficult to perceive, and so the listener is perhaps more likely to focus his attention on the small scale. Thus there is no reason to suppose that stochastic music will appear as disordered as its manner of composition might suggest. Equally, the order perceived by the listener is not necessarily that willed by the composer.

At the same time that he was developing stochastic music Xenakis was exploring the possibilities for musical analogy in another branch of the mathe- matics of indeterminacy: game theory. In his appli- cation two contestants reply in alternation to each other's tactics and are awarded points according to a scheme supplied by the composer. It is important to add that the tactics (passages of music) are chosen from those which the composer stipulates; the players are not free to invent new pieces for their game of musical chess. Xenakis's first compositions of this sort were orchestral (Strategie and Duel, 1959-62), with rival conductors directing their own groups; the only subsequent game work, Linaia-agon (1972), pits a solo trombonist against a duo of horn and tuba.

Thus from the early 1950s to the early 1960s Xenakis had been largely occupied with the intro- duction of disorder. Particular events were organized only in terms of probabilities in stochastic music (and the material for the tactics of the orchestral games was composed stochastically), or else they were submerged in 'sound masses' which could only be perceived as wholes. It is significant that the images Xenakis has used to describe his music have been drawn most frequently from mass phenomena: the behaviour of crowds of people, the distribution of molecular motions within a gas sample, the rocket bursts of a firework display. However, writing of such conceptions in 1967, he had come to feel that their 'exploration seems to be closed for all intents

3Xenakis: op cit, 23 4Xenakis: op cit, 144

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and purposes'.5 He had by then made his first essays in 'symbolic music', based on some of the ideas of and named after symbolic logic ('sound logic' might have been a more logical analogical title, but let that pass). This should not be confused with the attempts of Kassler6 to establish formalized languages proper to music; Xenakis's symbolic music has more the nature of a translation into sound of theorems of set theory.

In Herma for piano (1960-61), the first work in which this translation was put into practice, the sets were drawn from the pitches of the equal- tempered semitone scale, but Xenakis's theory7 can also be applied to other sound aspects. What the theory does not include, as Xenakis admits,8 is any analogue for signs of operation or relation, such as union (U), intersection (n) and equivalence (=). These are to be deduced by the listener. Thus if one hears a set S and then a set T and then a set con- taining all the pitches of S and Tand no others, one may deduce that the theorem stated is S U T = S U T. It is not difficult to see that the statement of more complex theorems is open to misunderstanding. This is to leave aside the question of whether the statements of theorems in this manner could be followed by a listener without a considerable degree of training; Butchers's argument9 that the method is valid because logic corresponds in some manner with 'the laws of thought' rests on an assumption that would not, perhaps, be upheld by many logicians. 1 However, in a later example of 'symbolic music', Nomos alpha for cello (1965-6), it is hard to (or more honestly, I cannot) find theorem translations in the manner of Herma, and so one must look elsewhere for the means of order.

Indeed, the most important (to his later works) result of Xenakis's experiences with set theory has been in his conception of 'outside-time structure'. By this he means a collection of values for one sound aspect, forming a sort of repertory for a composi- tion. Thus, for example, the outside-time pitch structure of a piece in C major consists of the pitches of the C major scale. He distinguishes from this the 'temporal structure', which is the rhythmic organiza- tion, and the 'in-time structure', which is the combination of the two.Il One of the criticisms he levels at western art music, from the invention of polyphony onwards, is that its range of outside-time structures is considerably impoverished by com- parison with that of ancient Greek, Byzantine or most folk musics. This is the area he has singled out for particular development in Nomos alpha and many later pieces.

It may prove worthwhile to describe how Xenakis generates an outside-time structure for Nomos alpha, since the presentation in his book12 is not ideally 5Xenakis: op cit, 182 6M. Kassler: 'A Sketch of the Use of Formalized Languages for the Assertion of Music', Perspectives of New Music, i/2 (1963), 83-94; and 'Toward a Theory that is the Twelve- note-class System', ibid, v/2 (1967), 1-80 7Xenakis: op cit, 155-76 8Xenakis: op cit, 172 9C. Butchers: 'The Random Arts: Xenakis, Mathematics and Music', Tempo (1968), no.85, p.3 10see, for example, A. J. Ayer: Language, Truth and Logic (London, 1936. 3/1971), 108 ilXenakis: op cit, 160-61 i2Xenakis: op cit, 230-32

clear and since the innovation is as significant as his more widely reported stochastic methods. First, the universe of discourse is the range of pitch with the equal-tempered quarter-tone as unit interval. The set in,, includes the pitch p, ti quarter-tones above an arbitrary origin, and all the pitches m quarter-tones above or below p. (For example, 244 with the origin at c would be the set of all Ds.) Other symbols have their conventional meanings. (For example, keeping the same origin, 244' is the set of all pitches which are not D, 244 U 2410 is the set of all Ds and Fs, 244 f 2410 is the set of all pitches that 244 and 2410 have in common, i.e. none in this case.) With these symbols it is possible to define any set of pitches in the universe of discourse; the reader may care to find the most economical notation for the pitches of a diatonic major scale. To generate sets which, like a diatonic major scale set, recur exactly every octave it is obviously necessary that all the values of S should be factors of 24. Xenakis has been more interested in producing sets which recur at non- octave intervals. For example, the outside-time structure of pitch in the first 45 bars of Nomos alpha is the union of the following two sets: ((133 U 135 U 137 U 139)' 12) U ((114 U 118)' U 139) with origin a ' and (130 U 13l U 136) with origin at'. (Note that there is a misprint in Xenakis's state- ment1 3 of the origin for the second subset.) Clearly the resultant set recurs every 143 quarter-tones, or about every six octaves. A segment from the set is shown in ex.1, where an asterisk indicates a pitch

Ex. 1

*9 t.* * - * '* _ * * * *

that is notated in the 45 bars under consideration. There may be doubts as to whether this outside-

time structure is audibly functional. The listener is not supplied with the whole repertory within the range of the music, and the lack of repeating pattern in the structure makes it difficult for him to be sure which are the pitches of it that are missing, while some pitches outside of the structure are heard as frequently as some of its members. Furthermore the music contains a large proportion of glissandos and other events in which pitch is uncertain. If there are doubts on this score, then the perceptibility of movements from one outside-time structure to another (and there are 12 such structures in the piece) must be in question. Even in this composition for a solo instrument, it would seem, the listener's attention is directed toward the general features of events, such as various types of tremolando, glis- sando or pizzicato, rather than to pitch and rhythmic aspects. In this regard the range of effects makes Nomos alpha an extraordinary virtuoso display.

That much is also true of Evryali for piano (1974). With the other more recent works, Nuits for 12 voices (1967-8) and Antikhthon for orchestra (1971), this represents an extension and development of the techniques which Xenakis introduced in the 1950s

13Xenakis: op cit, 232

and 1960s rather than an excursion into new areas. In Nuits, for example, one can find examples of the ill-defined 'clouds' which distinguish most of Xenakis's works, of the virtuoso use of a medium as in Nomos alpha, and of post-Herma symbolic music. Nuits is, however, exceptional, in that it is one of his very few vocal works; it was, indeed, his first not connected with an ancient Greek literary source (it is textless). Moreover the dedication, to political prisoners, and the exploiting of the emotive power of the voice make it a powerful defence, if it were needed, against the objection that Xenakis is simply a mathematician.

But it would be wrong to conclude that mathe- matics is not as intimately connected with music theory for Xenakis as it was for the Pythagoreans or Aristoxenos, a joint dedicatee of Nomos alpha. One of the main goals of his work has been the setting of music on a plane where 'it can be considered on the same level as the stars, the numbers, and the riches of the human brain, as it was in the great periods of the ancient civilizations'.14 Perhaps he has some- times overestimated those riches, perhaps he has sometimes been unsure about which particular riches he was drawing on. As 1 have suggested, the procedures about which Xenakis has been most forthcoming as a theorist may not be those that the listener finds most important to him in understand- ing the work of Xenakis the composer. Further study of the latter, and further performances, can only help in elucidating why it may be found to excite, to quote Xenakis quoting the Timaeus of Plato, 'a reasoned joy'. 14Xenakis: op cit, 178

English Bach Festival April 14- May 11

Two Bach recitals by

PETER HURFORD 'arguably the best Bach player in England'

Gramophone

Royal Festival Hall Sunday 20 April 3.15 p.m.

SCHWEITZER CENTENARY RECITAL

Passacaglia in C minor, 'Sei gegr[isset' Sonata in C major, Fantasia and Fugue in G minor

New College Chapel Oxford Saturday 3 May 3.00 p.m.

A TRIBUTE TO WALTER EMERY

Chorale preludes from the Orgelbuchlein, with sung chorales

Preludes and Fugues in G major and B minor

For details of all English Bach Festival concerts telephone 01-730 5925

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