xenakis, iannis - towards a metamusic.pdf

7/28/2019 Xenakis, Iannis - Towards a Metamusic.pdf

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TOWARDS METAMUSICby IannisXenakisToday's technocrats and their followers treat music as a message which thecomposer (source) sends to a listener (receiver). In this way they believe thatthe solution to the problem of the nature of music and of the arts in general liesin formulae taken from information theory. Drawing up an account of bits orquanta of information transmitted and received would thus seem to providethem with 'objective' and scientific criteria of aesthetic value. Yet apart fromelementary statistical recipes this theory-which is valuable for technologicalcommunications-has proved incapable of giving the characteristics of aestheticvalue even for a simple melody of J. S. Bach. Identifications of music with

message, with communication and with language are schematisations whosetendency is towards absurdities and desiccations. Certain African tom-toms can-not be included in this criticism, but these are exceptional. Hazy music cannot beforced into too precise a theoretical mould. It may be possible later, whenpresent theories have been refined and new ones invented.The followers of information theory or of cybernetics represent one extreme.At the other end are the intuitionists, who maybe broadlydivided into two groups:(a) the first ('graphist') group exalts the graphic symbol above the music(its sound) and makes a kind of fetish of it. In this group it is the done thing notto write notes but rather any drawing whatever. The 'music' is judged accordingto the beauty of the drawing. Related to this is so-called 'aleatory' music; thisis no more than an abuse of language, for the true term would be the 'improvised'music our grandfathers knew. This group is ignorant of the fact that graphicalwriting, whether it be symbolic, as in traditional notation, geometric or numer-ical, should be no more than an image that is asfaithful as possible to all theinstructions the composer is giving to the orchestra or to the machine.1 Thisgroup is taking music outside itself.(b) the second is that which adds a spectacle, in the form of extra-musicalscenic action accompanying the musical performance. Such composers, influence-ed by the 'happenings' which express the confusion of certain American artists,take refuge in mimetics and disparate occurrences and thus betray their verylimited confidence in pure music. In fact they concede certain defeat for theirmusic in particular.These two groups share in common a romantic attitude. They believe inimmediate action and are very little concerned with its control by themind. But since musical action, unless it is to risk falling into trivial improvis-ation, imprecision and irresponsibility, imperiously demands reflection, thesegroups are in fact denying music and taking it outside itself.

LinearThoughtI shall not say, like Aristotle, that the golden mean is the right path, for inmusic-as in politics-the middle means compromise. The path to follow islucidity and keenness of critical thought, in other words action, reflection andself-transformation by the sounds alone. Thus when it serves music, as with alli Cf. Gravesanerlatter,No. 29, ed. H. Scherchen,Gravesano,Tessin,Switzerland.( 1970 by Iannis Xenakis (Englishtranslation 1970 by G.W. Hopkins)


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TOWARDS A METAMUSIChuman creative activity, scientific and mathematical thought should amalgamatedialectically with intuition. Man is one, indivisible and total. He thinks with hisbelly and feels with his mind. I would like to propose what, to my mind, coversthe term 'music'.

. Firstly, a sort of comportment necessary for whoever thinks it andmakes it.2. An individual pleroma, a realisation.3. A fixing in sound of imagined virtualities (cosmological, philosophical

arguments . . .)4. It is normative, in other words unconsciously it is a model for being orfor doing by sympathetic drive.5. It is catalytic: its mere presence permits internal psychic or mentaltransformations in the same way as the crystal ball of the hypnotist.6. It is the gratuitous play of a child.7. It is a mystical (but atheistic) asceticism. Consequently expressions ofsadness, joy, love and dramatic situations are only very limited particularinstances.

Musical syntax has undergone considerable upheavals and today it seems thatinnumerable possibilities coexist in a state of chaos-a plethora of theories, of(sometimes) individual styles, of more or less ancient 'schools'. But how doesone make music? What can be communicated by oral teaching? (A burningquestion, if one is to reform musical education-a reform that is necessarythroughout the world.)It cannot be said that the informationists or the cyberneticians-and stillless the intuitionists-have posed the question of an ideological purge of thedross accumulated over the centuries and by present-day developments too. Ingeneral, they all remain ignorant of the substratum upon which they found thistheory or that action. Yet this substratum exists, and it is what will allow us toestablish for the first time an axiomatic system, and to bring to light a formalis-ation which will thus unify the ancient past, the present and the future; moreoverit will do so on a planetary scale, in other words embracing the still separateuniverses of sound in Asia, Africa, etc.In i9g4,2 I denounced linear thought (polyphony), and demonstrated thecontradictions of serial music. In its place I proposed a world of sound-masses,vast groups of sound-events, clouds and galaxies governed by new character-istics such as density, degree of order, and rate of change, which requireddefinitions and calculations using probability theory. Thus was born stochasticmusic. In fact this new mass-conception with large numbers was more generalthan linear polyphony, for it could embrace it as a particular instance (by re-ducing the density of the clouds). General harmony? No, not yet.Today, after more than ten years, these ideas and the realisations whichaccompany them have been round the world, and the exploration seems to beclosed for all intents and purposes. However the tempered diatonic system-ourmusical 'terra firma', upon which all our music is founded-seems not to havebeen breached either by reflection or by music itself.3 This is where the nextstage will come. The exploration and transformations of this system will herald2 Cf. GravesanerBlatter, Nos. i & 6, and the scores of Metastaseis(1954) and Pithoprakta(1956) (Boosey &Hawkes) recorded on Chant du Monde LDX A 8368, and Cardinal. See also sketches, designs and texts in my five-record album by Erato, Paris (i969). Pithopraktais also recorded on Nonesuch H 7I201.3 We are not here concerned with quarter-tone and sixth-tone music as currently practised, for these remainwithin the same tonal-diatonic field.

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a new and immensely promising era. In order to understand its determinativeimportance we must look at its pre-Christian origins and at its subsequentdevelopment. Thus I shall point out the structure of the music of Ancient Greece,and then that of Byzantine music which has best preserved it while developing it,and has done so with greater fidelity than its sister, occidental plainchant. Afterhaving demonstrated their abstract logical construction in a modern way I shalltry to express in a simple but universal mathematical and logical language whatwas and what might be valid in time (transverse musicology) and in space(comparative musicology).In order to do this I propose to make a distinction in musical architecturebetween 'outside-time'4 architectures or categories, 'in-time' architectures orcategories, and finally architectures or categories I shall call 'temporal. A givenpitch scale, for example, is an outside-time architecture, for no 'horizontal' or'vertical' combination of its elements can alter it. The event in itself, that isits actual occurrence, belongs to the temporal category. Finally a melody or achord on a given scale is produced by relating the outside-time category to thetemporal category. Both are realisations in-time of outside-time construct-tions. I have already dealt with this distinction elsewhere but here I shall showhow ancient and Byzantine music can be analysed with the aid of these categories,and how general this approach is since it permits both a universal axiomatisationand a formalisation of many aspects of all the music of our planet.Structureof AncientMusicOriginally Gregorian chant was founded on the structure of ancient music,pace Combarieu and the others who accused Hucbald of being behind the times.The rapid evolution of the music of Western Europe after the ninth centurysimplified and smoothed out plainchant, and theory was left behind by practice.But shreds of the ancient theory can still be found in secular music of the fifteenthand sixteenth centuries, witness the TerminorumMusicaeDffinitoriumof JohannisTinctoris.5 In order to look at antiquity the practice has been to look throughthe prism of Gregorian chant and its 'modes', which have long ceased to be under-stood. We are only beginning to glimpse other directions in which the modes ofplainchant can be explained. Nowadays the specialists are saying that the modesare not in fact proto-scales, but are rather characterised by melodic formulae.To the best of my knowledge only Jacques Chailley6 has introduced other con-cepts complementary to that of the scale, and he would seem to be correct. Ibelieve we can go further and affirm that ancient music, at least up to the firstcenturies of Christianity, was not based at all on scales and 'modes' related tothe octave, but on tetrachords and 'systems'.

Experts on ancient music (with the above exception) have ignored this fund-amental reality, clouded as their minds have been by the tonal construction ofpost-medieval music. However, this is what the Greeks used in their music: ahierarchic structure whose complexity proceeded by successive 'nesting', and byinclusions and intersections from the particular to the general; we can trace itsessential outline if we follow the writings of Aristoxenos.

4 Cf. My book MusiquesFormelles,ch.V (Richard-Masse, Paris).5 Johannis Tinctoris, TerminorumMusicaeDifflnitorium(Richard-Masse, Paris).6 'Le mythe des modes grecs', Jacques Chailley, in Acta Musicologicavol. XXVIII, fasc. IV, I956 (BarenreiterVerlag, Basel).7 Aristoxenosvon Tarent, Melik und Rhythmik,R.Westphal, Leipzig, I893, (Verlag von Ambr. Abel (ArthurMeiner) German introduction, Greek text.)

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TOWARDS A METAMUSICA. The primaryorder consists of the tone and its subdivisions. The wholetone is defined as the amount by which the interval of a fifth (the pentachord, ordia pente) exceeds the interval of a fourth (the tetrachord, or dia tessaron). Thetone is divided into halves, called semitones; hirds called chromaticdieseis; and

quarters, the extremely small enharmonicdieseis. No interval smaller than thequarter-tone was used.

B. The secondaryorder consists of the tetrachord. It is bounded by theinterval of the dia tessaron,which is equal to two and a half tones, or thirtytwelfth-tones, which we shall call Aristoxenian segments. The two outer notesalways maintain the same interval, the fourth, while the two inner notes aremobile. The positions of the inner notes determine the three genera of thetetrachord (the intervals of the fifth and the octave play no part in it). The pos-ition of the notes in the tetrachord are always counted from the lowest note up:I. The enharmonic genus contains two enharmonic dieseis, or 3 + 3 + 24=30 segments. If X equals the value of a tone, we can express the enharmonicas X114.X1/4.X2 - X5/2II. The chromatic genus consists of three types:(a) soft, containing two chromatic dieseis,4 + 4 + 22 = 30 segments, or Xl/3.X1/3.X1/3 + 3/2 = X5/2;

(b) hemiola (sesquialtera), containing two sesquialter dieseis,4.5 + 4-5 + 21 = 30 segments, or X(3/2)(1/4). X(3/2)(1/4). X7/4 - X5/2;(c) 'toniaion', consisting of two semitones and a trihemitone,6 + 6 + 18 = 30 segments, or X1/2. X1/2. X3/2 = X5/2;III. The diatonic consists of:

(a) soft, containing asemitone then three enharmonic dieseis then five enharmonicdieseis, 6 + 9 + 15= 30 segments, or X1/2. X3/4. X5/4 = X5/2;(b) syntonon, containing a semitone, a whole tone, and another whole tone,6 + 12 + 12 = 30 segments, or X'I2.X.X. = X5S2.

C. The tertiary rder,or the system, is'essentiallya combination of elements ofthe first two-tones and tetrachords, either conjunctival or separated by a tone.Thus we get the pentachord (outer interval the perfect fifth) and the octochord(outer interval the octave, sometimes perfect). The subdivisions of the systemfollow exactly those of the tetrachord. They are also a function of connexity andof consonance.D. The quaternaryrderconsists of the tropes, the keys, or the modes, whichwere probably just particularisations of the systems, derived by means of cadential,

melodic, dominant, registral and other formulae, as in Byzantinemusic, ragas, etc.This accounts for the 'outside-time' structure of Hellenic music. AfterAristoxenos all the ancient texts one can consult on this matter give this samehierarchical procedure. Seemingly Aristoxenos was used as a model. But later,traditions parallel to Aristoxenos, defective interpretations and sediments dis-torted this hierarchy, even in ancient times. Moreover, it seems that theoreticianslike Quintilian and Ptolemy had but little acquaintance with music.This hierarchical 'tree' was completed by transition algorithms-the


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TEMPOmetabolae- from one genus to another, from one system to another or from onemode to another. This is a far cry from the simple modulations or transpositionsof post-medieval tonal music.Pentachords are subdivided into the same genera as the tetrachord theycontain. They are derived from tetrachords but nonetheless are used as primaryconcepts on the same footing as the tetrachord in order to define the interval of atone. This vicious circle is explicable by Aristoxenos's determination to remainfaithful to musical experience (on which he insists), which alone defines thestructure of tetrachords and of the entire harmonic edifice which results com-binatorially from them. His whole axiomatic system proceeds from there and histext is an example of a method to be followed. Yet the absolute (physical) valueof the interval dia tessaron is left undefined, whereas the Pythagoreans defined itby the ratio 3 : 4 of the lengths of the strings. I believe this to be a sign of Aristox-enos's wisdom; the ratio 3 : 4 could in fact be a mean value.TwoLanguagesAttention must be drawn to the fact that he makes use of the additive oper-ation for the intervals, thus foreshadowing logarithms before their time; thiscontrasts with the practice of the Pythagoreans who used the geometrical(exponential) language which is multiplicative. Here, the method of Aristoxenosis fundamental since:

(a) it constitutes one of the two ways in which musical theory has beenexpressed over the millennia;(b) by using addition it institutes a means of calculation that is more econ-omical, simpler, and better suited to music;(c) it lays the foundation of the tempered scale nearly twenty centuriesbefore it came to be applied in Western Europe.The two languages: arithmetic (operating by addition) and geometric(derived from the ratios of string lengths, and operating by multiplication) have,over the centuries, always intermingled and interpenetrated so as to create spuri-ous and multiple confusions in the reckoning of intervals and of consonances, andconsequently in theories. In fact they are both expressions of group structure,having two non-identical operations; thus they have a formal equivalence.8A hare-brained notion that has been complacently repeated by musicolog-ical specialists in recent times runs as follows: 'It is said that the Greeks haddescending scales instead of ascending ones as is usual today.' Yet there is notrace of this either in Aristoxenos or in his followers, including Quintilian9 andAlypios, who give a new and fuller version of the steps of many tropes. On thecontrary, the ancient writers always begin their theoretical explanations andnomenclature of the steps from the bottom. Another hare-brained notion is thesupposed 'Aristoxenian scale', of which no trace is to be found in his text.?1Structureof ByzantineMusicNow we shall look at the structure of Byzantine music. This can contributeto an infinitely better understanding of ancient music, occidental plainchant,

8 Mathematiques y G. Th. Guilbaud, vol.I, Presses Universitaires de France, 1963.9 Aristidou Kointilianou, Peri MousikesProton, Teubner, Leipzig, 1963.o The Aristoxenian cale appears to be one of the experimental versions of the ancient diatonic which corres-ponds to the theoretical versions neither of the Pythagoreans nor of the Aristoxenians: X.9/8 . 9/8 = 4/3,6 + 12 + I 2 = 30 segments respectively. Archytas's version (X.7/8 . 9/8 = 4/3) or that of Euclid are signific-ant. On the other hand the so-called 'Zarlinian scale' is none other than the so-called 'Aristoxenian scale',which in fact goes back only as far as Ptolemy and Didymos.

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TOWARDS A METAMUSICnon-European musical traditions and the dialectics of recent European musicwith its wrong turnings and dead ends. We can also use it to foresee and build thefuture, having in view the distant horizons of the past as well as the electronicfuture. Thus new researches can be undertaken with a completely valid sense ofdirection. In contrast the deficiencies of serial music in certain domains and thedamage it has done to musical evolution by its ignorant dogmatism will beindirectly exposed.Byzantine music amalgamates the Pythagoreanand Aristoxenian, the multip-licative and additive means of calculation.1l The fourth is expressed by theratio 3 : 4 of the monochord, or by the 30 tempered segments (72 to the octave).'2It contains three kinds of tone: the major (9/8 or 12 segments), the minor (10/9or 10 segments) and the minimal (16/15 or 8 segments). But smaller and largerintervals are constructed and the elementary units of the primary order are morecomplex than in Aristoxenos. Byzantine music gives a preponderant role to thenatural diatonic scale (the supposed Aristoxenian scale) whose degrees have thefollowing ratios to the first note: 1, 9/8, 5/4, 4/3, 27/16, 15/8, 2 (in segments 0,12, 22, 30, 42, 54, 64, 72; or 0, 12, 23, 30, 42, 54, 65, 72). The degrees of thisscale bear the alphabetical names A, B, r, A, E, Z, and H. A is the lowest noteand corresponds roughly to G2. This scale was propounded at least as far back asthe firstcentury by Didymos, and in the second century by Ptolemy, who permutedone term and recorded the shift of the tetrachord (tone-tone-semitone),which has remained unchanged ever since. 13 But apart from this dia pason octaveattraction, the musical architecture is hierarchical and 'nested' as in Aristox-enos, as follows:

A. The primaryorder s based on the three tones, 9/8, 10/9, 16/15, a super-major tone 7/6, the trihemitone 6/5, another major tone 15/14, the semitone orleima 256/243, the apotome of the minor tone 135/128 and finally the comma81/80. This complexity results from the mixture of the two means of calculation.

B. The secondary rderconsists of the tetrachords, as defined in Aristoxenos,and similarly the pentachords and the octochords. The tetrachords are dividedinto three genera:I. Diatonic, subdivided into:first scheme,12 + 11 + 7 = 30 segments, or 9/8. 10/9. 16/15 = 4/3,starting on A, H, etc.;second scheme,11+7+12 = 30 segments, or 10/9. 16/15 . 9/8 = 4/3,starting on E, A, etc.;third scheme,

7 + 12 + 11 = 30 segments, or 16/15 .9/8 . 10/9 = 4/3,starting on Z, etc.Here we notice a developed combinatorial method that is not evident inAristoxenos; only three of the six possible permutations of the three notes areused.II Stichiodi Mathimata Byzantinis Ekklisiastikis Mousikis, Avraam Evthimiadis, O.X.A. Apostoliki Diakonia,Thessaloniki, I948.12 In Quintilian and Ptolemy the fourth is divided into 60 equal tempered segments.3 In Westphal op.cit. pp. XLVII etc. we have the formation of the tetrachord mentioned by Ptolemy: lichanos(16/1I) - mese (9/8) - paramese (Io/9) - trite (harmonics 2, I p.49)

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TEMPOII. Chromatic, subdivided into :14

(a) soft chromatic derived from the diatonic tetrachords of the first scheme,7 + 16 + 7 = 30 segments, or (16/15) (7/6)(15/14) = (4/3),starting on A, H, etc.;(b) syntonon, or hard chromatic derived from the diatonic tetrachords of the sec-ond scheme,5 + 19 + 6 = 30 segments, or (256/243) (6/5)(135/128) = 4/3,starting on E, A, etc.

III. Enharmonic, derived from the diatonic by alteration of the mobile notesand subdivided into:first scheme,12 + 12 + 6 = 30 segments, or (9/8)(9/8) (256/243) = 4/3,starting on Z, H, F, etc.;second scheme,12 + 6 + 12 = 30 segments, or (9/8)(256/243)(9/8) = 4/3,starting on A, H, A, etc.;third scheme,6 + 12 + 12 = 30 segments, or (256/234)(9/8)(9/8) = 4/3,starting on E, A, B, etc.PARENTHESIS

We can see a phenomenon of absorption of the ancient enharmonic by thediatonic. This must have taken place during the first centuries of Christianity, inthe struggle of the Church Fathers against paganism and certain of its manifest-ations in the arts. The diatonic had always been considered sober, severe andnoble, unlike the other types. In fact the chromatic genus, but still more theenharmonic, demanded a more advanced musical culture, as Aristoxenos and theother theoreticians had already pointed out, and such a culture was even scarceramong the masses of the Roman period. Consequently combinatorial speculationson the one hand and practical usage on the other must have caused the specificcharacteristics of the enharmonic to disappear in favour of the chromatic, a sub-division of which fell away in Byzantine music, and of the syntonon diatonic. Thisphenomenon of absorption is comparable to that of the scales (or modes) of theRenaissance by the major diatonic scale, which perpetuates the ancient syntonondiatonic.

However, this simplification is curious and it would be interesting to studythe exact circumstances and causes. Apart from differences, or rather variants ofancient intervals, Byzantine typology is strictly built upon the ancient. It buildsup the next stage with tetrachords, using definitions which throw unique lighton the theory of the Aristoxenian system; this is already expounded in some detailby Ptolemy.15The ScalesC. The tertiaryorder consists of the scales constructed with the help ofsystems having the same ancient rules of consonance, dissonance and assonance14 In Ptolemy the names of the chromatic tetrachords were reversed: the soft chromatic contained theinterval 6/9 and the hard or syntonon the interval 7/6, cf. Westphal, op.cit. p. XXXII.i Examples in Westphal, op.cit. p. XLVIII, selidion I, mixture of syntonon chromatic (22 : 21, 12: II,7 : 6) and toniaion diatonic (28 : 27, 8 : 7, 9 : 8); selidion 2, mixture of soft diatonic (2 : 2o, o1 : 9, 8 : 7)and toniaion diatonic (28 : 27, 8 : 7, 9: 8); etc.

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TOWARDS A METAMUSIC(paraphonia). In Byzantine music the principle of iteration and juxtaposition ofthe systems leads very clearly to scales, a development which is still fairly obscurein Aristoxenos and his successors, except for Ptolemy. Aristoxenos seems to haveseen the system as a category and end in itself, and the concept of the scale didnot emerge independently from the method which gave rise to it. In Byzantinemusic, on the other hand, the system was called a method of constructing scales.It is a sort of iterative operator which starts from the lower category of tetrachordsand their derivatives, the pentachord and the octochord, and builds up a chain ofmore complex organisms in the manner of biologic evolution based on genes.From this point of view, system-scale oupling reached a stage of fulfilment thathad been unknown in ancient times. The Byzantines defined the system as thesimple or multiple repetition of two, several or all the notes of a scale. By scaleis meant here a succession of notes that is already organised,

such as the tetra-chord or its derivatives. Three systems are used in Byzantine music:the octochord or dia pasonthe pentachord or wheel (trochos)the tetrachord or triphonyThe system can unite elements by conjunct (synimenon) or disjunct (dia-zeugmenon) juxtaposition. The disjunct juxtaposition of two tetrachords onetone apart forms the dia pason scale spanning a perfect octave. The conjunctjuxtaposition of several of these perfect octave dia pasons leads to the scales andmodes with which we are familiar. The conjunct juxtaposition of several tetra-chords (triphony) produces a scale in which the octave is no longer a fixed soundin the tetrachord but one of its mobile sounds. The same applies to the conjunctjuxtaposition of several pentachords (trochos).The system can be applied to the three genera of tetrachords and to eachof their subdivisions, thus creating a very rich collection of scales. Finally one mayeven mix the genera of tetrachords in the same scale (as in the selidia of Ptolemy),which will result in a vast variety. Thus the scale order is the product of a com-binatorial method-indeed, of a gigantic montage (harmony)-by iterativejuxtapositions of organisms that are already strongly differentiated, the tetra-chords and their derivatives. The scale as it is defined here is of a richer andmore universal conception than all the impoverished conceptions of medieval andmodern times. From this point of view, it is not the tempered scale so much asthe absorption by the diatonic tetrachord (and its corresponding scale) of all theother combinations or montages (harmonies) of the other tetrachords, that rep-resents a vast loss of a potential. (The diatonic scale is derived from a disjunctsystem of two diatonic tetrachords separated by a whole tone, and is representedby the white keys on the piano.) It is this potential that we are seeking here toreinstate, albeit in a modern way, as will be seen. The following are examples ofscales in segments of Byzantine tempering (or Aristoxenian, since the perfectfourth is equal to 30 segments):

Diatonic scales:Diatonic tetrachords;system by disjunct tetrachords,12, 11, 7; 12; 11, 7, 12,starting on the lower A,

12, 11,7; 12; 12,11,7,starting on the lower H or A;

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system by tetrachord and pentachord,7,12, 11; 7, 12, 11,starting on the lower Z;wheel system (trochos),11,7,12,12; 11,7,12,12; 11,7, 12, 12; etc.

Chromatic cales:Soft chromatic tetrachords; wheel system starting on H,7, 16, 7, 12; 7, 16, 7, 12; 7, 16, 7, 12; etc.Enharmonicscales:Enharmonic tetrachords, second scheme; system by disjunct tetrachords,

starting on A,12, 6, 12; 12; 12, 6, 12,corresponding to the white-note mode on D. The enharmonic scales produced bythe disjunct system form all the ecclesiastical scales or modes of the West, andothers too, for example: chromatic tetrachord, first scheme, by the triphonic

system, starting on low H:12, 12, 6; 12, 12, 6; 12, 12, 6; 12, 12, 6;Mixedscales:Diatonic tetrachords, first scheme + soft chromatic; disjunct system,

starting on low H,12, 11, 7; 12; 7, 16, 7;Strong chromatic tetrachords + soft chromatic; disjunct system, startingon low H, 5, 19, 6; 12; 7, 16, 7; etc.All the montages are not used. And one can observe the phenomenon ofthe absorption of imperfect octaves by the perfect octave by virtue of the basicrules of consonance. This is a limiting condition.

D. The quaternary rderconsists of the tropes or echoi (ichi). The echos isdefined by:(a) the genera of tetrachords (or derivatives) constituting it;(b) the system of juxtaposition;(c) the attractions;(d) the bases or fundamental notes;(e) the dominant notes;

(f) the termini or cadences (katalixis);(g) the apichima, or melodies introducing the mode;(h) the ethos, which follows ancient definitions.We shall not concern ourselves with the details of this quaternary order.Thus we have succinctly expounded our analysis of the outside-timestructure of Byzantine music.

ThemetabolaeBut this structure could not be satisfied with a compartmentalised hierarchy.It was necessary to have a free circulation between the notes and their subdiv-isions, between the kinds of tetrachords, between the genera, between the

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TOWARDS A METAMUSICsystems and between the echoi. Hence the need for a sketch of the 'in-time'structure, which we will now look at briefly. There exist operative signs whichallow alterations, transpositions, modulations and other transformations (meta-bolae). These signs are the phthorai and the chorai of notes, tetrachords, systems(or scales) and echoi.

Notemetabolae:(a) the metathesis: transition from a tetrachord of 30 segments (perfectfourth) to another tetrachord of 30 segments;(b) the parachordi :distortion of the interval corresponding to the 30 seg-ments of a tetrachord into a larger interval and vice versa; or again, transitinofrom one distorted tetrachord to another distorted tetrachord.Genusmetabolae:(a) phthora characteristic of the genus, not changing note names;(b) changing note names;(c) using the parachordi;(d) using the chorai.Systemmetabolae:Transition from one system to another using the above metabolae.Echos metabolae using special signs, the martyrikai phthorai or alterations ofthe mode initialisation.Because of the complexity of the metabolae, pedal points (isokratima)cannot be 'trusted to the ignorant'. Isokratima constitutes an art in itself, for itsfunction is to emphasize and pick out all the 'in-time' fluctuations of the outside-time structure that marks the music.

First CommentsIt can be seen that the consummation of this outside-time structure is themost complex and most refined thing that could be invented by monody. Whatcould not be developed in polyphony has been brought to such luxuriant fruitionthat to become familiar with it requires years of practical studies such as thosepursued by the vocalists and instrumentalists of the high cultures of Asia. Itseems, however, that none of the specialists in Byzantine music recognizes theimportance of this structure. It would appear that the interpreting of ancientsystems of notation has claimed their attention to such an extent that they haveignored the living tradition of the Byzantine Church and have put their names toincorrect assertions. Thus it was only a few years ago that one of them16 tookthe line of the Gregorian specialists in attributing to the echoi characteristicsother than those of the oriental scales which had been taught in the conformistschools. They have finally discovered that the echoi contained certain character-istic melodic formulae, though of a sedimentary nature. But they have not beenable or willing to go further and abandon their soft refuge among the manuscripts.Incomprehension of ancient music,17 of both Byzantine and Gregorianorigin, is doubtless due to the blindness resulting from the growth of polyphony,a highly original invention of the barbarous and uncultivated Occident followingx6 Egon Wellesz, A Historyof ByzantineMusicand Hymnography,O.U.P., I961, p.7I etc. On page 70 he too re-peats the myth of the ancients' descending scales.17 The same neglect can be found in the antiquising Hellenists, such as Louis Laloy in the classic Aristoxecmde Tarente, 1904, p.249 etc.

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TEMPOthe schism of the churches. The passing of centuries and the disappearanceofthe Byzantine state have sanctioned this neglect and this severance. Thus theeffort to feel a 'harmonic' language that is much more refined and complex thanthat of the syntonon diatonic and its scales in octaves is doubtless beyond thenormal capacities of a Western specialist even though the music of our own daymay have been able to (partly) liberate him from the overwhelming dominanceof diatonic thinking. The only exceptions are the specialists in the music of theFar East,18who have always remained in close contact with musical practice and,dealing as they were with living music, have been able to look for a harmonyother than the tonal harmony with twelve semitones. The height of error is to befound in the transcriptions of Byzantine melodies into19 Western notation usingthe tempered system. Thus, thousands of transcribed melodies are completelywrong! But the real criticism one must level at the Byzantinists is that in remain-ing aloof from the great musical tradition of the eastern church they have ignoredthe existence of this abstract and sensual architecture, both complex and remark-ably interlocking (harmonious), this remnant of and genuine achievement of theHellenic tradition. In this way they have retarded the progress of musicologicalresearch in the areas of:

(a) antiquity;(b) plainchant;(c) folk-music of European lands, notably in the East;20(d) musical cultures of the civilisations of other continents;(e) better understanding of the musical evolution of Western Europe fromthe middle ages up to the modern period;(f)the syntactical prospects for tomorrow's music, its enrichment, andits survival.

SecondCommentsI am motivated to present this architecture, which is linked to antiquity anddoubtless to other cultures, because it is an elegant and lively witness to what Ihave tried to define as an outside-time category, algebra, or structure of music,as opposed to its other two categories, in-time and temporal. It has often beensaid (by Stravinsky, Messiaen, and others) that in music time is everything. Thosewho express this view forget the basic structures on which personal languages,such as 'pre- or post-Webernian' serial music, rest, however simplified they maybe. In order to understand the universal past and present, as well as prepare thefuture, it is necessary to distinguish structures, architectures, and sound organismsfrom their temporal manifestations. It is therefore necessary to take 'snapshots',to make a series of veritable tomographies over time, to compare them andbring to light their relations and architectures, and vice versa. In addition,thanks to the metrical nature of time, one can furnish it too with an outside-timestructure, leaving its true, unadorned nature, that of immediate reality, ofinstantaneous becoming, in the final analysis, to the temporal category alone.8 Alain Danielou went to live in India for a number of years and learnt to play Hindu instruments. The samegoes for Mantle Hood with Indonesian music, and we should also mention Tan Van Khe, theoretician and com-poser practising traditional Vietnamese music, etc.19 Cf. the author cited in fn. 6, also the transcriptions by C. Hoeg, another eminent Byzantinist to have ig-nored the problems of structure, etc.20 The 'specialists' were surprised to discover Byzantine script in the notation of Rumanian folk-music; seeRapportsComplementairesu Xlle CongresInternationaldes EtudesByzantines,Ochride, 196I, p.76. Doubtless thesespecialists do not realise that an identical phenomenon exists in Greece.

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TOWARDS A METAMUSICIn this way, time could be considered as a blank blackboard, on which

symbols and relationships, architectures and abstract organisms are inscribed.The clash between organisms and architectures and instantaneous immediatereality gives rise to the primordial quality of the living consciousness.The architectures of Greece and Byzantium are concerned with the pitches(the dominant character of the simple sound) of sound-entities. Here rhythms arealso subjected to an organization, but a much simpler one. Therefore we shallnot refer to it. Certainly these ancient and Byzantine models cannot serve asexamples to be imitated or copied, but rather to exhibit a fundamental outside-time architecture which has been thwarted by the temporal architectures ofmodern (post-medieval) polyphonic music. These systems, including those ofserial music, are still a somewhat confused magma of temporal and outside-timestructures, for no-one has yet thought of unravelling them. However we cannotdo this here.ProgressiveDegradationof Outside-Time tructuresThe tonal organization that has resulted from venturing into polyphony andneglecting the ancients has leaned strongly, by virtue of its very nature, on thetemporal category, and defined the hierarchies of its harmonic functions as thein-time category. Outside-time is appreciably poorer, its 'harmonics' beingreduced to a single octave scale (C major on the two bases C and A), correspond-ing to the syntonon diatonic of the Pythagorean tradition or to the Byzantineenharmonic scales based on two disjunct tetrachords of the first scheme (for C)and on two disjunct tetrachords of the second and third scheme (for A). Twometabolae have been preserved: that of transposition (shifting of the scale) andthat of modulation, which consists of transferring the base onto steps of the samescale. 'Another loss occurred with the adoption of the crude tempering of thesemitone, the twelfth root of two. The consonances have been enriched by theinterval of the third, which, until Debussy, had nearly ousted the traditionalperfect fourths and fifths. The final stage of the evolution, atonalism, preparedby the theory and music of the romantics at the end of the nineteenth and thebeginning of the twentieth centuries, practically abandoned all outside-timestructure. This was endorsed by its dogmatic suppression by the Viennese school,who accepted only the ultimate total time ordering of the tempered chromaticscale. Of the four forms of the series, only the inversion of the intervals is relatedto an outside-time structure. Naturally the loss was felt, consciously or not, andsymmetric relations between intervals were grafted onto the chromatic total inthe choice of the notes of the series, but these always remained in the in-timecategory. Since then the situation has barely changed in the music of the post-Webernians. This degradation of the outside-time structures of music since latemedieval times is perhaps the most characteristic fact about the evolution ofWestern European music, and it has led to an unparalleled excrescence of temp-oral and in-time structures. In this lies its originality and its contribution to theuniversal culture. But herein also lies its impoverishment, its loss of vitality, andalso an apparent risk of reaching an impasse. For as it has thus far developed,European music is ill-suited to providing the world with a field of expression ona planetary scale, as a universality, and risks isolating and severing itself fromhistorical necessities. We must open our eyes and try to build bridges towardsother cultures, as well as towards the immediate future of musical thought, beforewe perish suffocating from electronic technology, either at the instrumental level

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or at the level of composition by computers.Reintroductionof the Outside-Time tructurebyStochastics

By the introduction of the calculation of probability (stochastic music) thepresent small horizon of outside-time structures and asymmetries was completelyexplored and enclosed. But by the very fact of its introduction, stochastics gavean impetus to musical thought that carried it over this enclosure towards theclouds of sound events and towards the plasticity of large numbers articulatedstatistically. There was no longer any distinction between the vertical and thehorizontal, and the indeterminism of in-time structures made a dignified entryinto the musical edifice. And, to crown the Heraclitean dialectic, indeter-minism, by means of particular stochastic functions, took on colour andstructure, giving rise to generous possibilities of organisation. It was able toinclude in its scope determinism and, still somewhat vaguely, the outside-timestructures of the past. The categories outside-time, in-time, and temporal,unequally amalgamated in the history of music, have suddenly taken on all theirfundamental significance and for the first time can build a coherent and universalsynthesis in the past, present, and future. This is, I insist, not only a possibility,but even a direction having priority. But as yet we have not managed to proceedbeyond this stage. To do so we must add to our arsenal sharper tools, trenchantaxiomatics and formalisation.

Sieve TheoryIt is necessary to give an axiomatisation for the totally ordered structure(additive group structure = additive Aristoxenian structure) of the temperedchromatic scale.21 The axiomatics of the tempered chromatic scale is based onPeano's axiomatics of numbers:Preliminaryerms. O = the stop at the origin;1 n = a stop; n' = a stopresulting from elementary displacement of n; D = the set of values of the partic-ular sound characteristic (pitch, density, intensity, instant, speed, order . . .)The values are identical with the stops of the displacements.First propositions (axioms):I. Stop O is an element of D;2. If stop n is an element of D, then the new stop n' is an element of D;3. If stop n and m are elements of D, then the new stop n' and m' areidentical if, and only if, stops n and m are identical;4. If stop n is an element of D, it will be different from stop O at the origin;g. If elements belonging to D have a special property P, such that stop Oalso has it, and if, for every element n of D having this property the element n'has it also, all the elements of D will have the property P.We have just defined axiomatically a tempered chromatic scale not only ofpitch, but also of all the sound properties or characteristics referred to above inD (density, intensity . . .). Moreover, this abstract scale, as Bertrand Russellhas rightly observed a propos the axiomatics of numbers of Peano, has no unitarydisplacement that is either predetermined or related to an absolute size. Thusit may be constructed with tempered semitones, with Aristoxenian segments

2 Cf. My text to the record, published by Chant du Monde, LDX A 8368. See also GravesanerBlatter, No.29,La Revued'EsthetiqueNo.2-3-4, I969 (Librairie Klincksieck, Paris) and my book already cited.i A stop = a point in musical space-time (translator's note)

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TOWARDS A METAMUSIC(twelfth-tones), with the commas of Didymos (81/80), with quartertones, withwhole tones, thirds, fourths, fifths, octaves, etc., or with any other unit that isnot a factor of the perfect octave.Now let us define another equivalent scale based on this one but having aunitary displacement which is a multiple of the first. It can be expressed by theconcept of congruence modulo m.Definition: two integers x and n are said to be congruent modulo m when mis a factor of x - n. It may be expressed as follows:x - n (mod m)Thus, two integers are congruent modulo m when and only when they differ bya whole (positive or negative) multiple of m, e.g:4 19 (mod 5), -3 - 13 (mod 8), 14 - 0 (mod 7)Consequently, every integer is congruent modulo m with one and with only onevalue of n: n = (0, 1, 2..., m-2, m-l)

Of each of these numbers it is said that it forms a residual class modulo m;they are, in fact, the smallest non-negative residues modulo m. x - n (mod m)is thus equivalent to x = n + km where k is an integer,keZ = (0, 1, -1, 2, -2,...)

For a given n and for any keZ, the numbers x will belong by definition to theresidual class n modulo m. This class can be denoted mn.In order to grasp these ideas in terms of music, let us take the temperedsemitone of our present-day scale as the unit of displacement. To this we shall

again apply the above axiomatics, with say a value of 4 semitones (major third) asthe elementary displacement.22 We shall define a new chromatic scale. If thestop at the origin of the first scale is a D sharp the second scale will give us all themultiples of 4 semitones, in other words a 'scale' of major thirds: D sharp,G, B, D' sharp, G', B'; these are the notes of the first scale whose order-numbersare congruent with 0 modulo 4. They all belong to the residual class 0 modulo 4.The residual classes 1, 2, and 3 modulo 4 will use up all the notes of this chromatictotal. These classes may be represented in the following manner:

residual class 0 modulo 4 : 4residual class 1 modulo 4 : 4residual class 2 modulo 4 : 42residual class 3 modulo 4 : 43residual class 4 modulo 4 : 40etc.

Since we are dealing with a sieving of the basic scale (elementary displace-ment by one semitone), each residual class forms a sieve allowing certain elementsof the chromatic continuity to pass through. By extension the chromatic totalwill be represented as sieve 10. The scale of fourths will be given by sieve 5nin which n = 0, 1, 2, 3, 4. Every change of the index n will entail a transpositionof this gamut. Thus the Debussian whole-tone scale, 2n with n = 0, 1, has twotranspositions:22 Elementary displacements are like whole numbers among themselves, in other words they are defined asthe elements of one and the same axiomatics.

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TEMPO20 -> C, D, E, F sharp, G sharp, A sharp, C ...21 - C sharp, D sharp, F, G, A, B, C sharp...

Starting from these elementary sieves we can build more complex scales-all the scales we can imagine-with the help of the three operations of the Logicof Classes: union (disjunction) expressed as v, intersection (conjunction) ex-pressed as A , and complementation (negation) expressed as a bar inscribed overthe modulo of the sieve. Thus: ...

2o v 2 = chromatic total (also expressible as lo)20A 21 = no notes, or empty sieve, expressed as 0

2o = 21 and 2t = 20The major scale can be written as follows:

(32A 40) V (3iA41) V (32A 42) V (30A 43)By definition, this notation does not distinguish between all the modes onthe white notes of the piano, for what we are defining here is the scale; modesare the architectures founded on these scales. Thus the white-note mode D,

starting on D, will have the same notation as the C mode. But in order to dis-tinguish the modes it would be possible to introduce non-commutativity inthe logical expressions. On the other hand each of the 12 transpositions of thisscale will be a combination of the cyclic permutations of the indices of sievesmodulo 3 and 4. Thus the major scale transposed a semitone higher (shift to theright) will be written:

(3oA 41) V (32A 42) V (30A 43) V (31A 4o),and in general:

(3n+2A 4n) V (3n+l V 4n+) V (3n+2A 4n+2) (3nA 4n+3)where n can assume any value from 0 to 11, but reduced after the addition ofthe constant index of each of the sieves (moduli), modulo the corresponding sieve.The scale of D transposed onto C is written:

(3nA 4n) V (3n+ A 4n+1) V (3nA 4n+2) V (3n+2A 4n+3)

MusicologyNow let us change the basic unit (elementary displacement, ELD) of thesieves and use the quartertone. The major scale will be written:

(88A 3n+i) V (8n+2A 3n+2) V (8n+4A 3n+i) V (8n+6A 3n)with n = 0, 1, 2 . . . 23 (modulo 3 or 8). The same scale with still finersieving (one octave = 72 Aristoxenian segments) will be written:(8n A (9n V 9n+6)) V (8n+2 A (9n+3 V 9n+6)) V (8n+4 A 9n+3) V (8n+6 A (9n V 9n+3))with n = 0, 1, 2... 71 (modulo 8 or 9).

One of the mixed Byzantine scales, a disjunct system consisting of a hardchromatic tetrachord and a diatonic tetrachord, second scheme, separated bya major tone, is notated in Aristoxenian segments as: 5, 19, 6; 12; 11, 7, 12, andwill be transcribed logically as:(8n A (9n V 9n+6)) V (9n+6 A (8n+2 V 8n+4)) V (8n+5 A (9n+5 V 9n+8)) V (8n+6 A9n+3) with n = 0, 1, 2. . . 71 (modulo 8 or 9).


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TOWARDS A METAMUSICThe Raga Bhairavi of the Andara-Sampurna type (pentatonic ascending,

heptatonic descending)23 expressed in terms of an Aristoxenian basic sieve(comprising

an octave, of periodicity 72) will be written as:Pentatonic scale: (8nA (9n V 9n+3)) V (8n+2A (9n V 9n+6)) V (8n+6A 9n+3)Heptatonic scale: (8 A (9n V 9n+3)) V (8n+2 A (9n V 9n+6)) V (8n+4A (9n+4 V9n+6)) V (8n+6A (9n+3 V 9n+6)) with n = 0, 1, 2... 71 (modulo 8 or 9).

These two scales expressed in terms of a sieve having as its elementarydisplacement ELD the comma of Didymos, ELD = 81/80 (81/80 to the power55 8 = 2), thus having an octave periodicity of 56, will be written as:Pentatonic scale: (7 A (8n V 8n+6)) V (7n+2 A (8n+5 V 8n+7)) V (7n+5 A 8n+l)Heptatonic scale: (7nA (8n V 8n+6)) V (7n+2 A (8n+s V 8n+7)) V (7n+3 A 8n+3) V(7n+4A (8n+4 V 8n+6)) V (7n+5A 8n+1) for n = 0, 1, 2... 55 (modulo 7 or 8).We have just seen how the sieve theory allows us to express any scale interms of logical (hence mechanisable) functions, and thus to unify our study ofthe structures of superior orders with that of the total order. It can be useful inentirely new constructions. To this end let us imagine complex, non-octave-forming sieves.24 Let us take as our sieve unit a tempered quartertone. Anoctave contains 24 quartertones. Thus we have to construct a compound sievewith a periodicity other than 24 or a multiple of 24, thus a periodicity non-congruent with k.24 modulo 24 (for k = 0, 1, 2 . . .). An example would beany logical

function of the sieve of moduli 11 and 7 (periodicity 11 x 7 = 77: k.24),(lln V lln+1) A 7n+6This establishes an asymmetric distribution of the steps of the chromatic quarter-tone scale. One can even use a compound sieve which throws periodicity outsidethe limits of the audible area; for example, any logical function of moduli 17 and

18(f[17, 181),for 17 x 18 - 306> (11 x 24).SuprastructuresBy stochastics.One can applyto acompound sieve a stricter structure or simplyleave the choice of elements to a stochastic function. We shall obtain a statist-ical coloration of the chromatic total which has a higher level of complexity.

Using metabolae. We know that at every cyclic combination of the sieveindices (transpositions) and at every change in the modulo or moduli of thesieve (modulation) we obtain a metabola. As examples of metabolic transform-ations let us take the smallest residues that are prime to a positive number r;they will form an Abelian (commutative) group when the composition law forthese residues is defined as multiplication with reduction to the least positiveresidue with regard to r . For a numerical example let r = 18; the residues 1, 5,7, 11, 13, 17 are primes to it, and their products after reduction modulo 18 willremain within this group (closure). The finite commutative group they form canbe exemplified by the following fragment:

5 x 7 = 35; 35-18 = 17; 11 x 11 = 121;121 -(6 x 18)= 13, etc.Moduli 1, 7, 13 form a cyclic sub-group of order 3. The following is a logical ex-pression of the two sieves having moduli 5 and 13:23 Alain Danielou, Northern ndianMusic,Halcyon Press (Barnet),1954, Vol.x I,p.72.24 This correspondsperhaps o EdgarVarese'swish for what he callsa spirallingscale = cycle of fifthsnotrelatedto the octave. Thisinformation,sketchythough t unfortunatelys, wasgivenme by Odile Vivier.

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L (5,13) = (13n+4 V 13n+5 V 13n+7 13n9)A5n+1 V (5n+2 V 5n+4) A 1 3n+9 V 1 3n+6

One can imagine a transformation of moduli in pairs, starting from the Abeliangroup defined above. Thus the cinematic diagram (in-time) will beL(5,13) L(11,17)-L(7,11)- L(5,)1 L(5,5) -... ->L(5,13)

so as to return to the initial term (closure).25This sieve theory can be put into many kinds of architecture, so as to createincluded or successively intersecting classes, thus stages of increasing complex-ity; in other words orientations towards increased determinisms in selection,and in neighbouring topological textures.Subsequently we can put into practice in-time this veritable histologyof outside-time music by means of temporal functions, for instance by givingfunctions of change-of indices, moduli, or unitary displacement-in other wordsencased logical functions parametric with time.Sieve theory is very general and consequently is applicable to any othersound characteristics that may be provided with a totally ordered structure, suchas intensity, instants, density, degrees of order, speed, etc. I have already saidthis elsewhere, as in the axiomatics of sieves. But this method can equally be

applied to visual scales and to the optical arts of the future.Moreover in the immediate future we shall witness the exploration of thistheory and its widespread use with the help of computers, for it is entirelymechanisable. Then, in a subsequent stage, there will be a study of partially-ordered structures such as are to be found in the classification of timbres, forexample, by means of lattice or graph techniques.ConclusionI believe that music today could surpass itself by research into the outside-time category, which has been atrophied and dominated by the temporal category.Moreover this method can unify the expression of fundamental structures of allAsian, African, and Europeanmusic. It has a considerable advantage-its mechanis-ation-hence tests and models of all sorts can be fed into computers, which willeffect great progress in the musical sciences.In fact, what we are witnessing is an industrialisation of music which hasalready started, whether we like it or not. It already floods our ears in many pub-lic places, shops, radio, TV, and airlines, the world over. It permits a consump-tion of music on a fantastic scale, never before approached. But this music is ofthe lowest kind, made from a collection of outdated cliches from the dregs of themusical mind. Now it is not a matter of stopping this invasion, which, after all,increases participation in music, even if only passively. It is rather a question ofeffecting a qualitative conversion of this music by exercising a radical but con-structive critique of our ways of thinking and of making music. Only in this way,as I have tried to show in the present study, will the musician succeed in dom-inating and transforming this poison that is discharged into our ears, and only ifhe sets about it without further ado. But one must also envisage, and in the sameway, a radical conversion of musical education, from primary studies onwards,throughout the entire world (all national councils for music take note). Non-25 These latter structures have been used in Akrata( 964), for 16 wind instruments (recorded on NonesuchH 71201), in NomosAlpha, for solo cello (1965), recorded on H.M.V., ASD 2441, and in NomosGammaforlarge orchestra (I968).

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TWO WORKS BY BENJAMIN LEESWO WORKS BY BENJAMIN LEESdecimal systems and the logic of classes are already taught in certain countries,so why not their application to a new musical theory, such as is sketched out here?This translation of 'Towards a Metamusic' will form part of the English edition of Mr. Xenakis'sbook FormalisedMusic, to be published later this year by the Indiana University Press. It is printedhere by courtesy of the editor of La Nef, in which the original French text was published, and ofthe Indiana University Press.

T W O WORKS B YBENJAMINE E S

by Niall O'LoughlinThe two works are the Piano Concerto No. 2 of 1966 and the Symphony No.3of 1968, which, like Lees's earlier works, continue to showample evidence of thiscomposer's ability to extend his traditional vocabulary in new and meaningfulways, without compromising his hard-won, individual style. Both are written ina sharp and incisive style that is full of tension. Vigorous and irregular rhythmsthat leave the listener no chance to relax his concentration are combined withpungent harmonies that are always imagined with a careful ear for the largercontext. Large-scale form is controlled with dramatic insight, especially in thesymphony, which is a work that breaks much new ground in its approach tosymphonic form, while the concerto is a full-blooded and compact virtuoso work,of great character and imagination, with a sharp and incisive wit and a symphonictautness.Written for Gary Graffman, who gave the first performance in I968, withthe Boston Symphony Orchestra conducted by Erich Leinsdorf, the concerto,which plays for just over twenty-five minutes, is remarkably full of incident.Its basic ideas are used with great economy, and overall organization is such thatscarcely a bar fails to make its mark. A 'standard' orchestra is used: triple wood-wind and a large number of percussion instruments, all used very sparingly. Therelationship between the piano soloist and the orchestra is one of fruitful co-op-eration, rather than titanic opposition.The plan of the first movement is one of a kind that Lees has made his own,and is obviously indebted to sonataform. With his individual approach to tonality,Lees does not find it necessary to recapitulate his subsidiary material, as it fulfils adifferent purpose from that of the traditional 'second subject'. His music is toodramatic always to fall naturally into traditional moulds. In the case of this work,there are three main parts to the exposition, an extended development, mainly ofmelodic fragments, a cadenza which continues the development, and a restate-ment of the opening. This opening unison statement, mainly on the strings, con-tains some of the germs of the movement (Ex. i). Its immediate elaboration bythe piano sets the tone of the movement, with much use of percussive chordalwriting. The figure x forms the rhythmic backbone of this section and of much ofthe development. A violent tutti, marked 'fiero', which combines x in the upperparts with its augmented inversion in the bass, leads suddenly to more lyricalmusic, in which rising melodic lines have an affinity with part of the opening(Ex. 2). The melodic smoothness is offset by the irregular time-signatures, which

( 1970 by Niall O'Loughlin

decimal systems and the logic of classes are already taught in certain countries,so why not their application to a new musical theory, such as is sketched out here?This translation of 'Towards a Metamusic' will form part of the English edition of Mr. Xenakis'sbook FormalisedMusic, to be published later this year by the Indiana University Press. It is printedhere by courtesy of the editor of La Nef, in which the original French text was published, and ofthe Indiana University Press.

T W O WORKS B YBENJAMINE E S

by Niall O'LoughlinThe two works are the Piano Concerto No. 2 of 1966 and the Symphony No.3of 1968, which, like Lees's earlier works, continue to showample evidence of thiscomposer's ability to extend his traditional vocabulary in new and meaningfulways, without compromising his hard-won, individual style. Both are written ina sharp and incisive style that is full of tension. Vigorous and irregular rhythmsthat leave the listener no chance to relax his concentration are combined withpungent harmonies that are always imagined with a careful ear for the largercontext. Large-scale form is controlled with dramatic insight, especially in thesymphony, which is a work that breaks much new ground in its approach tosymphonic form, while the concerto is a full-blooded and compact virtuoso work,of great character and imagination, with a sharp and incisive wit and a symphonictautness.Written for Gary Graffman, who gave the first performance in I968, withthe Boston Symphony Orchestra conducted by Erich Leinsdorf, the concerto,which plays for just over twenty-five minutes, is remarkably full of incident.Its basic ideas are used with great economy, and overall organization is such thatscarcely a bar fails to make its mark. A 'standard' orchestra is used: triple wood-wind and a large number of percussion instruments, all used very sparingly. Therelationship between the piano soloist and the orchestra is one of fruitful co-op-eration, rather than titanic opposition.The plan of the first movement is one of a kind that Lees has made his own,and is obviously indebted to sonataform. With his individual approach to tonality,Lees does not find it necessary to recapitulate his subsidiary material, as it fulfils adifferent purpose from that of the traditional 'second subject'. His music is toodramatic always to fall naturally into traditional moulds. In the case of this work,there are three main parts to the exposition, an extended development, mainly ofmelodic fragments, a cadenza which continues the development, and a restate-ment of the opening. This opening unison statement, mainly on the strings, con-tains some of the germs of the movement (Ex. i). Its immediate elaboration bythe piano sets the tone of the movement, with much use of percussive chordalwriting. The figure x forms the rhythmic backbone of this section and of much ofthe development. A violent tutti, marked 'fiero', which combines x in the upperparts with its augmented inversion in the bass, leads suddenly to more lyricalmusic, in which rising melodic lines have an affinity with part of the opening(Ex. 2). The melodic smoothness is offset by the irregular time-signatures, which

( 1970 by Niall O'Loughlin

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