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Musical form: From a model ofhearing to an analytic procedureJean‐Marc Chouvel a

a Associated researcher in the “Institut d'esthétique” ,University of PARIS I , 32, rue Sorbier, Paris, F ‐75 020,FrancePublished online: 03 Jun 2008.

To cite this article: Jean‐Marc Chouvel (1993) Musical form: From a model of hearing toan analytic procedure, Interface, 22:2, 99-117, DOI: 10.1080/09298219308570623

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Interface, Vol. 22 (1993), pp. 99-117 0303-3902/93/2202-0099 $25.00© Swets & Zeitlinger

Musical Form:From a Model of Hearing to an Analytic Procedure

Jean-Marc Chouvel

ABSTRACT

This text is an attempt to give a simple, efficient and sufficiently general definitionof a notion of musical form.It shows how an analytic procedure, also interpreted as a model of hearing, leads toa representation of musical works that makes intelligible their temporal structuresand gives a useful tool to the interpretation of composer's strategy of time.Examples are taken from the fugue number 16 in G minor from J.-S. Bach's firstbook of "Das Wohltemperierte Klavier", and the "Sonata Facile" in C, K.545, ofW.-A. Mozart. Those examples allow to illustrate the notions of discovery-frontand redundancy background in a very significant way. It is also shown how in thesame piece the temporal evolution can depend of the structure's levels involved.Important semantic conclusions are suggested.The end of the paper tries to give an illustrated definition of the relations in be-tween form and structure, and situate the notions of poi'etic and esthesic in thedynamic of the hierarchic representation introduced.

INTRODUCTION

The foundations of this musicological reflection rest on two main observations:firstly, in the plastic arts, if the notion of form can be derived from a mathemati-cal substratum such as geometry, in the field of music, no theory of time canprovide us with any help in the definition of form; secondly, time is difficult tothink about "in itself, and, all things considered, the art of the sounds attainsform only when it manages to inscribe itself in memory. Therefore we can speakof the living memory of the interpreter, the written memory of the score, theindividual memory of the listener. Obviously, these memories do not share eitherthe same medium and aim or the same degree of concreteness, but we are led tothe hypothesis that there is a coherence, at least for the composer, in between allthose levels of description; in other words, a composer writes in order to satisfyhis own idea of musical "perception". When we speak of composers as Bach orMozart, we can assume easily that something of this satisfaction is communi-cated. We will show that giving the elements of a theory of form based on time,it is possible to reconcile points of view usually regarded as having independentbearings on musical works. This unique and simple principle allows us to give areference to psychology and cognitive sciences, a particularly efficient analyticprocedure, and a viewpoint offering a broader perspective of the phenomena of

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music. The standpoint from which this theory is advanced might be more that ofthe analyst, and, while it may appear to frame the analyst within a perceptualperspective, it do not propose that it should bring to light what may be perceivedbut rather what may be there to be perceived.

After a short survey of most of the existing proposals for a description ofmusical structures, we will give the essential elements of such a principle. Wewill then see how it can be understood as a model of hearing, and we will test itsanalytical capacities on two famous works: the fugue number 16 in G minor fromJ.-S. Bach's first book of "Das Wohltemperierte Klavier", and the "sonata facile"in C, K.545, of W.-A. Mozart. We would like to make clear that the choice ofthose works has been done for didactic reasons, both because most of the readerswill already know these works, and because only a few novel observations con-cerning them might be expected. It by no means implies a limitation of themethod to classical or tonal music, especially as it has first been tested withcontemporary music.

HISTORICAL SURVEY

In order to provide some reference-point (even if it is impossible to be exhaustiveon so broad a subject) this chapter will try to give a summarised overview ofprevious attempts to represent or conceive musical structures. Most of them havelittle relation with what follows, either because of their specificity - which meansvery often a lack of neutrality towards the music involved - or because of theirvery low efficiency/complexity ratio, or because they do not take into accountthis fundamental idea that the elements of musical structures are ordered not onlyby spatial physical evidences, but also by time, or appearance order. What wewill call "formal diagram" has got no meaning if it is not conceived as "in time".

A review of the most important attempts to model musical structures has beenmade in West, Howell and Cross (1985). We will briefly summarise some oftheir conclusions: "A general model of musical structures must not depend on theanalytical capabilities and propensities of the model builder; perceptible dimen-sion of musical experience should be involved (...); any perceptible feature of asequence may play a role (...) a model must be able to frame musical structure interms of components and relationships that form part of the listener's compe-tence (...) it is necessary to model both horizontal an vertical structures, none ofthe models described have attempted to do this". They also remark that "listenersdo not necessarily perceive music in terms of a fully co-ordinated structure. Thestream of musical events may be left unstructured in places, and patterns de-tected may not be related coherently to each other". From the point of view ofthis paper, this is not taken to mean that psychology of perception does not needa reliable methodological reference giving the maximum possibility of determin-ing structure even if listeners' capacities are overestimated, or perhaps better, ifthis reference provides nothing more than a map wherein the listener finds hisown ways.

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MUSICAL FORM 101

Obviously, this reflection about form in music would probably not have beenpossible without the creative thought of I. Xenakis (1956, 1957, 1963). Specialattention must be paid to the works of Mesnage and Riotte (1988, 1989, 1991),especially concerning the notion of display ("deployment", Mesnage 1991). Re-lations have been suggested with proposals made by Simon (1972) and Clarke(1987).

In the field of the perception of form, the reader can refer to the works ofImberty (1981) and Deliege I. (1990-1991). Cognitive models for perceivingmusic have been proposed in Greussay (1985), in the spirit of artificial intelli-gence and Leman (1988) in a more "connectionist" way. A good summary of thepsychological models of memory can be find in Tiberghien (1991) where theACT (Adaptative Control of Thought) model and the auto-associative connexionnistmodel are presented and compared. The difference introduced in the ACT modelin between declarative memory and procedural memory - according to the tradi-tion of the computo-symbolic representation - seems to be very similar to thedistinction in between material and formal memory made here. However, in thecase of musical audition, a symbolic representation is not obvious to define. Thisdoes not mean that a sub-symbolic representation should be preferred, and per-haps will we be led to a "pre-symbolic" idea of music where each work definesits own processes of symbolisation.

Semiology may also provide some introduction to what follows, especially inRuwet (1972) and Nattiez (1975). Finally, general reflection about form in musiccan be find in Stoïanova (1978) and in Deliege C. (1988), for whom musical formis focused has a living experience through a phenomenological approach. In thefield of philosophy, Deleuze's work (1968) on "difference and repetition" can beconsidered of major interest in the discussion of the notions that follow.

THE NOTION OF MUSICAL FORM

When we speak about ABA form in music, we think in first place about thisidentity of two fragments across time, that allows the conclusive A to recognizethe introducing A as its equal, not dramatically, but intrinsically. We tend toforget that the first significant operation is to distinguish B from the initial A.Without this distinction, no form is possible, the come back of the A being afterall only one possibility among many.

Then we can say that there is, in ABA, two phenomena absolutely different:on one hand the distinction and possibly the recognition of two different ele-ments, A and B, on the other hand, the temporal distribution of those elements. Inother words, to the traditional notation ABA, it would be more convenient toprefer the one below:

A AB

But this written distinction is not sufficient if it is not taken to its logical

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extreme, that is to say, if the redundancy of A is remaining when its notation isno longer necessary as it had to be already there to be recognised in this actual-ised past named memory. We can then propose the following representation:

time

formalmemorised diagrammaterial

Fig. 1.

The axis of the "memorised material" could perhaps have been named "para-digmatic axis", according to the semiological vocabulary. It would not take intoaccount the idea that the representation of the "memorised material" is orderedaccording to the temporal rank of the events. This order is not necessarily anorder of availability of the memory, but the formal diagram can be a very helpfultool to define the "memory" distance and the "formal" distance, as well as someothers parameters such as redundancy. The formal diagram gives back to thetime its essential fabric, as time orders both its own stream and the discovery ofthe space of the "material". The "material" can be understood in a very broadway, aswe will see later. Discovery and repetition appear here, with the modula-tions of the temporal stream, as the two main elements of the formal dynamics.According to this observation, the formal diagram allows us to underline twofundamental directions of the form: what we will call the "discovery front" onone hand, and on the other, the "repetition background", as illustrated in thefollowing diagram:

time

repetition background

material

Fig. 2.

Those considerations are not indeed a specificity of music, and all arts basedon time, from the literature to the cinema, can be interpreted through thosecriteria. The music has the unique privilege of a relative independence of itsmaterial from its "meaning", independence that leads to a more natural transferof the meaning from the material to the structures.

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MUSICAL FORM 103

A MODEL OF HEARING

As well as providing a means for an effective visual representation of a temporalphenomenon, the intent of the foregoing is to coincide with an "in time" model ofauditory perception. As we already said, this is a model of listening while cer-tainly not being intended as a model of the listener. And, perhaps, first of allbecause the term "model" - that implies (e.g. Riotte 1988) the double constraintof economy and efficiency - is not pertinent when applied to an individual.Compared to a theoretical optimisation of a mechanism, a real listener will ap-pear at the same time less reliable and more imaginative.

The aim is not here to give a representation of the listener, but to understanda representation of the work. We can expect that even unconsciously, some ofthis ideal behaviour is represented in real perception, or at least that they sharethe same directionality, constrained by time. Moreover the way what follows canbe directly interpreted in the field of experimental psychology is unclear. Ofcourse tests on musical memory can be derived from it, and proposals have beenmade to the realisation of these, but it is impossible to obtain from a listener anamount of information that corresponds directly to the flow to witch he is listen-ing. It might even be more accurate to consider what follows as a proposal forlearning to listen analytically to music, as we can reasonably postulate thateducation has a determinant influence on our perception of musical forms.

The "model" fits within the following constraints: real time functioning, con-servation of the totality (or the greatest amount possible) of the information, andminimisation of the size of memory involved. We might imagine that those twolast constraints are contradictory. However, it is this contradiction that can helpto define an optimal solution to the problem. This one can be resumed briefly asfollows:

at each instant:hearing (acquisition) of the fragment;comparison(s) with the fragments already memorised;

if different: inscription in "material memory";in any case: inscription in "form memory".

Of course this is only a simplified algorithm, leaving undefined the words "in-stant" or "fragment", but it provides a way of taking into account the wholeassociative network generated by a work. Moreover, it is valid whatever the sizeand the nature of the fragments considered, even allowing to relate this size withan optimisation of the comparison criteria. It is worth noting that the problem ofvariation - which cannot be described with a dichotomous answer to the com-parison test, can be resolved by the introduction of the concept of "presencedensity", a concept itself corrected by perceptual thresholds of adhesion andmask-effects. Finally, we will see that the concept of material can resolve theproblem of its ontological definition through a recurrence of the intellectuallevels in a hierarchic interpretation of the work. The form is then something else

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104 JEAN-MARC CHOUVEL

than the abstract finalisation of the work, but a notion that actualises itself at anyinstant.

AN ANALYTIC PROCEDURE

If the former mechanism can appear to be far removed from the performance ofthe average listener, it is, on the contrary, perfectly suitable for the analystconfronted with a score. We will try to show, relying on two examples (Bach andMozart), first how what precedes applies, what is its particular interest for under-standing the formal strategy of a work and, furthermore, what relation it has withan idea of what "composing" means for its author.

We will first begin with an analysis of the first four bars of the fugue n° XVIof "Das Wohltemperierte Klavier". We will there encounter with some technicalproblems and then appreciate better the practical reality of the concepts we havepresented. The case of polyphony is in many ways distinct: firstly, because allanalytic methods that pretend to reduce the music to a linguistic linear chain areinapplicable to polyphonic textures, secondly because the basic material of thepolyphony, the melodic line, has the property of being in itself a function of thetime. Therefore, its representation in a formal diagram can be localised instead ofbeing treated globally as the A block in the ABA form we already studied,through a segment indicating a "bijective" relation in between the time of thework and that of the memory. Let us compare the two following documents: thebeginning of the partition and its transcription into a formal diagram: (Fig. 3,p.103).

The most evident feature of the formal diagram is its clarity compared to thatof the score. Two details need to be justified: the initial silence has not beentaken into account and the diagram has been reframed with a decay of a quaverrest; second order analogies, such as that between the beginning of the secondbar and its inversion in the third bar, are discriminated. This is an analyticalchoice, dealing with the idea of conserving the maximum quantity of informa-tion. In this case it is interesting to be able to visualise the utilisation of theinverted fragment in relation to the initial one, and to understand the formalimplication of each of them (e.g. : the superimposition and insistence of theinversion after the triple stretto in bar 30).

The information lost in the diagram is that of the tone situation of the melodicfragments, as it is assimilating the intervallic successions, that is to say thevalues derived from the pitch positions. To avoid this, the level of the transposi-tions can be written "over" the material memory or even on the formal diagramitself. All the others mutations can also be expressed in this way.

In fact, those precisions have little influence on the general aspect of theentire formal diagram and on the way it demonstrates the relations of a fragmentwith the whole work, both in the synchronie and the diachronic axis (both termsbeing understood in their primitive meaning and not in Saussure's perspective).The two following schemes intent to synthesise on one hand the enrichment of

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MUSICAL FORM 105

Fig. 3. The beginning of Bach's fugue in Gm from "Das Wohltemperierte Klavier" andits transcription into a formal diagram.

the fragment through its appearance in the work (Fig. 4) and on the other handthe enrichment of the work due to the fragment's role in it (Fig. 5).

All the useful elements of classical analysis are easy to read on the formaldiagrams: stretti, re-exposition, divertissements... Tonal analysis can also bedone according to those principles. We will now try to understand what, inaddition to thematic exhaustiveness, provides us with the material/time diagrams,

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106 JEAN-MARC CHOUVEL

roots

Fig.4.

DIACHRONY

enrichment by the•memory's resonance

restltuing pastexperiences

context

t

FRAGMENT(event)

SYNCHRONY

new situationenrichment byconfronting with theothers elements

context

DIACHRONY

superposition tothe impressions ofthe past andreactualisationof the memory

roots FRAGMENT(event)

SYNCHRONY

realisation ofthe contextparticipation tothe wholeactuation

Fig.5.

especially for it allows us to reach so easily the organisational thought of thecomposers.

REPETITION BACKGROUND AND DISCOVERY FRONT

If we have a look to the formal diagram of Bach's fugue in G minor (Fig. 6) thefirst conclusion that comes to the mind is that there is a tremendous redundancyof melodic material, redundancy mainly situated at the theme level or at the levelof the counter-subject motifs. The canon of the harmonic progression (bars 25 to27) is the only element that gives density to the discovery front, when this onereaches the last steps of its course, and as if the purpose was to emphasise the"formal distance" between this late discovery and the return of the theme withthe triple stretto that follows. We can find the same insistence on the. rupturebetween the ultimate and the initial in the first movement of the "Sonata Facile"

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Fig. 7. The formai diagram of Mozart 's Allegro of the "Sonata facile" in C, K.545. The second diagram takes into account the repetitions inthe score; it also shows how Mozart uses those repetitions to paradoxically avoid the repetitions in the formal sentence and increaseprogressively the temporal distance between two come backs of the theme.

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MUSICAL FORM 109

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K.545 of Mozart, insistence that is even accentuated by the different repeats. Butthe redundancy is not now situated on the level of an obsessive "repetition back-ground", but in a superficial embroidery of the "discovery front". In both cases,the curvature of the front means a saturation of the melodic material. We canwonder whether this saturation does not take into account the saturation of theperception. The formal diagrams deduced from the scores can be compared to anaverage "auditory performance", witch can be roughly defined as the ratio of theamount of information (material) that the listener can "process" in the unit oftime.

The curvature of the discovery front can take two directions in relation withthis average performance, if we except the case of a simple adaptation to its flow.Either, as we have just seen, it follows a reserve tactic, expecting that the combi-

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110 JEAN-MARC CHOUVEL

L S 10 15 ¿0 15 30

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Fig. 10. This diagram represents the way musical possibilities of composing melodicfragments are used by Bach throughout the fugue in G minor as a main recourse.A. voices accumulationB. harmonic modulationC. four voices polyphonyD. highest register of soprano's melodyE. most "dissonant" harmonic chordF. full quaver rhythmG. canon and harmonic progressionH. formal ruptureI. three voices strettoJ. conclusive cadence

natorial development, or the trance effect, will renew the interest, or it adopts aproliferation tactic, simulating an overflow of our perceptive capacities.

In fact, the two scores we have selected will provide us with examples ofmore subtle behaviours. First it would not be sufficient to limit music to onlymelodic means. To write a fugue, Bach has many others stylistic resources thanthematic accumulation alone. If we establish a list of such means, and if weconsider the apogee of their realisation in the score from the formal point ofview, we obtain a formal diagram oriented in a drastically distinct way from themelodic diagram (Fig. 10).

If we now consider the second movement of Mozart's sonata, we see that theAlberty bass, reduced to a simple line on the melodic diagram (Fig. 11), can beanalysed in terms of chords in the diagram of Figure 12. This elementary motif issupporting a harmonic evolution that overflows the melodic stream in successivewaves. It is also amazing to see how the structure of the first part, bars 1 to 8,gives a reduced image of the whole movement (Fig. 11 &12).

Both scores were written in the same century (1722 and 1788) within thesame tonal period. Nevertheless, they show fundamental differences of writingthat the formal analysis allows us to characterise with great precision. First, wecould imagine that there is an identity of project, something dealing with restric-tion and submersion, with the necessity of cohesion realised through a minimisa-tion of the objects used, and the intent of surpassing this limitation by an over-

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MUSICAL FORM 111

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flow in the other levels of the structure. Such a formal quality, such a compre-hension of musical possibilities, such an understanding of time, would certainlynot have been so obvious in any other work. We must remind ourselves that weare dealing with two exceptional composers, enamoured of a similar strictnesstowards music, but whose aesthetic motivations are quite different. For if Bachrealises a sublimated overflow, through formal means of higher level, providinga mastery of the melodic "body" by the musical "spirit", Mozart's formal dy-namic comes from below, it is the undercurrent of harmonic sensuality thatoverflows the melodic charm of the theme and becomes the main character of thescore's drama. The history of art could have told us such a thing in other words,but is it not remarkable to have been able to read it with such a precision in theworks themselves?

POLES OF THE MENTAL REPRESENTATION OF MUSICAL WORKS

It is obviously difficult to know to what point a composer is aware of its meansand what is the part of intuition in his work. In any case we should not be toonaive about this point, just as we probably underestimate the listener's ability to

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understand music. This should not be reduced to a mere segmentation, and evenif we can not assimilate hearing to the systematic "computerised" model wedescribed in the beginning, the essential movement of a work speaks to oursensibilities. The main vector of this comprehension is probably form. Besides,the word "form" should not represent only the fixed and posterior structure of themusic, but on the contrary, a network of links throughout time, related to thepleasure of memory and the dynamism of novelty-emergence.

Figures 13,14,15 and 16 show how the same movement of Mozart's sonata(Allegretto) can find different transcriptions into a formal diagram according tothe size of the segmentation window. The perfect strictness of the symmetries inthe Figure 15, both on the time axis and, less evidently, on the material axis,leaves us with few illusions as to Mozart's "innocence" concerning the construc-tion of a sonata. Here, the rondo structure can be observed at all levels in a kindof "fractalization" of the form. We would like to insist now, beyond the evidentexistence of those structures, on the relations they have with a mental representa-tion of music, both for the composer and the listener. We will attempt to situatethe composition and the hearing not only on the horizontal axis of a sociological-semiological information theory; for us in any case the work has to be faced in itsprincipal specificity: time. And the question we would like to illustrate could bethat one: what does "to organise the time" mean? It would certainly be tooambitious to believe there is one solution to such a problem, but we will try togive an other interpretation of the formal diagrams that can bring to the lightsome very important concepts about the possibilities of musical thought.

Figure 17 shows an attempt to represent in the same diagram all the possibleformal diagrams allowed by correct segmentation windows in the case of theAllegretto of Mozart's sonata. The reduced polyphony of this work demonstratesbetter the hierarchical implications of Figures 14, 15 and 16. Here, the positionsof the connections, being related to the "material" projection on the vertical axis,avoid the traditional boredom of the usual "flat" linear surface of these kind ofrepresentation. This drawing is amazingly reminiscent of one showing the tis-sues implied in the perception of auditory and visual form (Pitts and McCulloch

Fig. 13. The beginning of the Allegretto of Mozart's sonata "facile" in C K.545, showingthe size of the analysis window in Fig. 14 (the whole fragment), Fig. 15 (B) andFig. 16 (A).

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Fig. 14, 15 & 16. Three formal diagrams of Mozart's Allegretto of the sonata "facile"according to the segmentations described in figure 13.

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114 JEAN-MARC CHOUVEL

time

acoustical realitj

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abstract structure

Fig. 17. An attempt to represent Mozart's Allegretto of the sonata "facile" in C K.545according both to the conventional hierarchical representation and to the formaldiagrams. For reasons of graphic resolution, this representation stops to the levelof figure 15. If not, it would be supposed to continue until the horizontal dottedline.

1947) with their specific and associative afférents. But one must always bewareof seductive analogies, for if the formal diagrams finalise themselves in a spatialrepresentation, they must always be thought in "real time". This expression hashere a particular significance. Form appears to be a way in between two non-senses: the one framing the smallest analytic window, a mere note or even aperiod of the analogic signal, made only of repetition, and represented with thehorizontal "reality" dotted line that is also the time axis, and the one consideringthe piece as a succession of 4 parts, leading to the maximal "abstraction" line(45° dots). Musical form appears then clearly as an intermediary notion in be-tween time reality and space abstraction.

The terms of "poïetic" and "esthesic" can then be understood has the move-ment of objects of high level to their material decomposition (poïetic) and of

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elementary objects to formal abstraction (esthesic). The articulations of the mu-sical stream become the moments when intellectual levels are re-evaluated. Thosere-evaluations correspond to a transfer of information from the "material memory"to the "structural memory" (or the contrary in the case of esthesics). We can thenspeak, from inside the works, of a transfer from a-temporality to time (poi'etic) orfrom time to a-temporality (esthesic).

The difference between form and structure appears clearly on Figure 17. Theform, including the role of memory and its limitations, looks for an equitabledistribution of the information in between the material memory and the structuralmemory: we find that Mozart, with an exact intermediary (45°/2), expresses withthe strongest means the inflection of the discovery front that characterises theAllegretto. It seems that form coincides both with this average position and withthe maximum possibility of differentiation of the material (see Figure 13).

From the beginning, form has appeared as a transcription of memory's behav-iour, that is to say that the listener was passively recording the past. We can nowimagine how an expectation of the future is possible: it is not only related to akind of inertia of the discovery front or the repetition background; it is veryprobably provided, to say it in the most general terms, by the ability to forecastevents, ability that appears as an adequacy of temporal progression and spatialprogression, "spatial" being understood here both as physical order, such as thepitch height, and as structural implication, that is to say order in memory.

CONCLUSION

Form, understood as a dynamic concept, allows us to involve both the percep-tive, analytic and composition fields in the same common operative notions. Weare then led to define again some previously more restricted or neglected termswith a better precision, and at the same time a greater generality. The corre-sponding procedure is a matter for descriptive analysis, as it is a perfectly neu-tral, truly versatile, treatment of the musical information, evidencing the internalrelations of the works without hiding their temporal fabric.

Clarifying its links with perceptual mechanisms and compositional constraints,the analytic procedure we have described above leads naturally to an interpreta-tive analysis, dealing with fundamental notions of the temporal behaviour ofmusical structures usually passed over in silence. In other words, coming back tothe notion of time, form finds all its meaning again, and the choice of the forms,or what we could name the style, makes the relation of the acoustic object to themusical thought more explicit.

ACKNOWLEDGEMENTS

Special thanks to Nicolas Meeùs and André Riotte for their friendly reading of this textand their useful comments, and to Ian Cross for his help in the English translation.

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Clarke, (1987) edited by Sloboda in "Generative processes in music", Oxford UniversityPress, Oxford.

Cross, Ian, Howell, Peter and West, Robert (1985). "Modelling perceived musical struc-tures, in Musical structure and cognition", London, Academic Press Inc, p. 21.

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Jean-Marc Chouvel32,rue SorbierF -75 020 Paris (France)

Engineer, professeur agrégé d'art plastique, Doctorof the University of PARIS VIII with a thesis aboutform theory and its implications in contemporarymusical creation, he is composer and a foundingmember of the association "l'Instant Donné" cre-ated in Paris for the development of the new con-temporary music. He his the author of many worksand among them an electroacoustical piece "Dia-logue avec l'écho" and a string quartet "la cinquièmetentative" both created in Madrid where he spentone year in the Casa de Velàsquez, granted by theSpanish foreign office. He is actually working onnew interpretations of the harmonic phenomenonwith Boris Doval (LAFORIA-Jussieu). He is asso-ciated researcher in the "Institut d'esthétique", Uni-versity of PARIS I.

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