A Study in Semiological Analysis

Varese's 'Density 21.5': A Study in Semiological AnalysisAuthor(s): Jean-Jacques Nattiez and Anna BarrySource: Music Analysis, Vol. 1, No. 3 (Oct., 1982), pp. 243-340Published by: Blackwell PublishingStable URL: http://www.jstor.org/stable/854178Accessed: 21/06/2010 16:56

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JEAN-JACQUES NATTIEZ

VARESE'S 'DENSITY 21.5': A STUDY IN SEMIOLOGICAL ANALYSIS Translated by Anna Barry

CONTENTS

Introduction 244

I - PART I (bs 1-23) 1. The first five bars. 248 2. Digression on music semiology and the informational approach. 255 3. The progression to high G (b. 17). 259 4. The zone of B (bs 18-23). 270

II - PART II (bs 2X40) 1. The percussive section (bs 2X28). 272 2. Vertical falls and flights (bs 29-32). 273 3. The flights of 'Density'. 276 4. Permutations of B, F# and A (bs 29-32). 280 5. The end of Part II (bs 3g40). 282

III - PART III (bs 41-61) 1. Reprise of the opening (bs 41-43). 284 2. Permutations on B-D (bs 46-50). 285 3. The last segment (bs 51-61). 287

IV - RECAPITULATION 289

V- POIETIC ANALYSIS 1. The poietic problem. 301 2. Melodic poietics. 303 3. Harmonic poietics. 303

VI - ESTHESIC ANALYSIS 319

VII - COMPARISON OF ANALYSES 329

(O MUSIC ANALYSIS 1: 3, 1982 243


INTRODUCTION

Music analysis, as we understand it in the semiological perspective of attention to minute detail and clarification, does not lend itself well to exhaustive presentation.

In my book on music semiology (1975), I was able to give only a few examples of my approach, and the small number of semiologically inspired analyses published in periodicals is confined, for the most part, to parts of works.l I am, therefore, particularly grateful to Jonathan Dunsby for having offered to devote many pages of Music Analysis to the publication, in English, of this analysis of 'Density 21.5'. It first appeared in a French edition of 300 copies in 1975, and is long since out of print. I have made various changes in order to correct errors, to take into account the evolution of my theoretical ideas in relation to a text now eight years old, and to clarify my position on some issues.

The uncomfortable situation of analysis today can doubtless be explained by the difficulty experienced in drawing up and publishing written music analyses. When one ventures to reproach analyses for not coming to grips with the detail of a work and the multiple constituent variables which go to make it up, one is often told that analysis professors, in their classes, can 'go a long way' into a work. Could music analysis be an oral genre, or even an oral tradition?2 It must face the following problem: no analysis is truly rigorous unless written down (Granger), an epistemological elaboration of the adage 'Verba volant, ssnpta manent', since the record of the analysis enables it to be checked: once it is written down, it is possible to review, criticise and go beyond an analysis. Even with a very elaborate oral analysis, the listener has the physical problem of being unable to retain everything. If the teacher manages to give the impression of having penetrated the work deeply, the listener will be left with a positive 'aura', but a cumulative advancement of knowledge cannot be developed on the basis of impressions.

The present study therefore aims to urge musicologists interested in analysis to take the time to record their research and offers the first rather detailed analysis of an entire work from a semiological perspective. I am grateful to David Lidov for having understood this: 'This long study is an important complement to Fondements . . . It gives a much fuller picture than the latter does of the scope and force of the author's methods' (Lidov, 1977: 45). Written analysis enables us to take in all parameters, not that an oral analysis cannot do this, but it is extremely difficult to master the combination of all parameters in the absence of rules, tables and diagrams.

This analysis is also the first to illustrate links between the neutral level and poietic and esthesic dimensions, though it in no way claims to offer exhaustive poietic and esthesic analyses. It is not proposed to give a new presentation of the perspective from which I am working:3 it should suffice to remember that a neutral level is a descriptive level containing the most exhaustive inventory possible of all types of configurations conceivably recognisable in a score. The level is neutral because its object is to show neither the processes of

MUSIC ANALYSIS 1: 3, 1982 244

VARESE'S 'DENSITY 21.5: A STUDY IN SEMIOLOGICAL ANALYSIS

production by which the work unfolds (poietics) nor the processes of perception (esthesics) to which it gives rise. In this sense it provisionally neutralises

the poietic and esthesic dimensions of the piece. On the other hand, the neutral level provides the units in relation to which poietic and esthesic data in Sections V and VI will be examined.

Yet another justification of this neutral status is the use, from the beginning to the end of the study, of the neutral level as an analytical tool which is never called into question, the partitioning of the work into units according to abstract paradigmatic axes, that is, axes which group together identical or equivalent units from an explicitly statesl point of view. This technique is inspired by methods suggested by Ruwet (1972: Ch. 4), a continuation of the teachings of Jakobson and Levi-Strauss, but it is not followed blindly: the problems it presents are discussed elsewhere (Nattiez 1975: 239-356). The reader is referred to this same text for a complete presentation of the methodology used in the neutral description of 'Density 21.5'4.

The analysis proceeds 'from bottom to top', that is, from the smallest units to the largest, since Varese works with the differentiation of short units. Nevertheless, larger sections appear in the piece. As these are justified only later on, I shall begin by giving, without comment, a picture of the hierarchic structure of 'Density' in so far as it results from the complete analysis, so that the reader can see how the minutiae which are to be examined relate to broader phenomena.

Numbers in square brackets above the stave refer to the smallest units. Bar numbers are unbracketed:

MUSIC ANALYSIS 1: 3 , 1 982 245


Parts 1st part Sectl ons A

Sequences I II

units j r X 10 r 1 f -, 1 1[ 5 ^

246 MUSIC ANALYSIS 1: 3, 1982

VARESE'S 'DENSITY 21.5': A STUDY IN SEMIOLOGICAL ANALYSIS

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I - PART I (bs 1-23)

1. Thefirstfive bars Before going into the work in detail, I shall illustrate the principles of segmentation of the musical syntagm according to the various paradigmatic axes in an analysis of the opening of 'Density': Ex.1

I [1 ] A[2ad B[2b]

;; 1 ; 2 $ aS j 2 J $

3 , j | [4b], _3

rf , ' , '

>= p 9 ] w 3X, F ] ^ 3 j

f =P _ - J? p

First, the left of the example: bs 4 and 5 can be seen, in fact, in two different ways (A and B), showing straight away that the neutral level is not restricted to one manner of presentation, but on the contrary displays the diverse configurations possible, though without profession to unite them all. To avoid confusion, the numbers in square brackets correspond to the parti- tioned units; a letter following a square-bracketed number designates the particular paradigmatic alternative where there are several partitions. Roman numerals designate units regrouped at a higher level; these are discussed below.5

As far as pitch is concerned, the first three notes (unit [1]) are repeated at the end of b.3 ([3]). It is tempting to add to these the F#-G-F# ([5]) of b.4. The paradigmatic axis regroups units which are equivalent from a given point of view: this does not mean that they are homogeneous. For [1], [3] and [5], I use Molino's term block, [bloc]. The evaluation of affinity which allows us to make these associations, depends upon a mixture of separate criteria:

(1) The melodic identity of [1] and [3] (F-E-F#) (2) The rhythmic similarity of [1], [3] and [5]:

Ex.2 [1XfUlJuJ ,2

' - 3

t3]n l J

. '

[5g.r:' JS




The paradigmatic rhythmic theme, that is, the trait common to a group of units, consists here of two short values plus one long.6

(3) But this similarity is reinforced by two other factors: (a) the two short values form a kind of mordent (lower in [1] and [3], upper in [5]); (b) the long values are in every case an F#.

(4) Unit [5] begins on the same note that ends [1] and [5]. All this leads us to 'neglect' the difference between the two semiquavers ([1]

and [3]) and the two triplet semiquavers ([5]). The distrtbution of these elements permits the connection of the initial F's and F# of the three units, although they are different.

These two examples of assimilation, to which can be added the length of the final F's, illustrate what may be called equivalence classes. Within this class there is a whole range of relationships: because they are physically close, it is easier to assimilate the semiquaver and triplet semiquaver than the crotchet, and the compound value quaver tied to dotted minim tied to triplet semiquaver under the same category, 'long', since the feeling of length by opposition to the two short values does not prevent perception of the durational differences between the three (progressively shorter) final notes.

There is another reason for making a paradigmatic association of Ex. 1: from a wider distributional point of view, taking the broader context into account, [1], [3] and [5] initiate three larger segments, I, II and III.7

The three remaining units [2], [4] and [6] can be organised in two ways. In version A of Ex. 1, the criterion is essentially rhythmic:

Ex3

[2ag zP X J) 4 J S 3 1 3 & t

[4ad SJ : S J & 3 3

t I |

[64 ;. j J J 3 1 3 } 3

fP

Despite differences in detail, this paradigm has the pattern long-short-long8 as a common theme. There are, however, two anomalies to be considered: the inclusion of a semibreve in [2a] and the two equal values (triplet crotchets) which end [6a].

The semibreve is all the more important because it forms part of a procedure which is especially common in Varese: the constant lengthening of each new note within one musical segment.9 One phenomenon justifies the



paradigmatic isolation of the semibreve: the slur from the initial F to the second C of b. 2. The dynamics p-f-p also give the C conspicuous autonomy.

Obviously, strictly speaking one cannot claim the second triplet crotchet of [6a] to be longer than the note immediately preceding it, since they are of identical duration. The total paradigm, the block which takes all parameters into account, will in fact neutralise this anomaly with regard to the short-long- short pattern through the identical final pitches of [4] and [6]: C#-G. This means that, in making the paradigm, another variable (pitch identity) is hierarchically dominant in relation to the rhythmic variable.

For this reason layout B of Ex. 1 may be more pertinent: this paradigm is based on the identity of the final notes C#-G in all three units, not just two. Thus new relationships begin to appear: the C# in particular, initial only in [2b] and central in [2b], [4b] and [6b], plays the role of a pivot note. Because it always precedes the G, which ends not only the three units we are dealing with here but also segments I, II and III, the F# of [2a] is not unconnected, paradigmatically, with the F# of [1] and delays the arrival on G. These melodic relationships are obvious in the paradigm in Ex. 4:

Ex.4

['

< #2>-'J 3>

[2S_Xss

1;gL \ semitone

1 j J k [3] [4a]

03 [5,0 3

[6] , 3 }

zJ w1



It is no exaggeration even to call the F# a kind of leading note to G whose syntagmatic position and duration make it a 'polar note'. In so far as this table opens a separate paradigmatic axis for each new note, its configuration can be called oblique, established between lines 1 and 3 by the addition of G, one semitone above F#.(This detail will assume a certain importance later on.)

The final notes of [1], [3] and [6] have progressively shorter rhythmic values and the same is true here (semibreve, crotchet, triplet crotchet). The initial notes of [2b], [4b] and [6b], however, become progressively longer (triplet: dotted quaver, quaver tied to crotchet and dotted crotchet tied to crotchet respectively).

Layout B should not supersede layout A. Just as it is perfectly legitimate to emphasise the repetition of C#-G which concludes segments I, II and III, we must also take into account the caesura between G of b.2 and all that precedes it because of the slur. We have, therefore, to take account of a contradiction between the two paradigmatic choices. The neutral level shows, by its positioning in each parametric frame successively, that the melodic, then the rhythmic organisation both have their own logic; this can be seen only when the other variables are provisionally neutralised. By putting together all the information tabulated, the workings of a principle which appears to be characteristic of this piecel° is revealed: the principle of deception.

The whole of this opening passage will now be re-examined with the accent on the syntagmatic progression: one unit [1] of three notes with a chromatic rise (F-F#Sends on a long F#;the arrival on G is delayed by a second group of three notes (C#-F#-C#) which is connected to what precedes it by a slur. A rest follows. The initial unit reappears, with a slightly shorter final note, but this time goes directly to G which, in [4], frames the C#; here though, with the same rhythmic type (long-short-long), it is the C# which frames the F# ([2]). In the same breath, the motive from [1] returns in the form of [5] described above, here followed neither by C# nor G, but by an E ([6]) which precedes the final C#-G marked diminuendo. These three notes, at distributionally equivalent points, outline a diminished fifth chord (C#-E-G, suggested by the C#-G from b.2 onwards, and completed by the E of b.5 in exlremis). This means of delaying the G in b.2, with an intermediary unit of three notes, represents the principle of deception. The privileged distributional position of C#-G in [2], [4] and [6] strengthens the listener's impression of having been duped in [2]. Naturally, the listener does not consciously and discursively perceive the work exactly according to the process described here, but a detailed description of the neutral level can subsequently be used to describe a phenomenon which is functionally pertinent from an esthesic point of view; when we speak of deception, we are, after all, describing an effect upon someone.

Attention must be drawn to a feature of the method used up to this point: the research process consists of isolating units according to criteria of paradigmatic association. But a different point of view has now been adopted that of syntagmatic succession. Having begun with a relatively abstract



description of units regrouped in absentia to use Saussure's expression they are then projected onto the real axis where they link up.

Note that the description of the syntagmatic progression suggested above uses information and segmentation established from a paradigmatic point of view. Some music analyses have already used paradigmatic presentation,1l but in introducing to musicology the concepts of syntagm and paradigm, structural linguistics and semiology offer a thematisation of this distinction. A systematic search for the paradigms of related units effects regroupings and classifica- tions, thus advancing towards a typological knowledge of works and their constituent elements; the thematisation of the paradigmatic principle will require us to identify a category of upward-surging figures ('flights': see, below of II-2); the thematisation of syntagmatic organisation brings us to study their distribution. In previously published analyses,12 we emphasised paradigmatic decomposition of works because this aspect was not considered systematically in classical musicology. In the present study, we wish to concentrate on syntagmatic progression, but to take as a basis the infonnation yielded by the paradigmatic viewpoint. The order of discovery and of exposition does not necessarily coincide: in this paragraph we have stressed the method used (projection of the results of paradigmatic decomposition onto the syntagmatic axis); from now on we shall follow more closely the order in which the units appear. It is simply a matter of remembering that paradigmatics enabled them to be identified.

The paradigmatic and syntagmatic description of the numbered units in the score of 'Density' does not complete the analysis. In note 7 three reasons were given for regrouping the first six units into larger segments, labelled I, II and III. One might ask why, both initially and throughout this study, the small units appear to be used to establish the larger ones, whilst one hearing of the piece permits immediate identification of the largest sections.

This text could have begun by justifying, in broad terms, the division of the piece into three parts, the division of Part I into two sections-A and B, the division of A into three segments-I, II and III, the division of I into two units-[1] and [2], etc. But the procedure 'from bottom to top' is preferred because the hierarchically more important units are not identified according to the criteria in use for classical and romantic music: repetition of themes, of long phrases and of periods. Varese plays on subtle rhythmic differentiations and avoids strict melodic repetitions; it is these distinctions with which a scrupulous analysis should deal first. As Lidov rightly says, 'Varese has left all the a priori implicational relations of musical tonality behind. The unique system of the work is its only system. Instead of the tension between style (abstract) and example (concrete), the work takes its life and its energy from the complexity and ambiguity of its internally developed associations and contrasts. The taxonomy renders these explicit' (Lidov 1977: 4445). An emphasised note, a general melodic configuration, a rhythmic or intervallic contrast between two passages will define a large section. This is why it is easier to understand the large units when one knows from which smaller phenomena they are constructed.



At the end of this study the divergences in decision-making among musicologists regarding formal organisation will be examined, but what, in fact, is a form? To speak, for example, of the form ABA is to recognise a certain paradigmatic familiarity between two segments, separated by a third which is considered to constitute a separate paradigm. In analysis, the paradigmatic precedes the syntagmatic. If ABA is inflected as ABA', it is because the paradigmatic links between A and A ' are considered to be looser than those between A and A. In classic formal analysis the use of two identical symbols has never signified strict identity of A and A, but rather the neutralisation of differences considered to be negligible. In a work like 'Density', the identification of parts or sections greater than the small units is possible, even obvious, but only starting from a small number of variables which are hierarchically dominant in relation to others. If formal analyses of 'Density' differ from one researcher to another (see Section VII), this is because the dominant variables are not necessarily the same for everyone: here, the form is not the result of common practice among composers in the course of a given period of musical history, but is the consequence of a relative and mobile convergence of paradigmatic equivalence criteria between sections, for listeners and analysts alike. These divergences are neither dramatic nor regrettable, provided that we are conscious of the reasons behind them.

The regrouping of the first six units into three segments was justified by three criteria (see note 7). Other variables explain why these three segments each have a certain autonomy, but also a family resemblance which distinguishes them from what follows and enables us to look, to begin with, at the first six bars alone:

(1) Segment I presents F-E-F#-C#-G in succession. Segments II and III use the same notes, but in a different order.

(2) There is a certain analogy in the distribution of intervals in the three segments:

I [1] ld 2a 5a [2] 5a 5d 6a (2d)

II [3] ld 2a la [4] 6d 6a (ld)

III [5] la ld 2d [6] 3d 6a (6d)

The numerals designate the number of semitones contained in the interval (no distinction is made between an augmented second and a minor third): a and d mean 'ascending' and 'descending'; bracketed intervals mark 'joins' between segments. The interval between G and F in b. 3 is effectively neutralised by the rest, but the joining interval between [4] and [5] has greater weight, since the slur covers the whole of segments II and III. The criterion of a rest has not been used to identify [4] and [5]. A problem such as that of joining intervals shows clearly how the weight of each variable depends on many different factors which, themselves, change according to particular



contexts. Nevertheless, a constant can be drawn from the examination of these intervallic syntagms: every segment begins with a semitone and ends with a tritone, in accordance with a series of continuous augmentation:

I 1 2 5 6 II 1 2 1 6 6

III 1 1 1 2 3 6

(3) The study of intervallic directions discloses another syntagmatic principle:

I 1X,/(\) 2/\/ (X) II 3\ /(/) 4N/(<)

III 5/<(X) 6h/(X)

This abstraction of intervals joining [2] to [3], [3] to [4] and the descending movement of [5] and [6] (G-Ft-E-Ct), shows a certain predilection in the piece for systematic alternation of ascending and descending movement.

(4) Even if segments II and III are linked by the large slur of bs 3-5, each one of these units is distinguished by secondary slurs in [3] and [5] which isolate [4] and [6] by virtue of their difference.

(5) Rhythmic equivalence classes were discussed above.

Following an examination of the whole piece, the following rhythmic typology is proposed:

al : short plus long a2 : two regular shorts plus one long a3 : one (or two) short(s) plus one long plus one short b : constant augmentation c : one long plus one short d : one long plus one short plus one long e : regular rhythm.

These seven types can be grouped into three families on the basis of initial short, long or constant rhythm.

From this view point, the homogeneity of the three segments is perfect:

I [1] a2 [2] d + long (or triplet quaver + b)

II [3] a2 [4] d

III [5] a2 [6] 'd' (by assimilation)



2. Digression on music semiology and the informational approach'3

The analysis of the opening of 'Density' suggests several conclusions concerning the epistemological status of the neutral level. When Ruwet imagined 'a machine for identifying elementary units' (1972: 112), he provided a metaphor which evokes the explicit character of the method proposed, but which could be said to imply, rather dangerously, that these procedures are of an algorithmic character; the fact that analysis can begin 'from the top', and requires constant zig-zagging between 'small' and 'large units', shows quite well that explicit is not synonymous with algorithmic or mechanical.

The analysis of the first five bars combined two steps: the first gesture, intuitive in that it implies a confused collection of criteria, consists of defining what Molino calls blocks. It soon appears that in none of them are the criteria of paradigmatic association homogeneous. A study of the 'partitionings' (decoupages), no longer carried out by block, but parameter by parameter (pitches, then rhythms, then intervals etc.) shows that no single parameter dominates the constitution of the blocks, which are the result of a conflux of variables whose weight changes in each new context.

As a comparison, consider an approach of the informational type. At first, it would necessarily start at the bottom and proceed note by note. A scan would identify all identical units, taking the notes two, then three, then four at a time, and so on. At the level of pitch, the machine looks first for all the F- E's in the piece and finds them in b.3. Then, after scanning the sixty-one bars of the work, it starts again with E-F: which it finds in bs 3 and 54 etc. The procedure for units comprising three notes is simpler since we can state the following rule: if unit A (two notes) is followed by unit B (two notes) and the last note of A is the first note of B, A + B constitutes a three-note unit, provided that the same succession of notes is found elsewhere. Thus, there appear to be twenty-one three-note units in 'Density', for example F-E-Ft, bs 1-3,G-Ct-G, bs 4 and 5-6. Even then, in the latter case it should be recognibed that the machine does not take rests into account since the unit of bs 5-6 contains a triplet quaver rest. In this way, we can go up to the longest unit in the piece. It contains nine notes: B-Ft-A-F#-B-A-B-F:-A (bs 32-33, repeated in b.34).

This operation is repeated on the other dimensions of the piece: intervals, rhythm, dynamics, modes of attack, and slurs.

Note that the graphic representation of units defined in this way is not different from the system of paradigmatic notation proposed by Ruwet, though the tables are necessarily wider and shorter. The table of units formed by notes taken two by two comprises, syntagmatically, one hundred and thirty-six units of which fifty-one are repeated. The interval inventory begins:


JEAN-JACQUES NA1TIEZ

(1 = semitone, d = descending, a = ascending)

ld 2a 5d 5a 5d 6a 2d

ld 2a la 6a

ld la ld 2d 3d

6a 6d 6a

2a la 3d 3a 3d 3a

2a la ld la ld la etc.

This table clearly shows a pattern of units of 2, 3, etc. intervals: for example, ld-2a (bs 1 and 3), 2a-la (bs 6 and 8)

Such 'accounting' cannot be called analysis, but is rather a 'physical' inventory. Given that not all possible forms of transformaton are foreseeable, as soon as relationships are established between units that are not strictly identical we enter the realms of analysis, but it must be recognized that cultural and theoretical knowledge, a prioris and aural impressions affect

* u

c beclslons.

The difference between an inventory and actual analysis is that it does not appear to be possible to deduce the latter from the sum of the information provided by the former. In fact, partitioning carried out note by note and parameter by parameter does present problems:l4

(1) A certain number of variables is reduced, from the start, to the status of hapax, that is, they are not attachable to other variables, remaining isolated and unusable in the inventory. Rhythmic values are a case in point: with the exception of initial notes of 'phrases', characterised by two semiquavers (bs 1, 3, 9, 15, 21, 41, 43), there are very few strict repetitions in the piece. From the third note of the piece, the values of the F: in bs 1-2 and the F: of bs 3- 4 must be made equivalent in order to obtain an interesting result-a departure from the data. The algorithmic procedure lends too much weight to the note as the minimal pertinent unit. As Molino writes, it 'is an "amalgam" of heterogeneous characterisations: it indicates one absolute pitch, virtual intervals, degrees and function, and virtual durations which could, potentially, carry rhythms. This is why an isolated note could never constitute a unit: its most important properties (intervals, degrees and functions, rhythms) remain virtual until at least a second note is joined to it' (1975: 55). The example shows clearly how phenomena pertinent for analysis are present at the 'top'



but not at the 'bottom': the equivalence which the procedure demonstrates between physically different values is one such case, and it is hard to see how a computer could automatically establish an equivalence which depends on a judgement of similarity transcending concrete resemblances and differences.

(2) Even among repeated units, the inventory reveals physically identical phenomena which do not have the same significance. For example: the unit G-F of b.3 would be picked up by the machine because of its occurrence in b.43 (G-Et: we must allow enharmonic notes to be coded in identical fashion which is justifiable in this music). The same is true of the unit G-F# which appears twice in b.4 and reappears in bs 42 and 43. In view of the rests in both these examples, and also of the distance between and diversity of their contexts, these 'repetitions' do not appear to be of any use. Transposition is a typical case. The inventory dealing only with pitch will pick up the repetition of F-E-F: in bs l and 3, but will establish no relation between this and the E-D:-F from b. 15. The interval inventory, on the other hand, will pick out ld- 2a in bs 1, 3 and 15, but without being able to show, since pitch does not come into it, that the ld-2a of b. 15 is a transposition of the other two. Then the rhythmic element must be added: the units taken from bs 1, 3 and 15 have in common with the first three notes of b.9 the rhythm two semiquavers plus a long. The intervallic unit is not identical: Db-C-Db = ld- la, but the connection of the units through the intermediary of the rhythm gives la a presumed equivalence with 2a. We see, then, that the interest of a recurrence is not independent of its context: it depends on its insertion into a block, that is, a more or less homogeneous group constructed by the analyst on the basis of one or several criteria, dominant and convergent, that do not constitute all the criteria which could have been brought into the analysis. Until we have proof that the criteria are unsuitable, they justify the analytical choices; this is why they must be rendered explicit. For this reason, Molino calls them quasi- crzterza.

(3) Finally, the note by note inventory does not permit the identification of phenomena which normal musical competence isolates at once. We see in the paradigmatic tables that [4] introduces a G where before there was a Ct, and later we shall have an E. The importance given to these notes presupposes that all preceding material (i.e. the rhythmic figure, and the analogies between [1], [3] and [5]) and what follows (Ct-G) has been analysed. Does this mean that inventories of material are useless? Gilles Naud (1979) has proposed a method which, on a single table, 'reports' recurrences, parameter by parameter, variable by variable, together with information drawn from the completed analysis. There is, therefore, no limit to the number of possible columns.l5 There follows an example of what can be obtained with this method, applied to the beginning of the piece:



x = semiquaver

y- triplet quaver

z- crotchet

I do not intend to undertake a laborious inventory of units: it is possible, in a specific study, to take short cuts. From a methodological point of view, however, the comparison suggested by Naud to which we can refer for limited verification focuses our attention on three points: (1) It tells us, by comparison of physical data and analytical decisions,



which variables are strategic (Molino's term). For example, the unit G-F which joins the first two segments of Ex. 1, does not have the same value as the F-E or the E-F# of the opening. G-F assumes the status of a joining unit. All the joining units of the piece might be studied for certain common traits- an interval, perhaps, which is characteristic in relation to all the other intervals of the work. These comparisons should permit the identification of contexts and situations which give certain variables more weight than others.

(2) Secondly, Naud's inventory table is an aide-memoire, an instrument which confronts us with phenomena that the 'from the top' procedure of paradigmatic analysis might leave out. Once it is discovered that F-E-F#, with its characteristic rhythm, is an important 'thematic' element of the piece, the interval table can be scanned to see if ld-2a is found elsewhere. This succession appears in bs 10-11, in the trill of b.20 and in b.39, and makes it possible to show how these two characteristic intervals form a developmental thread, and at what privileged points (before the return of the real 'theme' at bs 15, 21 and 41).

(3) Finally, the table in columns shows up, on one page, conflicting segmentations, the identification of which would otherwise require comparison of paradigmatic tables, often spread over several pages. Here, a conflict arises between the melodic segmentation which isolates C#-G and the slur which isolates G. The partitioning finally adopted does not exclude the other, since it has been shown that it could be pertinent from another point of view.

3. The progression to high G (b. 17)

Concerning the first five bars in toto. What, in fact, is the beginning of 'Density'? Looking ahead as far as b. 17, we can call it a melody which remains faithful to romantic gestures, characterised by a crescendo. The crescendo is, of course, dynamic (thefofb.3 and b.8, thettofb.9, thetftofb.11, the crescendo of bs 13-14 and 16-17), but exists also in a more metaphorical sense: the progression of rhythms and intervals, and the melodic ascent which, at b.5, has only just taken off. A 'romantic' ascent, then, but one which constantly frustrates expectations conditioned by tonal dynamics: the same phenomenon is found in b.8, as in b.2, then between bs 8 and 9, and in b.16.

The function of section A (bs 1-5) is to introduce, after the motive which winds around F#, the tritone C$G three times. Momentum is generated C#-G reappears in b.6 and the upsurge of the melody ends only in b. 17 with the high G. For this new section, B, Ex. 5 shows the organization of the melodic progression (bs S8):


-J br r ( b*


Ex.5 ^ [63 , 3 j

3

3 p sud iho v ton e

Here, [6] is reintroduced as a memorandum. In the same way that C$-G of [7]

is a retake of [6], [8] contains the G-Bb taken from [7]. If in b.8 there were

not a clear slur over the Bb and C which divides them from the preceding

notes, one might say that Varese repeats G-Bb once more before going up to

C. But this is a new deception: the progression C#-G-A-Bb of [7], picked

out by the slur, the preceding rest and the breath-mark after the long Bb, is

repeated but only in respect to pitch, as G-Bb-C-Db:

Ex.6

But the Bb is isolated from the G by the slur which connects it to C, and the C,

subjected to distinctive dynamic emphasis, is separated from Db by duration

delaying its arrival, and by a breath which arrests the momentum: the move-

ment C#-G / C#-G-A-Bb delays G-Bb / G-Bb-C-Db. There will not be

perfect symmetry. The melodic analysis must be completed by a rhythmic description. Using

the typology presented above, we obtain:

[7] [8] [9]

b c or e al

Note that c or e for [8] indicates that the quaver of b.8 may be considered to

have a duration perceptively equal to that of the preceding triplet crotchets. In



any case, [8] is framed by two units whose rhythm is characterised by prolongation. It is already apparent that the rhythmic configurations are organised differently from those of section A. In b.6, something quite new begins.

The paradigmatic profile of Ex. 4 was oblique only because G was added on the right of the F: axis. This profile is primarily vertical (and later in Ex. 8 its oblique character is more marked). From b.9 to b. 11. it returns to vertical: D is the only note outside the play of permutations between Db and C which is spread over two bars:

Ex.7 melody rhythm

ka r E°] g _f

3 t1 1]

7 fi r r 5 6 brr mf subsoo mf Subito

H ,\, E2] H t semitonc

^ p1 3] k 3 ,,

¢ Stf t

'\ r

3 ' 3

: tf r

[10] must be paradigmatically associated with [1], [3] and [5]: it borrows their rhythmic type a2 (two semiquavers plus a dotted minim, as against two semiquavers plus a triple-dotted minim in [1]) and takes the form of a lower rather than an upper mordent as in [1]. In so far as [10] does not contain the intervallic succession ld-2a which would allow the introduction of Ft, one semitone higher than F, the intervallic pattern ld-la allows it, in contrast, to stay put. Thus, [10] and its prolongations ([11] and [12]) constitute a transition before [13] which introduces D, underlined by a crescendo and triple forte, according to a rhythmic (a2) and intervallic (ld---la) configuration precisely as in [1]. The pattern 'short-short-long' thus defines an equivalence class, just like the developmental procedure which consists of upward semitonal progression. Db appears thus to be a kind of preparation for the D in b.ll. In a sense, the play of permutations on two notes contributes to the principle of deception: it delays the appearance of a predictable event, the ascent to D.16



A study of the rhythmic paradigm (Ex. 7 right) shows how Varese introduces some variety into a melodic sequence which could have been monoto- nous. [10], distinctive because of its long final note, its crescendo and the semiquaver rest which separates it from what follows, belongs to the rhythmic type a2 as we noted; [11] takes the last two notes of [10], just as [9] did with [8]; type a1 (short-long) reinforces the parallelism between the Db's of [10] and [11], a2 being simply an extension left of a1 (or a1 a transformation of a2 by elision of the initial note); [12], the melodic inversion of [10], belongs to type b (triplet quaver plus quaver plus quaver tied to triplet quaver); [13] begins with the same two notes as [10] and has the same rhythmic type (a2), but ends on D. The crescendo which begins on Db and ends fff contributes to the emphasis of this unit. There is bias in the segmentation of [11] and [12]. They are separated neither by breathing nor slur. The following might be proposed:

Ex8 [11] 012]

1 F 3

b;

3 3 z3

v tr- r

The first unit of this example would become the melodic inversion of [10] and the second would borrow its first two notes. The length of Db in [11] makes this segmentation difficult to accept, especially since its repetition and length in-[10] give it decisive importance. The neutral analysis, however, must admit the configuration of Ex. 8 if only to foresee certain performers' choice of phrasing.

The following table (Ex. 9) for bs 11 and 12 is both melodic and rhythmic. It is melodic in showing the treatment of G#-D: G# in the lower octave in [15] then D in the lower octave in [16] followed by an expansion to the right of A-D#, semitones respectively above G#-D; in the upper octave, A, Bb and E too will be a semitone above the notes of [16]. Rhythmically, we find types a and b: more precisely, a1 in [14] and [15], prolonged by b in [16] and [17]. The rhythmic momentum follows that of the melodic progression.

In [14] the D was written in brackets. It would be possible to imagine a partitioning based not on the paradigmatic heading G:-D but on D-GS, as shown by the right-hand paradigm. I am reluctant to see it in this way, not because the D might belong to two units (we shall see in b. 36 that this is not a



Ex .9

P r5r C : > s

E5] 3 , 3

Sai r st s. . st

L<?e' 42$m

. , ,

54 ' 9 I tt

.- (st= semitone)

problem), but because the f on the G#, valid for the whole of b. 12, separates clearly by contrast [14] from [13] where thetfton D of b. 11 is the goal of the crescendo beginning at the end of b. 10. In addition, our segmentation throws into relief a rhythmic procedure which seems to be used frequently in this piece. But in the same spirit, it would be possible to partition [15] and [16] in the following way:

Ex.10

Rhythmically type a3 is followed by type b, and melodically a succession of three occurrences of the ascending tritone appears, prolonging the join [13]- [14] and the two descending tritones of [14], but emphasising the parallel movement of the units. Depending upon whether one chooses to lend weight to the variable 'rhythm in augmentation' or on the other hand to 'importance of tritones', the boundaries between [14], [15] and [16] may vary.

Music analysis is, like the works of which it attempts to give account, a symbolic phenomenon, since it is the result of human activity (and therefore has its own poietics); it leaves a 'trace' (the text of the analysis) and is subject to reading, interpretation and discussion (the esthesic pole). Thus, the direction of attention on one aspect of the work rather than another modifies the



organisation of things. What is appropriate to the neutral level is to make an inventory of all analytical possibilities and I shall show, as far as possible, the poietic or esthesic pertinence of some of these.

[17] was constructed from [16], using the familiar procedure of raising by one semitone. Unlike Ex. 9, the intervals are not identical, even though both units do end with a tritone, but in both cases the rhythm forms a continuous progression (type b: triplet semiquaver, triplet quaver, quaver, dotted quaver on the one hand, and quaver, quaver, dotted quaver, semibreve on the other) and the succession of directions is the same ( \ / / ).

Concerning the role of semitones and importance of tritones: as for section A, it is interesting to look at intervallic behaviour between units [7] and [17]. First, we shall group these units into larger segments:

B I [71 6a 2a la (3d) II [8] 3a 3d (3a)

[9] 2a (la) III [10] la la (ld)

[11] la (ld) [12] la ld (la) [13] ld 2a (6a)

IV [14] 6d (6d) [1S] 6a (6a) [16] Sd + 6a 6a (6a)l7

V[17] 12d l+a 3a (12d)

Unlike section A segments I and II of B show a tendency to diminution: 6 2 1 3 3 2. Then in the 'permutation' zone (segment III) Varese works only with semitones. The tone ending [13] is a step towards segment IV which is dominated by tritones (four occurrences before G of [16], then after a descending leap of a compound perfect fourth three tritones leading to A of [17]). Comparing [13] with previous intervallic sequences, we can see how Varese varies the paths from the semitone to the tritone:

A I: 1 2 S 6 II: 1 2 1 6

III: 1 2 3 6 B [13]: 1 2 6 (the inversion of 6 2 1 in [7

The tritone is a characteristic interval: it splits the tempered scale into two equal parts and is not without analogy to the semitone, which divides the scale into twelve equal parts. 18 This contributes to the tautness of the piece. As we can see, the intervals are distributed in privileged zones: the tritones end segments I, II and III of A, and dominate section IV of B, after the semitone zone. Segment V combines these two features: the initial A of [17] is a semitone above GS, a tritone above D: ; the interval between A and Bb is a



semitone and the last interval (Bb-E) a tritone. (There is one interval missing from the first thirteen bars and almost entirely absent from the rest of the piece, the major third. 19 This is probably because it is a particularly consonant interval, and 'Density 21.5' is articulated by seconds, tritones and sevenths.)

In [17] (segment V) there are intervals not heard before: a descending octave, a compound semitone which is followed once again by a tritone ending. With the compound fourth of [16] these are the largest intervals of this opening passage and they widen the ambit (starting from the semitone) in which the melodic curve can flower. But the progression, in evidence since [7], is not over. In b. 15, we find the melodico-rhythmic figure of the opening: [4] is transposed up a seventh (inversion by semitone), but reminders of the opening are not confined to [18]. [19] takes the notes of [18] in a gesture which is not without analogy to the passage from [1] to [2]: E-D#-F / D#-F here, and F-E / F# / C#-F# before. [20] and [21] contain the same notes re-ordered as the first few bars: E-F-F#-G.

Finally, as regards intervals, [18] to [21] show the same tendency noted for segment A and between [10] and [16], that is, the continuous broadening, after the celebration of tritones in [4] to [17], in tones, semitones and their compound revisions, which does not pass through the intermediary intervals 3, 5 and 6:

[18] 1 2 2 [19] 2 2 [20] 1 2 1+ (1) [21] 1++

Section B ends on the same note as section A, after which the rise to high G began (b. 7). The interval of [21] is a double compound semitone between F# and G, the two predominant notes of [1] and [2]. It would be an exaggeration to speak of a coda, but the composer is rounding off and summing up, and not only through allusions to the very opening of the piece. Here is the paradigm of bs 15-16: Ex11 i0180 S 4<

E9]e

kOa] 4;

9 0 st s

-; A

,



It is not unrelated to the preceding ones. Comparing Ex. 14 and Ex. 10, we see that [19] takes the last two notes of [18], just as [11] took those of [10]. Even rhythmically, the articulation of the two notes from [19] after the three notes of [18] resembles what happened in [11]. [20a] is relatively complex: the E, D# and F of [18] are permuted (in the same way that the Db and C of [10] were permuted in [12]); the F from [19] is transformed into an F#, just as [4] introduced a G after F#, [9] a C after Bb, [13] a D after Db, [16] a D# after D, and [17] an A, a Bb and an E after G#, A and D#. This last comparison enables us to draw together [16]-[17] and [20a]-21a].

Ex.12 560 1 +

Xg 4tS J) #r

[,5 f f

520ag 3 § t21b]H

7 $N, r LS

P *c

Just as in [16] a D# was introduced after a D with the characteristic directions / , so in [20a] Varese introduces an E# after E, with the same directions / X . Even if these intervals are not the same (ascending tritone and major second, decending compound perfect fourth and compound minor second), the leap which leads respectively to D# in [16] and to E# before the returning upward sweep (D#-A-D# / E#-F#-G) in [20a] is some justification for drawing these musical segments together. Finally, the culminating G in b. 17 has the same semitone relation (even in octave displacement) to F#, as the E# has to E, the Bb to A, and the A to G#.

It will be noted that the implicit partitioning in the three segments of Ex. 12 does not correspond to [16] and [20a]-[21a]. Recall that whilst in Ex. 9 rhythmic factors, indispensable for the analysis of bs 11-14, were taken into account, in Ex. 11 the accent was placed on melodic relationships. The break between [20a] and [2 la] cannot be retained because of the phrasing, dynamics and breathing. It should therefore be written:



Ex.13

D 20] <2= '

[21] -

L ,y C

Of course, this segmentation offers nothing, paradigmatically, in respect of [18] and [19], precisely because the F# and the G are new. It is this segmentation, however, which is retained as the slurs and the breathing impose a partitioning where it is only the melodic relationships that are no longer readable. This means that, once agairl, the joining of Ex. 1 1 and Ex. 13 on the neutral level throws into relief a new form of the principle of deception. Following the paradigmatic analogy of Ex. 15 the E# of [20] should go to the F# of [21] in the same motion which takes D# to A ([18]) or A to Bb ([17]). In fact, this natural momentum is arrested by the expansion of the phrasing after the first F# and the breath between E# and F#, in the middle of a crescendo.

Section B is autonomous because it displays, in its own right, traits which A does not possess. The rhythmic types in the first three units of B have been shown to be different from those of A. Let us look at B from this point of view:

B I [7] b II [8] c or e

[9] al III [10] a2

[1 1] al

[12] b [13] a2

IV [14] a [15] a [16] b

V [17] b VI [18] a2

[19] c [20] al a [21] a1

Except in [8] and [19], types a and b, those which go from the shortest to the longest duration, predominate. Type d, present in A, is peculiar to the beginning since it does not appear here at all. Its return from [22] to [26] will be all the more significant. The omnipresence of a and b implies that all these



units ([7]-[21]) end with a longer value than those already heard within the same unit. This is also the case with segments. One cannot help but establish a correlation between this tendency to lengthen durational values in each unit and each segment, and the melodic progression which is the object of B. Consider Ex. 14:

Ex.14a

I + JJ > nf f mff w f

E. L

3- , 3 '

m dF3 1)1 t L # w- ## Wu

P

(I)74/-

g J b p s6izo f

I r

m UfT

*g--

mf s>Eto 3 f*g Ex.14b

EH-e f cL"-'

- ^

fP w - _,ff9c -



The new notes from each unit actually form an ascending step-wise progression in relation to those preceding:

Ex.1 5

;> o .. bo w bo zqo Wo ̂ s o ct,,) #O o

Can we speak of tonality here? A scale, first minor (G, A, Bb, C, D) then chromatic (Dt, E, F, F,t, G) unfolds, but not every added note is felt as being in the key of G: on the contrary, at every transient point the tonality is uncertain since the notes in a unit or segment lean towards a polar note which is different each time and which, itself, leans towards another note that will predominate subsequently. Each transient point of these first seventeen bars delineates a moment which establishes itself in a precise ambit and is exceeded only by successive tones or semitones:

Ex.16 [1g-t2 [2,

@' o bo t.. o I

[1 ] [6] [72

o | U bo

t80 t9]

O bo l s

[lo]-[120 [13]

bo } bo I

01 4]-E 5] E 60 [1 7] n bt -

o I to I I

[18]-[20] t2og [2<

o t° $° 1 4" 1 $°' I

There is one note missing from the opening, B20, left, so to speak, in reserve for bs 1W23 (section C) where it evidently assumes the role of a pole of attraction.

MUSIC ANALYSIS 1 : 3, 1982 269

--L*S-

sJ -

[24Xff ft

Looking first at the left-hand paradigmatic axis, we see that melodically [22] is a lower mordent like [10], but rhythmically it belongs to type d of [2]. [23] transforms [22] by lowering A: one semitone, the familiar procedure, but operating downwards this time. G: is added, on the right, also a semitone lower. Contrary to what has happened up to b. 17, here the tendency is to descent.

In [24] there is the same melodic contour as in bs 1S17: Ex.1 8

,Q sl to 4;.w ("v) n

[24,

pt

o ts. to °


4. The zone of B (bs 18-23, [227-[287)

The zone of B maintains a number of links with what goes before: Ex.17

v l /

[2i

p tubito

>02603

I r27g

p wubito



It should also be noted that, whereas in [12]-[13] there is a slow trill ending a semitone higher (D), in [24] there is a rapid trill which follows the same intervallic pattern (descending semitone, ascending tone), the pattern of [1] which the phrasing of bs 19-20 (another use of deception) is not supposed to hide. [24] is a kind of response to [22]-[23]: these last descend chromatically towards G: with four notes (B-A-B-G:); [24] climbs up to D in inversion (B-Ct-B:-D): only B: alters the symmetry. But this note is important, heralding the B$t of [28]. [24], whose analogy to [1] has been mentioned, is a sort of transition to [25] which clearly borrows its rhythm (or that of [10]). Section C is thus attached to section A by two procedures: in [22] by the rhythm of [2], in [25] by the rhythm of [1], and in [24] by the intervallic pattern of [1]. [26] plays on the A already introduced which occupies, distributionally, the place of B and takes the rhythm of [2] and [22]. [27] uses only the first two notes of [26]. Since [25] we have not left the ambit A-B, just as [22]-[23] establish themselves between G: and B. [28] will play, an octave higher, on the C: and B: introduced by [24]. The appoggiatura is a sort of inversion of the lower mordent of [22] and [25], and section C ends a semitone above the polar B: a transition is assured.

This section is divisible into two 'moments', each of which may in turn be divided into two:

C Ia [22] 23 Ib [24] IIa [25] 26 27 IIb [28]

Ia-Ib and IIa-IIb are in fact symmetrical: Ia and IIa show a tendency to descent or to stasis and both are followed by an abrupt change of register. They are characterised by tones and semitones ([22]: 1 1; [23]: 2 2 3; [25]: 1 1 2; [26]: 1 1 2; [27]: 1). The two segments end by leaps of a tone, displaced up an octave, in the same way that the rises of [17] and [21] close segments V and VI of B.

Most musicologists21 agree that b. 23 is the end of the first part of the piece. Actually, from b. 24 onwards Varese uses completely different compositional procedures. We see section C more as a transition between what precedes and what follows: because in b.17 the climax on the high G completed a progression instigated at the beginning; because section C is the privileged zone of a note which we had not heard; because the alternation fall/rise-stagnation/rise does not show a clear picture as in the first two sections the important notes of the four segments first outline the chord B-G:-D, then the group B-At- B: which stays suspended but is identical to the intervallic pattern of [1].



II PART II (bs 24-40) Part II comprises section A (bs 24-28) which is characterised by percussive

use of the flute keys, section B (bs 29-32), section C (bs 32-36) and section D (bs 36-40). The third part (III) begins with a return to the motif from [1].

1. The percussive section (bs 24-28)

The segmentation of this section is obvious because of the numerous rests:

Ex.19

[29], 3 ,

[3 ] + + X

. [312 + [3i

#S 77S7'? , p P

t3i3 + +

[34] + +

nn [3i + 3 + '

03i +

,b:

Varese dwells first on E-Ct. Can [29] be drawn from the E-C: of [6]? It is tempting to note that bs 2X28 are cast in the ambit Ct-(E)-D which is that of the opening of the piece (bs 1-11) until the introduction by a play of tritones (b. 12) of a new mode of progression. In [31], Varese inverts the E-Ct. This last note, prolonged by the D of [32], becomes the pivot note ([33] - [34]) followed by D ([35] - [37]). The passage ends on an Eb, heard here for the first time, and stands above D as a compound minor second.

This whole section is, then, autonomous. The first unit ([29]) moves without a break to the last ([37]) by a play of successive additions and suppressions. In contrast to the first twenty-three bars, accented and dotted



notes are used systematically. The regular rhythm type is predominant and appears five times. The rests are longer and more frequent than in preceding sections. In Part I, sub-sections A and B are separated by a rest: there is a triplet crotchet rest between [6] and [7], [19] and [20]; semiquaver rest between [10] and [11]; quaver rest between [23] and [24], [24] and [25]. A dotted crotchet rest divides [17] and [18], before the return of the initial motive, and Part I ends on a minim rest. In Part I, these long rests enjoy a privileged distributional position: they precede the return of the initial motive ([3] and [18] in the two versions closest to [1] rhythm short-short-long; descending semitone, ascending tone).

In this we find: a minim rest between [29] and [30] and (approximately) between [32] and [33], a dotted crotchet rest between [33], [34], [35] and [36] (to the nearest triplet semiquaver), a crotchet rest between [30] and [31], [31] and [32], [35] and [36], [37] and [38], and a quaver rest between [36] and [37]: long rests predominate.

Finally, this is the only moment of the piece where Varese uses the percussive effect of the fingers on the flute keys. Because this was a new use of the instrument, these five bars have attracted most comment and sufficed to inscribe the piece in music history: it is really since then that certain purely technical properties of instruments have been used to musical ends.

2. Verticalfalls andflights (bs 29-32) With the tempo change in b. 29 and the descending leaps which the

composer uses systematically for the first time here, something different again begins:

Ex.20

t38,

[39]29

_[40Xt $+


--43 A; J4T'lQl

[432 t br^t

Ex. 21 bs 1-2

Ws' O tsZ t (# " ) O

[42] [43ai

o ("-) ^ O bo


[43> [43in ;-/st t [420 3 9

These leaps are in fact similar to those of [16] (G#-D#), [17] (A-A) and [20] (F#-E#), but this figure is combined for the first time with the characteristic rhythmic type a2 from [1] (short-short-long). The difference between the long F#'s of [38] and [40] is only a semiquaver, and this allows us to group them in an equivalence class.

In [38] to [40], we are struck by the play of minor and major seconds, simple, compound or inverted G-F#, F#-E#, E#-G (at the join), G-F#, F#-G, G-F#. Through this systematic preservation of small intervals, Varese makes a special use of clashes of seconds.

[42] and [43a] present a pattern reminiscent of the first bars of the work:

Here, the intervallic sequence is identical: an ascending tone, a descending perfect fourth, an ascending diminished fifth. The combination of intervallic directions with their continuous broadening makes each new note, on the ascending or descending slopes, higher or lower than the preceding one. This principle of motivic development is found elsewhere, even if the intervals are not identical. In Ex. 22 the two instances from Ex. 18 are reinserted to show the link between these four fragments which relies upon the alternation of directions and the play of simple or compound semitones, even though there is no interval identity:



EX.22 4b 12 S,} (o) - ZX

n to ) StZ

bs 16-17

.s go

(°} $0-----,<.

bs 20-21

O ta. t0 0

b.30

i,,, to n o - - - -_ _ _ _X..

The third section of Part II is divided into two 'moments': first, a series of descents, then, after [41] which rises and falls, the progression towards the high A of b. 32. Ex 23 shows how the ascent to A is effected:

Ex.23 4 /

. 3 [42] 3 .

n

[4i, ''s

b

[43io 4e

I

[43@ f

This is not as 'slow' or regular as the approach to high G: everything happens in a bar and a half. But the end of [42] introduces a G# and an A not contained in [41], the second note of [43a], Bb, is a semitone above the A of [42], the second note of [43b] is a semitone above the Bb of [43a], and the final A is a semitone above the G# of [43b]. The semitone is, once again, fundamental to this section: from [38] to [40] in the first segment of C, it frames the octave leaps; in [41] and [42] it slips in between the wide intervals (seventh and thirteenth) and follows them, but between all these intervals (10 11 12 1 +) the difference is always a semitone. On the rhythmic level [43a] and [43b] are of type a1 and [43] as a whole is of type b. All of section C, then, is filled



with progressive rhythmic types: type a from [38] to [42] and type b in [43].

3. Theflights of Density I shall now break away, exceptionally, from the principle followed from the

beginning of this analysis by depariing from the linear unfolding of the piece. [43] is compared with analogous phenomena already encountered or to come. I have given them the generic name 'flight':

Ex.24 bS 12-13

They are not all of one type: there are breaths in bs 13 and 16 with slurs straight afterwards in bs 13, 31, 58. But one could say that they articulate three types of 'crescendo': a non-metaphorical crescendo (< fff), augmentation of rhythmic values (bs 12-13: quaver, dotted crotchet, then quaver, dotted crotchet, semibreve; b. 16: crotchet, crotchet, semibreve; bs 31-32: semiquaver, dotted quaver, crotchet, dotted crotchet, dotted crotchet; b.44: quaver, dotted crotchet, dottedminim; bs 5W61: tripletcrotchet, triplet crotchet, crotchet, then quaver, crotchet) dotted crotchet and crotchet, minim, semibreve), and finally an unrelenting rise in pitch (towards E in b. 13, G in b. 21, A in b. 32, C: in b. 44, B in b. 60). These flights are difElcult to describe from a melodic point of view: in the five segments there is a feeling of functional similarity, but how is this to be made explicit? There is no obvious regularity in the interval series.22 All that can be observed is a certain prefer- ence for 'taut' and dissonant intervals, but if we look at the whole piece, there is nothing special in that.


t O b . R2

, .

R2, ' | { _

! bo .

* 7, R

b.44 _ bs 58-61

b- bo {b o \

. tone \ \ \t+22 R

s 1+R \ 1+R;

ts' ' . '

0 jR2 'R R2

_ '

80 #s'. q

(R signifies octave displacement and R2 double-octave displacement)

But the flights do not have only paradigmatic analogies. Syntagmatically, they have a characteristic distribution. First, and most obviously, they end an important section (in b. 13, segment B = IV of Part I; in b. 17, Part I; in b. 32, section B of Part II; in b. 45, section A of Part III; in b. 60, the entire piece). Equally, we see that the positioning of the echappees can be studied in relation to another melodic type already encountered - permutations. There

b. 1 6

follows a list of the two types concerned:


The constructional principle of the flights must be sought elsewhere, in the role played by semitones in the appearance of new notes:

Ex.25 bS t2-13 bs 31-32

Flights

[17] [21] [43]

[63]

[82] [83]

Permutations

[10] [14] [18] [25] [44] [54] [59] [64] [71] [79]

- [13] - [16] - [20] - [27] - [51] - [56] - [62] _ [70] _ [74] - [81]



In four cases out of five ([17], [21], [63], [82]-[83]) the flights are preceded by a permutation: in three cases out of five ([7], [43], [63]) except that since the last echappee ends the piece it is literally three cases out of four they are followed by a permutation (thus, two 'flights' are framed by permutations, [17] and [63]). Evidently, permutations constitute one aspect of the context of flights, just as flights constitute one aspect of the context of permutations: this is an important meloctic syntagmatic element of 'Density'.

Finally) flights have a third distributional characteristic, on the left this time: in four cases out of five, one of the two units contains a descending compound interval:

(1) before the flight of [17], descending compound fourth at the beginning of [16];

(2) before the flight of [21], descending compound semitone at the end of [20];

(3) before the flight of [43], descending compound semitone at the end of [41];

(4) before the flight of [82] and [83], descending compound major second at the beginning of [81].

Returning for a moment to the content of the flights, we recall that the first two ([17] and [21]) contain an ascending compound semitone in one- and two-octave displacement respectively, and we are tempted to see if intervals greater than an octave have some privileged relationship to flights. Every instance of descending compound intervals not in one of the two units preceding an echappee is a join ([24]-[25]: compound minor third; [28]-[29]: compound augmented fourth; [32]-[33] and [33]-[34]: compound semitone; [69]- [70]: major third with double octave displacement; [74]-[75]: compound minor sixth). With ascending compound intervals outside the flights, seven cases out of ten are also over joins ([27] -[28], [38]-[39], [80]-[81]: compound major second; [36]-[37], [39]-[40], [63]-[64]: compound semitone; [70]-[71]: compound perfect fourth). [24] and [77] contain a compound major second and [33] has a compound semitone. The inventory will be affected by the following observations:

(1) Every instance of a compound semitone, ascending and descending, from [33] to [37], is found in a section of Part II where because of the rests it is difficult to give the proposed partitioning of units an absolute value. Should [31] and [32] (b. 25) constitute one or two units, and [35], [36] and [37] (b. 28) one, two or three units ? We might hesitate. It is therefore interesting, whether or not joins are involved, to examine the importance of compound semitones in section B.

(2) The same type of problem is posed by the join [27]-[28], since [28] is to [27] as the C::-B: trill in b. 20 was to the preceding B. Moreover, the interval is the same. This assimilation is all the more appropriate in that [24] and [27]-[28] are not without analogy to the flights: leap to a


Before a Tritone [17] flight Semitone [20]

Semitone [41] Major second [81]

Within a Semitone [17] flight Semitone in

double-octave displacement [21]

In an Tone [20] assimilated Tone [27] - [28] flight Tone [77]

Part II, Semitone [33] Semitone [32] - [33] Section A Semitone [36] - [37] Semitone [33] - [34]


much higher note, crescendo, context of permutation. It will later be shown that unit [77] could have been considered an echappee were it not 'afflicted' with a diminuendo. If, therefore, 'assimilated flights' seems appropriate, it is interesting to note that [24], [27]-[28] and [77] all contain a compound major second. All other cases- except the debat- able example of [36]-[37] from section A of Part II - are clearly positioned at joins ([38]-[39], [39]-[40], [63]-[64], [70]-[71], [80]- [81])

(3) Once two of the six descending intervals have been located in section A, the other four ([24]-[25], [28]-[29], [69]-[70], [74]-[75]) are clearly at

a

olns.

On the basis of these observations and distinctions, a table of the distribution of compound intervals in the piece can be formulated:

Ascending intervals Descending intervals

Joins Semitone [39] - [40] [63] - [64] [38]- [39] [80]- [81] [70]- [71]

Minor third Tritone Augmented fifth Major sixth

[24] - [25] [69]- [70] [28]- [29] [74] - [75]

Tone

Perfect Fourth



It was [43] which motivated this digression. It is the third of the five flights and, in a way, it is central. The first flight ends on a high E, the second on a high G (b. 17), the highest note up to that point. G then disappears for eleven bars, and reappears as high G in b. 29. It is usurped in b. 31 by A. The last flight concludes the piece on a high B, the privileged note from section C of Part I, absent from the first two sections of Part II. G, A and B maintain their privileged relationship in terms of immediate context, of register and of distribution. The third section of Part II is in fact, a permutation zone where the intermediary F# inserts itself between B and A.

4. Permutations of B, F# and A (bs 32-36) A does not play a role comparable to that of G. Here it closes segment

[44]-[45] with a long note, but in [51] it is B that closes the second segment with the same value (a dotted minim). The zone of A is, in fact, intermediary: the rest of the piece prepares the climax of the work on high B.

After a brief passage in a slower tempo (crotchet = 60, bs 29-32) the initial tempo returns in [44]. The B-F#-A adopts, in the same register, the last two notes of the flight, filling them out, and, as Gilles Naud points out, the rhythm of the beginning of [44] is related to that of [43]:

[43] = semiquaver, dotted quaver, crotchet, [44] = triplet semiquaver, triplet dotted quaver, triplet crotchet

In [44] there is a triplet and in [43] the G# is a crotchet, but the articulation point of each note is, in both cases, proportionally comparable in relation to the preceding one. Thus, this opening section borrows its rhythmic and melodic components from the preceding echappee, and the whole of section C will be cast in the ambit of its last two notes. But consider the development from [44]:

Ex.26

s . ,

gr j

s . . . n

8 - n

t

280 MUSIC ANALYSIS 1 : 3, 1982


This paradigm is melodic, but the only rational partitioning must take into account the slur which links the last B of b. 32 to b. 33:

Ex.27 [440 i4

& . . 1

r

t460 3 9 g 3

& . ,

_#t T;

[48S 8 - 1

3 3

This is certainly the only place in the piece where a segment is repeated. There is, of course, a transformation (elision of the A) at the end of [47], but [48], which prolongs it, adopts melodically the last two notes of [44] and [46]. Hence the A from [47] in Ex. 27 iS in brackets: Varese cannot accept a strict repetition. The similarity between [45] and [46] iS SO great that, according to the principle of deception, it arrests the momentum tovvards A the second slur ends on the Ft. How, then, is the subsequent music to be analysed? The problem is identical to that encountered in bs 10-13. Rhythmic transformations will now be shown in relation to the melodic invariant:

E>c.28 X [Q]t

E i O------ n

L.ME [502 8 - - - - - - - - n

r#S 3 3

; 60 25 1] ,< ,

3ar^: G7'SS'' _ B



It is in extremis that [51] reintroduces the B. Note that, melodically, [51] adopts the descending succession A-F#-B which concludes b. 32, but as always the unit which creates this equivalence is imaginary, because of the slur which separates [44] from [45].23

Rhythmically and intervallically, this section is distinguished from the preceding and ensuing sections by the absence of semitones, the large number of minor thirds, perfect fifths and minor sevenths, and by types a1, b and e:

I [44] a1 a1 7a 3a 3d (7d) [45] b lOa lOd 7a 3a (lOa)

II [46] a1 a1 7a 3a 3d (7d) [47] b lOa lOd 7a (3a)

III [48] a1 3d (3a) [49] e 3d (3a) [50] e 3d (3a) [51] b 3d 7d

The breath between [45] and [46], like the repeat of the same units in [46]-[47], justifies the demarcation of segments I and II. Segment III, up to the falling fifth (F#-B), is dominated by minor thirds. The rhythmic types, then, have a characteristic distribution: a1 is always at the beginning of a section, b at the end and e in an intermediate position.

5. The end of Part II (bs 36-40) Section C has developed on the same three notes: C, a semitone above the

preceding B, seems to be an intruder. Armed with a triple forte, it is the culmination of the crescendo begun at the end of b. 35, but because of unit [53], it will be linked to E. This C can easily be accepted as marking the end of section C and the beginning of section D:

Ex.29

X ''7^t'lr .

[5! t9i

?

055] , 3 t >y

P SU6fb fp



, Lt , 7

- D

L572-- -I br

[56 SP vr 2^"

3 '2 ' fP

In the same way, it does not matter whether the Eb of [53] is the end of [53] or the beginning of [54]. We attach it to [53] since the diminuendo makes it a prolongation of C and of the first Eb. In contrast, thetton D isolates [54] with its own slur. The most important thing is that the rest of the paradigm shows how [54]-[58] is constructed in relation to Eb, D and Db by a play of permutations which can be read from the table above. Rhythmically and dynamically, [55]-[58] is characterised by the same procedure:24 the final note of the four units is always shortened; it is either piano and staccato ([55], [57], [58]) or of a shorter written duration ([56]) and follows an accentedforte note. In the section taken as a whole, the distribution of rhythmic types is rather scattered: al three times, b once, c twice, e once, and [56] which is the only example of b inverted.25 There are, therefore, roughly as many short-long types as long-short. The intervallic distribution, combined with the zones among which the different notes are divided (C-Eb / D-Eb / Eb-Db-B) delineates three segments in this section, where the semitone alternates with major sixths, where the semitone clashes with the tone and where the final interval is a descending diminished fourth. The whole is, however, dominated by a tendency to descent, felt from [51] of the previous section. The first segment ends on Eb, the second on D (a semitone lower) and the third on B the lowest note of section D: the B dominated section C but had not been heard again.

Part II can be described as full of contrasts and hesitations. Section A: two permutation zones (E-Ct-D and Ct-D-G:). Section B: three rapid falls, then flight to high A. Section C: permutation zone (B-F:-A). Section D: descent to B with permutation zone on Db-D-Eb. While Part I was characterised by the rise to high G, Part II, with its varied use of rhythmic types, dynamics and melodic directions, seems to be intermediary and this is partly due to the large number of 'permutation zones'. Evidently, three segmental types will have three functions in this piece: the permutation is stagnant, delaying the appearance of a new note which is generally a semitone higher; or oblique paradigms allow the piece to progress; or rapid flights lead to a climax. Between them, these types set up a dialectic: the permutation acts as a brake on development in relation to the oblique paradigms and the flights it favours a period of rest rather than moments of tension. Varese restores, on



another level, what the tonal system is no longer able to offer, by alternation of distinct functional types.

III PART III (bs 41-61)

1. Reprise of the opening (bs 4143) From b.41 there is a reprise. The initial tempo returns (crotchet = 72) while [59] is an exact transposition, a semitone above [1], of the first three notes of the piece, which permits a direct arrival on G without the 'suspensions' of [2]. This is probably why it is repeated (with durational shortening of the G) without intermediary development in unit [60]:

Ex.30 01 ]

4 F, _J_,> mf w

J J) S W 4 - - w

1 06$ i Lg b^t?t- i2xJ : 2sJ q; t I [

g ° t$s t(F"_,° r 1 °

[62] 3

- to ° v b. v j 4

r - - e

p 26i 3

p [610 3 3

1e1 2 P=r y

26i 3

This paradigm is simple: the notes in unit [61] are permuted, then the order at the head of the paradigm returns in [62]. It is interesting to trace the origin of the D and the Ab of [62]:

Ex.31 [2] 3 [lg



[62] is an exact melodic transposition of the melodic contour in the first two units of the piece. The rhythms of [59] to [62] are not identical to the opening. However, in [S9] and [60], type a2 of [1] occurs.

Interestingly, before the flight of [63], already analysed, the intervallic syntagm bears some analogy to section A of Part I, in the progressive growth of the same intervals, from the semitone to the tritone:

A I [S9] a2 ld 2a (ld) II [60] a2 ld 2a (2d)

[61] a2 la la (ld) [62] b ld 2a Sd 6a (la)

This intervallic progression, together with the tendency to rise which is initiated at the end of unit [62] with the introduction of an Ab (not yet heard), is reinforced by progressive rhythmic types, a2 and b: it is easy to see, here, how b is an expansion of a2

The analysis was begun, in respect of the flights in Ex. 24, on the D of [62]. But the slur and crescendo clearly isolate [63]. The caesura between Ab and A is another 'deception' in the rise beginning with D.

2. Permutations on B-D (bs 4S50) After the flight, Varese returns to his predilection for playing on two

alternating notes: this recalls the way Db and C are exploited in bs 9 and 10. The B and D at issue are positioned, respectively, a semitone above the notes of the flight (Bb - Ct). The D, moreover, is brought in by Ct, a compound semitone above it. This time, the D appears to predominate:

Ex.32 [6 i 8 - - - - - - n , 3 ,

t) , 3

L68-.---------------------,

3 3

t6 6] ; 3 - - - - - - - - - - - - - - - - '

f f ft' Y ,



2670 3 '

fS f [ , s ,

06! - ,

This is doubtless because of rhythmic types a and b which confer a long value on the final D's:

Ex.33

3 3 3

[ ] G 5 G r r, ; 3 3

r65] G G r (a2,

3 3 3

0660 : : r r (a2 ) 3 >s3

067] : r 5 (a1 ) 3 3

0680 r r G S (a1 ,

[69] G C f ( b)

Less important than G and B, D nevertheless plays a decisive role in certain intermediary moments: it closes the play around B in b. 21; we find it again in a privileged paradigmatic position in bs 25, 26 and 28; it will return before the final flight as the upper note of the permutations of bs 5S58. It owes its particular importance to the registral leap after C# of [63] which makes it the highest note of the piece.



This whole section of alternating B's and D's ends on a little melodic figure which is the inversion of the initial F-E-F# and outlines a final descent:

Ex.34 s t to ne

,2

X {w O #. 3

st tone

I j r 11-1 1 _ _ _ 5 . #

B P f P

[70] is treated as an appendix to section B for several reasons: first, because the openingfffis the end of the crescendo from [69]; second, because there is a breath only in the middle of the bar; finally because the character of the section beginning with [71] is quite different.

3. Thelastsegment(bsSI-61) This does not mean, as in previous cases, that there are no links between

this final 'phrase' and what precedes it. Between C and F# of [71] we have the same descending tritone as between D and Ab of b. 50 in double-octave displacement. But, above all, [71] bears some relation to the wide descending intervals in rapid rhythm of [38], [39] and [40], and, to a lesser extent, [53]:

Ex.35,

7

t539 ^ > '_ [54] -

4 ff

r - ff 070 s

[72]v. <:+

SfP rP r A: SL,4Jb; ()

3 3 , _



tW*-*s s [SS WS

t1 >

-

, (_) 077

. ' JR [d P w

3

br F: crcsc. rnolto

[29I'/ fE >

-ff4; : F [802

$ - 3 > I t82]e, te

Even though [71] begins with a C and [38] with a G, the fall to F#- E# on a practically identical rhythm and the demisemiquaver C make comparison with [38] legitimate. This paradigmatic table shows the rhythmic transformations in relation to melodic data, but the implicit units which emerge cannot be retained because the slurs have the upper hand. Numbers [74] and [75] mark the boundaries of more real units. Ex. 35 thus brings two contradictory levels of segmentation together. It will be noticed that [72] and [53] are immediately followed by a unit of the same rhythmic type ([73] and [54]), e, which also has the same number of notes.

From [75] onwards, the composer is sliding to another paradigmatic axis. Ex. 35 demonstrates that this entire concluding segment (from b. 51) forms a whole, since we progress towards the final high B by successive changes. These changes are of a particularly characteristic nature: in [75], E# (F) becomes E, a semitone below. The C on the left of the paradigm becomes C# (a semitone higher). [76] marks the first attempt at melodic flight: for the moment we stay on G, a semitone above the new F#, itself a tone above E. In [77], as in bs 5 and 6, Varese adopts two preceding notes and rises to A26 (a compound tone above the G). [78] introduces a Bb, the pitch-class a semitone above A of [77]. In [79], D, a seventh (inversion of the second) from E, is added. [80] and [81] play on C, E, D, [82] takes the Bb of [78] which thus fills in the interval E-D of [79]. There follows the final flight, whose construction



on plays of tones and semitones, is shown in Ex. 24, together with the way in which it ends on B.

This ending is thus a resume of all that precedes: the falls of bs 51 and 52, the play of permutations, the progression on G then A and B and the final flight are all procedures encountered previously. The tendency to return to events from the first two parts, which we noted for the two preceding sections, is confirmed here: after having recalled what goes before, Varese undertakes a rise to the last climatic point which G and A prepare.

IV RECAPITULATION The table on the following pages attempts to give a global picture of the

piece by assembling, in synoptic form, the essential elements which have underpinned this analysis. On the left, segmentation at four levels: parts (Part I etc.), sections, segments and units. Then the three principle data: rhythmic types, intervallic sequences and melodic pattern. On the right, a general characterisation of the syntagms in units: zones of permutation, of progression, of flight and of descent.27

This table could, in addition, have carried other descriptive variables: for example, slurs and dynamics, both of which are essential to the piece. But they arose above all, in the preceding pages, to delineate the units so that I consider it unnecessary to include them here.


o

Part I

Segmentation Rhythmic types Intervallic sequences Melodic Pattern

A I [1] [2]

II [3] [4]

III [s]

ld 2a (Sa) Sa Sd 6a (2d) ld 2a (la) 6d 6a (ld) la ld (2d) 3d 6a (6d)

a2

d a2

d a2

"d"

-

co

3 3

-

B [6]

I [7] II [8]

[9] III [10]

[11]

[12] [13]

IV [14] [15]

[16] V [17]

b c or e al

a2

al

b a2

al

al

b b

6a 2a la (3d) 3a 3d (3a) 2a (la) ld la (ld) la (ld) la ld (la) ld 2a (6a) 6d (6d) 6a (6a) Sd + 6a 6a (6a) 12d l+a (l+d)

C$G/A-Bb G - Bb

/C

Db -C

/D G#- D

/A- D: A-Bb - E

ation Rhythmic types Intervallic sequences Melodic pattern = : 5- w

VI [18] a2 ld 2a (2d) E - D# - F ,// 3 [19] c 2a (2d) / < [20] al al l+d (la) /F#-E#/ ////// S [21] la++ (8d)+ IG 9/// ,,

Ia [22] d ld la (O) B-A# g < [23] d 2d 2a 3d (3a) /A-G# , U [b [24] d 2a+ ld 2a (3+d) B+C# B# D / ////// 2: Ia [25] a2 ld la (2d) B-A# %///,/ t [26] d la la (2d) IA t%////% o [273 al la (2a ) /B+-C# g / °

I [29] e 3d (3a) E - C# ////,/2/ 2: [30] e 3d 0 (O) y///// <

[31] al 3a (lOa) /D , . A//////v: z

Segment

c :

Part II A

to semiquaver [33] e l+a (l+d)

C#-D-G#

[43]

Melodic pattern Segmentation Rhythmic types Intervallic sequences

/Eb ///X. z

G-F$E: W z

IG-G$A Y///// t E-Bb-G$B-A S////// N

ld 12d ld (2+a) ld 12d (l+a) ld 12d ld (O) lOd lla la l+d(la) la la (Sa) 6a lOa 3a lOa (lOd)

B I [38] [39] [40]

II [41] [42]

a3

a2

a3

a2

a2

b

KFFA C I [44] [45]

II [46] [47]

III [48] [49] [50] [51]

7a 3a 3d (7d) lOa lOd 7a 3a (lOa) 7a3a3d(7d) lOa lOd 7a (3a) 3d (3a) 3d (3a) 3d (3a) 3d7d(1a)

al al

b al al

b al

e e b


t: - 7

A= t ;

¢

Ct

-v CC

+

r r t r CC: v v ,/ m m

Ct Ct Ce Y Ct

r t Ct r Ct

t

ce ce ct D D

u)

Q - v

- r N 'e cC:

> < E-vt C -ooN t Q - Lo N

= r t 'e t 'e r t

o u) = Q u)

+

. - =

= - o

o

$ ce D X ce = t) t

_

N m t V £ S X = - , m m m m m n I . . .

o

-v s} -v

m o s} m

t 'e 'e 't

D ce ce ce

-t-t -t -t -t o o N m t v) £ t -

v)££££ ££££

I ffi I ffi

293

'e

. -

t # n m

v *

ct

bt

- - -

ct

Segmentation Rhythmic types Intervallic sequences Melodic pattern

[68] al [69] b

III [70] e

ct

3

-

A I [71] [72] [73] [74] [75]

IIa [76] [77] [78]

IIb [79] [80] [81] [82] [83]

6d ld (7a) 6d (ld) 7a 6d (la) 7a (8+d) 2d ld 3a 3d (3a) 2d la (3d) 3a 2+a (lld) 6d 6a (lOd) 4a lOa (lOd) 4d (2+a) 2+d 4a (6a) 4a 4a 7a (2a) 2a 2a 6a

SF$E:

/E-D-C: E-F$G

IA /Bb

CE-D

Bb-D-F$C: D$EtB

a3

al

e c

c + d a2

b d b al

e b b


The 'melodic pattern' column summarizes a phenomenon which has been discussed constantly here the progression of the piece by a succession of zones of privileged notes which are overshot by new added notes. This gives the following overall picture:

Ex.36, part one [1 ]-L63

g '>P ° [70

* ho

t82-t9]

, b.

E °d-[120 bo .,

L1 30 ,.,

E40-05] o

01 6,

.,

o

[170

bo

[180-[19]

0 >, o

[ ] 21]

o >

[2i-S2@ " g o

o ... 4O I o l

L2- [280 " t

o}^,4 ... I I

MUSIC ANALYSIS l: 3 , 1982 295


E>c.36, part two

[29]-032] [33]_[370

X 4° 14 #°

[38}[42] [43] -

°-$° o I o b l

[44]-05 1]

Xo I

[52]-[53] 0540-[570 058]

- b X o 1x^ bo | b*. <

Ex.36,part three

0590-2620 ,, bn "- vo ° ' g bsw 14vw

264]-0690 [700

° 40 1S4nW [ ]eLL [7SS [76, L7d 29] [8t] i2]-g83] #E"+Z_

The progression to high G in b. 17 is followed by a zone of hesitation which also belongs to B (bs 18-23). Part II, divided into four 'moments', comprises: play around E, C#, D, then C#, D, G#; successive falls then a climax on A; play around B, F#, A; play around C, Eb, D, Db. Part III contains: reprise, permutations on B;D and a final phrase.

Four syntagmatic families may be extracted from the melodic pattern: permutation, progression, flight, descent. Using a horizontal line ( ) for permutation, an oblique line ( / ) for progression, an arrowed oblique line ( / ) for flights, and a descending oblique line ( \ ) for descent, we obtain the following general picture of the piece:

PartI _/_/ /t /\/_/

PartII _/_/\//_

Part III _// / /



This plan shows clearly the tendencies of each part: Part I-ascent, Part II - hesitation, Part III ascent-descent-ascent.

Throughout the analysis, the sections and segments have been differentiated most often by contrast: A of Part I groups together rhythmic types a2 and d, whereas III of B in the same part is characterised by successions of semitones. During the course of the analysis dominant variables have been emphasised moment by xnoment. The complete table enables an assembly to be made of all the components of the musical material and consequently to replace dominant variables by those which are not dominant.

This kind of table also makes it possible to verify whether or not there is a correlation between two phenomena. By definition, the four segmental types on the right are correlated with the melodic patterns and intervallic sequences. Do they correspond to the rhythmic types? A hypothesis is that regular types will correspond to stagnant zones of permutation, that types a, b, c (augmentation) will correspond to zones of progression, etc. The partitioning of the four functional types is now projected onto the rhythmic types (noting that they do not necessarily correspond to segments and sections):

Permutation Progression Flight Descent a2 d a2 d a2 d b c or e al a2 al b a2 al al b b a2 c al al al dd

d a2 d al a2 e e al e e e a3 a2 a3 a2

a2 b al al b al al b a e e b al b e al al c c a2 a2 a2 b b b a2 a2 al al b e a3 al e c c d

a2 b d b al e b b

Try as we might, no particular correlation can be discerned between rhythmic and functional types. This negative result is still progress: unless there are gaps in our inventory of variables which could always be filled in by another researcher it shows that the impression of stagnation or progression is due only to modalities of the melodic line.

The value of the procedure followed is, in any case, obvious: without relying on the complete table, particularly looking at functions in terms of



variables, it could have been stated that rhythmic type b was absolutely characteristic of the flight (b five times, as opposed to a1 and d1 once), which is true in a way, but b is found six times in the permutations and four times in the progressions where it is as frequent as a2. It is impossible to claim, therefore, that b is the rhythmic variable which corresponds to the flight since it is found elsewhere, in combination with other variables. The same is true for e, mostly sited in permutation zones (except one case of descent) where we find all the other variables too. A variable is not pertinent in relation to a given function,28 in any absolute sense, and, except in the case of b, there do not appear to be any correlations strong enough to extract tendencies of any interest.

Here, the flight is, in fact, a special, distributionally privileged case (five occurences) in the progression: if we look at the kinds of units in which b is found, we can state that in twelve cases out of seventeen, these units end with one or several ascending intervals (in seven cases out of twelve, all the intervals ascend). It appears to be possible to establish, on anot)zer level, correlations between two or more particular variables entered in the table.

There is no recipe for finding interesting combinations: the analysis functions on the basis of hypotheses which can be neither confirmed nor refuted unless a taxonomic description is available which is as responsible and exhaustive as possible. This is done by 'churning' the data, that is, projecting the characteristics of one datum onto another, starting with those intuitively felt to be the most promising.29 It was shown, for example, that flights had particular internal and distributional characteristics (section II/3). Study of the table reveals that, within one segment, there are many permutations followed immediately by a progression. These are:

Part I B III [10]-[12]/[13] IV [14]-[15]/[16] VI [16]-[20]/[21]

Part III A I [59]-[61]/[62] C IIa [79]-[81]/]82]-[83]

With the exception of [25]-[27]/[28], these paradigms show an interesting rhythmic analogy:


5 o ^l % - - - - - g F ; r > v - Ea >

tS w r p

6 Ll; E 43 5

-z tf r 4 2tG 7

[59]-[620 [794-t820 [440_[45]

JJ$Sy >

3

1

3 3

..1d1 t #zi e-

A*m bJ r ]3.1 b9


E>t. 37

E 4]-016] E8]-t2lt

E00-E3q

r C- 5, | / s 3

Before the note which creates the progression, at the bottom of the paradigm, there is rhythmic acceleration through the use of the triplet after binary rhythms and an increase in the number of notes in the last unit (except in the first case). The same procedure will have been noted in [44]-[45], but there it does not lead to a progression. It is rare for one procedure to be encountered in 100% of cases: musical style is not a system.

Let us look at the permutations on two notes:

Db- C [10]-[13] B - A: [25]-[28] semitone E - C: [29]-[31]

F: [49]-[511 minor third L J | - J

D - B


[64]-[69]


The intervallic constant is in itself remarkable. It becomes even more remarkable when it is noticed that the first two follow passages which contain no intervals greater than a minor third. As for the other three permutations, they appear after segments where the perfect fifth is present, one of the rarest intervals in the piece, found only in [51], [73]-[74] and [82]. It is therefore no exaggeration to postulate a correlation between permutations on two notes, the interval of a minor third and a context in which the perfect fifth appears.

Still more little correlations could probably be found in this piece. Moreover, the global perspective of our neutral analysis in musical semiology is, we know, stylistic: this means that if other works by Varese are analysed, it should be possible to pick out from our table correlations or phenomena whose identification is not permitted by 'Density' alone. Returning to 'Density' on the basis of a wider field of works would be that much more efficient, given that we would already have drawn up a pretty detailed inventory of variables. The neutral level, it must be stressed, is only a 'moment' of analysis. A few general observations on the methodology used here:

(1) It is possible to partition parameter by parameter, but the global segmentations performed on the syntagmatic chain are the result of intuition about the convergence of different variables at a given moment and the hierarchic dominance of some of these. The collation of information, parameter by parameter, variable by variable, shows the constitution of the global segment. This global segment can be deduced from the sum of the individual segmentations, but only to describe texture. There is, therefore, no logical or necessary order according to which an analysis should be conducted and presented: here, as elsewhere, the order of reasoning does not coincide with the order of discovery.

(2) One particular consequence of this principle is manifest in the construction of equivalence classes: when the assimilation the of rhythms of [1], [3] and [5] was proposed, in spite of their physical differences, it was not only because these differences were minimal, but because, on another level, melodic identity (F-E-F#) induced an assembly of these rhythmic units. It is impossible to say, therefore, that the rhythmic analysis is purely rhythmic: in fact) other parameters are taken into account. The result of all this is a mixture.

(3) I have spoken several times of projectzon. This has a double role. First, it permits the evaluation of the role of one variable in relation to another, as in the previous case. But it also has an heurtstic value: it enables interesting correlations to be discovered between variables of different types. Consequently, there is no limit to the number of projections it is possible to perform from one variable onto another: this is essentially why analysis is endless, and even if one hopes to put one's finger on some characteristic correlations, there is still the conviction that other



equally valid relationships may have escaped our attention. There are, therefore, no rigorous rules for determining which projections will prove viable. We can procede only by trial and error.

(4) Thus, correlations are established between disparate characterisations of different types and levels: one example of this is the distribution of flights. This is identified, on the left, by a descending compound interval, a more concrete datum than rhythmic types in general, or than type b which, positioned at the end of a segment and with its play of semitones in the pitch structure, characterises its content.

(5) A great deal of intuition enters into the research, but although the objective of taxonomic clarification is confirmation, it may equally result in invalidation. This was the case, in the research phase, for rests: it was assumed that they could be entered in different classes according to the type of segment they delineated. A study of the taxonomic data reveals that the only interesting constant is the unusual presence of rests in the 'percussive' section of Part II one of its characteristics.

V POIETIC ANALYSIS

Several times, here and elsewhere, there has been occasion to recall the fact that neutral analysis is an essential but intermediary stage in the semiological approach to musical works. In this and the ensuing section the intention is to show how the data of neutral analysis relate to those of poietic and esthesic analysis.

1. The poietic problem There are two ways in which the phenomena catalogued by the neutral analysis can be considered poietically pertinent. To the extent that analysis deals with the score, it is directed at the only trace left by the composer. Therefore it is possible to consider that recurrent traits demonstrate the preferences of the composer for certain compositional procedures; they enjoy the presumption of poietic pertinence. This presumption is confirmed particularly if other works by the same composer contain these traits (don't we say 'he likes to do this, he likes to do that'?), or if our historical knowledge of the evolution of musical language establishes that, on the basis of the heritage received and experienced by Varese when he began to compose, he decided to orient his compositional practice in this or that direction.30

This poietic procedure may be qualified as inductive:

poietics; neutral level

But it is just as feasible to start with an external poietic element-a sketch, a rough draft, a commentary- and project it onto the work, either to direct the



analysis accordingly, or to reorganise the neutral level, constructed indepen- dently of the external datum:

; work

poietic data > or

neutral level

In the first, inductive perspective, which traits identified by the neutral analysis have a poietic presumption?

(a) The principle of deception: I have shown how it governed developmental procedures, at the same time frustrating the expectations of the Western listener. Thus, the procedures described stand out from the wide range of habits handed down by the dynamics of the romantic musical phrase, whose principles of progression and elevation Varese preserves but constantly contradicts.

(b) The principle of maximal differentiation: on the neutral level rhythmic equivalence classes were constructed (types a to e), but it must be remembered that Varese is careful to write values which are as distinct as possible from one another. One of his favourite procedures consists of multiplying the number of dots from the basis of a given value (as in Integrales). Slurs, triplet quavers or semiquavers introduce subtle differentiations from binary values. Equivalence groupings have, therefore, a presumption of esthesic pertinence above all in one particular case. This does not mean that the difference between a crotchet tied to a semiquaver and a crotchet tied to a triplet semiquaver is imperceptible, but it cannot be perceived on the same level. The discernment of a rhythmic progression (types a and b) and the discernment of a minimal difference between two values are not incompatible. Detailed experi- ments would in any case be required to establish the threshold below which two units are confused and above which they are distinguished from one another.

(c) The poeitic counterpart to the alternation tension/relaxation, that is, the methods chosen by Varese to create this esthesic effect: play of semitones, dissonant intervals, crescendi, the configuration of flights on the one hand, and the permutations of stagnant zones on the other.

But in the second perspective, there is a comment by Varese on 'Density', published, without reference to its source, by Hilda Jolivet, given here in its entirety:

Despite the monodic character of 'Density 21.5', the rigidity of its structure is defined overtly by the harmonic scheme carefully described in the



unfolding of the melody. 'Density 21.5' is based on two short melodic ideas, the first, modal and in binary rhythm, which begins and ends the piece, and the second, atonal and in ternary rhythm, which lends elasticity to the short developments separating reiterations of the first idea (Jolivet 1973: 110).

I shall use this external poietic document in a precise perspective: after makirlg an appraisal, its contents will be applied to the configurations of the neutral analysis in order to reorganise it. The objective is obviously to compare the two pictures of the work thus obtained and to define the status of neutral analysis in relation to this kind of poietic analysis.

2. Melodic poietics

We begin by examining the second part of this text which deals with melodic ideas. The qualifications 'modal' and 'atonal' will be set aside since, if the two ideas correspond as I believe to [1] and [2], it is not clear that the second should be more atonal than the first, or the first more modal than the second. One could observe sound philological principles and turn to other texts by Varese, in an attempt to shed light on the meaning of these words. But Varese was no theoretician, and nothing in his writings (1983) helps us to understand what he means, here, by 'modal'. With regard to 'atonality', these two quotations can be set off against each other:

[In contemporary music], whether we deny its presence or not, we sense a tonality. There is no need to have a tonic, with its third and fifth, in order to establish a tonality (1934). My language is naturally atonal, although certain themes, certain notes repeated in the manner of tonics, constitute axes around which sound masses appear to agglomerate. In this way, musical development grows, little by little, through the repetition of certain elements which are pre- sentedn each time, in a different aspect, and interest increases through the opposition of planes and the movement of perspectives. If themes reap- pear, they have a different function in a new medium: dynamics (1930).

All of this last text could be applied to 'Density 21.5', but it does not help to pinpoint the meaning of 'atonal' as opposed to 'modal': on the basis of these two quotations the piece could be qualified 'tonal' just as much as 'atonal'.

The binary/ternary opposition is more telling. I have therefore played along and constructed a paradigmatic table where axes of equivalence are defined by the rhythmic character of units, binary or ternary (Ex. 38 below). The first axis therefore contains [1] and its transformations. Notice that [5] has had to be placed in the second axis because its rhythm is ternary. [71] and [72] have been placed in the first axis because of their analogous distributional position and paradigmatic link with [1] through the intermediary [38]. The second axis contains all the units with at least one 'irregular' value. Therefore a third axis must be opened, one not mentioned by Varese, which groups together all units that are neither 'a return to the first idea', nor in ternary rhythm, that is,


LI t S 1D 1 o-

_ I

n X S Jt 04 ] L r -- 3

o a- 5 l

=FtU S : r f >

tf r Af r > 3 3

#2

L ] j -

ll

- L2 90 r 3 1

H;; [ MUSIC ANALYSIS 1: 3, 1982

Es.38 tr] t22

3 ,

[100 [93 3 ,

L ] v E g - s3 ossia 3

01 70

[1 8]

[2 3]

§ l l .

44r ' [27] ag

304

JEAN-JACQUES NA1YIEZ

ll - ww

[5 ,l; t

[ u Lf

_ 1 . _

vr _

t6 3 ,,

0 50 3 3

StJ77N,$ -

0340 +

L6]flC.e S I V rT, < \ = r \

bS

r560

' 3 >

[53 3 3

:

[580 3 =+G 7

Ex.38 cont.

2300 3

,

t]7 7P2: J 3 ,

[312 [3 2]

r w

L553 L3 '

305


5 L4528- r

t Lp " d: ,r n

L4 7 s- - - - - - ,

[532 r

L 0540

Ex. 38 end

t593

L6 1 l

>qi -X

t20g-J - -44 r'lF fi ..

Ve 3 _ . 3

[65] 8 t f

3

L 3 ° r- - & 3; rCC

8 - - - n t6 71 - t_^

J t _ Sr S 3-

1

068]ot [69]:5't'

s_3 T

tt- n ,621

[7O]JCo)z j_

[73] 3- [740 t

-- S S er Mu

[770 ; $ 5 j - nz J poJ.If

w tJ f

3

0750 3

079]

l - l

- - |

0800

[82] X tt

306


0630 bss"e

t7l; L _ 5

,

[72k#

t760 3 3

078] _

MUSIC ANALYSIS 1: 3, 1982


those that follow a ternary unit and precede the return of another ternary unit or of the first 'idea'.

What may be concluded from this example? First, it must incite us to approach poietic information with a measure of caution: this information defines a field of equivalence which is not systematically precise.31 We have already had to discard the designations 'atonal' and 'modal'. Varese says that the first idea ends the piece: in fact, it last appears in strict repetition at b. 42 of a sixty-one bar work. Moreover, he talks of 'short developments' between reiterations of two melodic ideas, but the second paradigm is by far the fullest (forty-five units as opposed to twelve for the first). If the hypothesis is adopted that the first basic melodic idea corresponds not to [1], but to segment I, the development is de facto shortened, but the first 'idea' is no longer short or binary. Finally, the third paradigmatic axis (twenty-four units) the rest is far from negligible: Varese does not mention it. In fact, it would be interesting to know the exact source of the text by Varese quoted by Hilda Jolivet: the composer's spoken words, programme note, personal message? Has it been correctly transcribed?

This poietic laxness allows a stretching of the principles set out above for the construction of axes: [1] and [60] contain an irregular value, but as [1] is precisely the one Varese qualifies as binary, it may be supposed that the ternary aspect (triplet semiquaver placed at the end) is negligible, since it has no influence over the moment when F: ([1]) is articulated. The same is true for the final D of [13], but hesitation, in this case, also comes from the fact that, as in [7], the long values belong to a unit defined as an entity.

The examination of the third axis is of great interest to our study. Indeed it shows that the projection of a poietic datum onto the neutral level can reorganise it, without the neutral dessnption losing its validity. That is not all: the poietic description would not be what it is had the neutral analysis not been performed first.

What goes to make up this third axis? The units it contains may be entered * ,* . n tour categorles:

(1) Descending units: [23], [49], [52], [53], [54], [75] and [80] where there is no privileged rhythmic type. The only descending units not in this axis are [38] to [40], [58] and [70]: the first three because I have decided to assimilate them, here, with the 'initial idea'; [58] and [70] are the only descending units in the second axis.

(2) All the units qualified as flights are in the third axis. These units were seen to be characterised by type b (rhythm in augmentation). The flights may thus be drawn together.

(3) Three ascending units, [9], [27] and [69], have a rhythm which is also progressive (a1 twice, b once).

(4) The last category groups together the other ascending units in the axis: five of these belong to the percussive section ([31], [32], [34], [36], [37]), the other, [74], is a hapax.



What might be concluded from this division? It confirms the final position of units of progressive types (a1 and b) and ascending direction, since material separating two ternary units or a ternary unit and a return to the initial idea is placed in the third axis. In as much as Varese does not speak of flights as specific procedures, one might wonder whether he was poietically conscious of them, but this distribution according to the three axes (of which one was obtained by differences) shows that the three principles of construction in this piece are: the initial binary idea, the ternary transition, and the ascending units with progressive rhythm whose final distribution constitutes a characteristic stylistic element of Varese's musical rhetoric.

By numbering units according to the paradigms in which they belong, the syntagmatic poietic organisation of the piece may be presented thus:

I II I (II)x9 III I II (II III)x2 III II III II II (II III)x3 I I I II II III (II)x5 (II III)x2 III III (II)x4 I I II II III (II)xS III II I I II (III)x2 (II III)x2 III II (III)x2

The following principles may be established: (a) It is not possible to go from I to III without passing through II. (b) I may follow itself between two and nine times. (c) The syntagm II III may be reiterated two or three times before the

return to I or II.

3. Harmonic poietics The first part of Varese's text states clearly that performing a monodic analysis is not enough to give an account of this piece. An harmonic analysis has not been included in the 'neutral' part of out study because there is already one in existence: this will be examined below. The harmonic description of 'Density' belongs as much to the neutral level as does melodico-rhythmic description. 'Neutral' implies, as I have said elsewhere (1975), 'neutralisation', since it is not true to say that melody and harmony should be analysed together: this leads to neglecting aspects which are strictly melodic or strictly harmonic. In saying that the unfolding of melody follows the harmonic plan and makes it explicit, Varese indicates clearly that the poietic analysis consists of showing, on the basis of two neutral analyses melodic and harmonic, how the monodic unfolding gives the harmonic structure its form.

The complete text of Marc Wilkinson's harmonic analysis (1957: 17-18) is set out below, interspersed with comments and critical remarks. I would like to say straight away that I consider this text quite remarkable, and the pernickety nature of my comments is due only to the explicit perspective I have chosen here. Wilkinson writes:



I shall concentrate on the harmonic and melodic structure, for the elements of rhythm and dynamics serve mainly to define harmonic areas, phrases, and motivic elements, and they are not particularly problematic.

This is a good example of what happens when a specific aspect of a work is given privileged status: other elements are taken as read or presented as secondary. I believe that I have shown, on the contrary, that interest in rhythm for its own sake (that is, the provisional neutralisation of other variables) shows up characteristics which are peculiar to rhythm and contri- bute to the global organisation of the piece.

The first phrase (bars 1-17) is composed melodically and harmonically around the notes E, C#, G. This relation of minor thirds to a central note (E-C#, E-G) is the harmonic element on which the whole piece is built. Bars 1-5 are the first and simplest statement of this relationship (with E as centre); F and F# are ornamental to the harmony, though by choosing to fill in chromatically the higher of the two structural minor thirds Varese prefigures the upward sweep of the melodic line throughout the piece. The little cadence E-C#-G in bar S merely recapitulates and clarifies the harmonic relations.

The author's description shows very well how the ambit C#-G is filled progressively, by F and Ft, then by the E which intervenes only in b. 5. It might be emphasised that the length of the E is partly responsible for its harmonic importance. It is not possible to speak of an ascending curve 'throughout the piece', since there are zones of stagnation and descent, but it is true to say that F and F: lean towards G. My analysis of the monody of the first five bars clearly shows how the rhythmic and melodic processes realise the harmonic plan, according to Varese's text, and thus how the poietic analysis is the result here of the combination of neutral analyses for each separate parameter. Wilkinson delays comment on bs S8 and continues:

Bars 9-17 are more complex, for two statements of the harmonic 'area' are telescoped into one, though in bars 12-13 the ambivalent notes are defined by octave transference. The first harmonic sphere is centred around C#; the minor third relations are traced in the notes C#< (bars 9-10) to Bb (bar 13), and in the notes C#-D-D# (bars 11-12) to EW (bar 13). The second sphere, centred on G, is presented in a melodic and harmonic form of retrogradation, for the G appears only at the end of the phrase (bar 17). The two minor thirds are built with the (G)-G#-A-Bb in bars 12-13 (notice the important role of octave doubling in making this motion clear), and with the notes E - F- F#- (G) in bars 15-16. The D# in these last two bars is held over from the other sphere (bar 12) and resolves to the EQ in bar 16. However, by the subtle use of phrase markings (such as the breath before the F# in bar 16), Varese suggests with this D# a new minor third relation (DtF#) as a secondary harmonic Eleld. Meanwhile the octave transference of E# and F# (bar 16), and the long duration values, define a cadence to G in bar 17.

Bars S8 link the two main sections of the phrase. The musical material



emerges by elision from the cadence in bar S; the upper note G becomes the lower note of a new minor third relation (G-A-Bb). This process repeats itself by a simple sequence upward (Bb-Ceelision to C# in bar 9). The rhythm and the dynamics strengthen this feeling of motion by minor thirds.

All of this analysis deserves attentive study. I have no objection to the account of bs S9. Wilkinson is explaining, in harmonic terms, the principle of deception which delays the arrival of Db, which I have emphasised throughout the piece. One wonders simply whether it is possible to continue from b. 6, to base harmonic explanation on a central note framed by an ascending and a descending minor third, as the position of E in b. 5 appears to imply. For bs 6-, Wilkinson does not refute it and this is doubtless why he makes this passage a linking zone between the two main sections of the phrase; but the explanatory principle returns in b. 9.

The author's idea is as follows: between bs 9 and 17 there are two inter- twined harmonic centres. The first is Db, the second, delayed until b. 17, is G. The descending minor third from Db explains the Bb of b. 13 and the ascending minor third accounts for E of the same bar. This Bb can be explained even by G of b. 17: this ascending minor third is filled out by G: and A of bs 11-13; the E of b. 13 is a minor third below G, filled out by the F#'s of bs 15-16. There remains the explanation of D: in bs 12 and 15: it is an appoggiatura to E, but its privileged position in b. 12 defines, with F# (doubtless the F# of bs 16 and 17), a secondary harmonic field.

I find this explanation a little complex, particularly because it establishes links between notes which are pretty far apart on the syntagm. Is it not rather contrived to go to b. 13 to find a Bb to form a minor third with Db when, as Wilkinson later says, the Db of b. 9 is a minor third above the Bb of b. 8? These difficulties can be avoided by explaining bs S17 as the expansion of the two thirds from the first five bars:

E>c.39 bS 1-5 , bS 6-9 , bS 9-17

Coming back to the principle of sliding by semitones (used for the monodic analysis), the rhythmic and melodic movement emphasises notes of harmonic importance: this was seen in the first five bars and is confirmed by the length of Bb in bs S7, the play on Db in bs 9-10 (C being its appoggiatura) and the position and duration of E (bs 13-14) and G (b. 17). Given this framework, Varese effects successive semitonal slides from G-Db until he arrives once again at Bb-E:



Ex.40 bs 6-10

rP w bo

o

bs 11-12 ,

t bo do v. .

b. 12 , . ,

.. #° 7

L @

W° a.

t J

This explanation has the advantage, in my opinion, of showing how the harmonic field fits, in accordarsce with a chromatic displacement shown up by my monodic analysis, into the tritone ambit prepared from b. 2, confirmed in b. 5 and reaffirmed in bs 11-13. The tritone culminates at the same time as the first stage of the melodic progression.

Bs 15-17 are cast in the interval of a minor third, E-G. If harmonic explanation is required here, the D# must be made an appoggiatura to E (b. 16) and F and F: considered elements which fill out the third; in as much as this 'phrase' (bs 15-17) ends on G like the first section (bs 1-5), it is interesting that these two notes have the same function. Resuming the idea of a secondary harmonic field (D:-Ft) the F would be interpreted, taking into account the melodico-rhythmic unfolding, as a suspension of F# which would itself appear, as soon as it is heard, to be a leading-note, or appoggiatura, to G. The interval of a tone between D: and F seems above all to prepare that between E: (thrown into relief by a breath) and G.

My explanatory hypothesis of sliding semitones is all the more acceptable since Wilkinson has recourse to it to explain what follows:

Varese has now established the notes E C$G as centres for both melodic and harmonic activity, and has consistently enlarged the overall sound area which this first phrase covers (the octave transference also serves this purpose, of course). But in doing this, he has automatically touched upon Bb as a centre (since Bb is a minor third distant from both C# and G). He does not use this note as a centre in the way we have already seen, however, but instead builds harmonic spheres from note centres placed symmetrically a minor second above or below Bb (on the notes BW and AQ), again using all the intervals of the minor third which can be derived (B-D, B-G# and A-C, A-F#).


I should prefer not to invoke the distance of a minor third from Ct, for reasons given above, but rather from G (expansion by minor thirds):

Ex.41

X 1, bo t.. a (t, bS 1B-23

"> _ "_ Q "t t_ "

Bars 18-23 establish the first of these secondary harmonic areas, and one notices that Varese has carefully avoided sounding the note BW in the first phrase, to prepare this entry.

Note here that it is the melodic and syntagmatic criteria which explain the harmonic centre and semitonal slide, and not the reverse.

Bars 18-19 outline B-G$; bars 2(}21 outline B-D, and bars 22-23 resolve to Bb (or A#) which connects, by a minor third relationship, with CtE in the next bar. The transference of the BW in bar 23 an octave higher separates it from the body of the preceding passage, or rather relates it only with the B$C$DW in the same register (bars 2(}21).

Here there is a problem in interpreting the score. In as much as Varese is careful to make the accidentals explicit where they could be ambiguous (for example the Au of b. 19), it seems difficult to imagine that he is following classical conventions of notation and that the accidental before the first B does not also apply to the triplet B in b. 23. If B# is retained, the analogy with notes of the same register in b. 20 works even better, and Varese does, as I have shown, like to end a melodic segment by a note a semitone above its predeces- sors. On this hypothesis, Wilkinson's explanation of the G# and D of bs 25-28 remains valid, because of its posiiion in relaiion to the doIxiinant B of bs 18-22:

Starting from this B, Varese juxtaposes in bars 2F28 a new (and now inverted) area B-D-G: with a return of the Ct-E relation, derived from the first five bars.

Wilkinson's explanation is represented as follows:

Ex.42 bs 18-23 b. 23

O qs

* bS 24-25 , bS 25-28

($A" A o t0



b - o to


Note that the author plays around with the thirds rather as he pleases since, if B really is the centre of the ambit D-G# (bs 25-28) and in this way conforms to the principle sustained for bs 1-5, 9-17 and 18-21, the third C#-E is, on the other hand, above the A#. This shows how the criterion of thirds around a pivot-note alternates with that of the expansion of superposed thirds (already used in bs S9).

It is interesting to notice that the first D (bar 25) is obviously related to the B in 23, that the D in bar 26 is related both to this B and to the C: (the rhythmic expression here is important), and the D in 27 to C: only (because of octave transference); while the G: completes the B-D-G: relation, and finally, in bar 28 both D's appear and the chromatic filling in of CtE is completed with an Eb. This Eb also links the passage with the next, for it is related to the F: at the end of the first phrase (bars 15-16). Both resolve to G as harmonic centre. The Eb serves, too, as a leading note to the E of the G-SBb harmonic area which follows. Moreover, it has been transferred an octave up, and out of its normal register, because it is to influence the harmonic motion a little later on (see bars 3640).

If the idea of Eb filling in C#-E (b. 28) is consistent with other moments of the analysis (bs 1-5, 15-17), it does not, however, have quite the same meaning, since Eb is heard three bars after the last C#-E and one octave higher Wilkinson feels, moreover, the need to justify this. Finally, it seems difficult to establish a link with the F# of b. 16 on the pretext that these two notes were linked together at that point (even then, only by the hypothesis of a secondary harmonic field).

On the basis of the idea that in b. 24 Varese takes the E-C# of the opening, the following hypothesis may be advanced: after the expansion of minor thirds which characterised the first seventeen bars, then the sliding to B which introduces, we note, a major third between G of b. 17 and B of b. 18, the composer takes a new chain of thirds, starting from the first third of the piece, but introducing a major third. The Eb is thus a semitone above the last note of the passage, as are B# above B, G above F#, E above D# and Db above C:

Ei.43 bS1-17 sm b

j- bo bo

bS 18-23 \ $

10 X- °

MUSIC ANALYSIS 1: 3, 1982 O1 o


Note that if one allows a B# at b. 23, it is possible, in Wilkinson's perspective, to see a relationship between this note and the Eb of b. 28, both final notes of a phrase.

Bars 29-31 are particularly interesting, for the harmonic areas fluctuate and resolve in a most complex manner. The basic area is G-LBb, suggested in the high and middle registers of bars 29-30. They are stated completely in the low register after the octave transference of the Gb (bars 3(}31).

I prefer to take the more straightforward basis of the E and Bb of b. 31 to establish the zone E-G-Bb, since it is not clear what might suggest these two notes in bs 29-30.

The middle register, while helping to bring about this octave transference, also outlines an harmonic motion by sequence of minor thirds (see also bars S8), from E: by way of F: (bars 29-30) to G: (bar 31), and from G: to B 4 with the A 4 transferred an octave higher to complete the minor third relation (FtA) begun with the first two notes of bar 29. It is worth noting how this is implied in the low register, at the beginning of bar 31, by the melodic arrangement and phrase markings. It is the first entry of this secondary harmonic area (A-Ft, A<).

The objective of this development is to explain GSB and A of bs 31-32 which are outside the zone G-E-Bb. In order to do this, a certain importance must be given to E# and F# of bs 29 and 30. Obviously, it is the insistence of F#-A of bs 32-35 which justifies the privileged extraction of these two notes in b. 31. The overall explanation is as follows:

Ex.44 bS 29-31

i e o t - bo ' b.29' \ b. 31

'S_ 4 \

b. 29 , b. 31 4 \

> - \ bS 32-364

40

It might be noticed that the zone G-E-Bb fits into the prolongation of the stacked thirds from the opening, if the latter had not deviated towards B in b. 18. As to G#-B-A which ends this passage, the explanation by play of semitones is preferred to the rather far-fetched relationship with E: and Ft,



as the developmental process of this flight is entirely comparable to the tritonal slides in bs 11-13, unless this passage is seen as sketching an alternation between E-G-Bb and G:-BD as possible prolongations beyond E of the stacked thirds of the first seventeen bars:

Ex. 45

bS 1-5 bS 6-9 bS 9-17

j ' bo bo 44° n \ \ bS 17-23

=1$o o

bS 23-28

# (,3 o b- o

bS 29-31

O bo o

\

o ho t0 t o l

b.32

4" In this last hypothesis, the permutation zone of bs 32-36, then the (; Eb of

bs 3S38, and the D of b. 38 to the return of B in b. 40, are explained thus:

Ex.46 bS 29-31

t0 0 bo

S 00 q ° (

bs 32-36

102v " "s bS 36-3B

"s

"s S{w

b5 3 B -40

o b. b to v



Wilkinson's explanation for this whole passage is somewhat different: Bars 32-35 offer a rhythmic elaboration of the new A-F: polarity, in relation to the already established centre, B. In bar 36, C completes the new area, and the following passage (bars 3640) relates the interval polarity SEb, in a similar way, to B. This Eb is derived from the Eb as leading-note in bar 28; the weakening dynamics, the unusual octave transference which makes a major sixth of a minor third, and the resolution, in bar 40, on to a B unrelated to its own sphere at this moment, are all intended to lessen the impulse and to keep the harmony floating.

One wonders why Wilkinson returns to the Eb of b. 28 to justify the Eb of b. 37, when he could explain it in terms of a literal minor third with C, in the same way that he explains C in relation to A, ignoring the fact that the actual interval is a descending major sixth. Note that the D of bs 38-40 is interspersed with rests.

The G-E-Bb relations in bars 4145 are clear, but the low DW (bar 43) and the rhythmically important Ab (bar 44) weaken this sphere, while being connected belatedly with the B 4 of bar 40. C: acts as a leading note to the next passage (bars 4649), which develops rhythmically the B-D relation.

D may be the result of a broadening towards the low register of the pattern E#-G-E of bs 3s3 1. But above all, Ex. 35 shows the intervallic pattern to be the same as in bs 3 and 4, which integrates Ab into the analysis. This is one point among others where it is necessary to resort to an autonomous melodic analysis to explain one note. If C# really is a leading note to the following D, nothing prevents us from explaining it by a minor third relationship with Bb according to the principle of expansion: the melodic development of bs 4145, whose analogies with the beginning of the piece are shown above, is thus reinforced by a stack of minor thirds, starting from the lowest, which is identical to that seen in bs 1-9. Note that the semitonal slide (Cl D) takes us into the second zone, G#-B-D:

Ex.47

bS 1-9

t O O bo b

bs 41-45

o b0 b°

bS 46-SO

ro 40 <

After the rhythmic development of the relationship B-D: The Ab in bar 50 is connected with this interval relation (=D, KG:), as



was the C with A4-F# in bar 36; but the Ab resolves to AW and GW (bar 51). A becomes part of the A-F$ A-C relation in the high register (bars 51-55) and G is related to the E-Ct, E-G sphere in the low register. The transference of AW an octave higher in bar 55 underlines these relationships. The E: serves as a link between the two harmonic areas, since it belongs by interval polarity to the lower register, but is in fact transferred an octave higher. It is worth noticing the use Varese makes here of the sequential pattern (whole tone, semitone) of bars S8 for he now employs it to establish strongly and for the last time the E<$G sphere (bars 53-55), instead of creating a link between two areas by a sequence of minor third polarities.

This commentary may be represented thus:

Ex.48 bs 1-9

@ O O bo bo , b.51 bS 51-53

b S 4 6 5 0 , q +

bS 53- 56

' bo F w °F n°

The E# has been explained, melodically, by paradigmatic analogy with b. 29. It seems possible, once again, to join the two harmonic zones to those already identified:

Ex.4s

O

n °$ot s

X ' , ts F

/ /

) ° w t

The final cadential phrase neutralizes the relations established during the piece and covers the full range of registers. It becomes a resolution of all centres and all polarities, and dissolves the tendencies toward harmonic motion.

This remark is particularly interesting and goes well with the spirit of my



comments: apart from the fact that it is the first time in the piece that Varese uses low C, this note is a major third from E in the prolongation 'to the left' of my second harmonic zone. Bb belongs to the first zone, D and F# to the second, C# to the first and D#, E# and the final B to the second:

Es.50

j O bto bo 4ffi}

'>, tl <, b_

, 8 \ I

, , AK . . . .

,'' ,' X , '

' 1, ' ° j

There is a conspicuous neutralisation of harmonic divisions into two distinct zones. The first zone is made up entirely of minor thirds; the second mixes major and minor thirds, which explains why it contains both E and Eb and why to go beyond Eb by a major third would effect a return to the first zone. If zone 2 were prolonged by a minor third it would lead to F, then to the G# already heard. With C-E and D-F#, what I define as a second harmonic axis is not 'pure'. Why this mixture of major and minor? The octave is divisible into three major thirds and into four minor thirds. By introducing semitonal slides in the succession of thirds, mixing major and minor thirds, Varese creates a developmental process which avoids touching on notes already heard. Perhaps he is applying in this piece one of his poietic principles which Xenakis is said to have heard from Odile Vivier: 'Varese's wish for a spiral scale, that is, a cycle of fifths which would not lead to a perfect octave' (1971: 266). The explanation of the move to B as a polar note in b. 18 should doubtless be maintained here. Once the cycle of G is completed, Varese inaugurates a new one, a semitone higher, from G#. And once the zones of G, then B, are established, he constructs the development with recourse alterna- tively to one or the other.

It will have been noticed that my attempt at harmonic explanation relies on the principle of development by thirds proposed by Wilkinson for bs S8 and 29-31; for all that, the principle of development by descending or ascending minor thirds around a pivot-note has not been excluded. Why this exchange? It must be quite clear that Wilkinson's explanation, based essentially on this last principle, has the same status as mine: that is, an attempt to assemble the maximum number of phenomena around a common principle. My own



exegesis attempts to join up with another poietic datum: the spiral. Re-reading this last section in detail, it will be noticed that certain notes are incompletely, or badly, integrated into this explanation and that the door is open for other hypotheses.

Throughout this section I hope it is clear, in any case, how the poietic point of view reorganises the neutral analysis, but also how some of its elements cannot be integrated in the poietic explanation and thus retain their right to autonomous existence.

VI - ESTHESIC ANALYSIS

1. The esthesic problem In the same way that there are two main approaches to poietics, depending

upon whether one proceeds from the work to the poietics or from a poietic datum to the work, attempts at perceptual analysis may equally be classified in two categories. The first, inductive as before, consists of starting from the musical text and picking out configurations to which a perceptual pertinence is accorded on the basis of esthesic hypotheses:

| work | g esthesics |

It is supposed that a given picture of the work corresponds to what we perceive. This is the method followed by Leonard Meyer in the second part of Explaining Music (1973) and in a stimulating article by Thomas Fay (1971). The other method would start from an external esthesic datum, that is, a document which bears witness to perception- from Proust's pages on the sonata by Vinteuil, several aspects of his privileged perceptual strategies may be deduced- or an experiment performed on listeners. As in the case of poietics, esthesic analysis practiced in this way would probably lead to the modification of some aspects of a neutral analysis undertaken first, or would be projected onto the work:

| work | l or | < j esthesicdata |

| neutral level | )

In one way, experimental esthesic analysis could be considered a validation of inductive esthesic analysis.

All this is conditional because there are few, even no convincing esthesic analyses which take this direction (with the possible exception of work by Frances (1958) and Imberty (1979, 1981), but these concern the more special-



ised sector of musical semantics). It is doubtless because the experimental approach to works from a perceptual point of view is no more than embryonic that, most often, music analyses which are advanced as perceptual operate in

* @ an lnc uctlve manner. Here, both approaches will be considered. I feel that the interpretation of the

work by a flautist constitutes a testimony to his or her perception of it. This standpoint may be contested. I have often been reproached for proposing 'score analysis', but the status we give to the score in its musical realisation does not appear to have been clearly understood. In Western music, it seems absolutely obvious that the score is the composer's means of pinning his work down; it also guarantees the identity of the work from one performance to another. The score is, therefore, a symbolic fact which is absolutely essential to its transmission. The performer may thus be seen as playing an intermediary but decisive role between the written score the trace of the composer's intentions and the listener, and in this sense is the first to perceive the work, that is, to make a series of choices on the basis of the composer's text.32 Following an enquiry of that kind, I shall examine the analysis proposed by James Tenney, inductive this time, which has already been referred to in this study.

2. Interpretation as an esthesic document Naturally, musical semiology does not pretend to tell a performer how he

must play a work. If it took on a normative status it would be turning its back on its scientific intentions. Semiology can come only post festum to describe what has happened and not justify it. Since the techniques of the neutral level, when applied to a monody, define units, their result is not without interest for the definition of phrasing, above all when there are no such indications in the score. From this perspective, I have compared four interpretations of 'Density 21.5' which will be called the Zoller, Gazzelloni, Debost and Craft versions.33

I shall draw attention to a few passages where differences between score and performer, or between performers, are most apparent. Letters designate precise points in the score.

(1) [1] and [2]:

Ex.51

[1] @ 02 >

@eg J 19&1 J l

mf f mf p - S

(a) Gazzelloni ignores the secondary phrasing which delineates [1] and joins all the notes up to the last C: in a single legato phrase.



(b) Zoller and Debost make F# of [2] seem longer than it should be. (c) Debost makes only a light break between the second C# and the G. (2) [5] and [6]:

Ex.52

3_ @ 3_

XpZ J I S;. ;01 1 f fP

(d) Debost connects E of [6] to F# of [5] in the same phrase (e) and interprets the f on C# of [6] as the sign of a break which isolates

C#-G from what precedes. Gazzelloni, on the other hand, breaks the phrase between F# and E, just as indicated by my caesura between [5] and [6], but C#-G is linked to the preceding E.

(3) [8]-[9]:

Ex.53 _3 (!) @ t ]

6 j b<45> P subito Cf

(f) in the Craft version, [8] and [9] are linked like [7], which seems to underplay the difference between the phrasing of [7] on the one hand and [8]-[9] on the other.

(g) Gazzelloni slurs G and Bb of [8] and [9], probably influenced by the repetoon of G-Bb. The interest of the principle of deception lies precisely in showing how Varese frustrates the melodic dynamism with which we are now familiar: if one does not abide by the slur on Bb-C ([9]), is this not a contradiction?

(4) [11]-[16]:

Ex.s4 [110 b 10 [120 Li b.11 0140 b 12 01i L160

e Z r r tS ,3,r t s 3 t < it

Gazzellonl +6 | r r tC: 32 . i::

MUSIC ANALYSIS 1 : 3, 1982 321

@r tffltC1: tr?;3*=

-Xa t b- -r r? r r r^ltf: t 1Ce


Zo11er

Debost

9 V t 3 t 1, j . . j 3 i 3 J <

Here, the same musical fragment has been copied four times with the phrasing of each performer. Although all 'respect' [11], their choices then diverge: Debost and Craft split the music up according to [12]: Zoller, on the other hand, isolates the second C of b. 10. As for bs 11 and 12, we may say that the four versions illustrate all possible segmentations: Craft and Debost on D of b. 10, Gazzelloni on G#, Zoller on D of b. 11. This is a perfect illustration of the different weighting of variables since each of these choices means attributing a specific value to certain notes: in Craft and Debost the length of D in b. 10; in Gazzelloni, this D is treated as a pivot-note around which the low and high G#'s turn; in Zoller it is, on the contrary, the importance given to the repetition of G#-D which seems to determine the phrasing. This particular section of research shows clearly how different partitionings of the neutral level describe potentialities of esthesic pertinence.

(5) [24]: With the exception of Debost, who makes a slight break between the loud B# of b. 20 and the D of b. 21, the three other flautists join D to B# as if the slur from bs 19-20 were prolonged until b. 21. In making this passage one single segment, I have opted for the same interpretation. This is without doubt because of the trill (C#-B#) which demands to be followed by something, and also because of the move from f to ff which prolongs the crescendo begun on B.

(6) [25]-[27]: Gazzelloni ignores the three slurs which led us to distinguish three units. Note that, on a higher level, as we have seen, these three units form a whole, separated from the previous context by a quaver rest and from the ensuing context by change of register. The homogeneity of the passage is guaranteed also by the play on the same three notes: A, A#, B. We see, therefore, that even though Gazzelloni does not follow the score to the letter (the first two bars show clearly that Varese is capable of indicating two levels of phrasing if he wishes), he has opted to outline large sections (cf. his phrasing of bs 1n11 and 12) rather than small segments.

(7) [43] - [47]: Gazzelloni proceeds in the same way with bs 31-34. While Craft joins E of [43] to the legato of [42], Gazzelloni, allowing himself to be



carried by melodic dynamics, begins the legato of b. 43 on the E. In the same way, [44] and [45] are included in a single phrase as with [46] and [47], without undermining Varese's phrase-marks which are specific only in [45] and [47]. Gazzelloni continues in this vein in b. 54 by joining the last C of [75] to the slur of [76]. But on the other hand C and E of [81] will be separated from the preceding D and from [82]. Debost uses the same gesture for [44]-[45] and [46]-[47].

(8) [38]: All the performers play [38] like [40], that is with a slur up to E$t. But why then has Varese taken the trouble to establish, from this point of view, a difference between bs 29 and 30?

(9) [48]-[51]: Here there is the same problem as in [11]-[16]: the absence of phrasing gives the performers a certain freedom. Zoller includes [48]-[50] in a single phrase. Debost, just like Craft, anticipates, from [50], the phrasing of [51].

Unlike these two 'ambiguous' passages, [64]-[69] are interpreted in accordance with the partitioning proposed. The importance of D, rests and breathing doubtless combine to suggest to the different performers a common solution.

Within this list, two types of comment may be discerned: mistakes pure and simple, and divergences in interpretation on the basis of a matrix of possibilities opened up by the score. It is difficult and dangerous to establish the boundary between the two: Gazzelloni has his reasons for grouping [25]-[26] together, for example, and his interpretation in general reveals that such a choice reflects an overall bias.

In expressing my surprise concerning certain choices, for example the identification of [38] and [40], I have almost slipped from descriptive semiology into what might rightly be called music criticism. But which music- ologist can really claim to capture the 'spirit', the 'essence' of the work, for example the so-called principle of deception that appears to me to be fundamental? Then again, the following presupposition would have to allowed: any self-respecting interpretation must reflect the composer's intentions.

Influenced by the combination of New Criticism in literature and the creative renewal of the artistic avant-gardes, there has appeared a new attitude which allows that the musical interpreter, like the critic, has a right to free interpretation in every sense of the word of the works he tackles. But where is the truth, if it does in fact exist? Are not the choices in this matter oriented for the most part by the spirit and the tastes of the time? There was little negative criticism when Boulez 'renewed' the interpretation of the Sacre, turning it into something 'less Russian' than would an Ansermet or a Marke- vitch. On the other hand, not everyone agrees with the anti-romantic bias of Boulez's Parsifal. If semiology must abstain from making judgements, it is not in the name of a desire for objectivity which one might denegrate as 'positivist'. Semiological discourse, like that of criticism and all human activ-



ity, is itself a symbolic fact accountable to a semiology. The neutral level of analysis is there only to facilitate the comprehension of musical phenomena and to provide a basis for comparison.

Critical judgements which one might be tempted to make on the basis of our analysis about one interpretation or another must also be tempered. Phrasing, the issue here, is only one aspect of the variables which play a part in the performers' choices. Craft's version is perhaps the most faithful to Varese's text, and Gazzelloni sometimes takes surprising liberties,34 but the latter interpretation is perhaps the most lively and lyrical by comparison with Craft's rather dull version this said subjectively.

These remarks call not only for a semiology of music criticism, but for a semiological study of the variables at work in an interpretation; this study would constitute an important link in the chain of a methodology for esthesics which has still to be elaborated.

3. An inductive esthesic of 'Density' It is fortunate that the composer and theoretician James Tenney chose

'Density 21.5' as one of the works analysed in his article 'Temporal Gestalt Perception in Music' (1980). This title speaks for itself regarding the orientation of the article. In addition, the author systematically compares his analysis with my own. There are therefore two reasons for examining it here.

Tenney's perceptual approach can be summarised thus: When we perceive a piece of music, its temporal 'continuum' is divided into 'a hieratchically ordered network of sounds, motives, phrases, passages, sections, movements, etc.'. Tenney calls these perceptual units 'temporal Gestalt units' or 'TGs' (1980: 205). A detailed study of all the theoretical and methodological implica- tions of Tenney's diverse propositions would require an entire article. For our purposes, the following aspects will be retained:

(1) His objective is to 'predict [my emphasis] where the TG boundaries will be perceived' (:206). This really is, then, an inductive step, starting from hypotheses about musical perception.

(2) Tenney mainly takes the following perceptual datum as a basis: 'The perceptual formation of TGs at any hierarchical level is determined by a number of factors of cohesion and segregation, the most important of which are proximity and similarity' (:208).

(3) More concretely, and calling on the ordinary experience of a listener, he considers that in a monodic piece temporal and pitch-class intervals which are greater than those immediately preceding or following them create the TG boundaries (:208-9). To these two criteria Tenney adds the role of dynamics.

(4) The model takes into account neither 'harmonic relations between pitches or pitch classes' nor 'motivic/thematic relations' (:217).

(5) The author is conscious of the fact that each variable has a specific weight, but notes the impossibility of providing an adequate measure



of it: 'We have no way of knowing, a priori, the relative importance of one parameter verSus another, in its effects on TG-formation. As yet, no clear principle has been discovered for determining what the weights should be' (:211-12). And he refers, not without reason, to the need for acoustic psycho-research. But his model is dealt with by computer. Explicitly, his method becomes algorhythmic. He has thus had to neutralise the problem of weight by simply taking the sum of temporal, pitch-class and dynamic distances.

The comments Tenney devotes to his analysis and mine, and also the table where he compares our two segmentations, are quoted below. Note that circled numbers in his analysis designate the number of each event (notes and rests are divided up individually):

The segmentation given by Nattiez for this piece is shown in the lower portion of Ex. 55, so that a direct, point-by-point comparison can be made. Here the correlations between the two partitionings are quite close

especially at the clang- and sequence-levels although the two are not identical, of course, and the similarities diminish at higher levels. In fact, some 81% of the clang-initiations in our results, and 85% of the sequence- initiations (but only 44% of the segment-initiations) coincide with the corresponding boundaries in Nattiez's segmentation. There are no coinci- dences at any higher level. Some of the discrepancies between the two segmentations are fairly trivial, as where one of two 'models' simply interpolates an extra clang-break between two otherwise coincident boundaries (as at elements 8, 25, 54, 109, 117, 118, 140, 179, 224, 226, 233, and 241). A few differences result from the fact that Nattiez does not prohibit one-component TGs, as our model does. These occur in his segmentation in the form of 'one-element clangs' beginning at elements 109, 117, and 118, and as sequences containing only a single clang, beginning at elements 22, 52, 74, and 97.

Even if we disregard such discrepancies as these, however, there will still remain a number of places where the two segmentations differ. Some of these probably have to do with the fact that neither harmonic nor motivic factors are considered by our algorithm. For example, the high- level TG-initiation which Nattiez locates at element 188 is clearly determined by the fact that the initial motivic idea of the piece suddenly returns at this point, and a model which included some consideration of motivic relations might well yield a result here more like Nattiez's. On the other hand, the strong element of surprise that this return of the initial motive evokes in my perception of the piece suggests that this motivic factor is here working very much 'against the grain' of most of the other factors of TG-organization, and that an important part of the musical effect of this even in the piece depends on the fact that motive recurs at a point which would not otherwise be a high-level boundary.

After all of the foregoing reasons for the differences between the two segmentations have been accounted for, a few discrepancies will remain which suggest that our weightings may not be quite 'optimum' after all, or that they are simply different from those unconsciously assumed by Nat-


11 13 16

l ls -7 1@ 1S 1t §7

r r

tft < P f ;J tJ ti Xt p izte

HIERARCHICAL j |R 13 4 5 6 7 |!

STRUCTURE I In m X ar

FROM NATTIEZ A B (1975) lSt¢t

N 6 7 16 8° 0}s 1 123 p>+ 1s 1}& 1Sz 1j! 1{9 1nO t; te

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tiez, or even that some aspect of our algorithm may need refining. Finally, however, I must say that I think our segmentation represents the perceptual 'facts' here more accurately than Nattiez's at certain points. These would include the clang-initiations at elements 13, 20, and 75, and the sequence-initiations (and perhaps even the segment-breaks) at 177 and 238 (Tenney 1980: 221).*

Ex.55

SECTIONS-

S EGMENTS:

SEQUENCES:

Cl.ANGS - 1,

A I ,s, X 1, I X ;:79" ;

* The Editor is grateful to Professor Martha Hyde for permission to quote text and music (Ex. 55) extensively from the 70urnal of Music Theoty, Vol. 24, No. 2.


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At the end of his commentary Tenney says: 'Finally, I must say that I think our segmentation represents the perceptual 'facts' here more accurately than Nattiez's at certain points' (:221). It is the five contentious points then enumerated by Tenney which will now be examined in detail. It must be emphasised first of all that the author does not appear to have taken account, in his criticism, of my methodological premises, particularly the distinction between the neutral and the esthesic: my segmentation is not necessarily supposed to be esthesically pertinent.

(1 and 2) Elements 13 and 20 in Tenney: [4] and [6] in Nattiez

On his graph, Tenney retains only one of the two alternative segmentations I have proposed (cf. Ex. 1, B above). Nevertheless it is interesting to note that Tenney's different criteria for the segmentation of Cit-G are no less legitimate. On the other hand, I should find it difficult not to register the return of the melodico-rhythmic figure of the opening in b. 4, in [5].

(3) Element 75 in Tenney: [24] in Nattiez From an immediate perceptual point of view, the ascending compound major third leap and the f obviously create a break. Nevertheless, the slur which links the trill to B does constitute an argument for the existence, at this point, of a unit. Ex. 17 shows, iough, how this trill is outside the paradigm. The analogy I have indicated with [28] could also be invoked to make Cit-B:-D an independent unit. The confrontation here of these two analyses demonstrates unequivocally how analysis as symbolic fact depends on the respective weight given to different variables.

(4) Element 177 in Tenney: [55] in Nattiez The divergence with regard to [55] brings to light the difference in our approaches. [55] belongs, with [54] and [56], to a larger sequence II. For Tenney, on the contrary, this is the beginning of a segment, a unit higher in the author's hierarchy. What justifies this strong partitioning? The contrast between ff and p and the length of the rest (crotchet) have 'increased' the weight of the break in the quantification of variables. It is here that we realise there is doubtless no direct relation between the quantification of weight of parameters and real perception. Throughout this passage, despite both the rest and the dynamic change, is not Varese's play on permutation of the same notes (D- Eb-Db) the dominant criterion for the homogeneity of [54]-[56]? On the contrary, how is it possible to miss perceptively the return to the initial unit in [59] ? How can the two rests of this unit be stronger than the repetition of [60], to the extent that Tenney's element 59 is more cut off from what precedes than his element 58? 'It now appears that such optimum weightings are slightly different for each piece analysed' (:212). There is every reason to think they vary from one point to another in the same work.



(S) Element 238 in Tenney: [75] in Nattiez

Here, his criteria lead him to establish a huge break on the fortissimo E. Certainly, [75] is a unit of transition, but for the same reasons as in point (4) we are unable to separate it from what precedes ([71]-[74]) hence segment I at a higher level since [75] ends the fall from high C to low Ci$: begun in [72], delayed but also reinforced by the tension of the E#'s in [71], [73] and [74] (F enharmonically). The paradigm reveals, besides, the role of the axis EtE (a semitone lower), then the function of C# in relation to C a semitone higher ([71], [72], [73] and [74]) which thus attaches [75] to the four preceding units.

What may be learned from this confrontation of approaches? There is no doubt that Tenney's basic criteria are perceptual. But the neutralisation of the relative weight of variables and the non-consideration of motivic relationships do not permit acceptance of his segmentation as more esthesically pertinent than mine. Our two approaches have a crucial point in common: clarification, But they diverge on a no less decisive epistemological question. Tenney takes the 'objectivity' of the informational approach as a basis, whereas I feel it necessary to adopt a semiological approach: perceiving is a symbolic act, exactly like music analysis. The idea of a block with which I began, less rigorous but more sensible, turns out to account more adequately for the changeable character of the hierarchy of variables in the course of a piece. Paradigmatisation takes on the flexibility of the symbolic. From now on, the most important thing is not to carry out an analysis by computer, but to make the analytical criteria explicit not excluding others which might be possible which account for the symbolic character both of perception and of analysis.

What of an inductive esthesic analysis? Still possible, starting from a neutral analysis, but only once general perceptual strategies are better known (through experimentation): in this way we shall be able to interpret the data from the neutral analysis more adequately.

VII - COMPARISON OF ANALYSES The neutral level is not simply a fundamental datum for poietic and esthesic

approaches. In as much as music analysis is, itself, a symbolic activity, it is epistemologically crucial to compare different analyses of the same work.35 What right has the neutral level to serve as a basis for this comparison? It is distinguished from most common procedures on the one hand by its goal of exhaustivity or, more exactly, its refusal to look at things from an a przorz privileged point of view, and on the other hand by the integration into its text of the most searching clarification possible of the method used; this has the effect of combining the results of the analysis itself with a level of methodological metalanguage that may be projected onto it.

From a fairly general point of view, I believe it is possible to classify the analyses of 'Density' in two categories:



(1) If the circumstances of composition36 and realisation are set aside all this is more the concern of music history than of analysis commentaries are mostly 'synthetic' and often quite short. On record sleeves, for example, it is a question of grasping a few characteristic traits of the work without being able to go into detail: this is no reproach properly done, such notes presuppose a great deal of insight. In addition to the texts which accompanied the performances examined in the previous section, I shall review the pages devoted to 'Density' by Odile Vivier in her article 'Innovations instrumentales d'Edgard Varese' (1955: 193) and in her book on the composer (1973: 114- 17), Halbreich's brief analysis in the conversations of Varese and Charbonnier (1970: 150-51), and the remarks by Milton Babbitt in his article 'Edgard Varese: A few Observations on his Music' (1966: 18). In the context of an 'analysis of analyses', these texts are of great interest because they show which particular traits have been privileged in order to capture the dominant characteristics of 'Density'.

(2) The other type of analysis is the kind of detailed study which follows the work step by step, just as I have done here. This type of published analysis is relatively rare for reasons given at the beginning of this study. Apart from Wilkinson's work already quoted and discussed, there is only one other article on the whole piece: 'Versuch an Varese Density 21.5' by Martin Gumbel (1970).

The study of diverse commentaries provides an answer to a question which is often asked: what does an analysis in the semiological perspective but one might just as well say any music analysis which is fairly precise tell us that we did not already know? This question thus concerns the cognitive value of this type of approach. The analytical elements will be classified according to some points tackled by the different authors.

(1) Melodic progression

Vivier (1955: 193 and 1973: 114): 'It is a pure melody which grows around certain pivot-notes, cast alternately in mirror chromaticisms and very disfunct intervals'.

For Halbreich, the pivot-note of the opening is Ft: 'Everything is born of the first bars: the intervals broaden progressively either side of the pivot Fi$:, sound moving off in quest of its rightful pitch, and then its dynamic level' (1970: 151).

When Vivier talks about mirror chromaticisms, she is alluding to the inversions F-E-F: / Ft-G-F or B-At B / B:n:-Ci$:-B:n:. The words 'very disfunct intervals' refer to all that is not conjunct chromaticism. There is certainly a broadening of intervals, as Halbreich says, and a conquest of wider and wider spaces (high notes; maximum range with the introduction of C in b. 56). There is therefore nothing wrong in all that. The attribute of a precise taxonomic method, or, in Gumbel's case, of a statistical analysis this presents difficulties, but I shall return to the problem later is to show the



wherefore of these general statements. In my analysis the paradigms show all the transformations based on [ 1], that is, show the process of melodic progression; in Gumbel (1970: 37) a diagram shows how the progression in use of dynamics is parallel to the evolution of the ranges used.

Can it be said, however, that 'everything is born of the first bars'? This is true only on the basis of the developments of the initial motive, but the technique of systematic clarification of the development shows that not all the relationships between units are transitive, which Halbreich's phrase, if taken literally, might imply: he expresses an overall feeling of expansion and progression from the basis of the initial cell. Finally, one might ask why the initial pivot-note should be Ft: the paradigmatic tables show, on the contrary, the importance first of Ct, then G.

In her book, Vivier shows specifically how the progression is organised: 'The fundamental structure, the skeleton, is composed of a chromatic ascent, always underlined by a long value, but with a freedom greater than in the chromatic descent of Octandre' (1973: 114). With the aid of these long 'prop' notes, the author reconstructs the following chromatic scale:

Ex.56

'G: is out of order in the scale', she continues, 'but appears in a high register and is repeated in an arpeggio movement after the held D; it also prepares Dt, which is presented only as a dotted crotchet, but emphasised by a sforzando and followed by a breath it thus reverberates longer in the ear'. The missing B appears only in b. 18. 'Already in Octandre', she adds, 'the note missing from the twelve sounds makes its absence felt'. Note that this progression is described only up to b. 23. Short analyses sometimes give the illusion

this general complaint is not directed particularly at Vivier of having captured the 'essence' of the piece in describing an important moment, while the work taken as a whole presents other modes of progression. If, in a two-page text which does not claim to be exhaustive, Vivier's analysis is adequate, it is because the passage described is symptomatic of a general tendency to ascent. Note also that the 'scale' she proposes is not quite the same as mine. Perhaps influenced by serialism, she attempts to show the presence of twelve notes, whereas I have produced a kind of mixed scale: diatonic for bs 1-8 (G-A-Bb-C), then chromatic up to b. 17. Ft, therefore, did not figure in the beginning: had I inscribed it, it would have had the status of an appoggiatura, or a leading note to G.

(2) Rhythmic diversity Vivier: 'Rhythmic values, of great variety, set short notes and long held notes



in opposition with each other, making subtle use of the triplets which have been falsely called "irrational values" ' (1955: 193; 1973: 115).

Halbreich: 'What suppleness, what variety in rhythmic invention, brought to life by the very breath of the life-pulse' (1970: 151).

Certainly, but the important thing is to show how. The rhythmic typology I have proposed comes in to complete or to fill out Vivier's comment on shorts and longs. It has been seen how the poietic point of view could assign a place to the triplets.

(3) Processes of variation Babbitt:

There are, I believe, no two identical measures in Density. The durational succession associated with the attack points of the initial three pitches occurs, in the same metrical orientation, only at two further places in the work, and at those places is associated with the opening interval succession also, but the pitch succession is altered in each case by transposition (1966: 18).

A precise remark, and a correct one, which taxonomic description completes for the rest of the piece. In his article, Babbitt is concerned simply to capture Varese's stylistic tendency to diversity. He adds:

Varese is one of those composers . . . whose music has necessarily directed our attention to the inadequacies of our analytical concepts with regard to rhythm, by decreasing compositional rhythmic redundancy, by increasing the number of rhythmic configurat ons, and the dimensions in which these configurations are made to appear(1966:19).

Hopefully, typological classification of rhythms has accounted essentially for this rhythmic specificity, which eludes the usual analytical techniques.

(4) Register and the 'polyphonic' aspect of the piece Vivier: 'The use of the different registers of the flute is remarkable because

they are combined with different modes of intensity and dynamic levels. In certain passages the mode of attack and the dynamic change on every note. An echo effect, or, more precisely, a feeling of expansion and relief between distanced planes is created by changes of register linked to opposing dynamics: a high register fortissimo is succeeded by a medium register piano subito, or again a medium-low register "enfle" forte is followed by a high note, piano subito, repeated three times after ornamentation. Several instruments seem to be answering one another, several instruments, not several flutes, since certain percussive effects go beyond the sound world which we might customarily have expected from a flute' (1955: 193; 1973: 115).

Halbreich: 'With the aid of this unique instrument, Varese conquers a new sound space, combining oppositions of register, dynamics and agogics in such



a way as to give the illusion of several instruments answering one another. The percussive effects of bs 2F28, often imitated since, open the way for a genuine flute equivalent to violin pizzicato' (1970: 151).

The link Vivier establishes between registers and dynamics is illustrated by two examples. It therefore constitutes both the expression of a dominant impression on the part of the analyst and a hypothesis for study: it would suffice to compare two separate columns (for the definition of register and the enumeration of dynamic levels) which we would add to our general table, in order to verify up to what point this correlation is correct.

(S) Formal organisation In Vivier's book the piece is divided into three parts: the first goes to b. 23,

that is until the appearance of B has completed the first eleven sounds introduced by long values. Bs 2F28 (percussive use of keys) constitute a 'central interlude'. Then 'three very fast arpeggios announce the return to normal breathing in a melody which becomes more and more ascetic with its wide intervals. A modified reprise of the initial element (b. 41) brings in the final part, particularly disjunct, which ends its development in ascending movement over nearly three octaves in a crescendo tofortissimo' (1973: 115).

For Halbreich, there are three periods, 'the first two of which are separated by the strange "percussive" interlude of bs 2X28. The second (bs 2940), intensely exultant, throws out an unutterable summons to night, flames of light rising against the temptations of despair. The third, a cumulative synthesis, typical of Varese, takes the quintessence of the first two and culminates, once again, in a desperate fortissimo in the upper extreme of the register' (1970: 151).

The authors agree on the essenfial division into three parts and on the intermediary role of the percussive section. My own analysis coincides with these views.

Martin Gumbel presents a quite different but subtle point of view. His objective is to show that traditional analysis cannot account for this piece and that the statistical approach picks out a more fundamental aspect. 'The question will not be addressed here whether the use of inadequate analytical metilods can determine, iniluence or even quite falsify the result of an analysis in an unreliable way' (1970: 31). One might ask also whether the traditional formal analysis Gumbel proposes is not a little contrived (1970: 31-32). This is the overall segmentation he proposes:

MUSIC ANALYSIS 1: 3, 1982


Parts Formal Bars Function Overall form sections

Part I

1 a 1 Exposition 0 A 2 a1 2, 337 Evolution, Exposition

variation 3 b 6, 1 Contrast 5

transformation I 4 a1 9, 1 Exposition A 5 b1 15, 1 Evolution, Evolution

* .

varlatlon varlatlon

6 b1 19, 4 Contrast transformation

7 24, 1 ) B Contrast Transformation

Part II

8 a2 29, 1 Exposition (?) i A 9 a2' 32, 3 Variation (?)

Evolution (?) 10 b2 36, 2 Transformation (?) 11 a3 41, 4 Exposition ) A1' 12 a3' 46, 1 Variation (?) 5 Evolution 13 b3 50, 3 Transformation (?)) Variation (?) 14 53, 1 j B1

5 Contrast and 8 transformation (?)

Gumbel discovers an exposition classically divided into three, according to the pattern a-a'-b. The first large section of his Part 1 ends on C of b. 8: everything depends on the principle adopted for segmentation and the chosen privileged variable. For me, the first five bars formed a whole because of the stagnation on G. Gumbel privileges the return of the initial motif in b. 9. What appears more difficult to sustain is the parallelism he establishes between Parts 1-3 and F6 and for the entire piece between A-A-B and A'-A1'-Bl. The author's basis is essentially the returns of the initial motive. As this is not supported by a detailed analysis of the relationships between units, it is obvious that 'contrast and transformation' cannot have the same

MUSIC ANALYSIS 1: 3 , 1982 334


meaning for b1and B or B1 of sections 6,7 and 14, etc. But he himself doubts the pertinence of this segmentation (see the numerous question marks). He considers the pattern a-a'-b too simple and is not sure that the ostinato sequences in bs 32-35 or bs 4g50 are transformations of the opening motivic material (1970: 32). Perhaps, in grafting a historically dated pattern onto 'Density', the author has taken a rather facile example to demonstrate the legitimacy of the stylistic approach. This does not mean that the latter is not pertinent, but the demonstration would have had more force if the segmentation had respected the 'natural articulations' of the musical text. Moreover, Gumbel's position remains ambiguous: despite his misgivings about formal partitioning, he considers the division into fourteen parts as 'quite practicable' and it is in relation to the units thus defined that he proceeds to make a certain number of statistical calculations.

If the functional scope of his partitioning and the more or less precise criteria which motivated it are ignored, there remains a framework whose legitimacy might be questioned: why, for example, does Part 5 cover bs 15-19? The rise to high G and the first appearance of B are totally neglected. It is also difficult to understand why Part 13 begins in the middle of b. 50. This is relatively serious, since a statistical calculation, like every type of description, is always relative to its given field. The partitioning in units is fundamental in that it conditions the validity of all that is said about the piece.

The principle of Gumbel's analysis is, in itself, perfectly legitimate: he draws up diagrams of the evolution of pitches, intervals, durations and dynamics to show the progression of these parameters and the correlation between pitches, range and dynamics. His analysis shows 'a form of development . . . according to a differentiated and complex process which cannot be reduced to letters or verbal denominations' (1970: 38). In pointing out the difference between conventional systems (like serialism, which Varese rejected) and principles, Gumbel's analysis ties up with the words of Varese at a conference in Princeton in 1959: 'Form is a result, the result of a process. Each of my works determines its own form . . . My music cannot be put into any traditional musical-box' (Charbonnier 1970: 85).

We are now in a better position to evaluate the scope of these different analyses. They are rarely wrong in any literal sense; they simply do not have scientific status-nor do they claim this because it would be impossible to reconstruct the work from their proposed characterisations.38 If the ideas of 'mirror chromaticisms' or 'very disjunct intervals' are to be meaningful to the listener, the piece must already be known. What taxonomic analysis provides is not the overall conclusions which an intuitive approach might often enable us to pinpoint, but the wherefore of these conclusions.

It is possible that we have here one of the semiological characteristics which distinguishes the different metalanguages of musicology: without description, conclusions are like words deprived of their referents. Because they are too general, these appraisals become empty. This is why they are on record sleeves, or directed on a relatively simple level, at music lovers who have



personal knowledge of a piece. The 'known' work is the object of such appraisals, whereas in any scientific analysis the referent is not the work as it is experienced, in amorphous form, but the work as it is organised, already distinct from its immediate perception through characterisaiions which are explicit at different levels.

My critical observations do not seek to discredit the work of anyone, but I am opposed to the mixture of genres, and, while it may be obvious that a study is not addressed to record-buyers or interested music-lovers buying a book of information or to professional musicians, epistemology nevertheless has the right to a definition of the differences between 'musico-graphic' commentary and rigorous analysis. We have insistently evaluated the level of truth of the general characterisations examined, and since truth there is, let us insist on the fact that they may very well mention important facts which escaped the taxonomic decomposition. In this sense, they may offer hypotheses which are verifiable by systematic confrontation with the detailed analysis.

The epistemological point of view adopted here, and the semiological perspective I have attempted to illustrate, do not therefore seek to make watertight divisions between different modes of analysis, but rather to suggest a framework which specifies their respective scopes and merits. While there may be no unique and glorious road to the attainment of musical knowledge, it is still true to say that the description, classification and distinction of the phenomena studied, like the techniques used to account for them, allows the introduction of order and clarity where confusion may reign. This, in the end, is the objective I am pursuing. I believe that its pedagogical value is self- evident.

REFERENCES

Babbitt, M., 1966: 'Edgard Varese: a Few Observations on his Music', Perspectives of New Music, Vol. 4, No. 2, pp. 14-22.

Charbonnier, G., 1970: Entretiens avec Edgard Varese, Paris, Belfond. Deliege, C., 1975: 'Webern: Op. 10, No. 4; un theme d'analyse et de reflexion',

Revue de musicologie, Vol. 61, No. 1, pp. 91-112. Fay, T., 1971: 'Perceived Hierarchic Structure in Language and Music', 3rournal of

Music Theory, Vol. 15, Nos 1-2, pp. 112-37. Frances R., 1958: La perception de la musique, Paris, Vrin. Gardin, J.C., 1974: Les analyses de discours, Neuchatel, Delachaux et Niestle. Guertin, M., 1981: 'Differences et similitudes dans les Preludes pour piano de

Debussy', Revue de musique des universites canadiennes, No. 2, pp. 5S83. Gumbel, M., 1970: 'Versuch an Varese Density 21.5', Zeitschrift fur MusiEtheorie,

Vol. 1, No. 1, pp. 31-38. Halbreich, H., 1970: 'Etude de l'oeuvre d'Edgard Varese', in Charbonnier, 1970,

pp. 12147. Herndon, M., 1974: 'Analysis: Herding of Sacred Cows?', Ethnomusicology, Vol. 18,

No. 2, pp. 21942.



Imberty, M., 1970: 'Polysemie et coherence du language musical - I: La polysemie dans les reponses verbales associees a la musique et la construction d'une echelle circulaire des expressivites musicales', Sciences de l'Art, Vol. 7, Nos 1-2, pp. 75-90.

1979: Entendre la musique, Paris, Dunod. 1981: Les ecritures du temps, Paris, Dunod.

Jolivet, H., 1973: Varese, Paris, Hachette. Lidov, D., 1977: 'Nattiez's Semiotics of Music', The Canadian3rournal of Research in

Semiotics, Vol. 5, No. 2, pp. 13-54. Meyer, L.B., 1973: ExplainingMusic, Chicago, UniversityofChicago. Molino, J., 1975: 'Fait musical et semiologie de la musique', Musique en3reu, No. 17,

pp. 3742. 1982: 'Un discours n'est pas vrai ou faux, c'est une construction symbolique',

L'Opinion, January 8 & 15, 1982. Nattiez, J.-J., 1973: 'Quelques problemes de la semiologie fonctionnelle', Semiotica,

Vol.9.No.2,pp. 157-90. 1974a: 'Sur les relations entre sociologie et semiologie musicales', International

Review of the Aesthetics and Sociology of Music, Vol. 5, No. 1, pp. 61-75. 1974b: 'Pour une definition de la semiologie', Languages, No. 35, September,

pp. 3-14. 1975: Fondements d'une se'miologie de la musique, Paris, 10/18 (English translation

. . \

n preparatlon). 1979: 'A propos de Schoenberg: les problemes de la construction du modele

poietique en semiologie musicale', 2nd Congress of the Association Internationale de Se'miotique (Vienna, July 1-6), 'Actes du Congres', to appear. 1982: 'Problemes de la poietique en semiologie musicale: quelques reflexions a

propos du "De Natura Sonorum" de Bernard Parmegiani', in L'atelier du sonore, Paris, Buchet-Chastel.

Nattiez,J.-J. andHirbour-Paquette, L., 1973: 'Analysemusicaleetsemiologie: a propos du Prelude de Pelleas', Musique en 3reu, No. 10, pp. 4249.

Naud, G., 1979: 'Pour une methode d'analyse des analyses', Address to 1st Congress of the Association Internationale de Semiotique (1974), in Chatman, Eco, Klinken- berg, eds: Panorama semiotique, The Hague, Mouton, pp. 1015-18. 1975: 'Apercous d'une analyse semiologique de Nomos Alpha', Musique en 3reu, No. 17, pp. 63-72.

Ouelette, F., 1966: Edgard Varese, Paris, Seghers. Ruwet, N., 1972: Langage, musique, poe'sie, Paris, Seuil. Stefani, G., 1974: 'Sur l'approche fonctionelle des pratiques musicales', International

Review of Aesthetics and Sociology of Music, Vol. 5, No. 1, pp. 77-82. 1976: Introdazione alla Semiotica della Musica, Palermo, Sellerio.

Tenney, J., 1980: 'Temporal Gestalt Perception in Music', Zournal of Music Theory, Vol. 24, No. 2, pp. 20541.

Varese, E., 1930: 'E. Varese y la musica de vanguardia', interview by Jose Andre, La Nacion, Buenos Aires, April 20.

1934: 'Varese and Contemporary Music', Trend, Vol. 2, No. 3, May-June, pp. 125 ff.

1959: Conference at Princeton, in Charbonnier, 1970, pp. 83-86. Fuller version in Liberte' 59, Vol. 1, No. 5, September-October, pp. 276-83.

1983: Ecrzts, Paris, Bourgois (in press).



Vivier, O., 1955: 'Innovations instrumentales d'Edgard Varese', Revue musicale, No. 226, pp. 188-96.

1973: Varese, Paris, Seuil. Wilkinson, M., 1957: 'An Introduction to the Music of Edgard Varese', The Score,

No. 19, pp. 5-18. Xenakis, I., 1971: FormalizedMusic, Bloomington, Indiana University.

NOTES

1. See, for example, Gilles Naud's article on Nomos Alpha (1975) or Marcelle Guertin's study of Debussy's Pre'ludes (1981).

2. Perhaps like the passing on of the teachings of Oliver Messiaen and Nadia Boulanger, hardly recorded at all except in the memory of those who heard them.

3. On semiological tripartition cf. Nattiez (1974a, 1974b, 1975), Naud (1975), Molino (1975, 1982).

4. This analysis was the subject of several seminars in the M.Mus. Semiology course at Montreal University in 1974. I thank all the students, the members of the Groupe de Recherches en Se'miologie Musicale, especially Gilles Naud who gave me concrete help at the beginning of this study, and colleagues Louise Hirbour- Paquette and Jean Molino for critical comments which contributed to modifying the content of this analysis. James Tenney's analysis published in 1980, and the fruitful conversations with him in June 1982, led me to extend Section VI devoted to esthesic analysis which now contains his analysis. Following a suggestion by David Lidov (1977: 44), I have removed from the text all reference to Pike's distinction between 'etic' and 'emic' units. It is not that these terms have no place in musical semiology, simply that their presence is, in this context, superfluous. These two words deserve, furthermore, a profound conceptual study which will be undertaken elsewhere. The first edition of the present study, in French, was part of a project in semiological music analysis sponsored by the Conseil des Arts du Canada (No. S73-1826).

5. The score above will give an initial overall view of the different levels of segmentation.

6. The numeral 3 below the last semiquaver of [1] indicates that it belongs to a triplet.

7. This is an example of the necessity underlined by Ruwet (1972: 114) for performing an analysis both from bottom to top and from top to bottom. The presentation of the analysis could have begun by delineating segments I, II and III on the basis of three criteria: (a) the similarity of the initial notes of [1], [3] and [5], (b) the identical final notes of [2], [4] and [6]: C#-G, and (c) the rest between [2] and [3].

8. The 'short' and 'long' values in this paradigm can obviously not be put onto the same footing as the shorts and longs of Ex. 2.

9. This can be seen in characteristic fashion in the opening of Integrales (cf. Nattiez 1975: 285-97). One additional comment: in Fondements I advocate seriation, while this monograph is devoted to a single work. The example of rhythmic type which concerns us here clearly shows how one trait cannot be considered peculiar to a single work unless the field of works studied is widened. It is obvious that here we have touched on one of Varese's stylistic traits.


VARESE'S 'DENSITY 21.5: A STUDY 1N SEMIOLOGICAL ANALYSIS

10. It is up to future research to show whether it is found elsewhere in Varese. 11. Analyses of Le Sacre du Przntemps by Boulez, of the Prelude a l'apres-midi d'un

faune by Austin and of Hungarian folk monodies by Erdely, and see Nattiez 1975. 12. Discussion of analyses of the Prelude to Pelleas (Nattiez-Paquette 1973),

Debussy's Syrinx, beginning of Varese's Integrales and Brahms's Intermezzo, Op. 119, No. 3 (Nattiez 1975).

13. It is not my intention to make a systematic comparison of musical semiology and the use of the computer in analysis. We are talking here of a hypothetical attempt at automatic analysis of monodies alone. All of this section II is found in the original text of my analysis (1975). Tenney's computer analysis shows evidently that it is possible to ask the machine to segment other than on the basis of paradigmatic association. But I believe that everything said here explains why it was impossible (or at least difficult) to integrate the recognition of identical or analagous motivic units into his model. See particularly, below, the ideas of block, amalgam, quasi-criteria and mixture.

14. The rest of this paragraph is a summary of verbal observations made by Jean Molino (May 1974).

15. For example, dynamics could be added here. 16. On the other hand, the paradigmatic connection of [13] and [1] isolates [13] from

[12]. 17. The sign + indicates that the interval is compound. 18. This is perhaps an argument in favour of Deliege (1975: 93), who considers the

tritone to be a chromatic interval but does not explain why. 19. Two exceptions: in bs 57-59 this final 'localisation' is, in fact, characteristic

and at the join [37]-[38] in compound form. 20. This fact has been noted by musicologists. Cf. in particular Vivier 1973: 114. 21. Cf. the extracts from Vivier and Halbreich quoted here in Section VII. This is not

the case for Tenney, whose model does not pick up motivic analogies. 22. In numbers of semitones: bs 12-13: 6 6 6 12 1 3; b. 16: 1 1++; bs 31-32: 6 10 3

10;b.44:733;bs58-61:46447226. 23. More simply, one may say that section C ends with the descending movement

A-F#-B while it had begun ([44]) by the ascending movement B - Ft A. 24. This process explains why [55] is not divided into two units of two notes like [57]

and [58]. 25. That is, a unit of more than two notes which goes from the longest to the shortest

value. 26. This unit has not been described as a flight because, unlike the cases mentioned

above, it is characterised by a diminuendo. 27. Note that this table conforms to paradigmatic principles, since the syntagmatic

succession of all the types may be found by reading everything from right to left and from top to bottom.

28. Gino Stefani (1974: 82) stated that he took from Nattiez 1973 the idea of it being necessary to go from the material studied to the function and not the contrary. Later Stefani insisted on the functional aspect of analysis (1976).

29. We come to the same conclusion here as does Imberty when he projects semantic characteristics, obtained experimentally, onto sound material: 'The traits of musical structures are, in the end, only pertinent for a given factor, although they may be present in others' (1970: 92).

30. Although couched in different terms, this is the idea of the 'conventional matrix'



expounded by Jean Molino as one of the elements of poietic knowledge. 31. Varese would doubtless agree: 'The role of creation, in every art, is to reveal a

new world, but the creative act itself escapes analysis. The composer is no wiser than anyone else as to where the substance of his work is coming from and it is only as a craftsman that he can speak coherently about it' (1959: 283). But this is no reason to ignore or to dismiss the poietic dimension of works as semiological fact. It should simply be expected that poietic analysis (which, unlike neutral analysis, deals with processes and not structures) will encounter specific difficulties and above all will not resemble neutral analysis, if only because of the gaps in its data. On this subject see Nattiez (1979) and (1982).

32. For more details on these problems see Nattiez (1975: Pt. 1, Ch. 5, and pp. 109-17).

33. Karlheinz Zoller, HMV, C 061-28950; Severino Gazzelloni, Virgo, 89836; Michel Debost, Angel, S-36786; Robert Craft (the name of the flautist is not mentioned), Columbia, MG 31078.

34. For example, the two quavers of b. 13 are played as double-dotted quavers, influenced, it would seem, by the long-short rhythms of the previous bar.

35. I have already devoted a systematic study to this type in the Prelude to Pelleas (Nattiez-Paquette 1973). An identical tendency is coming to light in other sectors of musicology (cf. Herndon 1974).

36. For 'Density', see Ouelette (1966: 147-148) and Hilda Jolivet (1973: 109-11). 37. A figure after a comma in Gumbel's table indicates the division of the bar in

crotchets. 38. On this epistemological criterion of analysis, cf. Gardin (1974: 107-14).


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A Study in Semiological Analysis

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