Ligeti Triads

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    I

    A remarkable feature of Gyo rgy Ligeti's music dating from the late 1970s isthe return of triads, seventh chords and other traditional tertian harmonies.In the 1960s, he received acclaim for works such as Atmosphe res and theRequiem, which stressed the textural and timbral shaping of clusters at theexpense of harmony as it is traditionally understood. In contrast, his latest

    compositions make frequent use of triadic sonorities, albeit placed inunfamiliar contexts (one would be hard-pressed to describe his recent musicas neo-tonal).1 The triad, previously scorned for its recollections of tonalpractice, is now tolerated. Manifestations of this tendency can be seen inworks such as Hungarian Rock, Passacaglia ungarese, the piano e tudes Arc-en-ciel and Cordes a vide, and the Horn Trio. Perhaps the clearest instance ofhis new approach to harmony occurs in the opening section of his fourthe tude, Fanfares (shown in Ex. 1).2

    The harmonies in phrase 1 ofFanfares are a result of the coincidence of the

    ostinato in the left hand and the more melodic dyads in the right hand. Theostinato (which runs throughout the entire piece) is composed of a diatonictetrachord and its transposition by a tritone the first tetrachord begins on c(c, d, e, f), the second on f (f, g, a, b). This simple bipartite structureconflicts with the asymmetric metre, which divides the bar into three unevenbeats of 3+2+3 quavers (these units are brought out by accents in theostinato), accenting the notes c, f and g. The right hand's dyads coincide withthese accented ostinato notes. From the perspective of set theory, the resultantsimultaneities are trichords of set-class 311. Later, the addition of a fourth

    note to the texture expands the harmonic palette to include 427 tetrachords inphrase 3, 310 trichords in phrase 4 and 420 tetrachords in phrase 5. From theperspective of traditional harmonic practice, these chords are nothing otherthan a series of triads and seventh chords major triads in phrase 1, minortriads in phrase 2. In phrase 3, the addition of an extra note to various chords(placed a third below the root of the triads) introduces minor seventh chords.In phrase 4, diminished triads first appear. And in the final phrase in thissection (phrase 5), major seventh chords enter.

    Although these verticals may be identified easily enough, it is more difficult

    to determine how they function in the present context. Reference to tonal

    Music Analysis, 22/iii (2003) 283 Blackwell Publishing Ltd. 2003. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK

    ERIC DROTT

    THE

    ROLE OF

    TRIADIC

    HARMONY IN

    LIGETI'S

    RECENT

    MUSIC

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    practice is oblique at best. Fanfares engages with, as Mike Searby writes, `thevocabulary but not the syntax of tonal music'.3 What is retained of this

    `vocabulary' is the constitution of the harmonies themselves. They are alltertian sonorities; or, as the excerpt from Fanfares indicates, they are all chordsbuilt upon the foundation of the triad. This is evident in the expansion of themajor triads of phrase 1 into minor seventh chords in phrase 3. In phrase 1,when the dyads are situated above the ostinato, the resulting triads areexclusively major (C, F, D, A, etc.); in phrase 2, when the dyads are belowthe ostinato, the resulting triads are exclusively minor (D, F, A, B, etc.). Inphrase 3, the dyads return to the upper register, and the chords are major onceagain at least until the entry of the C minor seventh chord in bar 23. The

    conceptual origin of this simultaneity is clear: the C minor seventh is theproduct of an E major triad plus c.

    f g a b f a g f d c c

    d f a b c a c

    D A F B A F E G

    C F D A E F B A

    Ex. 1 Fanfares, bars 117

    E tudes pour Piano. Premier livre. Schott Musik International GmbH & Co. KG, Mainz. Reproduced by kind permission.

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    Further complicating the role of the triads and seventh chords in Fanfares isthe consistent stratification of the texture into two seemingly independentlayers. The contrast in character and function between the two parts isapparent: the rigid and unvarying ostinato sets off the more fluid, apparently

    improvisatory succession of dyads. The simultaneities produced from theconjunction of these two parts are not pronounced; the fact that they only comeinto being by straddling two separate textural strands weakens their sense ofcohesion. Nevertheless, there is a covert relationship established early in thepiece between the two parts, which delimits the fund of harmonies available atany given moment. Since the dyads coincide with accented ostinato notes, andsince the ostinato is (by definition) not subject to alteration, only a handful ofharmonies is feasible. According to this constraint, the dyad falling on thedownbeat of the notated bar in phrase 1 may harmonise with the c in the

    ostinato in three ways, forming a C major, an A major or an F major triad. Inphrase 2, the same note in the ostinato yields a C minor, an A minor or an Fminor triad. Beyond this, there are no hard-and-fast rules that determinewhich particular harmony `should' or `should not' be used. (The onlysuccession that Ligeti consistently avoids is direct repetition of a chord. Eventhough the ostinato's c and f can both be harmonised with an F major triad,there is no occasion when both notes are harmonised with the same chordconsecutively.) This method of arranging the harmonies ensures not only thatthe music will avoid clear-cut tonal progressions, but also that the musical

    surface, taken as a whole, will be more or less chromatic.One way of accounting for the incongruity between diatonic harmonies and

    the overall chromatic pitch space they occupy not to mention the incongruitybetween the presence of triads (the quintessential markers of tonal music) andtheir resolutely non-tonal setting would be to place Ligeti's late music underthe rubric of the postmodern. Certainly, these works represent a renunciationof the (late) modernist concern for stylistic consistency and purity, leaninginstead towards a more inclusive musical language. Yet this way of under-standing his recent music may turn out to be trivial, if postmodernism is taken

    as a periodising term; by this measure, everything written in the past twenty-five to thirty years is, in one way or another, postmodern. Furthermore, it isvital not to use the term `postmodern' simply as shorthand for musicalsyncretism that is, it is vital not to efface what is distinctive about thisrestorative gesture.4 Saying that the use of triadic harmony is postmodern doesnot say much, given the breadth and vagueness of this term. To do Ligeti'sharmonic practice justice, one needs to situate it more precisely. For, asFrederic Jameson has pointed out, there are `as many different forms ofpostmodernisms as there were high modernisms in place, since the former are

    . . . specific and local reactions against these models'.

    5

    Jameson's observationsuggests that the best way to approach Ligeti's `rediscovery' of triadic harmony

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    is to read it through the lens of the specific, local modernisms to which itresponds. And this in turn suggests that there is not a single exclusionary logicthat explains the prohibition of triadic harmony during the ascendancy ofatonal and serial music in the 1950s, but that there are a number of logics,

    which must be explored in their own particularity. To this end, I will begin bydiscussing two arguments against the use of the triad in modern music thoseposited by Pierre Boulez and Theodor Adorno before returning to Fanfaresto see how it may counter such arguments.6

    But two disclaimers are in order one terminological, the other historical.Firstly, I will use the term `triadic harmony' throughout this article to refer tothe entire range of tertian sonorities that have their conceptual basis in thetriad. In many cases, this usage reflects the actual processes at work in Ligeti'slater music, where seventh and ninth chords are the products of adding notes

    onto a triadic substrate (such is the case in Fanfares). Furthermore, I will usethis term in a deliberately loose manner, since the harmonies he employs are asmuch symbolic entities as they are generative or syntactic ones. His late worksmay allude to functional harmony, without invoking functions as such: they areempty referents. But although clear-cut tonal function may be lacking, thesechords play upon (and against) an essential quality shared by triads andseventh chords their relative harmoniousness. As Boulez's and Adorno'scomments on triadic harmony will demonstrate, the problem posed by triads(and, by extension, seventh chords) is not simply their potential to invoke tonal

    convention, but their ability to fuse individual pitches into a single,supervenient entity. It is the ease with which triads and seventh chords areable to `coalesce', to come together as single perceptual units relative to non-tertian sonorities, that is at issue in Ligeti's later works an ease that is as muchthe product of the acculturated familiarity of these harmonies as of psycho-acoustical factors.

    Secondly, it should be noted that Ligeti's belated response to the positionsstaked out by Boulez and Adorno is indirect in nature. His reaction to themodernist aesthetic typified by these two authors takes shape as a reaction to

    the modernist impulse found in his own works of the 1950s and 1960s. Bysubjecting his own compositional past to a form of self-reflexive critique, he isable not only to create a new compositional style, but also to sever his lingeringassociation with Darmstadt. In an essay written in the 1980s, he muses on the`aging' of modern music: `the modernism and the experimental avant-garde ofthe fifties and sixties, don't they seem to be passe , part of history, part of theacademy as well?'7 He then qualifies this thinly veiled attack on what hasbecome a safely canonical modernism by applying the same critique to his ownformer style. He recognises that if the avant-gardism of the 1950s and 1960s

    has become senescent, then his own music must be rejuvenated as well to avoidthe same fate. Rather than repeating proven formulae, he declares himself in

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    favour of `a modernism of today': `For me, this signifies in the first place adistancing vis-a -vis the total chromaticism and the dense polyphonic tissuesthat had characterized my music at the end of the fifties and the beginning ofthe sixties.'8

    Ligeti's attempt to transcend modernism can be read as an attempt totranscend his own prior domination by modernism's various taboos, andthereby to assert his independence as a composer. But to achieve thisindependence, he is compelled to distance himself in turn from what heconsiders to be the empty fashion of postmodernism (which he, like manyGerman critics, tends to equate with neo-romanticism in music):9

    We live in a period of artistic pluralism. While modernism and even the

    experimental avant-garde still exist, `post-modern' artistic movements are

    increasingly manifest. `Pre-modern' would, however, be a more accurate way of

    describing these movements, for the artists who take part in them are interestedin restoring historic elements and forms: naturalism in painting, columns,

    cupolas and tympana in architecture, and, in music, the recovery of both

    tonality and rhythmic-melodic figures impregnated with expressionistic pathos.

    The syntax of the nineteenth century is present in all the arts. 10

    This disdain for the postmodern the notion that it represents a callowrestoration of the past explains in part Ligeti's desire to institute a`modernism of today' (even though his stark opposition of the modern andpostmodern betrays a continuing loyalty to the sort of binary logic that

    postmodern thought allegedly replaces). It is my contention that Ligeti's use ofthe triad in his music dating from the late 1970s onwards exemplifies thisdesire, a desire to negotiate a position between what he sees as the totalising(and thus reductive) claims of modernism and postmodernism a positionbetween a blind affirmation or an equally blind negation of convention.

    II

    In his lectures on composition given at Darmstadt in 1960, Boulez accounted

    for the exclusion of triads in serialism over the course of an extendeddigression. The reason for avoiding such harmonies, he argued, has little to dowith their stylistic or structural inconsistency within an atonal context. Rather,Boulez's primary justification for excluding the triad has to do with itspotential to interfere with the perception of serial structures. He begins bydescribing the problem posed by octaves:

    . . . octaves create a weakening or hole in the succession of sound relationships by

    way of reinstating a principle of identity denied by the other sounds, so that

    they are at variance with the principle of structural organization in the world in

    which they appear; . . . octaves must be completely avoided, at the risk ofstructural nonsense.11

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    In other words, he claims that the identity of pitch-class in octaves is so strongthat it conflicts with the serial ordering in which the octaves are embedded.The relative salience of the octave sets it apart from the rest of the texture,which undermines the intended serial groupings. Thus, octaves create simple

    relationships that cut through the complex pitch structures of serial music.Boulez continues by including triads in his list of proscribed sonorities:

    . . . the same applies to common triads, not only when they appear in their own

    vertical dimension but also if they are the products of the superposition of

    several horizontal structures. As with octaves, they reinstate the principle of

    identity denied by all the other sounds . . .12

    He gives an excerpt from Webern's String Quartet, Op. 28, to illustrate thispoint, without providing any commentary. Ex. 2 is a reproduction of this

    excerpt. Based on his discussion, it appears that his dissatisfaction with thepassage stems from the two chords he marks (F major and D major), whichostensibly distract the listener from the work's motivic structure. Thepertinent `information' contained in this passage, for him, resides in itsmotivic structure; the triads are nothing but noise.

    Adorno's treatment of the triad charts similar territory, but in a morenuanced way. In his early writings, he is open to the idea that conventionalmusical language might still be employed in a critical fashion by ironising orsubverting it. He does not exclude the residue of musical tradition a priori, but

    rather deems it to be of value so long as it is defamiliarised by the composer andthus turned into a vehicle for social and musical critique. Thus, in his essay,`On the Social Situation of Music', he views the music of a composer such asWeill in a generally positive light.13 By the time Adorno had written Philosophy

    pizz. arco

    pizz.

    pizz. arco

    poco

    Violin I

    Violin II

    Viola

    Cello

    ( )

    ( )

    ( )

    pizz.

    rit.

    F major D major

    [ 0 1 3 ]

    [ 0 1 3 ]

    [ 0 1 3 ]

    [ 0 1 3 ][ 0 1 3 ]

    Ex. 2 Excerpt from Webern, String Quartet, Op. 28 (taken from PierreBoulez, Boulez on Music Today, Ex. 10)

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    of Modern Music, however, his position had become more rigid. Here he treatsthe triad as particularly susceptible to ideological distortion, arguing that thesense of accord intrinsic to this sonority, while tempting, presents a speciousimage of reconciliation to listeners. In a thoroughly antagonistic society, one

    that not only promotes but also feeds upon alienation and social fragmentation,such an audible token of social harmony becomes false. Furthermore, theabsorption of discrete notes into a single, fused harmony suggests that thesense of accord manifest by the triad is grounded in a regressive form ofdomination. This latter point hinges upon his striking phenomenologicalreading of the sonority:

    The more dissonant a chord, the more sounds contained . . . the more

    `polyphonic' is this chord . . . The predominance of dissonance seems to

    destroy rational, `logical' relationships. Dissonance is nevertheless still morerational than consonance, insofar as it articulates with great clarity the

    relationship of the sounds occurring within [the chord] no matter how complex

    instead of achieving a dubious unity through the destruction of those partial

    moments . . . through `homogeneous' sound.14

    Like Boulez, Adorno notes that the triad possesses a force of attraction, but hegoes further in claiming that the cohesion of the triad effectively effaces itsconstituent pitches. The audible unity of the triad is purchased at the expenseof the individual note. Hence, the image of social harmony that the triad

    projects is reactionary in character, a throwback to a historical situation wherethe unity of the social organism was vouchsafed by a transparent form ofdomination (feudalism). Instead of a progressive surpassing of alienation, then,the triad is emblematic for Adorno of a wholly regressive subordination of theindividual with respect to the totality; instead of confronting the socialantagonisms inherent in a class-based society, the embrace of the triad bymodern composers would disguise these contradictions at the aesthetic level.In contrast, Adorno sees dissonant sonorities as symbolically resisting such aregressive form of domination, since they do not absorb discrete notes into a

    single, unified sonority. Not only is the dissonant sonority valorised for itsintrinsic `polyphony'15 i.e. its refusal to subvert the impulses of theindividual notes in favour of a single, global impulse but also because it betterreflects the social fragmentation and alienation endemic in advancedcapitalism, and in this regard is considered to be more honest to the structureand experience of modernity.16

    Let us return now to Fanfares, to evaluate it in relation to Boulez's andAdorno's arguments. What is striking about this and other e tudes is theiradamant emphasis on the horizontal, rather than the vertical, dimension.

    Despite the presence of triadic harmony, whose intrinsic tendency is to fusethese two polyphonic layers together into a single consonant sonority, the

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    layers manage to maintain a strong degree of independence from one another.The source of this independence lies in the conflicting rhythmic organisationof the two layers (as Ex. 3 shows). Whereas the ostinato is three `beats' long(the `beats' being uneven groups of two and three quavers), the dyads' phrases

    are five `beats' long, keeping the two parts out of alignment. As a result, thedyads' rhythmic patterning is irregular, constantly changing with respect to theostinato. The dyads' first melodic unit has the rhythm 3+2+3+3+2; thesecond has the rhythm 3+3+2+3+3; and the third has the rhythm2+3+3+2+3. It is only with the fourth melodic unit that the rhythmicpatterning of the dyads begins to repeat, completing the cycle.

    The two parts lack a sense of coordination at the level of the phrase, a factthat is compounded by the repetitions of the ostinato. The lack of variation inthe ostinato reinforces the sense that the two parts proceed along separate

    tracks, diminishing the salience of any vertical formed by the coincidence ofthe two layers. Although the parts do come into alignment once every five bars,the asynchronous phrase structure of the parts denies a sense of texturalcoherence. In this respect, Fanfares is representative of Ligeti's later works, inthat the simplicity of its harmonies stands in contrast to the complexity of itsrhythm. Indeed, this excerpt shows how the use of triads intersects withanother preoccupation of his recent work, namely the exploration of rhythmicand metric dissonance. The two tendencies seem to go hand in hand: harmonicconsonance offsets rhythmic/metric dissonance and vice versa. It is hardly a

    coincidence that Ligeti's restoration of the triad in the late 1970s occurs atroughly the same time as his first attempts to synthesise the dual influences of

    3

    (3

    3 3 3

    3333

    (3 2 3)

    3 (3 3 322)

    etc.+ +

    ++

    3) (2 3 2 3) (3 2 3 3 2)

    5 52 3

    5 5

    + + + + + +

    ++++++++

    Ex. 3 Stratification in the opening ofFanfares (bars 17)

    E tudes pour Piano. Premier livre. 1986 Schott Musik International GmbH & Co. KG, Mainz. Reproduced by kind permission.

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    Conlon Nancarrow's Studies for Player Piano and the polyrhythmic intricaciesof sub-Saharan music.17 The result of this conjunction is that metric andharmonic dimensions are placed in conflict with one another. The twodimensions pull in opposite directions: the consonant harmonies tend to unify

    the texture, while the multiplicity of rhythmic patterns stratifies it.This conflict between horizontal and vertical dimensions is hardly new to

    Western music; it is simply that Ligeti's works exacerbate a feature present inall polyphonic music. Both his later music and more traditional contrapuntalgenres play off the perceptual difficulties involved in focusing one's attentionon multiple simultaneous structures. As the theorist James Wright and thepsychologist Albert Bregman have asserted, auditory processes possess theGestalt principle of `belongingness', which consists in `the tendency for aparticular piece of sensory evidence to be allocated to one or another perceived

    object, but not to both at once'.18 The consequence of this principle is that itsets simultaneously unfolding dimensions at odds with one another, so that ourability to perceive coherence in one dimension curtails our ability to perceivecoherence in the other. That is to say, it is easy to hear a note as part of a line oras part of a chord, but it is difficult to hear a note as part of both at the sametime. This sheds light on Adorno's criticism of the triad: because of itsacoustical consonance and cultural familiarity, it always threatens to outweighthe polyphonic impulse of the individual notes. Wright and Bregman assertthat

    if the factors favoring the sequential grouping of the components are stronger,

    the fusion of simultaneous elements may be inhibited. Similarly, if the factors

    favoring simultaneous fusion dominate, the components may be restricted from

    grouping sequentially with other components that resemble them.19

    Bearing this in mind, it is easy to see how Ligeti's organisation of rhythm andharmony at the opening of Fanfares addresses the listener's inability to attendfully to both line and chord at the same time. Both present the listener withstructured perceptual objects. The rhythmic and textural independence of the

    parts is conducive to stratification, whereas the sheer familiarity of the triad isconducive to cohesion. Since it is difficult to concentrate on both structures atthe same time, listeners will tend to focus on one more than the other: oneplane is foregrounded, and the other is `heard' (if not `listened to' as such). Andsince the factors favouring auditory streaming more often than not outweighthose favouring vertical cohesion in his recent works, it is more likely thatlisteners will direct their attention towards the horizontal dimension. Theconsonance of the triad is present, but it is relegated to a secondary position inthe perceptual field.

    Even if Ligeti's music from the late 1970s onwards manipulates some of thesame characteristics that guide conventional polyphony, it is nonetheless clear

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    that his music uses them for an entirely different purpose. Wright andBregman, whose article addresses the rules of traditional counterpoint, notethat the conflict between horizontal and vertical structures is often used as ameans of lessening the phenomenal impact of dissonance. For instance, the

    stepwise preparation and resolution of dissonances in strict counterpointaffords enough cohesion in the horizontal dimension that the discord present inthe vertical dimension tends to be weakened. From this perspective, thecontinuity provided by something as simple as a passing-note helps to softenthe impact of non-harmonic notes by deflecting attention away from whateververtical sonorities they may incidentally form with other voices.20 In mostpolyphonic compositions, this principle is extended to the formal level, so thatthe relative independence of the lines within phrases is redeemed by the factthat they come together in stable, fused sonorities at points of syntactic import,

    such as cadences. These points of harmonic cohesion thus take on rhetoricalsignificance, in that they affirm the centripetal, unifying force of the triad overthe centrifugal force of the line. But, as the excerpt from Fanfares indicates,Ligeti turns the same attribute of conventional polyphony on its head. Here,the factors that favour horizontal integration do not serve as a means ofsuppressing dissonance, but work instead as a means of suppressing one'sperception of the consonant sonorities themselves. From this perspective, hisuse of triads can be seen as a compromise between a modernism that proscribesconventional harmonies and a tradition that demands their presence. By

    situating the triad in a context that is designed to fragment texture intoseemingly unrelated layers, he is able to undercut the triad's function as ameans of unifying the texture into a whole.

    III

    Rhythmic and metric conflict is perhaps the most effective way of splittingmusical texture into discrete horizontal layers. But it is not the only way. Threeadditional strategies undermine the fusion of vertical sonorities in Ligeti'srecent music, dividing the texture by means of harmonic collection, melodicprocess or the use of repetitive figures. In this section, I shall discuss excerptsfrom three works (the tenth piano e tude, Der Zauberlehrling; the Alla marciamovement from the Horn Trio; and the second piano e tude, Cordes a vide) inorder to illustrate these additional strategies.

    Der Zauberlehrling

    Beginning in bar 66 of Der Zauberlehrling, Ligeti uses the harmonic differen-tiation of voices to undercut the impact of triadic harmony. Throughout this

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    passage, the chromatic aggregate is divided into two discrete collections (asseen in Ex. 4). The right hand plays melodies based on the white-note diatonicscale, and the left hand plays melodies based on the black-note pentatonicscale. This partitioning of the chromatic, reminiscent of Barto k, has been a

    mainstay throughout Ligeti's career, serving to dilute the neutrality of totalchromaticism by distributing it into distinct collections (see, for instance, bars1322 of Atmosphe res (1961), where he partitions the aggregate in this way inorder to lend colour to an otherwise harmonically neutral cluster). Thisprocedure allows him to have it both ways, as it were, reconciling the ostensiblemodernity of chromaticism with the ostensible conventionality of diatonicism.The situation is complicated in this passage by the presence of triads. Eachhand presents descending scalar fragments, their entrances staggered withrespect to one another, and these overlapping lines are arranged so that a triad

    is formed at the beginning of each fragment, where a new line intersects withthe end of a line in the other hand. As in Fanfares, the perception of the triadsis weakened by the segregation of the layers on the horizontal dimension. InDer Zauberlehrling, however, it is the staggering of the lines and, moreimportantly, their organisation according to two discrete harmonic spaces thatdisrupts the ability of the vertical sonorities to bridge these spaces.

    The Alla Marcia from the Horn Trio

    In both Fanfares and the passage from Der Zauberlehrling discussed above,rhythm plays a vital role in stratifying the horizontal dimension. In the middlesection (bars 31103) of the Alla marcia, the third movement of the HornTrio,21 such rhythmic differentiation of the layers is altogether lacking (theopening bars of this section are shown in Ex. 5). Rhythmically, this sectionconsists of a virtually unbroken succession of crotchets; texturally, it ishomophonic, set in two voices for the most part, breaking into a thicker textureonly towards the end of the section.

    There is a reason for this general lack of rhythmic and textural variation. As

    it turns out, two different ordered arrangements of the chromatic aggregateform the harmonic framework for the section. The rhythmic and texturaluniformity ensures that each of these orderings is presented as a fixedsuccession of six dyads. The first of these orderings of the aggregate (Ex. 6a)recalls the distorted `horn-call' motif that opens the first movement of thework, transposing the motif down a major third. Similarly, the second of theseorderings (Ex. 6b) anticipates the opening of the final movement, transposingthe first three dyads of the Finale down a minor second. The two orderingsthus serve as connective tissue, linking the opening and closing movements of

    the work. But the two orderings also serve to parse the section into even, four-bar groups: two statements of ordering A are followed by two statements of

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    ordering B, which are followed by two further statements of A, and so forth.Irregular slurring and frequent changes in timbre prevents this section frombecoming monotonous. The phrasing contradicts the articulation of theunderlying harmonic structure.

    Towards the end of the section, another factor contributes to the obscuringof the underlying pitch structure the `filling-out' of the texture with triadsand other harmonies. Extraneous pitches begin to appear during the latter halfof the section, supplementing those that comprise the two ordered aggregates.

    The first of the extraneous pitches appears at the end of bar 57, with the C inthe horn. The addition of this pitch realises the latent harmonic potential of the

    simile8

    (simile)

    38 39

    Klv.sempre legatissimo

    espressivo con delicatezza

    Horn in F

    con sord.

    dolce, espressivo con tenerezza

    sempre poco in relievo

    (poco)

    Vn

    . = 76 (ganze Takte / whole bars)

    Pi mosso, ohne Akzente, sehr gleichmig und flieend / without accents, very evenly and fluid

    31

    con sord., legato, espressivo con delicatezza

    32

    (poco)

    33 34 35 36

    (simile)

    37

    40 41 42 43 44

    Pno

    Ex. 5 Horn Trio, third movement (Alla marcia), opening bars of the middlesection

    2001 Schott Musik International GmbH & Co. KG, Mainz. Reproduced by kind permission.

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    dyad with which it coincides (EG) by forming a C minor triad. Otherstructurally `superfluous' pitches surface over the next twenty bars or so,although it is only around bar 79 that the accretion of pitches begins in earnest.

    From here until the end of the section in bar 103, more and more notesaccumulate around the series of dyads that make up the alternating orderings ofthe aggregate, forming at first triads, then seventh chords and then ninthchords. Ex. 7 is a harmonic reduction of bars 7997. The remainder of thepassage (bars 99103) continues the accumulation of pitches, burying all tracesof triadic harmony beneath the welter of notes. Diatonic clusters, followed bychromatic clusters, gradually take the place of the triads and seventh chords.

    In the Alla marcia, the horizontal and vertical dimensions still compete withone another, albeit in a manner that is altogether different from the conflict

    encountered in Fanfares and Der Zauberlehrling. On the one hand, the verticaldimension is more cohesive than in works such as Fanfares, insofar as the extrapitches not only harmonise with the fixed succession of dyads, but also fail tostray rhythmically from the other part. Neither rhythm nor register slice uppitch space into discrete, horizontal layers. There is nothing in the setting, inthe arrangement of the parts, that works against the audibility of the resultingchords. On the other hand, there is a more subtle force that serves todistinguish the component pitches of each harmony from one another, a forcethat runs counter to the cohesion of the vertical dimension. It is not the way

    that the chords are presented that leads to their fragmentation, but theharmonies themselves the pitches that comprise them that deny their simple

    Ex. 6a Ordering A of the aggregate, middle section of the Alla marcia

    Opening dyads of Horn Trio, IV:

    3 6 8

    Ordering B:

    3 6 8 4 6 3

    Ex. 6b Ordering B of the aggregate, middle section of the Alla marcia

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    cohesion. The distinction between structural and non-structural pitches,between the notes that make up the chromatic framework of the section and thenotes that decorate this framework, is enough to subvert the chordal texture.Indeed, the sheer repetitiveness of the two aggregates works against the fusion

    of the chords, since this not only establishes but also reinforces the twoorderings in the listener's ear.23 A makeshift hierarchy is set up within the

    79

    83

    87

    91

    95

    Note: All accidentals affect the note that immediately follows. The notation reflects (roughly) the score of the Trio in spacing andin spelling of chords. For the purposes of identifying chords, the analysis assumes enharmonic equivalence where necessary(i.e., AD EA in bar 79 is identified as an A major seventh chord, despite the unusual spelling).

    A

    B B

    AA

    B B

    AAordering:

    E 7 D 7 AM7 [0146]

    [0347]

    B M/m f E 7 D 7 d A 7 g g

    B G7 E B 7 [026] F [025] f E B [016] am 7

    A 7 g 7 c 7 am7(+g ) cm7 f m7 A M7 G7 c m7(+g) f (+g) gm7 DM7(+g)

    f 7 G7 EM7 B 7 g 7 FM7 g m7 [0246] c (+d ) cm9 A9 am7

    [014579][012468t]E 9AM7gm7(+c )[0127]c m7G9F9

    Ex. 7 Harmonic reduction, Alla marcia, bars 7997

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    sempre espr.

    3 3 3

    (con ped.)

    Piano

    m.g.

    sempre legatiss.

    dolce, espr.

    Andantino con moto, molto tenero = 120

    espr.sempre333

    3

    33

    33333

    Ex. 8 Cordes a vide, bars 116

    Horn Trio. 2001 Schott Musik International GmbH & Co. KG, Mainz. Reproduced by kind permission.

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    horizontal dimension between the invariant dyads that form the core of thesection and the mutable notes that accompany them. It is this hierarchy ahierarchy that is not a product of a pre-existent musical syntax, but simply theproduct of repetition that derogates vertical sonorities, fixing them in a

    secondary position.

    Cordes a vide

    Like the middle section of the Alla marcia, the opening bars of Ligeti's secondpiano e tude, Cordes a vide,24 are homophonic (see Ex. 8).25 Up to bar 11, bothhands play even streams of quavers, although differences in phrase length,slurring and accentuation add a degree of rhythmic tension to the passage.(Following bar 11, the piece introduces increasingly complicated polymetric

    relationships between the two voices.) But unlike the middle section of the Allamarcia, Cordes a vide does not differentiate the horizontal layers by means of aquasi-hierarchical distinction between structural and non-structural pitches.Rather, stratification arises in this work out of melodic directedness: thepattern of descending perfect fifths that gives this piece its title (open strings)recurs with such regularity that it stresses the linear dimension over theharmonic. This strong horizontal impulse disguises the intervallic andharmonic relationships created by the counterpoint of the two voices namelythe series of imbricated seventh chords that are a by-product of the ubiquitous

    melodic fifths. Ex. 9 labels the seventh chords present in bars 14. Like theremainder of the section, this four-bar excerpt is rife with such intimatedharmonies, which overlap to such a degree that multiple harmonic inter-pretations are possible.

    The left hand's line, in particular, is more predictable than the looser material

    presented in the right hand. Its phrases present a sequence of seven quavers, all

    of which consist of a series of descending fifths, with the final quaver serving as a

    link to the following phrase. There is, in addition, a regular pattern that guides

    the large-scale trajectory of the left hand's groups. The accented notes that

    initiate each phrase form an ascending line (as seen in Ex. 10). At first, each

    cm

    A M7 gm7bm7e m7F M7E BB M7

    dm7 E M7(no 3rd)

    C M7 b 7 C 7 em7 DM7 [027] d7 E 7 A M7 E M7 f m7 EM7

    Ex. 9 Harmonic progression, Cordes a vide, bars 14

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    group begins a semitone above the previous one; later, when the line deviates

    from the stepwise motion, there is still some attempt to fill in the gaps. The

    motion from c2 in bar 4 to d2 in bar 5 is followed by a compensatory downwardmotion to d2 in bar 6, before the ascent resumes its course. Likewise, the groups

    skip over f2 at first, proceeding from e2 in bar 7 to f2 and then g2 in bar 8, before`correcting' the mistake by touching upon f2 in bar 9. The only truly radical

    break in this gradual ascent comes at the very end of the passage, with the skip

    from f2 in bar 9 to b2 in bar 10, a skip that effectively marks the close of the

    e tude's opening section.

    In contrast, the right hand represents the melodically free voice during thissection, presenting the perfect fifth as a mobile motivic cell. Listening to thepiece, there is little to suggest a logic guiding the choice and juxtaposition ofthese fragments; there are sudden shifts in registers, motifs break off withoutwarning and the transposition of motifs seems arbitrary. Analysis suggests,however, that harmonic concerns play a role in determining the path followedby the right hand's motifs. A case in point is the repetition of the opening motifin bar 2, which is broken off prematurely. Ex. 11a compares the first statement

    of this motif with its repetition in bars 2 and 3. At the beginning of bar 3, themusic unexpectedly leaps up to c3 instead of returning to a2, as occurs in the

    1 2 3 4 5 6 7 8 9 10 12 13

    Ex. 10 Left hand's ascent, Cordes a vide, bars 113

    1 2

    Ex. 11a

    2

    parallel octaves

    Ex. 11b

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    initial statement of this motif in bar 1. Had the motif continued as before, theseries of seventh chords would have been disrupted, as Ex. 11b demonstrates;indeed, the result would have been parallel octaves between right and lefthands.

    Similar harmonic considerations guide the right hand's motion elsewhereduring the first eleven bars, with sudden breaks in the melodic line acting topreserve the succession of seventh chords formed between the two voices. Buteven if latent harmonic concerns are a motivating factor in the organisation ofthe right hand's motifs, the texture ensures that these harmonies occupy asecondary position in the listener's perceptual field. By rendering the two linesso distinct, so independent of one another, Ligeti relegates the harmonies tothe phenomenal background, concealing the influence that harmonicconsiderations exercise on the right hand's line.26 Instead of determining the

    course of the lines, the vertical relationships sound more like the inadvertentresult of the coincidence of the lines.

    *

    In all of these works Fanfares, Der Zauberlehrling, the Alla marcia andCordes a vide horizontal and vertical dimensions are set at odds with one

    another. But in each, Ligeti effects this opposition differently throughconflicts in rhythm and phrase structure in Fanfares, through thedifferentiation of harmonic collections in Der Zauberlehrling, throughostinati in the Alla marcia and through melodic process in Cordes a vide.As a result, each of the works responds in its own way to the arguments ofBoulez and Adorno. For Boulez, the triad is problematic because of itstendency to interfere with other structures. In Fanfares and other suchworks, however, the extreme polyphonic stratification reverses the situation.It is now the horizontal forces that interfere with the perception of the triad.

    For Adorno, on the other hand, it is the tendency of the triad to obscure theparticular that casts doubts on its aesthetic and ideological legitimacy.Clearly, the textural stratification encountered in these works forestalls thetriad's ability to conjoin the various pitches into a homogeneous unit. Yet inorder to accomplish this feat in order to dissolve the force bindingsimultaneous notes together it is necessary to strengthen the force bindingsuccessive notes together. It is the use of regular metres, predictable ostinatiand contrasting harmonic collections that ensures that the cohesive force ofthe lines is commensurate with that of the harmonies. In Ligeti's works,

    individual notes may escape from the domination of the triad that Adorno soabhorred, but only by being absorbed into horizontal structures instead.

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    IV

    The interdiction on the triad in Adorno's and Boulez's writings can be tracedback to Schoenberg. Yet in his writings on the subject, Schoenberg voices

    fewer objections to the consonance of the triad than either Boulez or Adorno(which is not to say that the avoidance of consonance had no role to play in hisaesthetic considerations). Nor was his objection to the use of the triadpredicated on ideological grounds (at least not explicitly). Writing in 1926,early on in his deployment of the twelve-note technique, Schoenberg wassensitive to the residues of tonal practice that triads possessed. Thus, hisargument against the presence of triads in post-tonal music rests as much onthe syntactical implications these harmonies would have as on their purelysonorous or perceptual qualities. The triad, according to Schoenberg, was

    more than a euphonous collection of notes; it was a token of the tonal system asa whole, with all of its syntactical norms and expectations. To incorporate asingle triad into an otherwise non-tonal context was to run the risk ofdeforming this context, as the triad brought into play an entire system ofmusical relationships. For this reason, he claimed that

    My formal sense (and I am immodest enough to hand over to this the exclusive

    rights of distribution when I compose) tells me that to introduce even a single

    tonal triad would lead to consequences, and would demand space which is not

    available within my form. A tonal triad makes claims on what follows, and

    retrospectively, on all that has gone before.27

    Schoenberg's comments could be reformulated as follows: listeners steeped intonal music cannot help but hear triads as expressing tonal functions.Composers who indulged in placing such structurally unsupported harmoniesin their works were guilty of misleading the listener, making promises theycould not keep.

    One way of avoiding the problem outlined by Schoenberg would be toexpunge any trace of triadic harmony from post-tonal music. Yet Schoenberghimself did not feel that such harmonies ought to be restricted solely to strictly

    tonal compositions (a fact that is borne out in a number of his later works, suchas The Ode to Napoleon): `even standing where I do at the present time, Ibelieve that to use the consonant chords, too, is not out of the question, as soonas someone has found a technical means of either satisfying or paralyzing theirformal claims'.28 He allows the possibility of including triads and other familiartertian chords in non-tonal contexts only if the historical and conventionalassociations that these chords carry with them can be neutralised that is, if theinternal organisation of a work could somehow defuse their residual tonalfunctions. In some respects, his fears seem over-inflated. One may doubt

    whether the presence of a triad in a twelve-note work, say, could single-handedly overturn its internal structure and institute a quasi-tonal hierarchy.

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    His fears may be accounted for in part by his perception of the historicalsituation at the time he made these comments; at an early stage in theestablishment of the twelve-note technique as an alternative to tonality, hejustifiably saw the continuing sway of convention as a tremendous threat.

    Besides, his fears seem more plausible when reformulated along more moderatelines. He is surely correct to claim that triads and seventh chords are notneutral entities, but are charged with associations that are difficult, if notimpossible, to avoid. Indeed, it is hard to imagine a method that would becapable of severing totally the connection that listeners make between suchharmonies and tonal practice. It is more likely that some residue of their tonaluse will linger on for most listeners, even if such harmonies are arranged in amanner that consistently disappoints habitual expectations. A single triad maynot be able to transform a twelve-note piece into a tonal one all by itself, but it

    may influence how one hears its immediate musical context.Schoenberg's comments are suggestive of the interpretation of triads and

    seventh chords in Ligeti's later works. By deflecting attention away from thesechords, Ligeti effectively weakens their `formal claims'. If such harmonies arethemselves subordinated perceptually, then whatever residual tonal impulsethey possess will be subordinated as well. Yet even if these impulses aremarginalised, this does not mean that they disappear altogether. Rather, theirimpact on the perceived surface of these works is indirect. Just as the triads andsevenths, taken by themselves, are `heard' without being `listened to', so too are

    the successions they form. Progressions that conform to the listener'sexpectations be they the ingrained expectations of tonal syntax or thoseraised by the immanent constraints of the work will give the music asemblance of coherence; those that fail to do so will give a semblance ofunpredictability.

    Once again, Fanfares will serve as the point of departure. An argumentcould be made that the triads used in the opening phrase (F major, B major, Cmajor, A major, etc.), in addition to the trichord outlined by the accentednotes in the ostinato (C, F and G, which are enharmonically equivalent to an F

    minor triad), project F as some sort of centric pitch. But the chord-to-chordsuccession resists a straightforward tonal interpretation. Likewise, the presenceof harmonies that fall outside the putative tonal centre of F (such as the Emajor triads in bars 3 and 7) are hard to reconcile with a tonal reading. Fromthis perspective the perspective of inculcated tonal expectations thesuccession of resultant harmonies is unpredictable. Although listeners may notattend to this succession as the most salient dimension of the musical fabric, itwill nonetheless colour their perception of the piece. At the same time, thepredictability of the ostinato works against this implicit sense of harmonic

    unpredictability. The work may exploit one's expectations vis-a -vis tonality,but it manages to satisfy other expectations at the same time. Despite the

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    relative flexibility in the choice of harmonies (each accented ostinato pitch canbe harmonised with one of three triads), the simple fact that the dyads mustharmonise with C, F or G, depending on metric position, limits the fund ofchords available at any moment, as well as in the phrase as a whole (in the first

    phrase, the harmonic vocabulary is limited to C, F, A, D, B and E triads).29

    Lacking a conventional syntax, the music offers up an ersatz one. The result isa succession that is both predictable and unpredictable at the same time: thechords are predictable to the extent that they invariably harmonise with theaccented notes of the ostinato, unpredictable to the extent that the progressionsresulting from this constraint escape the residual demands of tonal syntax.

    The same ambiguous situation holds in many of the other works discussedin this article. There is an interplay between the linear impulse, whichtypically guides the harmonic motion and provides it with its immanent

    logic, and the inability of the resulting succession to fulfil whatever tonalimplications the triads might suggest. Consider Cordes a vide. In the openingsection (Ex. 8), the relative regularity of the descending fifths in the left handmaps out a restricted set of possible sonorities. The intervallic consistencyand predictability of the horizontal dimension compensates for the absenttonal syntax. Each sonority created out of the falling fifths pattern in bothhands will sound `justified' as a consequence. This offers Ligeti theopportunity to play the expectations created by the internal structure of thework off the residual tonal implications of the underlying harmonies. For

    instance, following the d3

    that begins bar 4, the right hand's line breaks off,its next phrase beginning some two octaves lower, on c1 (see Ex. 12). Thisdisruption in the right hand's line coincides with the low point of the lefthand's line, marking the moment as particularly salient. The harmonies thatensue gain relative prominence as a result: in particular, the A majorseventh harmony formed by the parallel tenths between the hands comes tothe fore. Yet the registral break that occurs at this moment is mitigated bythe way it is prepared: the dyads immediately preceding the leap down to thec1 spell out an E dominant seventh. It is unlikely that listeners will directly

    perceive this harmony; it is more likely that this gesture will indirectly

    d7 E 7 A M7 E M7

    Ex. 12 Quasi-cadential progression, Cordes a vide, bars 34

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    influence how the salient musical features are understood. Elsewhere inLigeti's late works the disjunction between the presence of tertian harmonyand the absence of tonal syntax gives the music a sense of consistentunpredictability. Here, the momentary coincidence of intrinsic expectations

    (the falling fifths) and extrinsic ones work to smooth over the palpabledisjunction in the melodic line.

    This passage is vital for understanding the dialogue established betweenLigeti's triadic pieces and the tonal system to which they implicitly refer.Chords and (as is the case in this passage) brief progressions associated withcommon-practice tonality are not assumed; they are generated by the systemsset up by Ligeti. For the most part, this means that the demands of tonal syntaxare partially negated. But the result of this denial of expectations is not so mucha feeling of frustration as one of defamiliarisation. The vocabulary stays the

    same, along with occasional fragments of a familiar syntax; it is simply that theharmonic substructure is arranged according to an alien logic, one that soundssensible in its own right, but that is strange nonetheless.

    V

    By way of conclusion, I should like to discuss a passage in Ligeti's sixth piano

    e tude, Automne a Varsovie, which epitomises the dichotomy between triadic

    harmony and linear stratification that I have sketched thus far. Of the first set of

    e tudes, the sixth is by far the most complex, unfolding in some passages as manyas four distinct metric layers at the same time. Yet the most striking moment in

    the piece comes approximately halfway through, when the proliferating lines

    come to a momentary halt. At first there is a brief interlude that presents the

    theme the `lamento' motif that recurs in many of Ligeti's recent works in

    parallel tritones, at the registral extremes of the piano. Following this interlude,

    the music returns to the simple melody plus accompaniment texture of its

    opening, with the left hand performing the lamento motif while the right hand

    picks up the tritone as an accompanimental motif (shown in Ex. 13). This

    simplicity is short-lived, however. Almost immediately, in bar 63, the middlevoice begins to move in contrary motion to the descending lamento theme,

    creating melodic wedges (as shown in Ex. 14). As the passage continues, the two-

    voice polyphony in the left hand is filled out: notes begin to accumulate around

    the dyads, providing harmonic support. The lower voice introduces a third note

    to the lamento line in bar 68, which is eventually joined by a fourth note in bar 72.

    The addition of these notes transforms the line from a succession of dyads into a

    series of descending seventh chords.

    In this passage, the way that the seventh chords are formed provides their

    sequence with a semblance of logic. By gradually adding notes to the lamento

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    68

    8

    dim.

    66

    8

    64

    8

    8b

    con ped.

    62

    8

    8b

    58

    8

    Ex. 13 Automne a Varsovie, bars 5879

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    sotto

    (cresc.)

    76

    8

    (cresc.)

    74

    8

    crescendo poco a poco

    72

    8

    sim.

    70

    8

    cresc. poco a poco

    78

    8

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    theme, the succession of chords takes advantage of the predictability of thistheme. As in Fanfares and Cordes a vide, whatever logic the series of chords hasderives from the force of the lines that comprise it. But before long, thefragmentation that had been temporarily held at bay begins once again. In bar 73,the top voice in the left hand's incomplete D minor seventh chord splits off to

    form a new pulse stream. Whereas the left hand's succession of chords hadadhered to a pulse made up of seven semiquavers, the new line beginning withthe C presents a pulse made up of multiples of four semiquavers (the pulse isnotated by crotchets and semibreves on the score). The same phenomenonoccurs in bar 77. Here, the lowest note splits off, forming a pulse stream whosebasic unit consists of multiples of five semiquavers.

    This process of fragmentation distils in a single gesture the primarycharacteristics of triadic harmony in Ligeti's recent music. As with the otherworks discussed thus far, in Automne a Varsovie triadic harmony ceases to

    serve as an emblem of cohesion, but instead becomes radically unstable,incapable of being held together. But unlike the other works, here thestratification of parts is dramatised. The chords unravel before us, underpressure from the centrifugal force of the polyphonic layers. The ability oftriadic harmony to unify the musical texture, to bridge the space separatingdifferent polyphonic threads, dissolves. Harmonies are built up only to splinterinto independent lines.

    If, for Schoenberg, the problem with using triadic harmony in post-tonalmusic was the danger of awakening inappropriate expectations in the listener,

    then the solution to this problem posited by Ligeti's recent works is to use thestrong linear impulse to overwhelm such expectations. If, for Boulez andAdorno, the problem with triads lay in their tendency to cut across the musicalwork, interfering with the `proper' perception of its intrinsic, autonomouspitch structure, then Ligeti's response is to explore the possibility of reversingthis tendency, so as to cut across the triad itself. Yet the restoration of the triadin the works discussed above, a compositional manoeuvre that opposes themodernist desire to escape or obliterate convention, is hardly an absoluterejection of modernism. Instead of simply affirming the tradition that gave rise

    to triadic harmony, Ligeti's music reaches a similar end by pursuing an entirelydifferent route: it strives to negate the modernist tradition of negation. 30 In

    62

    Ex. 14 Melodic wedges, Automne a Varsovie, bars 627

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    other words, his desire for a `modernism of today' can be understood asapplying the critical, exclusionary aesthetic of modernism to modernism itself(although this way of transcending the modernist aesthetic may approach apostmodernist ethos too much for Ligeti's liking).31 His use of the triad does

    not simply depart from the serial goal of creating a self-contained musicalstructure, free from the weight of tradition. Rather, what is truly distinctive inhis deployment of the triad in Fanfares, Cordes a vide, Automne a Varsovie andother such works is that it challenges the cohesive function of triads, even inthe process of restoring them.

    NOTES

    1. This return of the triad may be understood as an extension of the `restorationof interval' described by Jonathan Bernard in `Ligeti's Restoration of Intervaland Its Significance for His Later Works', Music Theory Spectrum, 21/i (1999),pp. 131.

    2. In the examples, I identify major and minor triads with upper and lower caseletters (e.g. C and c for C major and c minor triads), respectively; 7 by itselfdesignates dominant sevenths (e.g. B7); M7 designates major seventh chords; andm7 designates minor seventh chords. On occasion, a harmonic designation will begiven to incomplete `chords', so that a 38 trichord may represent, for instance,an incomplete dominant seventh. Even though this expands the range of possible

    tertian sonorities considerably to the point where it may seem that the termceases to serve as a useful distinction it is almost always the case that suchincomplete chords will be situated in a series of unambiguous triads and sevenths.In other words, context makes it quite clear whether a 34 actually represents anincomplete major seventh chord, or whether it is just a 34.

    3. Mike Searby, `Ligeti's ` Third Way'': ``Non-Atonal'' Elements in the Horn Trio',Tempo, 216 (2001), p. 17. Along similar lines, Richard Steinitz remarks that theuse of triadic harmonies in a non-tonal fashion `generate[s] a less definable``supertonality'' (Ligeti has called it ``consonant tonality'')'. See Richard Steinitz,Gyorgy Ligeti: Music of the Imagination (London: Faber and Faber, 2003), p. 291.

    4. For discussions of Ligeti's relationship to postmodern aesthetics, see Searby,`Ligeti the Postmodernist?', Tempo, 199 (1997), 914; Be atrice Ramaut-Chevassus, Musique et postmodernite (Paris: Presses Universitaires de France,1998), pp. 9095; and John Cuciurean, `A Theory of Pitch, Rhythm andIntertextual Allusion for the Late Music of Gyo rgy Ligeti' (PhD diss., SUNYBuffalo, 2000), pp. 13982 (esp. pp. 18082). There has been a strong inclinationamong writers on Ligeti's music, however, to dispute the `postmodernity' of hislate style: see, for instance, Searby, `Ligeti the Postmodernist?'; HermannDanuser, `Zur Kritik der musikalischen Postmoderne', in Das Projekt Moderneund die Postmoderne, ed. Wilfried Gruhn (Regensburg: Gustav Bosse Verlag,

    1989), pp. 6983; Constantin Floros, Gyorgy Ligeti: Jenseits Avantgarde undPostmoderne (Vienna: Lafite Verlag, 1996), pp. 22931; and Steinitz, Gyorgy

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    Ligeti, pp. 3289. Attempts to distance Ligeti from the term `postmodern' may bea result, in large part, of his own expressions of disdain for its aesthetic (see n . 9).

    5. Frederic Jameson, `Postmodernism and Consumer Society', in Hal Foster (ed.),The Anti-Aesthetic (Port Townsend, WA: Bay Press, 1983), p. 112.

    6. A more comprehensive survey of Ligeti's harmonic language would account for theintermittent appearance of triads and seventh chords in his music of the late 1960s

    and early 1970s. These harmonies, after being more or less suppressed in the music

    he wrote immediately following his departure from Hungary in 1956, gradually

    reappear in the later 1960s (following the Requiem), and usually adopt one of two

    different guises. Firstly, these chords may occur as individual links within a longer

    process of gradual harmonic transformation. The chords that emerge in this case are

    evanescent entities; furthermore, they are typically masked by other, concurrent

    sonorities. For instance, beginning in bar 110 of Melodien, the cello unfolds a Dminor chord, at the same time as the viola unfolds a C major chord and the second

    violin a G major chord. As a result of their overlap, none of these chords is audibleas such. These harmonies have a purely notational existence. (Other instances of the

    same general approach occur in Melodien, bars 8891, first and second violins;

    Coulee, p. 11, fourth system; and Ramifications, bars 81ff.) Secondly, triads and

    seventh chords occasionally crystallise in the form of the oft-cited `Ligeti signal'

    the diatonic sonorities that emerge when the massed chromatic clusters so

    characteristic of his music of the 1960s disperse. Usually, such Ligeti signals

    consist of some combination of major seconds and minor thirds (such as [024], [025]

    or [036]). Occasionally, however, a triad or seventh will emerge. An interesting case

    in point can be seen in bar 30 ofRamifications, where a B seventh chord results from

    the combination of a BF dyad (played by the second group of strings) and a DAdyad (played by the first group of strings). Notationally, this appears to be a B

    minor seventh chord. Yet because the first string group is tuned a quarter-tone

    higher than the second, the resulting chord lies somewhat awkwardly between a B

    minor seventh and a B major seventh. The aim, it would seem, is to defamiliarise the

    chord. Other, similar `cameo' appearances by triads or seventh chords can be found

    in bar 90 of Lux Aeterna (the D minor chord in the basses) or bars 8791 ofContinuum (which features successive B major and minor triads).

    7. `. . . le modernisme et l'avant-garde expe rimentale des anne es cinquante ou encoredes anne es soixante, n'appartiennent-ils pas aussi au passe , a l'histoire, a

    l'``acade mie''?' (Ligeti, `Ma position comme compositeur aujourd'hui',Contrechamps, 12/13 (1990), p. 8).

    8. `. . . je me declare pour un modernisme d'aujourd'hui. Pour ma part, cela signifieen premier lieu une prise de distance vis-a -vis du chromatisme total et des densestissus micropolyphoniques qui caracte risaient ma musique vers la fin des anne escinquante et au de but des anne es soixante' (ibid.).

    9. Largely absent in the early reception of postmodernism by German-languagemusic critics is an appreciation for what Hal Foster has termed the `post-modernism of resistance'. Rather, in German writings on contemporary music a

    suspicion is more frequently encountered that what passes for postmodernism isaesthetically (if not ideologically) regressive. This has much to do with the

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    equation of postmodernism with neo-romanticism problematic from anAmerican perspective, but comprehensible from a German one, given theprominence of composers such as Wolfgang Rihm, Hans-Ju rgen von Bose andWolfgang von Schweinitz during the 1970s and 1980s. Helga de la Motte-Haber,

    for instance, has noted that, in Germany, `The term ``postmodernism'' spread likewildfire and throughout the early 1980s was more or less universally defined as aneo-conservatism which seemed to characterize the Neo-Romantic language of anew generation of composers' (`Postmodernism in Music: Retrospection asReassessment', Contemporary Music Review, 12/i (1995), p. 77). The generalscepticism towards postmodernism in German critical circles may also be a resultof the profound influence of Frankfurt School thinkers such as Adorno andJu rgen Habermas. Adorno's defence of modern music, and Habermas'srepudiation of certain strands of postmodern thought as the work of `youngconservatives', serve to fuel whatever suspicions German-language writers oncontemporary music might bear towards the idea of the postmodern. See

    Habermas, `Modernity: an Incomplete Project', in Foster (ed.), The Anti-Aesthetic, pp. 1215.

    10. `Nous vivons dans une pe riode artistique pluraliste. Alors que le modernisme etme me l'avant-garde expe rimentale existent encore, des mouvements artistiques``post-modernes'' se manifestent de plus en plus. ``Pre -modernes'' serait toutefoisun mot plus juste pour designer ces mouvements, car les artistes qui en font parties'inte ressent a la restauration d'e lements et de formes historiques: le naturalismeen peinture, les colonnes, coupoles et tympanons en architecture et, dans lamusique, une tonalite retrouve e ainsi que des figures rythmiques-me lodiquesimpre gne es de pathos expressionniste. La syntaxe du XIXe sie cle est pre sente

    dans tous les arts' (Ligeti, `Ma position', p. 8). Elsewhere, he has said that `Mytaste runs counter to the neo-movements. I hate neo-Expressionism and I can'tstand the neo-Mahlerite and neo-Bergian affectations, just as I can't stand post-modern architecture' (Tunde Szitha, `A Conversation with Gyo rgy Ligeti',Hungarian Music Quarterly, 3/i (1992), p. 17). See also his comments onpostmodernism in `Wohin orientiert sich die Musik? Gyo rgy Ligeti im Gespra chmit Constantin Floros', O sterreichische Musikzeitschrift, 49/i (1994), pp. 78.

    11. Pierre Boulez, Boulez on Music Today, trans. Susan Bradshaw and RichardRodney Bennett (London: Faber and Faber, 1971), p. 46.

    12. Ibid., p. 48.

    13. Theodor Adorno, `On the Social Situation of Music', in Essays on Music, selectedwith introduction, commentary and notes by Richard Leppert, trans. Susan H.Gillespie (Berkeley: University of California Press, 2002), pp. 391436. Observethat the only thing separating Weill's `surrealistic' music from Stravinsky's neo-classicism in Adorno's opinion is its conscious connection of aesthetic alienationto social alienation. Hence, already at this early stage of his work on the sociologyof music, Adorno detects a risk in the appropriation of convention for criticalends: `It is beyond question that Weill's music is today the only music of genuinesocial-polemic impact, which it will remain as long as it resides at the height of its

    negativity; furthermore, this music has recognized itself as such and has taken its

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    position accordingly. Its problem is the impossibility of remaining at this height. . . The further course of events reveals another danger in ambiguity: illusionblends into false positivity, destruction into communal art within the realm of thestatus quo, and behind this mocking primitivity, conjured forth by its bitterness,

    comes into view the nave-credulous primitivism of a reaching-back . . .' (pp. 41011). At this point, he exempts Weill from this potential backsliding (he notes that`it is hardly to be expected that he will fall victim to the dangers of theundangerous'), but such comments set the stage for the later hardening of hisposition vis-a -vis attempts to integrate tonal means into modern music.

    14. Theodor Adorno, Philosophy of Modern Music, trans. Anne G. Mitchell andWesley Blomster (New York: Seabury Press, 1973), p. 59.

    15. Elsewhere, Adorno opposes the stability of triadic harmony to the instability ofatonal harmony harmony that has become latently polyphonic: `With theincreasingly dissonant character of harmony, the tension in individual sonorities

    also increased. No sonority was self-contained, like the old consonance, the``resolution''. Every sonority seemed to be laden with energy, to point beyonditself, and every one of the distinct individual notes contained within it requiredan independent ``melodic'' continuation of its own, instead of there being asuccession of one synthesized overall sonority after another' (`Function ofCounterpoint', in Sound Figures, trans. Rodney Livingstone (Stanford, CA:Stanford University Press, 1999), p. 127).

    16. By the time of his Aesthetic Theory (ed. Gretel Adorno and Rolf Tiedemann,trans. Robert Hullot-Kentor (Minneapolis: University of Minnesota Press,1997)), Adorno's position had evolved further. What is remarkable in this laterwork is his attempt to rescue harmony from its prior stigmatisation. This is atwofold process. Firstly, he generalises harmony by resurrecting the classicalaesthetic idea that harmony, broadly construed, concerns the balance betweenpart and whole: `inspiration, if it is to count, must gel; that tacitly presupposes anelement of organization and coherence, at least as a vanishing point' (p. 157).Secondly, he maintains that modern art, despite the appearance of disorder andincoherence, still possesses a faint link with this broader conception of harmony,insofar as such art stands in a relationship of determinate negation with respect tothe classical notion of harmony. Even if there is no immanent unity in a givenwork, this unity persists insofar as the active denial of harmony depends upon a

    memory or trace of some antecedent notion of harmony: `What speaks for thesurvival of the concept of harmony as an element is that artworks thatremonstrate against the mathematical ideal of harmony and the requirement ofsymmetrical relations, striving rather for absolute asymmetry, fail to slough off allsymmetry. In terms of its artistic value, asymmetry is only to be comprehended inits relation to symmetry' (p. 158). What proves problematic, then, is the specificform that harmony takes in the triad, since the chord presents an imbalance in therelationship of part to whole. It is not harmony that is to be discarded, but aparticular manifestation of harmony: `The mistake of traditional aesthetics is thatit exalts the relationship of the whole to the parts to one of entire wholeness, to

    totality, and hoists it in triumph over the heterogeneous as a banner of illusorypositivity' (p. 157). The spectre of this mistake haunts the broader notion of

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    aesthetic harmony as well, to the extent that the attempt to coordinate the variousparts of a work risks lapsing into a form of domination: `With regard to itselements, such aesthetic harmony is negative and stands in a dissonant relation tothem: they undergo something similar to what individual tones once underwent

    in the pure consonance of a triad' (p. 157). What is intriguing here is Adorno'sclaim that the aims of the whole do not accord with the aims of the elements,which introduces a point of conceptual dissonance to the otherwise perceptualconsonance of the triad.

    17. See Ligeti's comments in `On My E tudes for Piano', Sonus, 9/i (1988), pp. 37;```Eine unglaublich direkte Emotionalita t'': U ber Conlon Nancarrow',MusikTexte, 73/74 (1998), pp. 614; and in his interview with Denys Bouliane,`Stilisierte Emotion: Gyo rgy Ligeti im Gesprach', MusikTexte, 28/29 (1989),pp. 5262.

    18. James K. Wright and Albert S. Bregman, `Auditory Stream Segregation and the

    Control of Dissonance in Polyphonic Music', Contemporary Music Review, 2(1987), p. 70.

    19. Ibid., p. 72.

    20. The preceding discussion is drawn from Wright and Bregman (ibid., p. 74).

    21. A brief discussion of this section can be found in Ulrich Dibelius, `Ligetis HornTrio', Melos, 46/i (1984), pp. 545.

    22. Ligeti's fascination with just intonation and alternative tuning systems incipientin works like the Horn Trio and the Piano Concerto has assumed a more

    prominent position in his musical output during the 1990s, in works such as theViolin Concerto and the Hamburg Concerto. This fascination may be traced backto a number of sources, including Harry Partch (whose music Ligeti encounteredduring his sojourn in California in the early 1970s); the late works of ClaudeVivier; the music of sub-Saharan African and Balinese gamelan music (the latterexplicitly cited in the eighth e tude, Fem); and, most recently, the music of theFrench Spectralists (hence the title of the fifth movement of the HamburgConcerto `Spectra').

    23. In fact, this passage can be seen as manifesting what Wright and Bregmandescribe as the `principle of repetition' in auditory streaming: `. . . the obstinate

    repetition of the same melodic unit, from which the ``ostinato'' derives its name,clearly contributes to the perceptual segregation of simultaneous tones inpolyphony.' They continue by asserting that the `ostinato seems to have the effectof capturing each tone into a larger linear structure, the repetitive sequence oftones. The sequence itself seems to act as a unit, whose repetitions groupsequentially with one another, rather than with simultaneous tones' (`AuditoryStream Segregation', p. 80).

    24. In its original published form, the e tude was entitled Cordes vides. However,Ligeti changed the title in the so-called `final' edition of the e tudes to Cordes avide. The change in title may have something to do with the latter's richer

    potential for wordplay. No longer does the title simply refer to `open strings', but

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    it suggests in addition a movement `into the void'. It may even connote `avid'strings. (My thanks to Sarah Raff for pointing out these various implications ofthe new title.)

    25. Observe that this e tude, like the first of the set, Desordre, is dedicated to none

    other than Pierre Boulez.

    26. Thus Hannes Schutz observes that a `clear functional distinction between the twohands is produced from the outset, in spite of all their similarities (continuousmotion in quavers; short, accented phrase groups)'. See `Wiedergeburt der Arssubtilior? Eine Analyse von Gyo rgy Ligetis Klavieretude Nr. 2 Cordes vides', DieMusikforschung, 50/ii (1997), p. 208. Intervallic content abets the segregation ofthe lines, as Schutz notes; the line in the right hand includes tritones and sixths inaddition to fifths, as a consequence of its freer motion, whereas the left hand isrestricted within the first section to perfect fifths and fourths and theirtranspositions.

    27. Arnold Schoenberg, `Opinion or Insight' (1926), in Style and Idea: SelectedWritings of Arnold Schoenberg, ed. Leonard Stein, trans. Leo Black (Berkeley:University of California Press, 1984), p. 263.

    28. Ibid.

    29. John Cuciurean has also analysed the chord progressions in Fanfares in somedepth, extending neo-Riemannian concepts to explain some of the more unusualprogressions in the opening (`A Theory of Pitch', pp. 12037).

    30. Ligeti's own comments motivate my interpretation. Describing the composition

    of his opera Le Grand Macabre, he remarks that his original intention of writingan anti-opera (such as Mauricio Kagel's work Staatstheater) dissipated as he cameto the realisation that `the time of anti-operas is over. To use a witty phrase, Icalled Le Grand Macabre anti-anti-opera and the double negative results inaffirmation' (Gyorgy Ligeti in Conversation with Peter Varnai, Josef Hausler,Claude Samuel and Himself, trans. Gabor J. Schabert et al. (London: EulenbergBooks, 1983), p. 68).

    31. In his interview with Szitha, Ligeti complains that `My rejection of avant-garde music also lays me open to attacks and accusations of being a postmoderncomposer. I don't give a damn' (Szitha, `A Conversation with Gyo rgy Ligeti',

    p. 15).

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