A new theoretical paradigm of considerable power was developed by David Lewin in the early 1980s and formalized in his Generalized Musical Intervals and Transformations (1987). The theory speaks of points in a conceptual musical 'space', and of distances between such points. These points are called 'elements', and the distances 'intervals'. 'Space' is conceived broadly to denote any of three musical dimensions: pitch, time, and timbre. Pitch space denotes any collection of pitches or pitch-classes arranged in scale order; Lewin's examples cite diatonic and chromatic scales, each either in just intonation (extending infinitely upwards and downwards, hence its elements are 'pitches') or in equal temperament (recycling with each octave, hence elements are pitch-classes mod-7 and mod-12); but any other kind of scale applies equally well -- pentatonic, whole-tone, octatonic, or any of the church modes (mod-5, mod-6, mod-8, mod-7 respectively, in recycling). Temporal space is marked off by pulses called 'beats', or if time is considered as circular, 'beat-classes' (mod-4, mod-8, mod-16, etc.), or by 'durations' or 'duration-classes', by 'ratio-classes', even by 'tempo-classes'. Timbre space comprises a collection of steady-state timbres, defined by their harmonic spectra (i.e. absence or presence of individual harmonics, and strength of those harmonics present). In all of these cases, the elements in the spaces are conceived as having 'intervals' between any two -- pitch-, rhythm-, or timbre-intervals -- the intervals in each space having the properties of a mathematical 'group', and capable of being formalized as a single 'Generalized Interval System' (GIS), and giving rise to the operations of transposition, inversion and interval-preservation.

Lewin proceeds to generalize certain aspects of unordered set theory (including Forte's 'interval vector', 'complex' and 'subcomplex' -- see above) so as to be invoked within the theory of GIS. He next moves to a higher music-organizational level. Instead of starting with elements in a space and examining the operations that occur between them, he starts with the operations themselves, and conceives them as forming GIS structures. This renders the theory more intuitive in that listeners, for cultural reasons, tend to hear in terms of such operations (transposition, modulation; we might add augmentation, diminution, change of tempo, etc.) rather than of individual notes, time-points, durations, etc. Lewin thus subsumes GIS theory within a broader transformation theory of musical relations. The latter, although cast in mathematical form, in turn connects interestingly with existing theories such as motivic transformation and tonal prolongation. In particular, Lewin invokes tonal function theory, as explicated by Hugo Riemann (1893 and later: see above §11), using 'Klang' for consonant triad, and treating 'dominant', 'subdominant', 'relative', 'parallel', etc. as transformational functions. Thus the form '(C,-)MED = (Ab,+)' reads 'C minor become the mediant of Ab major', and so forth. Lewin incorporates also Riemann's 'leading-tone exchange' (Leittonwechsel) relations, thus '(E,-)LT = (C,+)', i.e. 'E minor becomes the leading-tone exchange chord of C major'. He develops a method of graphic analysis to present such transformations, and applies them analytically to passages from Wagner's Ring and Parsifal, while also formalizing them mathematically. In Musical Form and Transformation: 4 Analytic Essays (1993), Lewin applied his transformation theory to works by Debussy, Webern, Dallapiccola and Stockhausen.

The final part of this theory itself spawned a new line of enquiry known as 'Neo-Riemannian theory', which goes back to Lewin's first article on the topic (JMT, 1982). Dissertations by Hyer (1989) and Kopp (1995) made important contributions, another by Mooney (1996) explored further Riemann's Tonnetz (network of tonal relations), bringing Riemann's theoretical evolution and the theories of Hauptmann (1853) and Von Oettingen (1866) into the framework of the new theory, as also did Harrison's reinterpretation of dualism (1994). To these, Cohn (1996, 2000) has drawn in the progressive harmonic theories of Weitzmann (1853). Elements of later 19th-century tonal theory have, in this way, been revalidated and formalized as tools for examining late 19th- and early 20th-century harmonic practice.

The period after 1970 saw analysis emerge as a recognized discipline within musical studies, comprising a number of approaches and methods. This was reflected in a spate of new journals, some of them edited by graduate students in America, and in the formation of new societies of music theory. Theory and Practice, the organ of two student societies in New York State, began in 1975; so too did In Theory Only, journal of the Michigan Music Theory Society; and Indiana Theory Review followed them in 1978. All three contained analytical articles. Most important among the new organizations was the Society for Music Theory, founded in 1977, and its journal, Music Theory Spectrum, launched in 1979. Music Analysis, founded from King's College, London, in 1982 by Jonathan Dunsby, was the first periodical (at least since the days of Schenker) to be devoted specifically to analytical matters, representing a wide range of theories and approaches; and early issues of Contemporary Music Review, launched from Nottingham in 1984, featured largely analysis of contemporary music. A series of monographs, Studies in Musical Genesis and Structure, connecting source- and sketch-studies with analysis, was launched in 1985 with Lewis Lockwood as editor. In Germany the periodical Musiktheorie was inaugurated in 1986. A regular series of British conferences on analysis began in London in 1984 and was taken over by the newly formed Society for Music Analysis in the early 1990s.

Initially, the circumscription of analysis through its methods brought new levels of theoretical rigour and operational refinement in the wake of virtually autonomous development. By the mid-1980s, two methods had achieved the widest currency and were seen respectively as core systems for the analysis of tonal and atonal music. These were (1) codified Schenkerian techniques, and (2) techniques using pitch-class set theory. (A sharp distinction was thus drawn between tonal and atonal analysis.) A little later, as those engaged in analysis sought to define their discipline in relation to broader currents in contemporary thought, and those outside analysis correspondingly sought to understand its purposes and outputs, the perceived self-sufficiency of method-driven analysis diminished dramatically. Analysis in the late 1990s was seen as a critical activity, drawing on established ways of writing that came in part from its historical accumulation of methods, but with a significant shift from the model of forensic examination towards the construction of interpretations.

Many developments in the period can be understood against the context of three broad lines of inquiry that were followed in relation to Schenkerian analysis. One, reflecting a widespread belief in the potential of methodical analysis to offer insight into music of virtually any kind, involved the application of extended kinds of Schenkerian analysis to musical repertories that had not been Schenker's own concern, including music composed around the turn of the 20th century, early music, popular music and non-Western music. Another, reflecting a school of thought that valued formalism in both theory and method (and which considered analysis as the practical application of theory), saw a number of scholars seeking to rationalize and codify Schenker's approach into a rigorous combination of music theory and analytical method. A third line of development, mainly occurring towards the end of the period and reflecting the growth of a new musical criticism, saw studies that treated Schenker's work as a body of critical writing situated in the cultural and philosophical network of its time, and examined the reinterpretation of its language and ideas in the decades following World War II.

Aspects of set theory entered the theory of musical composition with J.M. Hauer's theory of tropes (1925), and are evident in the writings of Rene Leibowitz, Josef Rufer, George Perle, George Rochberg (1955, 1959) and Pierre Boulez (1964, chap.2; 1966, part ii). The proper formulation of a set theory of music was the work of Milton Babbitt (1955, 1960, 1961, 1972), Donald Martino, David Lewin and John Rothgeb (JMT, iii--v, x, xi). But while Babbitt's work, using particularly the mathematical concept of the group, dealt with harmony and with the functions of melodic and rhythmic configurations in 12-note music, and also with the interaction of components over longer spans of time, it belonged to the realm of compositional theory rather than analysis.

The most significant analytical contribution was made by Allen Forte (1964, 1965, 1972--3, 1973). Forte established a numerical notation for musical pitches by disregarding the octave in which they were sounded and treating enharmonically equivalent pitches as identical. All Cs (together with all B#s and Dbbs) were assigned to pitch class 0, all C#s and Dbs to pitch class 1, and so on to pitch class 11 (all B?s and Cbs). He reduced the number of possible collections of pitch classes to manageable proportions by classifying sets as identical if they could be reduced to the same 'prime form' by transposition and/or inversion, and established a system of labelling by which any prime form, and thus by extension any set, could be identified. These labels had two main elements, the first of which (the cardinal number) was simply the number of pitch classes in the set. The second element (the ordinal number) was, strictly speaking, arbitrary -- it simply referred the user of the system to a list of prime forms published by Forte (1973) -- but was also to some extent based on an assessment of the intervals that arose within the set (i.e. among its constituent elements). In a few cases, distinct types of set could by such assessment be observed to possess an identical 'interval-class content'; such sets were said to be 'Z-related', and 'Z' was incorporated into the names of both set classes. Examples of set class names include 3--1, 4--Z15, 4--Z29 and 9--1: of these, 4--Z15 and 4--Z29 are Z-related, and 3--1 and 9--1 are complementary, which is to say that for each set of type 3--1 there is a set of type 9--1 which includes exactly the pitch classes that the 3--1 set omits from the 12 available.

Analysis using set classes had to reckon with the fact that the theory of pitch class sets deals almost exclusively with sets of from two to 12 elements, so that a musical work containing perhaps many hundreds or thousands of notes had to be divided up into small units amenable to the categorization and comparison that the theory allowed. This 'segmentation' process remained dependent on informal musical judgment, despite receiving some attention from theorists such as William Benjamin (1979) and Christopher Hasty (1981). By contrast, the abstract relations among sets and set classes were the subject of considerable theoretical examination, for example by Robert Morris (1979--80, 1987), John Rahn (1980), Charles Lord (1981) and Eric Isaacson (1990). It is impossible, however, to observe a water-tight distinction between the processes of musical segmentation and the assessment of pitch class set relations, nor can it be maintained that one is firmly the province of analysis and the other of theory.

Forte extended basic pitch class set theory to include the association of sets within 'set-complexes' (K) and 'subcomplexes' (Kh) -- a 'complex' being an array of all the sets that are related by inclusion and/or complementation to any one given central class of sets (the 'nexus set'). This established a type of organization that made possible the elucidation of atonal coherence in large-scale musical structures (Forte, 1978, on Stravinsky's The Rite of Spring; Janet Schmalfeldt, 1983, on Berg's Wozzeck; James Baker, 1986, on sonatas and orchestral works by Skryabin). A further extension of the theory was made by Forte in 1988 through a theory of pitch class set genera, which was based on a classification of all sets with four or more elements according to their trichordal subsets. 12 genera of varying sizes were thus constituted, with several large set classes being assigned to two or more genera. Each genus was linked informally with descriptions such as 'chromatic', 'diatonic' and 'atonal'. The associated analytical procedure involved making a segmentation of a musical passage and tabulating those set classes to which it could thus be reduced, and then noting the genus (or genera) to which each set class in the analysis belonged and calculating the statistical significance of each genus's representation in the sample. On this basis, weakly represented genera that were superfluous to the overall generic profile of the analysis were omitted. This typically left a generic profile based in quantified proportions on two or three genera, in a way that had been sensitive to the analytical context.

Fig.24 is taken from Forte's analysis (1981) of the first of Schoenberg's Three Piano Pieces op.11. This uses a diagrammatic notation in parallel with a passage in score: the segmentation into pitch class sets is shown by enclosing pitch class letter-names in rectangles, circles and ovals, the result leaving space for set labels, bar numbers and other items. It can be seen that the analysis proposes three levels of structure in bars 1--8. In level (a), six of the hexachords (i.e. six-note sets) which are to be of great importance for the structure of the piece are identified (6--Z10/Z39, 6--16, 6--21, 6--Z42 and 6--Z44) along with several of the pentachords, which together with their seven-note complements are destined to be significant. Level (b) shows new ways of segmenting the musical fabric; level (c) shows in particular some of the trichords (three-note sets) which play on the surface of the music. The recurrent hexachords and pentachords are seen as the basic harmonic vocabulary of the piece, whereas the trichords are the foreground through which the analysis must penetrate. Among the hexachords, the importance given in the analysis to the Z-related pair 6--Z19/44 illustrates the interdependence of pitch class set theory, segmentation and critical analysis. One pitch class set of type 6--Z44 [10, 11, 0, 3, 4, 7] corresponds to those letters of Schoenberg's surname which may be interpreted as letter-names for musical pitches (SCHBEG, extracted from 'Schonberg'). Although this set class is represented only once in level (a) and the segmentation would be no less complete without it, Forte found significance in its appearance in a number of Schoenberg's early works (1978). Elsewhere, the Z-relation allows him to assert the presence of Schoenberg's 'signature' through sets of type 6--Z19 as well.

Other writers identified a number of specific pitch class collections as being of particular importance in early 20th-century music. The most prominent of these was the 'octatonic' collection -- comprising, for example, the pitch classes C, Db, Eb, E?, F#, G, A and Bb (two further transpositions of the collection are available) -- which had been discussed in analyses of music by Skryabin and Stravinsky by Perle (1962) and Arthur Berger (1963--4). The role of octatonic collections in the music of Debussy, Webern, Bartok and Skryabin was further discussed by Forte (1991, 1994), Richard Cohn (1991) and Wai-Ling Cheong (1993). The fullest investigation was carried out by Pieter van den Toorn (1977, 1983), who showed in exhaustive detail how interactions between this collection and the diatonic collection could be charted in much of Stravinsky's music, taken from all the commonly recognized stylistic periods of the composer's career. In doing so, van den Toorn indicated that the octatonic collection could be articulated into a network of characteristic harmonic and melodic features, including triads, dominant 7th chords and scale fragments (fig.25).

This renewed concern of analysis with compositional method reflected a general expectation that analysis of 20th-century modernist works would assist in their broader comprehension. Appeals to biographical data and the composers' own writings frequently underpinned analyses of works by Schoenberg, Berg, Webern, Hindemith, Messiaen, Boulez and others who had given clear indications of their own technical procedures. Without necessarily showing greater technical rigour, these analyses achieved greater acceptance in the early part of the period than, for example, Roy Howat's assertion (1983) of Golden Section proportions in Debussy's works. Later in the period analyses that were not dependent on biography for their credibility came to be more widely valued. Among these were studies of Debussy by Richard Parks (1989), Bartok by Paul Wilson (1992) and Berg by Dave Headlam (1996). Other notable analytical studies of individual 20th-century composers included those on Britten by Peter Evans (1979), Berg by Douglas Jarman (1979), Bartok by Elliott Antokoletz (1984), Hindemith by David Neumeyer (1986), Varese by Jonathan Bernard (1987) and Webern by Kathryn Bailey (1991). Bernard's graphic notation (see also 1981, 1983, 1987) was a notable individual innovation in presentation whose similarity to display formats routinely available in MIDI sequencing software by the mid-1990s suggested one possible route to future computer-assisted analysis.

Robert Morgan (1976) and James Baker (1983, 1986) were prominent in developing Schenkerian principles to reflect the changes in harmony and form evident in progressive music composed during the mid-19th century through to the turn of the 20th. These extensions of the technique found wider acceptance than had earlier comparable analyses of Stravinsky, Schoenberg and Bartok by Roy Travis (1959, 1965--6) and of Wagner by William Mitchell (1967). A number of writers including Craig Ayrey (1982), Anthony Pople (1983, 1989), Joseph Straus (1987), James Baker (1990) and Edward Pearsall tackled the question of how Schenkerian chord prolongation might be said to occur in post-tonal or atonal music, but little consensus was reached on this topic. Another approach to music of the early 20th century which seemed to have characteristics of both tonality and atonality (according to the common understanding of those terms) was to present linked analyses based on the application of both Schenkerian and pitch class set methods to the same musical passages. There was generally little attempt at a hybridization of analytical method or of underlying theory: the aim was rather to provide demonstration of the co-existence of tonal and atonal features through established methods. Notable examples of such writing were studies of music by Schoenberg and Berg by Forte (1978, 1985) and Janet Schmalfeldt (1991; see also Schmalfeldt, 1983).

Applications of adapted Schenkerian principles to early music, for example to Lassus by Mitchell and Isaac by Saul Novack (both 1970), and to Monteverdi by Salzer (1983), were no less contentious theoretically than were Schenkerian analyses of post-tonal music; they also failed to achieve ready acceptance among scholars and musicians with a lively historical interest in this repertory (see the discussion by Mark Everist, 1992). A number of later studies responded to this difficulty of assimilation by examining voice-leading in early music less rigidly in line with Schenkerian archetypes and more evenhandedly in juxtaposition with technical matters familiar from the history of theory and composition -- such as cadence formation, modality, compositional process and text structure. Examples included work by Daniel Leech-Wilkinson and Sarah Fuller on Machaut (1984, 1987, 1992), and by Cristle Collins Judd on Josquin (1985, 1992). These writers were perhaps more successful in achieving a conceptual synthesis than were many of those engaged concurrently in analysis of the post-tonal repertory, though this was not necessarily recognized at the time.

Explicit codification of Schenker's principles went beyond the earlier objective of bringing his work to the fuller attention of English-speaking musicians, something which was largely accomplished by the translation of Der freie Satz by the Schenker pupil Ernst Oster as Free Composition (1979). Together with subsequent translations, including those of Kontrapunkt (1987) and Das Meisterwerk in der Musik (1994, 1996, 1997), this largely displaced the need for texts such as Forte's presentation (1959) of Schenker's approach as a conception of musical structure, which had been instrumental in drawing the Austrian's work into the discipline of analysis as practised and understood by professional scholars in the English-speaking world, particularly the USA. The demands of music theory pedagogy, however, led to a need for textbooks in Schenkerian analysis which was initially met by those of Forte and Gilbert (1982) and David Neumeyer and Susan Tepping (1992). Examples of the use of Schenkerian techniques in analysis were too numerous to be usefully cited here. Among recurring concerns of such analysis were large-scale parallels across sonata-form movements made evident at a middleground analytical level, often involving voice-exchanges and/or the enharmonic reinterpretation of chromatic pitches. Examples were to be found among the writings of Eric Wen (1982), Carl Schachter (1983), David Beach (1993), and Roger Kamien and Naphtali Wagner (1997).

A second kind of codification of Schenker's work may be distinguished, more far-reaching than the first, in which theorists sought explicitly to refine and reinterpret Schenkerian principles along scientific lines into an explicit and rigorous body of theoretical constructs and analytical practices. Two such reinterpretations were undertaken by Gregory Proctor and Herbert Lee Riggins (1988) and Richard Littlefield and David Neumeyer (1992); an earlier, less explicit reformulation was given by Jonathan Dunsby (1980). All of these were apparently independent of the computer-based reformulations of Schenkerian concepts and procedures undertaken in the 1960s and 70s by Michael Kassler (1967) and Stephen Smoliar (1976--7).

Schenkerian thought was also one catalyst for the broad theory, based on analogies between linguistics and analysis, of Fred Lerdahl and Ray Jackendoff (1977, 1981, 1983). Under Noam Chomsky's influence, questions such as 'Does music have a deep structure?' and 'Do universals exist in music?' had fascinated musicians in the 70s. The series of lectures given under the title 'The Unanswered Question' by Leonard Bernstein in 1973, later televised, and published in 1976, raised these in a challenging fashion. In a series of articles beginning in 1977 and culminating in A Generative Theory of Tonal Music (1983), Lerdahl and Jackendoff, composer and linguist respectively, evolved a theory whose central purpose was to elucidate the organization that the listener imposes mentally on the physical signals of tonal music. Using principally music by Bach, Haydn, Mozart and Beethoven as their 'idiom', they compiled a grammar that 'models the listener's connection between the presented musical surface of a piece and the structure he attributes to that piece. Such a grammar comprises a system of rules that assigns analyses to pieces'. The system was thus like Chomsky's grammar in that it was 'mentalistic', that is, concerned with mental processes rather than with end-products, and in that it was at heart an analytic procedure in which the generative function of the theory was a system for deriving or testing analyses.

The theory of Lerdahl and Jackendoff had an outward resemblance to Chomsky's in that it was a set of rules operating on four components. These four were 'dimensions' of musical structure, and all were hierarchical: 'grouping structure'; 'metrical structure'; 'time-span reduction' (reduction in the Schenkerian sense, but based equally on pitch and rhythmic criteria); and 'prolongational reduction', which took account of the intuitive sense of tension and relaxation in music. Two types of rule governed each category: 'well-formedness rules' which controlled the making of possible structural descriptions of pieces; and 'preference rules' which determined which of a number of possible descriptions corresponded to a listener's preferences.

Three graphic conventions were adopted for demonstrating the operation of these rules. Grouping structure was shown as horizontal braces, and metrical structure as lines of dots, both below the staff and reiterated vertically to express hierarchy; reduction was shown as branching trees above the staff. Fig.26 gives a small-scale example, which is a local time-span reduction. For prolongational reduction, the three diagrams acquire solid and void circles at the nodes to indicate the type of prolongation that the branching represented; reductive levels were displayed on separate staves below, adopting Schenkerian conventions for solid and void noteheads and solid and dotted phrase-marks. Fig.27 shows the St Anthony Chorale analysed in this way (the original is already slightly reduced).

Lerdahl and Jackendoff claimed that much of their grammar was 'idiom-independent' (i.e. it held good whatever the musical style), and thus that certain of their rules constituted 'universals' of musical perception and could be taken to represent innate aspects of musical cognition. Their work was published at a time when points of contact between analysis and the cognitive sciences were gaining significant ground. The journal Music Perception, founded in 1983, included in its early issues a number of articles that reflected this, such as a group of investigations into hierarchical structures in music by Eugene Narmour, Allan Keiler, and Lerdahl and Jackendoff (all 1983--4), studies of synaesthesia in Skryabin and Messiaen by Peacock (1985) and Bernard (1986), and of the perceptibility of polytonality and serial organization by Krumhansl and Schmuckler (1986), Sandell and Sergeant (1987).

Many researchers in musical cognition took the results of musical analysis as a point of departure. Their focus was on such questions as whether the structures typically proposed by analysis were perceptible by others. To some extent this was a consequence of the difficulty of finding a proper role, in such empirical enquiry, for one of the characteristic aspects of analysis, namely the analyst's personal reflection on a musical work. For example, the large team of specialists built up in the late 1970s by Boulez at IRCAM in Paris sought to place acoustical and psychoacoustical research at the service of composers: in doing so it encouraged the analysis of the materials of music rather than of musical works themselves. Ironically, this general difficulty may have contributed to the development of an enhanced critical dimension in analysis, both by subjecting typical products of structural analysis to direct scrutiny and by making it seem that if analysis were transformed into a cognitive science it might surrender its capacity to foster new interpretations of pieces of music. This did not prevent the cognitive sciences from making a significant impact on musical thoery, however, as was shown, in addition to the theory of Lerdahl and Jackendoff, by Narmour's exceptionally detailed theory of melodic structure (1992), and Robert Gjerdingen's application of schema theory to phrase patterns in classical music (1986, 1988). The impact of these theories on analytical practice, however, was limited, whereas the Lerdahl--Jackendoff theory could be regarded as sufficiently predictive to warrant empirical investigation of the validity of its rules, for example by Irene Deliege (1987).

Ruwet's analysis triggered a dispute among semiologists as to whether in such a mechanized procedure the analysis should begin with musical units of large proportions and work towards a microscopic finished analysis, or begin with a microscopic segmentation and gradually construct the larger formal units by the recognition of equivalence between phrases that were distinguished only by differences of detail. The immediate dispute was won by the second school of thought, and the Montreal-based scholar Jean-Jacques Nattiez (1975) aroused remarkable interest with intensive analyses that proceeded from small-scale segmentation. Despite the variety of Nattiez's means of presentation (which included the tree-structure diagram, the lexicon of items and the table of distributions), and despite the apparent limitation of distributional analysis to monophonic music, or at any rate to a single melodic line, the distributional principle was for a while adopted as if it were an analytical method to be set alongside Schenkerian and set-based methods. The comparison of uninterpreted 'unities' was seen as a productive strategy in the analysis of otherwise intractable modern scores (e.g. Nattiez's analysis of Varese's Density 21.5, 1975; Eng. trans., 1982; fig.28) and in music whose interpretation seemed prey to ambiguities. Among the subsequent products of this line of work were a full-scale analytical study of Berio's Sinfonia by David Osmond-Smith (1985) and Nattiez's study of the 1976 Bayreuth performance and production of Wagner's Ring cycle (1983; see also 1985).

Nattiez's work followed a branch of semiotics that proposed a partition of the semiological space into three levels: the 'poietic' level, which concerned the relations between the score and the composer (or performer, if the object of study was a performance); the 'esthesic' level, which concerned the relations between the score (or other musical object) and its interpreters; and between these two the 'neutral level' (niveau neutre), a supposedly uninterpreted domain within which distributional analysis was to take place (see Molino, 1975; Eng. trans., 1990). The neutral level was soon acknowledged as problematic, however, since even the most basic division of an object into 'unities' demanded some rudimentary understanding, and this implied that semiosis had taken place. Once the neutral level had been revealed as a methodological convenience it was clear that the concerns of semiology, properly understood, were not so easily to be brought into conjunction with the practice of analysis. Nattiez's later writings (e.g. 1985, 1990) addressed this difficult relationship with responsibility; others such as Simha Arom (1969), David Lidov (1975), Raymond Monelle (1992) and Eero Tarasti developed musical semiotics along more secure and conventional semiological lines. But the encounter with the directly analytical semiotics of Ruwet and Nattiez served to encourage analysts including Agawu (1991) and Robert Samuels (1995) to develop wide-ranging arguments that assisted in broadening the perspectives of the discipline: the former presented a rich synthesis of viewpoints on the Viennese Classical style, and the latter an analysis of Mahler's Sixth Symphony that dealt equally in close motivic analysis and the work's wider cultural resonances.

The semiological focus of these writings notwithstanding, both Agawu and Samuels readily incorporated Schenkerian analyses in their work, thereby reflecting the dominating impact of Schenker's ideas on professional analysts. Some writers, however, exhibited a concern that these ideas had been transmuted in the course of being assimilated into a mainstream of analytical practice. Returning to Schenker's own writings and re-reading them in the context of his own time, they drew attention to considerable disparities that had thus arisen, although their criticisms were often directed against earlier interpreters of Schenker rather than against their own contemporaries. Prominent among these writers were William Rothstein (1986) and Robert Snarrenberg (1994).

The rise of such criticism reflected the growth of analysis into an academic discipline with sufficient maturity and self-awareness to question its own assumptions and practices. The early part of the period had seen the publication of analysis symposia, notably in the Journal of Music Theory, in which musical works were discussed by two or more authors using contrasting analytical approaches. A collection of these symposia, through which Schenkerian analyses of pieces by Mozart, Schubert, Beethoven and Brahms were contrasted (by implication favourably) with analyses that followed a variety of other methods, appeared under the editorship of Maury Yeston in 1977. By around 1980, however, this practice had largely fallen away, leaving Schenkerian analysis and pitch class set methods supremely prominent, with the additional consequence that repertories central to the development of those methods were taken as points of reference in the discussion of other musics.

It was at this time that an influential attack on the discipline by Joseph Kerman (1980) advised musical scholars 'how to get out' of analysis. Perhaps the most substantive of Kerman's observations were that analysis tended to concentrate on 'masterworks' and, concomitantly, that it took the aesthetic value of its musical objects of study as a given. In his view this discouraged the exercise of a properly critical faculty that was the first duty of a musicologist. The new florescence of critical writing encouraged by Kerman's wider commentary on musicological disciplines (see Kerman, 1985) did not have a great impact on the practice of analysis until it addressed issues of theory. When it did so it brought post-structuralist critical theory to the attention of musical scholars at a time when the influence of semiology was receding. Derridean thought centred on ideas of deconstruction was powerful in undermining the de facto definition of analysis as a constellation of methods among which a mere two methods were overwhelmingly prominent.

Joseph Straus (1990) and Kevin Korsyn were among analysts who made direct use of the literary critic Harold Bloom's theory of the 'anxiety of influence' in comparing musical works written by composers of different generations. Korsyn (1991) measured Brahms's Romance op.118 no.5 against Chopin's Berceuse op.57 through a close comparison of musical detail and correspondences of design, concluding that 'Brahms incorporates Chopin's text into his own, but ... then breaks with [it], resisting influence, choosing himself rather than the precursor' (p.57). Straus's examination of compositions by Bartok, Berg, Schoenberg, Stravinsky and Webern maintained that tonal allusions observable in them were traces of an inescapable past that these composers had nonetheless mastered, and argued firmly that the musical coherence of such works should be addressed through atonal, serial or other post-tonal analytical techniques.

Another important development during the 1980s and 90s was a reconsideration of the history of analysis, notably by Ian Bent (1984, 1994, 1996). In particular, Bent identified precedents for the critical-analytical hermeneutics of the 90s across a wide variety of 19th-century writings on music, not all of which had previously been considered analytical in nature or intention. A number of other scholars, such as Alastair Williams (1997), found a different impetus towards an analytical criticism that exposed ideologies, while dealing rigorously with musical detail, in the writings of Theodor W. Adorno.

A concern with immediate perception also motivated developments in the analysis of phenomena of timbre, melodic contour and aspects of rhythm and metre. Robert Cogan and Pozzi Escot (1976) took phonological analysis, as performed in the field of linguistics, as a model for investigating what they called 'sonic design' -- the way in which sound-spectra are shaped in musical space. Their contention was that compositions are just as much formations of basic sonic stuff as formations of tonal or rhythmic materials, and that composers and eras of music often bear recognizable sonic 'fingerprints'. They made resourceful use of graphs to carry out tone-colour analysis of single instrumental sounds and ensembles and also made use of sound-spectrum analysis, a technique developed by IBM which was capable of photographing the sonic 'content' of a whole composition. Cogan provided an analysis of Stravinsky's The Rite of Spring in his journal Sonus (1982), and in 1984 presented a series of analyses together with a theory of tone-colour strongly influenced by the linguist Roman Jakobson. Studies of melodic contour were less spectacular in their presentation, frequently being expressed in numerical or symbolic terms reminiscent of pitch class set analysis, though with the important difference that the order of terms was crucial in defining a sequence of directed intervals or of changes of direction. Foundations for future work were laid by Michael Friedmann (1985), Elizabeth West Marvin and Paul Laprade (1987), Larry Polansky and Richard Bassein (1992), Robert Morris (1993) and Ian Quinn (1997).

Like much work in the analysis of contour, Lewin's broad phenomenology (1985--6) was presented in formal terms suggesting the use of Artificial Intelligence as a metaphor:

I propose as a provisional model for 'a musical perception' this basic formula:
p = (EV, CXT, P-R-LIST, ST-LIST)
Here the musical perception p is defined as a formal list containing four arguments. The argument EV specifies a sonic event or family of events being 'perceived'. The argument CXT specifies a musical context in which the perception occurs. The argument P-R-LIST is a list of pairs (p_i, r_i); each pair specifies a perception p_i and a relation r_i which p bears to p_i. The argument ST-LIST is a list of statements s1, ..., s_k made in some stipulated language L.

Similarly, Lewin developed structural analysis through a theory of 'generalized musical intervals and transformations' which was expressed in terms of the formal apparatus of mathematical lemmas and theorems (1987). The underlying principles of his approach were, first, that 'intervals' of some kind are found in many musical domains, including most obviously pitch and rhythm (each of which can be measured in a number of ways), and that their means of measurement can be modelled by simple mathematical operations; and, secondly, that any interval, for example the interval of two semitones between middle C and the D above it, can be expressed equivalently as a particular 'transformation' of the C into the D or vice versa (in this case a transposition upwards or downwards by two semitones). The second principle is harder to grasp than the first, but its potential to refocus musical reflection is correspondingly greater: it allows something that is intuitively thought of as a transformation of one musical object into another, such as a statement of a theme and a statement of a variant of part of it, to be brought within the same mathematical framework as something that is intuitively conceived as a relationship between two musical objects, such as the pitch interval between two notes, or the interval of time between successive statements of a fugue subject.

Lewin's theory of transformations was based on mathematical group theory, a powerful development of set theory which does not merely deal with collections of objects but associates those collections with specific operations on the objects. In analytical applications, for example by Lewin himself (1993) and by Henry Klumpenhouwer (1992; see also Lewin, 1990, 1994), it resulted in descriptions that reflected musical complexity as networks of transformations (and of transformations of transformations etc.) each of which was defined mathematically and could be labelled in a way that reflected the phenomenological import of the transformation (fig.29). The application of group theory to late 19th-century harmonic practice was also more directly explored by both Lewin himself (1983--4) and later writers including Cohn (1991, 1996).

During this period analysis may be said variously to have come into intimate contact with the cognitive sciences, semiology and critical theory, to have taken on the style if not the substance of applied mathematics, and to have defined itself as an academic discipline before dissolving that definition in favour of a millennial uncertainty. Of all the cognate disciplines, it is criticism that is presently the hardest to differentiate clearly from analysis. Both tend firmly towards the formalist side of what can be identified as a formalist/historicist divide, but if this has focussed 'formalism' as a sensitive issue, it must be acknowledged that those engaged in the new musicological criticism of the 1980s and 90s were less happy to be seen as formalists than were those whose primary concern was to develop structural analysis. Indeed, the concern of analysis with structure might still be taken as a defining characteristic -- were definition itself not shunned by many of its practitioners today. In these circumstances it seems right to acknowledge that structures are now understood to be asserted rather than discovered, that the analyst is more inclined than ever to see his or her work as the writing down of interpretations from a personal perspective, and that charting the discipline historically has been one catalyst in making the languages of analysis a focus of self-awareness for those who read and write with them.

Fundamental to most software synthesis systems is the provision of a basic library of functions that may be used as the building-blocks for a particular sequence of synthesis operations. Many of these functions simulate the hardware components of a traditional analogue studio, such as oscillators, filters, modulators and reverberators, although an increasing number of more specialist functions have been developed over the years to model particular instrumental characteristics, such as the excitation of the human voice-box or the vibration of a string. In the case of musicn programs, each integral grouping of these components is identified as an 'instrument', broadly analogous to the individual instruments of a traditional orchestra. These 'instruments' collectively form an 'orchestra', ready to receive performance data from an associated 'score'.

Since these instruments are simulations that are no more than ordered statements of computer code, the opportunities for varying their design and application are extensive. The only real constraints are general ones imposed by the computing environment itself, for example the maximum number of instrumental components that can be accommodated in the memory at any one time, and the overall processing performance of the system. It is possible, for example, to synthesize finely crafted textures by directly specifying the evolution of each spectral component in terms of its frequency, amplitude and duration. Such a strategy involves considerable quantities of score data and the simultaneous use of a number of instruments, one for each component. Alternatively, highly complex instruments can be constructed with the capacity to generate complete musical gestures in response to a simple set of initial score commands.

Although software synthesis methods are not nearly as well known to the music community at large as the custom-designed hardware systems that predominate in the commercial sector, their significance should not be underestimated, given the steadily increasing power and availability of the personal computer. With the rapid development of information systems such as the Internet, an increasing number of powerful synthesis programs can be located and downloaded for local use by means of a simple modem and telephone link. Since many of these facilities are being made available at little or no charge, their impact on future activities, professional and amateur, is likely to be considerable.

What has distinguished the commercial MIDI synthesizers from all-software synthesis methods such as those described above is the set of functional characteristics associated with each design. One of the earliest all-digital synthesizers, the Yamaha DX7, which appeared in the same year as MIDI, relies exclusively on the techniques of frequency modulation for its entire repertory of sounds. These techniques are based on research originally carried out by John Chowning at Stanford University in the 1970s using a musicn software synthesis program. The use of a custom-designed processor facilitated the registration of patents that forced other manufacturers to develop rival hardware architectures, each associated with a unique set of synthesis characteristics. Methods employed have ranged from additive synthesis, where composite sounds are assembled from individual frequency components, to phase distortion techniques that seek to modify the spectra of synthesized material during the initial process of generation. The latter shares some features with FM techniques, where one wave-form is used to modulate the functional characteristics of another.

The synthesis of sounds from first principles is subject to a number of constraints. Although particularly evident in cases where hardware features limit the choice and use of synthesis methods, such difficulties are also encountered in software-based environments, even those that permit skilled users to write their own synthesis routines from first principles rather than relying on library functions provided with the program. The root of the problem lies in the character of many natural sounds which can prove exceedingly hard to replicate by formulaic means, such as the transient components associated with the attack of an acoustic trumpet or oboe. In the commercial sector, the ability to imitate instrumental sounds is especially important, and impediments to the production of a realistic repertory of voices have inspired a number of manufacturers to pursue an alternative method of synthesis known as sampling. This is essentially a three-stage process of sound capture, optional intermediate processing and re-synthesis, starting with the selection of suitable source sounds that are first digitized and then loaded into a memory bank as segments of numeric audio data. A variety of processing techniques may then be employed to control the processes of regeneration, ranging from the insertion of a simple loop-back facility to allow sounds to be artificially prolonged, to sophisticated facilities that allow multiple access to the data for the purposes of transposition upwards or downwards and the generation of polyphonic textures. Although commercial samplers, like synthesizers, incorporate custom-designed hardware to meet the specifications of individual manufacturers, their general architecture comes very close to that encountered in a conventional computer. Whereas the methods employed in the design of the digital synthesizer clearly developed from earlier work in software synthesis, the progression in the case of sampling techniques has undoubtedly been in the reverse direction. As a result, many software synthesis programs, including the musicn family, now provide sophisticated facilities for the processing of externally generated sound material, and such modes of operation are gaining in popularity.

The blurring of a clear distinction between systems that rely on proprietary hardware and those that do not becomes even more evident when consideration is given to the wider spectrum of digital tools that have become available for manipulating and processing sound material of any origin, natural or synthetic. These range from simple editing facilities, which are little more than the digital equivalent of a razor-blade and splicing block, to more complex tools, which enhance the content of sound material by added reverberation, echo or chorus effects, or directly modify its spectral content by means of subtractive techniques such as filtering. The resources available for such applications range from self-contained processing units, which can be manually operated by means of controls on their front panels, to sophisticated computer-based facilities, which make extensive use of interactive computer graphics.

The measurement and analysis of musical tones is not easy. For a steady, unchanging tone the quantities that are most useful are the predominant frequency associated with it, the intensity or loudness, and the relative amplitudes and frequencies of the other components of the complex. If, however, the note is changing with time, then the way in which all these separate quantities change must also be recorded.

Intensity, or loudness, is usually measured by means of a microphone, amplifier and meter, but careful calibration is necessary and the relative positions of the microphone and source, the surroundings and many other factors affect the result. Sound level meters are available with built-in filters that have a frequency characteristic resembling that of the average human ear, but for an accurate estimation of the loudness of a sound it is necessary to measure the amplitude at a series of frequencies over the audio spectrum.

The ready availability of cheap and powerful computers and microprocessors has led to the almost universal adoption of digital techniques for the analysis of rapidly varying waveforms. An analogue-to-digital converter samples the magnitude of the disturbance at intervals that may be as short as desired (usually around 40,000 per second), yielding a sequence of numbers. Once in digital form, the signal can be processed by a mathematical technique known as Fourier analysis (see §8(ii)) to show how the amplitude of each frequency component changes over the duration of the signal. Information about the frequency content of a signal can be displayed in a number of ways. One is the sonagram, a two-dimensional diagram with frequency on the vertical scale, time on the horizontal scale, and intensity represented either by colour or by a grey scale. Fig.17 shows a sonogram of the first five seconds of a harpsichord note. Each of the vertically equidistant horizontal bars represents one of the almost exactly harmonic frequency components of the harpsichord sound; the different rates of decay of the components can clearly be seen.

The sonagram is generated by dividing the digital sound sample into a series of short time slices, on each of which the Fourier analysis is performed. The frequency spectrum of each slice can be individually displayed if desired, and the information contained in the frequency spectrum can be used to compute the loudness of the sound or the predominant frequency at the chosen time. Fig.17b, c and d show the frequency spectra at the beginning, middle and end of the harpsichord sound sample in Fig.17a.

With a suitably fast processor, the frequency spectrum can be displayed on a screen within a small fraction of a second of the data capture, and the display can be updated several times a second. The system is then described as a real-time analyser. Using a real-time analyser, a performer can see immediately how a change in the method of sound production affects the frequency spectrum of the sound.

Other modern but non-electronic techniques are also used in studies of musical sounds. High-speed cinematography can reveal a great deal of useful information and has played an important part, particularly in understanding the behaviour of reeds and of vibrating strings. The modern optical technique of holography is playing a part in revealing the way in which the body of a violin or other string instrument is vibrating. The patterns produced are something like those of the Chladni plate, but to produce sand figures on violin back plates large vibrators are needed and, though useful, measurements probably do not correspond to the behaviour of the instrument when it is played normally. Holographic techniques show up the vibration patterns even when the notes being played are extremely quiet. Fig.18 shows holographically produced vibration patterns for a violin back plate.

But a single note from a piano or harpsichord, for example, has no part that is strictly periodic, since the amplitude after the initial transient section is decaying all the time, and indeed different components, as has been observed, decay at different rates. Fig.20 may help to show how the analysis can be extended to cover this problem. If the diagram was drawn with three components which were the 200th, 201st and 202nd harmonics, and if the same frequency were used for the first component (i.e. 6?00 Hz), the second component would have a frequency of 6?03 Hz and the third 6?06 Hz. It is not difficult to see that the waveform would now repeat with a fundamental of 0?03 Hz. Thus by making the harmonics very close together it is possible to take care of a wave that repeats only after long periods; and making the harmonics infinitesimally close will enable one to deal with a wave that never repeats precisely. So the same technique of analysis can be used for non-periodic waves, provided one takes harmonics that are so close together that they form a continuous sequence. This kind of analysis of transients, using the digital techniques mentioned, is yielding important information about the transient behaviour of real instruments. In this form it is usually termed 'Fourier transform' or 'Fourier integral' analysis.

Any musical interval or difference of pitch distinctly smaller than a semitone. Some writers restrict the term to quantities of less than half a semitone; others extend it to refer to all music with intervals markedly different from the (logarithmic) 12th part of the octave and its multiples, including such scales with fewer than 12 pitches as are used, for example, in south-east Asia.

The use of microtones in Western art music is essentially a 20th-century phenomenon, though Julian Carrillo had experimented with his 'sonido 13' system of equal-tempered quarter-tones in the 1890s. Two basic approaches may be distinguished. Either microtonal intervals are introduced as finer divisions within regular 12-note equal temperament, or they arise as a necessary condition of different tunings. The former sort of microtonal composition was practised by Carrillo, by Charles Ives early in the 20th century and by Alois Haba and Ivan Vishnegradsky from the 1920s onwards. Haba also used smaller intervals, particularly the sixth-tone, and wrote a great many microtonal works, from piano pieces and string quartets to a full-scale opera. The capacity of string instruments to play microtones is limited only by the player's ear, but Haba's music required the construction of special instruments, including quarter-tone pianos, harmonium, clarinet and trumpet, all made in the 1920s and 30s. In the 1950s Carrillo had pianos built to play in every integral division of the whole tone down to the sixteenth-tone. Ives and Vishnegradsky wrote instead for multiple pianos differently tuned.

The foremost problem in microtonal music -- beyond the making and tuning of instruments -- is perhaps that of harmony; this may have encouraged composers to look in other directions from the 1970s onwards. Another prompt would have come from the new availability of quarter-tones on woodwind instruments, facilitated by unusual fingerings. On the one hand, quarter-tones became normal in music where high virtuosity and speed are in the foreground and natural consonance is not an issue, such as that of Brian Ferneyhough and Chris Dench. On the other, they found a cogent place in the upper treble for composers who based chords on harmonic spectra, for example Tristan Murail, Gerard Grisey and Claude Vivier.

The harmonic question is differently settled, of course, when microtones are conceived not as additions to the equal-tempered chromatic system but as basic intervals in other tunings -- tunings which have customarily been developed not in order to make available intervals smaller than a semitone but to find better approximations to just intonation than 12-note equal temperament can deliver. Harry Partch pioneered this approach, employing a 43-interval octave which made available frequency ratios involving the primes up to 11 and their multiples (his ascending scale begins 1:1, 81:80, 33:32, 21:20, 16:15, 12:11), and building his own instruments. Just intervals have also been achieved with a range of instrumental and vocal resources, notably by Eivind Groven, Lou Harrison, Ben Johnston, La Monte Young and James Wood.

Other composers have developed alternative equal temperaments, especially those which offer good simulacra of just intervals, as do the temperaments with 19, 31, 41, 53 and 72 intervals to the octave. Joseph Yasser argued for a 19-interval octave as a logical evolution from the 12-interval one of convention, just as the latter evolved from the 7-interval diatonic scale. Adriaan Fokker commissioned a pipe organ in 31-interval equal temperament, allowing accurate renderings of the just major 3rd and natural 7th. This organ has been used by other Dutch musicians such as Henk Badings and Hans Kox, and its tuning system has been applied by composers in the USA. Groven and Johnston have used 53-interval equal temperament, the 'Mercator system' favoured by earlier theorists. Easley Blackwood has composed with alternative equal temperaments on the piano, ranging from 13 to 24 intervals per octave.

Electronic music allows any kind of tuning without the need for virtuoso technique, retuning or the physical construction of new instruments, and without the complications of new notational systems, of which several kinds exist for equal-tempered quarter-tones, the most commonly used microtonal intervals. However, only in a few cases have composers set out to explore a defined microtonal system with electronic means. Examples include Stockhausen's Studie I (1953), in which there are 25 equal-tempered intervals within each stretch of two and a half octaves, and Krenek's Spiritus intelligentiae, sanctus (1955), with 13 equal-tempered intervals per octave.

C. Ives: 'Some ''Quarter-Tone'' Impressions', Franco-American Music Society Bulletin (25 March 1925); repr. in Essays Before a Sonata and other Writings, ed. H. Boatwright (New York, 1962), 107--19

J. Yasser: A Theory of Evolving Tonality (New York, 1932)

I. Vishnegradsky: 'La musique a quarts de ton et sa realisation pratique', ReM, clxxi (1937), 26--33

H. Partch: Genesis of a Music (Madison, WI, 1949, 2/1974)

A.D. Fokker: Neue Musik mit 31 Tonen (Dusseldorf, 1966; Eng. trans, 1975)

C. Gamer: 'Some Combinatorial Resources in Equal-Tempered Systems', JMT, xi (1967), 32--59

L. Gerdine and others: 'Microtonal Music in America', Proceedings of the American Society of University Composers, ii (1967), 77--122

Xenharmonikon: an Informal Journal of Experimental Music (Houston, 1974--)

D. Ellis: Quarter-Tones: a Text with Musical Examples, Exercises and Etudes (Plainview, NY, 1975)

B. Johnston: 'Rational Structure in Music', Proceedings of the American Society of University Composers, xi--xii (1976--7), 102--18

G. Secor: 'New Scales', Interval, i/1 (1978), 4--6

J. Schneider: 'The Microtonal Guitar', Guitar & Lute, no.16 (1981), 42--6; no.17 (1981), 32--4; no.19 (1981), 28--31; no.20 (1982), 20--22; no.21 (1982), 33--4; no.25 (1982), 14--17

E. Blackwood: The Structure of Recognizable Diatonic Tunings (Princeton, NJ, 1985)

D. Leedy: 'A Selective List of Source Materials on Alternative Tuning Systems in American Music', Interval, iv/4 (1985), 7--9

R. Karpen, D. Keisler, J. Kohl, eds.: 'Forum: Microtonality Today', PNM, xxix (1991), no.1, pp.172--262; no.2, pp.72--211

I. Zannos: Ichos und Makam: Vergleichende Untersuchungen zum Tonsystem der griechisch-orthodoxen Kirchenmusik und der turkischen Kunstmusik (Bonn, 1994)

PAUL GRIFFITHS, MARK LINDLEY, IOANNIS ZANNOS

About 1960 a vocoder was incorporated into the Siemens Synthesizer, which probably offered the first opportunity for musicians to use the device. Since around 1970 several manufacturers, including Bode Sound, Moog, Korg, Roland, EMS, Sennheiser, Synton, Eventide and Doepfer, have produced vocoders (some of which are controlled from a keyboard) primarily for use in electronic music studios or in live performances of rock music; the 'harmonizer' is a similar device. Such vocoders permit the timbre and articulation of one sound source (usually a voice) to control another. Simpler devices for producing 'talking (or singing) instruments', particularly in films, include the Sonovox (c1938) and, from the early 1970s, various 'voice boxes' ('voice tubes') used in rock music, which impart vocal qualities with the larynx and mouth respectively. The 'phase vocoder' (1966) is a software program developed at Bell, which has found a role in computer music since the late 1970s, primarily for time compression or expansion.

H. Dudley: 'Synthesizing Speech', Bell Laboratories Record, xv (1936), 98--102

H. Dudley, R.R. Riesz and S.S.A. Watkins: 'A Synthetic Speaker', Journal of the Franklin Institute, ccxxvii (1939), 739--64; repr. in Speech Synthesis, ed. J.L. Flanagan and L.R. Rabiner (1973)

H. Dudley: 'The Vocoder Remakes Speech', Proceedings of the Institute of Radio Engineers, xxviii (1940), 1--47

J.L. Flanagan and L.R. Rabiner, eds.: Speech Synthesis (Stroudsburg, PA, 1973)

T. Rhea: 'Harald Bode's Frequency Shifters and Vocoders', Contemporary Keyboard, vi/2 (1980), 86 only

C. Roads and others: The Computer Music Tutorial (Cambridge, MA, and London, 1996), 148, 444--6, 549, 566--77, 1094--9

HUGH DAVIES

Timbre is the auditory attribute that distinguishes two sounds presented in a similar manner and having identical pitch, loudness and duration. This formal definition leaves a wealth of possibilities that resisted scientific experimentation until the late 20th century. Timbre is now understood to have two broad characteristics that contribute to the perception of music: (a) it is a multifarious set of abstract sensory attributes, some of which are continuously varying (for instance, attack sharpness, brilliance, nasality), others of which are discrete or categorical (the 'blatt' at the beginning of a sforzando trombone note or the pinched offset of a harpsichord sound), and (b) it is one of the primary perceptual vehicles for the recognition, identification and tracking over time of a sound source (a singer's voice, a clarinet, a set of carillon bells) and thus involves the absolute categorization of a sound (McAdams, 1993; Hajda and others, 1997).

This first approach concerns relative perception: the ways in which and the degree to which sounds are perceived to differ. Early research on the perceptual nature of timbre focussed on preconceived aspects such as the relative weights of different frequencies present in a given sound, or its 'sound colour' (Slawson, 1985). A voice singing a constant middle C while varying the vowel being sung, or a woodwind player holding a given note while varying the embouchure and mouth cavity shape, both vary the shape of the spectrum. Helmholtz (2/1885) invented ingenious devices for controlling spectral weighting to explore these aspects of timbre. However, the real advances in understanding the perceptual representation of timbre had to wait for the development of powerful multi-dimensional data analysis techniques in the 1960s.

Multi-dimensional scaling has no preconceptions about the physical or perceptual structure of timbre. Listeners simply rate on a scale from very similar to very dissimilar all pairs from a given set of sounds that are equalized in terms of pitch, loudness and duration. The resulting judgments are then analysed with a computer program that fits the dissimilarity ratings to a distance model in which sounds with similar timbres are close together and those with dissimilar timbres are far apart. The basic model is expressed in terms of continuous dimensions that are shared among the timbres. More elaborate models also include dimensions or features that are specific to individual timbres ('specificities') and different perceptual weights accorded to the dimensions and specificities by individual listeners or classes of listeners (McAdams and others, 1995). Such techniques have been applied to synthetic sounds (Plomp, 1970; Miller and Carterette, 1975), resynthesized, imitated or simulated instrument sounds (Grey, 1977; Wessel, 1979; Krumhansl, 1989; McAdams and others, 1995; Roussarie, McAdams and Chaigne, 1998), recorded instrument sounds (Iverson and Krumhansl, 1993) and even dyads of recorded instrument sounds (Kendall and Carterette, 1990--91).

Independent acoustic correlates have been determined in many cases for the continuous dimensions (Krimphoff, McAdams and Winsberg, 1994), which is important if these results are to be applied to sound synthesis or the search for sounds in large audio databases. The most common correlates include spectral centroid (representing the relative weights of high and low frequencies), attack time (distinguishing 'continuant' instruments that are blown or bowed from 'impulsive' instruments that are struck or plucked), spectral flux (the degree of evolution of the spectral shape over a tone's duration which is high for brass and lower for single reeds) and spectral irregularity (the degree of jaggedness of the spectral shape, which is high for clarinet and vibraphone and low for trumpet).

Specificities are often found for complex acoustic and synthesized sounds and represent the presence of a unique feature that distinguishes a sound from all others in a given context. For example, in a set of brass, woodwind and string sounds, a harpsichord might have a strong specificity due to the return of the hopper which creates a slight thump and quickly damps the sound at the end; no other sound has such a feature (McAdams and others, 1995).

Individual and class differences are modelled as weighting factors on the different dimensions and the set of specificities. Some listeners pay more attention to spectral properties and ignore temporal aspects while others have the inverse pattern (McAdams and others, 1995). It has yet to be demonstrated that such individual differences have anything to do with musical experience or training. It may be that since timbre perception is so closely allied to sound source recognition in everyday life, everybody is an expert to some degree.

The timbre space models that result from this approach have been useful in predicting listeners' perception in situations other than those specifically measured in the experiments. This suggests that they do in fact capture important aspects of timbre representation and have the most important feature of a scientific model: the ability to predict new phenomena. By exchanging the spectral envelopes on pairs of sounds that differ primarily along the spectral dimension, these sounds have been found to switch positions in the space, as would be predicted by the model (Grey and Gordon, 1978). The timbre space representation has been used as a basis for defining timbral intervals (by analogy with pitch intervals) in terms of directional vectors in the space (Ehresman and Wessel, 1978; McAdams and Cunibile, 1992). Musical transposition is equivalent to a spatial translation of the vector, keeping constant the degree of change along each of the shared dimensions. The difficulty with applying this notion to orchestral timbres is that it does not take into account the specificities of individual timbres that would 'distort' the vector in some sense, and the timbre space available with acoustic instruments and their blends is often full of holes where no instrument exists, limiting considerably the possible transpositions. However, this approach would be quite useful for a palette of synthesized timbres without specificities that were distributed homogeneously in the perceptual space.

Timbre space representations also predict aspects of the phenomenon of auditory streaming -- the assignment of successive events to a coherent mental representation -- on the basis of which melody and rhythm are then perceived. The further apart the timbres of two instruments are in the perceptual space, the more likely it is that the melodies they are playing will segregate into separate streams (McAdams and Bregman, 1979; Gregory, 1994--5; Iverson, 1995; Singh and Bregman, 1997).

The second approach to timbre concerns absolute perception, the sound being represented in reference to a particular category. Categorization is a primary reflex in the perceptual process and is particularly evident in the processing of pitch and duration in music. One reasonable hypothesis is that the sensory dimensions that compose timbre serve as indicators used in the categorization, recognition and identification of sound events and sound sources (McAdams, 1993). This is perhaps the more neglected aspect of timbre and brings with it advantages and disadvantages for the use of timbre as a form-bearing dimension in music (McAdams, 1989).

One of the advantages is that categorization and identification of a sound source may bring into play perceptual knowledge (acquired by listeners implicitly through experience in the everyday world and in musical situations) that helps them track a given voice in a complex musical texture. Listeners do this easily and research has shown that timbral factors may make an important contribution in such voice tracking (Culling and Darwin, 1993; Gregory, 1994--5), which is particularly important in polyphonic settings.

The disadvantages may arise when a composer seeks to create melodies across instrumental timbres, as in the Klangfarbenmelodien of Schoenberg. The predisposition to identify the sound source and follow it through time would impede a more relative perception in which the timbral differences were perceived as a movement through timbre space rather than as a simple change of sound source. For cases in which such timbral compositions work, the composers have often taken special precautions to create a musical situation that draws the listener more into a relative than into an absolute mode of perceiving.

Timbre perception is at the heart of orchestration, a realm of musical practice that has received relatively little experimental study. The creation of new timbres through orchestration necessarily depends on the degree to which the constituent sound sources fuse or blend to create the newly emerged sound. Sandell (1995--6) has proposed three classes of perceptual goals in combining instruments: timbral heterogeneity, in which one seeks to keep the instruments perceptually distinct; timbral augmentation, in which a single instrument embellishes another one that perceptually dominates the combination; and timbral emergence, in which a new sound results that is identified as none of its constituents. Blend appears to depend on a number of acoustic factors such as onset synchrony of the constituent sounds and others that are more directly related to timbre, such as the similarity of the attacks, the difference in the spectral centroids and the overall centroid of the combination.

Timbre is also an important component in the perception of musical groupings, whether they are at the level of sequences of notes distinguished by changes in timbre (Deliege, 1987) or of larger-scale musical sections delimited by marked changes in orchestration and timbral texture (Deliege, 1989).

Work on the perception of musical tension and relaxation has focussed on the role of pitch and rhythm in carrying such structures. Timbral modulation can also play an important role in these large-scale expressive aspects of musical experience. Comparing orchestrated music with direct piano transcriptions of the scores for both tonal/metric and non-tonal/non-metric music, Paraskeva and McAdams (1997) demonstrated a modulating role of orchestration on tension and relaxation profiles measured across the excerpts.

Taken together, these areas of research into timbre perception are moving in the direction of creating a true theory of orchestration and timbral control in sound synthesis.

Computer technology exerts a powerful and ever-increasing influence on the world in which we live. The personal computer in particular holds the key to a wealth of processing possibilities that could scarcely have been envisaged less than a generation ago. In terms of music applications the sheer diversity of digital functions makes it increasingly hard to present a balanced perspective within a brief dictionary article. In the following sections distinction will be made between applications that have an essentially passive role in the communication of music information, such as the conventional audio compact disc, and those such as the CD-ROM which involve a more conscious process of musical interaction.

PETER MANNING (I--III), ELEANOR SELFRIDGE-FIELD (IV, VI), SUZEL ANA REILY (V), ANTHONY POPLE (VII, VIII)

Capital city of Argentina. Sacred music was first used in the church of the Jesuit College about 1611 and at the cathedral, which was completed in 1622. In that year a primitive organ was installed at the Jesuit College church; the cathedral had a similar one. The missionary Jesuit priests taught music, and from the late 17th century their church had a rudimentary choir of black American slaves. The cathedral choir was supplemented by an orchestra of 14 players, and the earliest local compositions are those of the first known organist and maestro de capilla Juan Vizcaino de Aguero (b Tucuman, Argentina, 1606); they date from 1628, the year of his arrival in Buenos Aires. He was succeeded at the cathedral by his pupil Juan de Caceres y Ulloa and later by Francisco Vandemer (1756), Antonio Beles (1775--90), Bernabe de San Gines (1775), Francisco del Pozo (1780), Mario Cabral (to 1785), Teodoro Guzman (fl 1750--1820), Ambrosio Belarde (fl 1760--1815) and Tiburcio Ortega (1759--1839). The Portuguese musician Salinas de Lima played there in 1806 and the Italian Gaetano Lino Loforte in 1810. From 1785 to 1813 the organist was Bautista Goiburu, a noted teacher who guided local musical activity during this period. Music books of the period 1617--1809 are in the Universidad del Litoral, Santa Fe, the Museo Historico, Cordoba, the Biblioteca Nacional, Buenos Aires, and the Archivo General de la Nacion, Buenos Aires. The early colonial works clearly show Spanish influence, whereas the more elaborate settings of the mass written after 1800 attest to some Italian influence.

The first public theatre, the Teatro de Operas y Comedias, opened in 1757; its productions included puppet shows, tonadillas escenicas (musical intermezzos) and a few short plays. The Teatro de la Rancheria (1783) presented tonadillas, zarzuelas and later the major Italian operas of Rossini, Bellini, Donizetti, Mercadante and others. The period of the viceroyalty (1776--1810) was characterized by lively musical activity in the salons of the colonial mansions, where parts of tonadillas and zarzuelas were staged. Italian opera became overwhelmingly popular in the early 19th century and during the century led to the opening of many more theatres, including the Coliseo Provisional (1804, where the first complete opera heard in Buenos Aires, Rossini's Il barbiere de Siviglia, was performed in 1825 -- the Argentine national anthem being performed for the first time on the same occasion), the Teatro de la Victoria (1838), the Teatro del Buen Orden (1844) and the Teatro de la Federacion (1845). The first Teatro Colon, built in front of the Plaza de Mayo, was inaugurated with La traviata with Sofia Vera Lorini and Enrico Tamberlik in 1857. It had 2500 seats but a short life as an opera house; the building became the Banco de la Nacion Argentina in 1888. The leading opera season was transferred to the Teatro de la Opera (1872), which held the Buenos Aires premieres of Verdi's Don Carlo (1873) and Gomes's Il Guarany (1874), and the world premiere of Pampa (1897) by the Argentine Arturo Berutti. Caruso sang there for the first time in Giordano's Fedora in 1899; the theatre's highpoint was in 1901 when he sang in Tosca with Hariclea Darclee, conducted by Toscanini. The new Teatro Colon, now Buenos Aires's principal opera house (see illustration), opened in 1908 with Aida under Luigi Mancinelli. The house seats around 4000 (2500 at the time of its opening). The new Colon became the biggest and most prominent opera house in Latin America and one of the leading in the world. It has staged 58 premieres of Argentine operas, among them Hector Panizza's Aurora (1908), Felipe Boero's El matrero (1929), Juan Jose Castro's Bodas de sangre (1956), Alberto Ginastera's Don Rodrigo (1964) and Mario Perusso's La voz del silencio (1969) and Escorial (1987). As well as the traditional repertory the Colon has given the South American premieres of operas by such composers as Berg, Stravinsky, Janacek, Dallapiccola, Pizzetti, Schoenberg, Milhaud and Poulenc. A highpoint was the completed version of Manuel de Falla's Atlantida sung in Catalan (1963); a rarity was the production in 1982 of the earliest surviving Spanish opera, Celos aun del aire matan (1660) by Juan Hidalgo. The Colon is also used for symphonic concerts, solo recitals and ballet seasons.

Concert life began likewise in the colonial salons, where the favourite form was the popular song with guitar accompaniment. Simple song genres of Spanish origin ranged from the salve (song of praise), saeta (hymn to the Virgin), alabanza (chant of praise) and rogativa (chant of supplication) to lovers' songs and Christmas carols. The most notable salon was that of the composer and statesman Amancio Alcorta (c1850).

The musical life of the new capital was dominated by the composers Juan Bautista Alberdi and Amancio Alcorta, who wrote mainly chamber works for piano and some Romantic songs, and Juan Pedro Esnaola, who wrote works for voice and for piano and some chamber and orchestral music, also in the European Romantic idiom.

Throughout the 19th century philharmonic societies were founded to perform symphonic and chamber music; more than 20 remain, of which the most important are the Sociedad Filarmonica (1822), the Escuela de Musica y Canto (1822), another Sociedad Filarmonica (1823) and the Sociedad de Mayo (1854). The repertory of chamber music alternated with symphony concerts and concertos performed by visiting foreign soloists; small instrumental concerts were given by local players, supplemented when necessary by players from elsewhere. The Orquesta Filarmonica of the Asociacion del Profesorado Orquestal was founded in 1919; its first conductors were Ernesto Drangosh, Ferruccio Cattelani, Georges Zavlawsky and, in 1924, Ernest Ansermet. It performed the symphonic repertory, giving first performances in Buenos Aires of works by Stravinsky, Honegger, Falla, Malipiero, Debussy, Ravel and Prokofiev and by young Argentine composers. This association and others like it (the Asociacion Wagneriana, 1912, the Amigos de la Musica, 1946 and especially the Grupo Renovacion, 1929) did much to promulgate new works and unfamiliar genres (e.g. lieder) as well as the standard repertory. The present orchestras are that of the Teatro Colon (1924), the Orquesta Radio El Mundo (1930), the Orquesta Sinfonica Nacional (1948), the Orquesta Filarmonica (1949) and the Orquesta Radio Nacional (1951), all of which have had native and foreign conductors. Smaller groups formed in the 1960s include the Camerata Bariloche (1966, founded by the violinist Alberto Lysy), the Ensemble Musical de Buenos Aires (1968, founded by Pedro Calderon) and the Orquesta de Camara Juvenil (1970, founded by the composer, cellist and conductor Washington Castro). Other orchestras are the Sinfonieta Omega Seguros, founded and conducted by Gerardo Gandini; the Orquesta San Isidro Labrador, under Charlotte Stuijt; and the Orquesta de Mayo and the Orquesta Juvenil (founded 1995), both conducted by Mario Benzecry.

The concert-promoting organizations, the Asociacion Amigos de la Musica, Asociacion Amigos del Arte, Asociacion Argentina de Compositores, Asociacion de Conciertos de Camara, Asociacion de Jovenes Compositores de la Argentina and Asociacion del Profesorado Orquestal, all organize annual concert seasons (from April to October) including solo recitals, chamber music cycles and symphony concerts. The Agrupacion Nueva Musica is directed by Francisco Kroepfl and Lucia Maranca. Many organizations (e.g. the Asociacion Amigos de la Musica) have their own concert halls; others use rooms in the Museo Nacional de Bellas Artes, the Museo de Artes Decorativas, the Biblioteca Nacional and the great hall of the Facultad de Derecho (school of law, cap. 1500), the hall of the Editorial Argentina de Musica, the Teatro Comico (cap. 1200) and the Municipal S Martin which has three halls, Martin Coronado (2000), Lugones (1000) and Casacuberta (700).

The first public institution to provide music instruction was the Colegio Real de S Carlos, now the Colegio Nacional, founded by Juan Jose Vertiz (viceroy 1778--84); the influential Juan Bautista Goiburu was professor there. The Spanish priest Jose Antonio Picazarri founded the Escuela de Musica y Canto in 1822 and the Argentine composer Juan Pedro Esnaola was the first president of the Escuela de Musica de la Provincia founded in 1875. The city also has a Conservatorio Municipal de Musica and the Conservatorio Nacional de Musica. The municipal radio broadcasts cultural programmes. The Centro Latinoamericano de Altos Estudios Musicales at the Instituto Torcuato di Tella, founded in 1962 and directed by Ginastera, no longer exists. The Facultad de Artes y Ciencias Musicales of the Universidad Catolica Argentina (UCA) was established in 1958 with Ginastera as dean. It has departments of composition, musicology and criticism, sacred music, music education, orchestral conducting and choral conducting. Roberto Caamano was dean until his death in 1993; he was succeeded in 1994 by Marta Lambertini. The UCA also has a symphony orchestra and centres of electro-acoustic music (directed by Pablo Cetta), contemporary music (Gerardo Gandini) and early music (Clara Cortazar). In 1994 the UCA and the Academia Nacional de Bellas Artes together established the Premio Nacional de Piano Roberto Caamano. The Centro de Experimentacion de Opera y Ballet (CEOB) was created by Gandini in 1994 as an experimental group, with regular performances in three small halls: the Centro Cultural Recoleta, an auditorium of the Teatro S Martin, and one of the secondary stages of the Teatro Colon. Each year one of the productions is selected to have its premiere on the principal stage of the Colon.

O. Schiuma: Musica y musicos argentinos (Buenos Aires, 1943)

N. Slonimsky: Music of Latin America (New York, 1945, 3/1949/R)

O. Schiuma: Musicos argentinos contemporaneos (Buenos Aires, 1954)

V. Gesualdo: Historia de la musica en la Argentina (Buenos Aires, 1961, 2/1978)

H. Dianda: Musica en la Argentina de hoy (Buenos Aires, 1966)

R. Caamano, ed.: La historia del Teatro Colon (1908--1968) (Buenos Aires, 1969)

R. Arizaga: Enciclopedia de la musica argentina (Buenos Aires, 1971)

E. Valenti-Ferro: Las voces del Teatro Colon 1908--1982 (Buenos Aires, 1983)

SUSANA SALGADO