Complexity and interaction

Horacio Vaggione is a contemporary of the composers of New Complexity music, and while his own music is also complex, its complexity works in reference to scientific theories of complexity, such as that by physicist Jean-Marc Lévy-Leblond, who says, “A system will be said to be ‘complex’ if it exhibits reciprocal couplings between different levels or parts.”¹ In Vaggione’s understanding of complexity, interaction is key. He writes:

The emergence of an approach based on generalized interaction (internal to the musical work) today allows us to envision both the existence of possible pathways among time’s disjointed dimensions and the nature of the nonlinearities that follow from their interaction. The problem, for a composer who is interested in expanding a syntax to all dimensions of time, is finding ways to articulate this complexity.²

It might be added that this type of complexity is anti-reductionist: it is impossible to trace back to a single cause or principle, because in a generalized interaction, there is no linear chain of causality.

There are at least three fields in which the question of interaction takes on a distinctive dimension in Vaggione’s work. The first is that of computers. In its long early years, computer-assisted composition was dominated by the ideal of automated composition — composers such as Hiller, Barbaud, Xenakis, and König come to mind. If composers wished to intervene in the process, they could only do so using the output, making selections from or transforming results produced by an algorithm. The alternative, according to Vaggione, was imagining “a multitude of different operations, rather than a single algorithm.”³ This alternative would allow composers to work in constant interaction with algorithmic calculations, through direct actions (or “interventions” or “compositions”), sometimes called “manual” actions. Explaining his work on phase decorrelation⁴ with delays created one at a time, Vaggione notes: this possibility

is neither as laborious nor as naïve as it seems: writing music “manually,” note by note, or overtone by overtone, or grain by grain, is something that composers do. This is one possibility of their craft, and there is nothing to be ashamed of in using it.⁵

In pursuing this notion of craft, Vaggione is actually addressing the question of formalization: the interplay of direct composition and algorithmic calculations reflects the interplay of craftsmanship and formal structure. Instead of being in conflict, these two concepts complement each other through their interaction.

The second area in which interaction takes on special value for Vaggione is that of the local and the global. This schema underlies the interaction between direct action and algorithmic calculation. Even though Vaggione uses algorithmic calculation only in parts of his compositional process, he still defines it as a global approach — it is even the sole definition he gives to the notion of the algorithm. He links direct action to the local. He thus conceives of the interaction between direct action and algorithmic calculation as an interaction between local and global: “It is possible for a localized action of writing to be integrated into an algorithmic process. By the same token — symmetrically — the outcome of an algorithmic process can be transformed locally by the direct action of writing.”⁶

The third field in which the notion of interaction comes into play is one in which Vaggione excels: the intermixing of acoustics and electronics. He brings about this interaction in several ways, as we will see in other analyses. The most obvious is his creation of “augmented” instruments. In Thema, for example, the solo saxophone is amplified and extended by electronics.⁷ Additionally, among Vaggione’s more recent pieces, Trek (2018, for timpani and electronics; also titled Timpani Trek) consists in “the construction of an ‘augmented’ instrument whose timbres and registers go significantly beyond the sound universe of the timpani, by projecting it onto other acoustic planes, while at the same time preserving its original morphological traits.”⁸ See Example 1.

Example 1.
Horacio Vaggione, *Trek*: score, opening. © Horacio Vaggione, composer’s edition. Published courtesy of the author.

Time

In his writings, Vaggione discusses time. Among his most important points is that his music, in parallel with that of spectralist musicians, integrates the modern idea of time as irreversible. In this context, sound is no longer conceived of in terms of periodicity and repetition, according to Helmholtz’s model, but as an energetic, dynamic phenomenon. One of Vaggione’s major points of reference is the work of the physicist Ilya Prigogine, who developed a theory of dissipative structures. Vaggione refers to it in his description of sound structures as “dissipative structures of sound energy.”⁹ And his pieces, in part or in full, sound like exactly that. Sudden “gestures” emerge from nowhere, like brusque blazes of energy, and these new gestures are interrupted by slow entropic changes. Often — as in Thema, Tar (1987, for bass clarinet and electronics), Scir (1988, for bass flute and electronics), Till (1991, for piano and electronics), Tahil (1992, for piano), Kitab (1992, for bass clarinet, piano, double bass, and electronics, with one part in real time and one part recorded), Myr-S (1996, for cello and electronics), Nodal (1997, electroacoustic music on a multitrack device), and Harrison Variations (2002, electroacoustic music on a multitrack device) — the entire work builds from an original gesture, something like a big bang, to borrow the expression Philip Mead uses to describe Till.¹⁰ Or, as Jean-Claude Risset puts it, Vaggione’s music

often begins with a fairly sudden impulse, a disruption, a kind of explosion, followed by the emergence of a kind of resonance or after-effect with more sustained behavior. This sort of progression might recall the link, in physics, between the free behavior of a linear dissipative system near equilibrium, in which one can assume linear relations between the flux and forces of the system, and the system’s response to excitation, its sensitivity to constraint. […] Is this a conscious or unconscious metaphor for Vaggione? No doubt an acute awareness of irreversibility and its consequences — since the objects used in digital music can be analyzed systemically and (thermo)dynamically, where Ilya Prigogine has introduced the notion of order by fluctuation.¹¹

Vaggione is also intrigued by the infinitely small — that is, what has often been presented as sound’s “inner life.” This topic has been a focus for a significant lineage of composers, starting with Varèse (if not earlier) and moving through Stockhausen and then the “microsounds” of the 1990s and 2000s.¹² On this subject, Vaggione sets himself apart in at least three ways. First, he does not adhere to the image of “inner life” and its corollaries that tend to recall space (“immersion,” “auscultation,” or even “diving into the infinitely small”). To him, the issue is time: he describes a “descent” into micro-time.

Second, beyond his fascination for things that happen at the micro-time scale — a fascination shared by all who have experimented with it — the crucial question for him is articulation, how to piece together these phenomena. The title of a 1996 article sums up these two facets: “Articulating Micro-Time.”¹³

Last, while “inner life” can evoke fascination with a supposed original matrix, a Oneness — whence the mystical tendencies of Scelsi, Stockhausen, Harvey, or even Grisey¹⁴ — Vaggione, by contrast, sees an opportunity to discover pluralism:

For musicians, descending into micro-time is a means to discovering phenomena they ignore when they are satisfied to dabble on sounds’ surfaces, without taking into account their substrata. […] As Bachelard said, “our temporal institutions are still quite poor, summed up in our intuitions of absolute beginnings and continuous duration.” We must therefore “find the pluralism beneath the identity” and “break identity beyond immediate experience, which is too quickly summed up in one facet of the whole.”¹⁵

This is the wellspring of Vaggione’s interest in granular synthesis, his specialty: his goal is to locate the “pluralism” (the grains) underlying “the identity” (the resulting sound). Moreover, he highlights the contrast between describing sound in terms of particles versus waves, linking back to the concept of time as irreversible. The granular approach “makes it possible to work with complex morphologies in a space-time in which irreversibility is dominant: from ‘dissipative’ structures that emerge in an oriented, directional space-time, and not smooth continuities, sub specie æternitas.”¹⁶ For this reason, in Vaggione’s music, the granular approach is much more than a synthesis technique.

Continuing with the question of time, some of Vaggione’s writing develops a multi-scalar approach. Among others, Xenakis and Grisey had evoked such an approach before, but Vaggione brought new developments. The first of these is his pragmatic observation that in both musical tradition and human perception, a threshold exists delimiting two orders of time scale: micro- and macro-time. In terms of musical tradition, this threshold is the note that is the basic unit of instrumental music; macro-time “encompasses all possible scales” above it and micro-time all those below.¹⁷ This seemingly innocuous observation has considerable implications: it makes it possible to reformulate and temper the divide between instrumental and electroacoustic music. Seen from this angle, the distinction between the two is not a difference in “nature” (of the material, for example); it consists of a change in (temporal) scale. This manner of thinking came into being with the advent of digital electroacoustics, which made it possible to compose in micro-time. From then on, composers could think about both sides of the threshold, micro- and macro-time, as part of the composition — articulating and connecting each. This is not to say that the threshold was abolished. In terms of perception, the threshold is located between 50 and 100 milliseconds. This threshold means that with fewer than ten to twenty sounds per second, the ear perceives the grains as entities; any more, and it hears them as part of an overall texture. By applying this model to both sound synthesis and instrumental music, it becomes possible to unify them without erasing their differences. In the case of synthesis, one perceives the granular nature of the resulting sound — which, indeed, is not entirely one sound. Conversely, with instrumentals, even when the threshold to micro-time is almost being crossed — as is often the case in Vaggione’s scores, in which musicians play thirty-second notes, or even smaller, when the quarter note equals 100 beats per minute. See Example 2, which is an excerpt of Phases (2001, for clarinet, piano, real-time computer, and other electronics), which remains a segregative flow. For this reason, the granulation should not be taken literally when it is applied to instrumental music: Vaggione is not seeking “instrumental granular synthesis.”¹⁸

Example 2.
Horacio Vaggione, *Phases*: page 1. © Horacio Vaggione, composer’s edition. Published courtesy of the composer.

Pieces such as Tar, Ash (1990, multitrack electroacoustic music), Till, Atem (2003, for horn, bass clarinet, double bass, piano, and multitrack digital device), and Gymel (2003, multitrack electroacoustic music) are representative of play with time scales.¹⁹ Schall is

a purely electroacoustic work using only piano sounds sampled and processed by a computer through a variety of techniques, such as cross modulation, granular synthesis, phase vocoder, etc. The sound palette is intentionally limited [compared to other compositions]. It is centered on a few figures of varying size that repeat, changed and unchanged, at various points throughout the process. Underlying this is a concern for the interlinking of micro-events, as well as for the integration of typically piano-like elements — for example, the use of diatonic glissandi, which are taken as sound objects and carried past their ordinary connotations, toward electroacoustic integration.²⁰

As has been noted, Vaggione, while differentiating between microscopic and macroscopic time scales, does not attempt to erase the threshold linking them. While it is possible to bring the two levels together, their difference is maintained: one can move from one level to the other only through transposition. To use Vaggione’s terminology, there is a nonlinearity between the levels of time, and one cannot be reduced to the other. Perhaps this is where his theory becomes original. Several of the other musicians who explored this question before or in parallel with Vaggione, often using scales, leaned more toward the principle of transposition. This is true for Xenakis in pieces designed for the GENDYN program, where everything is automatically derived from the waveform. It also applies to Grisey, who used the same waveform shape on several time scales (e.g., Vortex Temporum). Certain uses of fractals in music also come to mind.

In a reworked version of his article “Dimensions fractionnaires en composition musicale,” Vaggione noted: “My latest pieces, such as Tahil (1992) and Kitab (1993), or Schall (1994) and Rechant (1995, multitrack electroacoustic music), explore interaction, convolution, withdrawal, the passage between liminal and turbulent sonic states, from which figures emerge that do not function equally at different time scales.”²¹ However, he often points out that he became interested in nonlinearity in the mid-1980s, in works such as Them, Ash, and Till. For him, the nonlinearities among time scales can be fertile ground for musicians:

Recognizing the reality of these disjunctions is not paralyzing in any way; to the contrary, recognition opens the option to explore passageways between the levels, allowing us to connect them in a syntactical network that covers the entire spectrum of composable relations.²²

Morphology

Unifying electronic and instrumental music (while still taking into account their nonlinear connection) also means bringing together material and form. Both the material and the form can be composed, linked, and articulated. From Vaggione’s perspective, there is no fundamental difference between these two concepts, only a difference in (time) scale. This is why the duality can be set aside: instead of using two terms, we could use one word that varies based on the scale. This is the role that “morphology” appears to play in Vaggione’s writing. For him, all time scales are connected to morphologies: a sound sample (in granular synthesis), a waveform, a figure composed of a few notes, or the overall structure (macro-form) of a piece — all of these can be understood as “forms” that evolve over their specific time scales. The morphological approach succeeds in unifying the field of sound, by bridging the gulf between material and form.

In addition, the morphological approach makes three other contributions. First, it postulates that sound material is not neutral. There are no minimal units, or “bricks,” that can be freely combined to produce “abstract” forms (that is, forms autonomous from the material).²³ Second, the resulting morphological approach differs from the parametric approach. If we were to analyze a sound form (the morphological approach is not necessarily a holistic one), we would speak of its “qualities,” “aspects,” or “parts,” rather than its “parameters.” Nevertheless, Vaggione does not reject parametric treatment: he suggests that both approaches complement each other.²⁴

Finally, the morphological approach makes it possible to think of sound forms as dynamic movements or processes. In Vaggione’s terms, this approach is “transformational,” meaning that qualities or parts of a sound morphology change in relation to their context.²⁵ We know that process-based thinking can “orient” or “vectorize” musical discourse.²⁶ It also offers solutions to the problem of combining different sound sources: we can connect the instrumental and electroacoustic worlds “by constructing both sources from the same musical situation” or through a “shared vectorization,” Vaggione notes with regard to Tar.²⁷ He prefers to speak of a “transformational approach” rather than a “process.” He uses the latter term in a more general sense, likely to distinguish himself from spectral composers who have widely used it. Moreover, it is worth noting that the word “vectorization” is also used by Tristan Murail.²⁸

Let us briefly examine an example of the morphological approach in Vaggione’s treatment of space. As is true of many other composers, Vaggione posits that space itself can be composed. However, unlike musicians working in serialism, he does not see space as just a parameter of sound: space is part of sound’s morphology. While it has some autonomy, it is as a morphology “that will modulate and be modulated by other morphologies.”²⁹

These two qualities of space explain why Vaggione does not use standard techniques like reverb and panning. These techniques do not relate to the morphology of the sound they are “spatializing”; they are simply added on and can end up flattening the sound’s distinctive features (morphology).³⁰ Instead, Vaggione uses the micro-temporal decorrelation techniques from signal processing. He may, for example, duplicate a waveform and desynchronize their phase relations at a micro-temporal scale, creating a sense of space that is closely tied to the sound’s morphology.

A simple example of this is Example 3, where

a single monophonic sound object is confronted with its “dry” replica, with only one decorrelation value from beginning to end […]. We have three successive events (A, B, C), arranged in ascending order of frequency (as clearly indicated by the speed of the three respective waveforms). Because the object is very brief, the time scale can be adjusted to show both the beginning and end. The decorrelation between the source and its replica is set arbitrarily to 31 milliseconds. This decorrelation is achieved by simply cutting 31 milliseconds from the source channel’s track before the signal starts. Even in this basic case, listening reveals that the musical figure seems to occupy a “bigger” space than the original on its own, as well as a directionality (movement or trajectory — here oriented in a single direction — within the azimuth plane).³¹

Example 3.
Horacio Vaggione (in Anne SEDES [ed.], *Espaces sonores: Actes de recherches* [Paris, Éditions musicales transatlantiques, 2003], p. 26): simple example of micro-temporal phase decorrelation.

In his compositions, Vaggione’s use of micro-temporal decorrelation is far more complex than this example. Moreover, this process can now be created in real time, as in Arches II (2019, for electric guitar, alto saxophone, piano, cello, percussion, and multitrack electronics). This piece

falls within, among other things, a certain heterophony, produced by the superposition of similar phrasings, slightly delayed using decorrelations of the respective wave forms, at the micro-temporal level, which facilitates the construction of a composed spatial landscape, as the instrumental sounds are amplified and subject to the same type of controlled decorrelations (continually variable in time as well as in the vertical layer) as the electronic sounds.³²

Singularities

The morphological approach centers focus on a crucial notion: singularity (or singularities). Vaggione’s reading of René Thom was fertile. Thom wrote in a voice that is almost Vaggione’s own:

The primary duty of any morphological performance is to identify the discontinuities within a morphology and the stable parts of these discontinuities. In this interpretation, the concept of singularity emerges, with the discontinuity being a specific instantiation of it.³³

Morphology holds special interest in music because it connects traditional notions of material and form. The idea of “morphological singularities” adds precision to this connection: it challenges both the neutrality of the materiality and the universality of forms. Morphological singularity implies that, as the material evolves over time, certain features stand out at some point. This is why “salience” is a synonym for “singularity”: as Vaggione writes, “It is the study of a morphological salience (the brilliance of the brass sounds and its perception in terms of temporal evolution) that led Risset […] to express a model [of the] dynamic character” of sound.³⁴

The singularities are assembled, held, placed in tension, framed by (in) something: to be noticeable, they must be inscribed in a context that is not neutral (or just a background). Nor is it a sum of singularities, though it also includes less salient elements. The problem could be stated differently: a “direct action” has the possibility of causing a singularity, which means it also contains elements that are less singular. If the singularity constitutes a “catastrophe” (Thom’s word), it can only be measured against the continuum it shatters.

The frame in which singularities emerge is called a figure. In one sense, the notion of figure as Vaggione uses it, should be understood in its traditional musical sense: a set of a few notes (combining pitches, rhythms, nuances, and modes of play) that form a unit. Since Thema, Vaggione has used such figures when mixing electroacoustic and instrumental media. For each work, he begins by composing figures for the instruments. Then, he records and analyzes them to understand their morphological singularities. Finally, he composes both the electroacoustic and instrumental parts by transforming these figures so that their morphological saliences are amplified or projected in other places. This composition technique is called “composing-processing” and it functions as a “prism.”³⁵

Example 4 is an excerpt from the electronic part of the computer composition Shifting Mirrors (2016, for alto saxophone and multitrack electronics). The electronic part was

created using very brief alto saxophone sounds played by Pedro Bittencourt [who premiered the piece], sampled, and then multiplied, transformed, and articulated by computer. [It] is presented on a digital medium with, depending on the version, six to thirty-two independent channels. This electronic part contains figures that are sometimes very close to those played by the solo instrumentalist, sometimes simultaneous, and some of which have delays of a few thousandths of a second, as well as others in which the delays create morphological traits very far from the original sounds. Taken together, these composed delays form the spatial texture of the work. The title, Shifting Mirrors, is a reference to a network of “decorrelated mirrors” in which the instrumentalist moves.³⁶

Example 4.

Shifting Mirrors: electronic part. © Horacio Vaggione. Courtesy of the composer.

In the second meaning of the word “figure,” the concept can be applied at any time scale. Vaggione notes that the morphological transformations he creates “generalize a ‘figural’ work that can be projected across the most diverse time scales.”³⁷ This generalization seems to apply particularly to micro-time, where one can, for example, consider the granulation of a sampled sound as a figural work, or where, thanks to the technique of micro-temporal decorrelations, Vaggione implements “composed spatial figurations.”³⁸

Networks of objects

A figure “can be paired with the concept of an object, the latter being a category that makes it possible to include and to circulate figures in a network of compositional operations.”³⁹ In a sense, Vaggione’s compositions yield singularities by building musical structures on three levels: figures, which we have just discussed, objects, and networks. It is important to note that these levels do not describe fixed (temporal) orders of magnitude, as they may change scale.

Vaggione’s concept of an object is borrowed from computer science: it is connected to the object-oriented programming languages that emerged in the 1980s as an alternative to linear programming. In both his music and his theoretical work, Vaggione radically appropriates the computer science notion of an object. For instance, his first article on the subject was titled “A Note on Object-Based Composition.”⁴⁰ Objects may be “functions (algorithms), lists of parameters (scores), scripts (sequences of actions to be carried out), or sounds (both products and sources).”⁴¹

Aside from this reference to computer science, it is significant that Vaggione uses the concept of an object: in electroacoustics, this term refers to the musique concrète tradition conceptualized by Pierre Schaeffer. But Vaggione completely sets apart his own work from this tradition. First, Schaeffer’s “sound objects” are in micro-time, whereas Vaggione’s appear at any time scale. Second, Schaeffer’s are “opaque” because they are made on magnetic tape, while Vaggione’s objects, being digital, are always sugject to a form of notation.⁴² Moreover, Vaggione carefully differentiates his notion of the object from the philosophical opposition between subject and object.⁴³

Vaggione notes that the use of this notion in composition makes it possible to “encapsulate” both sound and score, in the sense used above.⁴⁴ In musical terms, the “score” can also mean structure (or syntax). Thus, by using the concept of the object, Vaggione bridges the gap in the traditional duality of sound and structure — just as morphology made it possible to bridge the gap between material and form. The concepts of morphology and object make it possible to overcome these dualities because they allow the concepts they oppose to be thought of in terms of composition, connection, and notation.

The idea of the “network” is tightly linked to that of the object. Vaggione explains, “Every object is […] a network, and every network is made up of objects.”⁴⁵ However, it can be distinguished from the object in that a network represents a level higher. Thus “the concept of the network applies to all types of relationships possible between object ensembles and subensembles (classes and subclasses).”⁴⁶ Moreover, the network can also be understood as an association among different types of representation, as with a hypertext.⁴⁷

One might think that Vaggione would use a computer science reference to define networks, but he cites a book by Michel Serres from 1968, Hermès 1: La communication. The concept of the network

pertains to a situation where there is “a plurality of points (summits) linked together by a plurality of branches (pathways),” where “no point is univocally subordinate to any other: each has its own power […] or zone of influence, or original determinant.” At the same time, a network can always be reconfigured: “it represents any given state of a dynamic situation.”⁴⁸

In a footnote, Vaggione adds that he gives this definition, which is “relatively old and non-technical, first to show the lasting nature of the concept of the network, and then because this definition is not biased by more recent connotations that tend to trivialize it.”⁴⁹

Example 5 provides a 40-second excerpt of computer composition taken from 24 Variations (2001, electroacoustic music on multitrack media), showing the timeline created with the IRIN program.⁵⁰ Each rectangle represents a “clip” or sound sample (the vertical position of a sample within a track is not significant — it does not indicate pitch). With IRIN, figures can be encapsulated within tracks and represented as isolated fragments, which makes possible a hierarchical construction of the meso-structure. The resulting visual is typical of composing networks of digital objects on a given time scale. Regarding the piece itself, Curtis Roads notes, that of Vaggione’s electroacoustic compositions, this is one of

the most gracefully poetic […]. In order to appreciate this, [we] recommend listening […] at a moderate volume in order to savor its subtleties. One is drawn in not by the expectation of spectacular climaxes, but by the originality and virtuosity of the articulations as they pass by.⁵¹

Example 5.
Horacio Vaggione: *24 Variations* (IRIN software), opening. © Horacio Vaggione. Published courtesy of the composer.

In his compositions and his theory, Vaggione defines music as a network of digital — that is, composed — objects, with networks that are also composable. From micro-time to macro-time, the work is a process that requires writing.

We conclude with the words of the composer himself:

This presupposes […] an extension of syntactical control at the level of micro-time — which, in turn, implies recognizing the formal relevance, from a compositional point of view, of articulating all possible temporal dimensions or scales (that is, everything postulated in a given work), certainly including those handled by conventional notation. Taking on musical composition that manifests at multiple levels of articulation means, in the end, considering that one of the tasks of this composition is the determination of the interactions between atoms and multiples, it being understood that each object, of whatever temporal scale, is to be viewed simultaneously in both of these ontological modes. To do this, this composition must be broken down into different sets of symbols, each with the pertinence needed to be operational at each proposed level.⁵²

---

^{1. Cited in Horacio VAGGIONE, “Son, temps, objet, syntaxe: Vers une approche multi-échelle dans la composition assistée par ordinateur,” in Antonia SOULEZ and Horacio VAGGIONE (eds), Musique, rationalité, langage. L’harmonie: du monde au matériau in the journal Cahiers de philosophie du langage n° 3 (1998): p. 171. ↩}

^{2. Horacio VAGGIONE, “Autour de l’approche électroacoustique: situations, perspectives,” in Esthétique et Musique Électroacoustique, Actes de l’Académie internationale de musique électroacoustique, vol. I (Bourges, Éditions Mnémosyne, 1995), p. 102. ↩}

^{3. Horacio VAGGIONE, “Composition musicale et moyens informatiques: questions d’approche,” in Makis SOLOMOS, Antonia SOULEZ, and Horacio VAGGIONE (eds), Formel/Informel: musique-philosophie (Paris, L’Harmattan, 2003), p. 97. ↩}

^{4. In order to generalize a spatialization arising from the morphology of sound, a concept to which we will return later on in this article. ↩}

^{5. Horacio VAGGIONE, “Décorrélation microtemporelle, morphologies et figurations spatiales du son musical,” in Anne SEDES (ed.), Espaces sonores: Actes de recherches (Paris, Éditions musicales transatlantiques, 2003), p. 24. ↩}

^{6. VAGGIONE, “Vers une approche transformationnelle en CAO,” p. 24. ↩}

^{7. Cf. Elsa JUSTEL, “Un ‘Thema,’ une image,” in Makis SOLOMOS (ed.), Espaces composables: Essais sur la musique et la pensée musicale d’Horacio Vaggione (Paris, L’Harmattan, 2007), p. 235-269. ↩}

^{8. Horacio VAGGIONE, performance notes. ↩}

^{9. VAGGIONE, “Composition musicale et moyens informatiques,” p. 102. ↩}

^{10. Philip MEAD, “Horacio Vaggione et le piano: Une introduction au style pianistique de sa maturité,” in Espaces composables, p. 231. ↩}

^{11. Jean-Claude RISSET, “Horacio Vaggione: vers une syntaxe du sonore,” in Espaces composables, p. 15. ↩}

^{12. Curtis ROADS, Microsounds (Cambridge, Massachusetts, MIT Press, 2001). ↩}

^{13. Horacio VAGGIONE, “Articulating Micro-Time,” Computer Music Journal vol. 20 n°1 (1996): p. 33-38. ↩}

^{14. Cf. Chapter 4 in Makis SOLOMOS, De la musique au son: L’émergence du son dans la musique des XXe-XXIème siècles (Rennes, Presses Universitaires de Rennes, 2013). ↩}

^{15. Horacio VAGGIONE, “Transformations morphologiques: quelques exemples,”Actes des Journées d’Informatique Musicale (JIM) 1998 (LMA-CNRS, Marseille, 1998), p. 98-118. ↩}

^{16. Horacio VAGGIONE, “Notes sur Atem,” in Espaces composables, p. 124. ↩}

^{17. VAGGIONE, “Son, temps, objet, syntaxe,” p. 172. ↩}

^{18. Cf. Vaggione in Pascale CRITON, Paul MÉFANO, Makis SOLOMOS, and Horacio VAGGIONE (2007), “Entretien autour d’Atem,” in Espaces composables, p. 140. ↩}

^{19. For Till, see Martin LALIBERTÉ, “Pistes analytiques pour Till de Horacio Vaggione” (2004), in Espaces composables, p. 161-226. For Gymel, see Renaud MERIC, “L’appréhension spatiale de l’écoute: un mouvement entre imagination et perception. L’exemple de la musique électroacoustique,” doctoral dissertation, Université Montpellier 3 (2009): p. 465ff. ↩}

^{20. Horacio VAGGIONE, liner notes to the album Musiques pour piano et électroacoustique, Chrysopée électronique-Bourges, LDC 278 1102 (1995). ↩}

^{21. Horacio VAGGIONE, “Dimensions fractionnaires en composition musicale,” unpublished, 13p. (text taken from “Modelle der Unvollkommenheit in der Computer Musik,” a presentation at Symposium Chaos und Ordnung, Steirischer Akademie, Graz, 1989). ↩}

^{22. Horacio Vaggione in Osvaldo BUDÓN, “Composer avec des objets, réseaux et échelles temporelles: une interview avec H. Vaggione,” in Espaces composables, p. 111. Original: Osvaldo Budón: “Composing with Objects, Networks and Time scales: An Interview with Horacio Vaggione,” Computer Music Journal vol. 24 n°3 (2000): p. 9-22. ↩}

^{23. Cf. Horacio VAGGIONE, “L’approche morphologique,” in Musique Électroacoustique: expérience et prospective, Actes de l’Académie internationale de musique électroacoustique, vol. IV (Bourges, Éditions Mnémosyne, 1999), p. 144. ↩}

^{24. In BUDÓN, “Composer avec des objets,” p. 109. ↩}

^{25. Cf. Horacio VAGGIONE, “Vers une approche transformationnelle en CAO,” Actes des Journées d’Informatique Musicale (JIM) 1996, Les cahiers du GREYC (CNRS-Université de Caen, 1996), p. 24-34. ↩}

^{26. Cf. ibid., p. 26. ↩}

^{27. Horacio VAGGIONE, “Description d’une approche compositionnelle (à propos de Tar, pour clarinette basse et ensemble),” unpublished, 14 p. (text from a class given at the Technische Universität Berlin in 1987). ↩}

^{28. Tristan MURAIL, Modèles et artifices, texts collected by Pierre Michel (Strasbourg, Presses Universitaires de Strasbourg, 2004), p. 56. Original: Tristan Murail, “Question de cible,” Entretemps n°8 (1989). ↩}

^{29. Horacio VAGGIONE, “L’espace composable: sur quelques catégories opératoires dans la musique électroacoustique,” in Jean-Marc CHOUVEL and Makis SOLOMOS (eds), L’espace: musique-philosophie (Paris, L’Harmattan, 1998), p. 154. ↩}

^{30. Cf. VAGGIONE, “Décorrélation microtemporelle.” ↩}

^{31. VAGGIONE, “Décorrélation microtemporelle,” p. 27. ↩}

^{32. Horacio VAGGIONE, notes on the piece. ↩}

^{33. René THOM, Paraboles et catastrophes (Paris, Flammarion, 1983), p. 91. It is worth noting that other composers have also been inspired by Thom, notably with respect to the notion of morphogenesis: Hugues Dufourt, François Bayle, Aurèle Stroë, etc. ↩}

^{34. VAGGIONE, “Composition musicale et moyens informatiques,” p. 92. ↩}

^{35. Cf. VAGGIONE, “Composition musicale et moyens informatiques,” p. 104. ↩}

^{36. VAGGIONE, notes for the piece. ↩}

^{37. VAGGIONE, “Transformations morphologiques.” ↩}

^{38. VAGGIONE, “Décorrélation microtemporelle,” p. 27. ↩}

^{39. VAGGIONE, “Transformations morphologiques.” ↩}

^{40. Horacio VAGGIONE, “A Note on Object-Based Composition,” in O. LASKE (ed.), Composition Theory, Interface vol. 20 n°3-4 (1991): p. 209-216. ↩}

^{41. VAGGIONE, “Son, temps, objet, syntaxe,” p. 187. ↩}

^{42. VAGGIONE, “Son, temps, objet, syntaxe,” p. 188-192. ↩}

^{43. In BUDÓN, “Composer avec des objets,” p. 108. ↩}

^{44. VAGGIONE, “A Note on Object-Based Composition,” p. 212-213. ↩}

^{45. VAGGIONE, “Son, temps, objet, syntaxe,” p. 187. ↩}

^{46. Vaggione in BUDÓN, “Composer avec des objets,” p. 106. ↩}

^{47. Cf. VAGGIONE, “Son, temps, objet, syntaxe,” p. 189-190. ↩}

^{48. M. Serres cited in VAGGIONE, “Composition musicale et moyens informatiques,” p. 99. ↩}

^{49. Ibid., p. 114. ↩}

^{50. Cf. Carlos CAIRES, “Micromontage in a graphical sound editing and mixing tool,” in Proceedings of the ICMC 2004 (San Francisco, International Computer Music Association, 2004). ↩}

^{51. Curtis ROADS, “L’art de l’articulation: la musique électroacoustique d’Horacio Vaggione,” in Espaces composables, p. 85. ↩}

^{52. VAGGIONE, “Son, temps, objet, syntaxe,” p. 195. ↩}