TEMPO CURVING AS A FRAMEWORK FOR INTERACTIVECOMPUTER-AIDED COMPOSITION

Jean BressonUMR STMS: IRCAM-CNRS-UPMC, Paris

bresson@ircam.fr

John MacCallumCNMAT - University of California, Berkeley

john@cnmat.berkeley.edu

ABSTRACT

We present computer-aided composition experiments re-lated to the notions of polyrhythmic structures and vari-able tempo curves. We propose a formal context and sometools that enable the generation of complex polyrhythmswith continuously varying tempos integrated in composi-tional processes and performance, implemented as algo-rithms and user interfaces.

1. INTRODUCTION

Tempo variations in musical performances significantly in-fluence musical and rhythmic perception. Expressive mu-sical timing and tempo curves (or time maps) are the objectof previous studies in the field of computer music [1,2]. Ingeneral the timing of beats and musical events is computedby the integration of tempo curves [3], and the composi-tional challenges are concentrated on the joint specificationof these curves (or other expressive timing controls) and ofa certain level of synchrony between simultaneous voices.

As a slightly different approach, we concentrate here onthe notion of rhythmic equivalence in the context of time-varying tempos. Rhythms are prescriptive structuresdenoted by sequences of durations (also called temporalpatterns [4]) when associated with a given (and possiblyvarying) tempo. Intuitively, it is possible to imagine thattwo different rhythms produce an equivalent temporal pat-tern if played following adequate tempo curves. Consid-ering a rhythm as the convolution of another rhythm and atempo curve, or as a superimposition of other rhythms andtempo curves, can be attractive musically as different rep-resentations of the same musical material can be suggestiveof different interpretations in performance. In this paper,we explore and actualize this idea through computer-aidedcomposition tools and techniques.

2. PRELIMINARY DEFINITIONS

In this section we introduce simple conventions that willbe used throughout this paper. Our intention is not to dis-cuss or overlap with the rich literature on rhythm theoryand formalisation, but to provide keys for the reading andunderstanding of the subsequent parts.

Copyright: c©2015 Jean Bresson et al. This is an open-access article distributed

under the terms of the Creative Commons Attribution 3.0 Unported License, which

permits unrestricted use, distribution, and reproduction in any medium, provided

the original author and source are credited.

2.1 Rhythmic Figures / Durations / Tempo

It is important to first distinguish the notated/compositionalform of a rhythmic structure (e.g. ♩. �) from the cor-responding “phenomenological” rhythm, that is, the result-ing sequence of durations or temporal pattern (e.g. 2s, 1.5s,0.5s). These two forms are related functionally by a tempovalue (in the previous examples, ♩ = 60, i.e. a quarter notecorresponds to 1s).

We will use the upper-case character R to identify no-tated rhythms and lower-case r for the rendered temporalpatterns. We note ⊗ the rendering operation associating atempo τ to a rhythm R, yielding to r : R⊗ τ = r.

2.2 Equivalence

We call equivalent two rhythms yielding equal temporalpatterns and note this equivalence R1 ≡ R2. In otherwords:

∀τ,R1 ≡ R2 ⇔ R1 ⊗ τ = R2 ⊗ τ. (1)

Recent works have delved into formalisms that allow oneto search for equivalent notations for a given rhythm R,that is, Ri 6= R such that Ri ≡ R [5]. Most of the time,the tempo τ is used to convert rhythmic figures into actualdurations, or conversely, to guide the search for the rhyth-mic notation that will best match a given temporal pattern(rhythmic quantification [6]). In both cases, it is the sameon the two sides of the equality (as in Eq. 1).

In order to integrate the tempo as a variable parameter,we will group the rhythms and tempos in pairs (R, τ) andnow call equivalent two pairs (Ri, τi) and (Rj , τj) whichverify Ri ⊗ τi = Rj ⊗ τj . We also note this equivalence(Ri, τi) ≡ (Rj , τj). Given a pair (Ri, τi), a limited num-bers of rhythms Rj 6= Ri will verify (Ri, τi) ≡ (Rj , τj) ifτj 6= τi. 1

2.3 Polyphony

In order to work with polyphonic rhythmic structures, wealso introduce the operator ⊕ which perceptually mergesseveral rhythms or temporal patterns into a single one. Wewill use it for instance to compose complex rhythmic linesfrom simpler ones, or conversely to find sets of rhythms{R1...Rn} which verify: ⊕ni=1Ri ≡ RT (where RT iscalled a “target” rhythm). 2

1 Rhythms verifying this property are equivalent rhythms (Rj ≡ Ri)modulo a “speed factor” τi/τj (e.g. ♩ ♩ and ��♩ ).

2 We simplify the notation here using Ri for expressing rhythms ingeneral, that is either (Ri, τi) pairs or ri.

mailto:bresson@ircam.frmailto:john@cnmat.berkeley.eduhttp://creativecommons.org/licenses/by/3.0/

Note that the ⊕ operator is hard to define and implementin the notation domain (see Figure 1a), but it is trivial inthe “real” time domain, where there is a direct mappingbetween the clock time and the notes’ onsets and durations(see Figure 1b).

(a) (b)

Figure 1: Merging rhythms a) in the notation domain andb) in the time/durations domain.

2.4 Varying Tempo

We now note τi(t) the function giving the value of thetempo τ at time t.

Considering the tempo as a variable function of time, wecan assume that for any pair (R, τ) there exist an infinityof (Ri, τi(t)) which verify (Ri, τi(t)) ≡ (R, τ), and as acorollary, that for any given rhythms R1 and R2 and forany tempo τ1 there exist a tempo function τ2(t) such that(R1, τ1) ≡ (R2, τ2(t)). Conversely, given a target rhythmrT = (RT , τT ) and a tempo curve τ(t) there must exist arhythm R which verifies (R, τ(t)) ≡ (RT , τT ).

Finding τ(t) or R here, or a combination of rhythms andtempos (Ri, τi) such that ⊕ni=1(Ri, τi) ≡ (R, τ), is an ap-pealing challenge from a musical point of view: it will al-low us to render predetermined target rhythms (RT , τT )using poly-temporal structures computed from time-varyingtempo curves (see next section).

This problem is hard to solve with purely formal or al-gorithmic methods, and the search gets even more com-plex when combinations of rhythms are involved. As wewill see below, supervised/interactive tools and heuristicsprovide interesting opportunities for compositional explo-ration.

3. COMPOSITIONAL CONTEXT

Musical polytemporality has been explored by many com-posers throughout the 20th and 21st centuries, however, thechallenges involved in the composition and representationof polytemporal music have prevented many from progress-ing beyond experimentation to the development of a praxis.Gérard Grisey (Tempus ex machina), Iannis Xenakis(Persephassa), György Ligeti (Kammerkonzert, 3rd mvmt.),and Conlon Nancarrow (Studies for Player Piano) all pro-duced works that explored the textures that become avail-able to a composer when the clock that unifies performersis removed. However, the limited number of polytemporalworks produced by these composers is representative ofthe challenges of constructing compositional systems andintuition in this domain. Dobrian [7] recently publisheda survey of compositional and technical issues related to

polytemporal composition. In this section, we present re-cent works by John MacCallum that serve as a case studyhighlighting the need for a set of compositional tools thatfacilitate experimentation, exploration, and situated action.

3.1 Motivation

The conceptual motivation behind the works described be-low is predicated on the idea that synchrony between mul-tiple musicians is a fictional construct of music notation.The concept of a musical “now” is not an infinitesimal,but rather, a small window, the width of which varies con-tinuously between the imperceptibly small and the unac-ceptably large. As performers use the visual and auditorycues around them to negotiate and approximate a commontempo, they construct a system that is not synchronous, butrather, plesiochronous in nature, i.e., nearly together, orclose enough for the intended purpose. One’s attention israrely drawn to this fundamental feature of performance,except in the most extreme moments when the system be-gins to diverge from plesiochrony and approach true syn-chrony or diverge off to asynchrony. The works belowforeground the human aspect of performance, albeit in arepresentative way, and push Platonic ideals inherent inmusic into the background.

3.2 Virtual and Emergent Tempos

In MacCallum’s recent works, performers listen to a click-tracks that vary smoothly in tempo over time, indepen-dent of one another. The compositional challenge in theseworks is to construct musical material that unifies the dif-ferent parts that are no longer bound by a common clock.aberration for percussion trio, 3 is an investigation into theuse of composite rhythm and the emergence of a “virtualtempo” as a means of producing coherent ensemble ma-terial. In this work, tempo curves τi(t) were chosen us-ing a random process and, in many sections of the piece,the rhythms Ri are chosen using a simulated annealingalgorithm with the goal of producing ⊕3i=1(Ri, τi(t)) ≡(RT , τT (t)) where RT represents a sequence of 1/16th

notes in a tempo τT (t) that can be imagined as the idealtempo continuously “running in the background” that themusicians are trying to approximate.

The form of aberration was constructed largely indepen-dently of τi(t), and the material itself was algorithmicallyderived and then altered to draw the listener’s attention tocertain features of the relationship between the tempos ata given moment. The result is a complex rhythmic tex-ture with a number of emergent properties unforeseen bythe composer at the time of its creation. It is largely theseunderdeveloped textures that became the focus of a moreintuitive and less systematic/process-oriented explorationin Delicate Texture of Time for eight players.

3.3 Methods

The composition of aberration relied heavily on a care-fully constructed plan that was designed to project a smallnumber of textures of interest to the composer who had, at

3 http://john-maccallum.com/index.php?page=./compositions

http://john-maccallum.com/index.php?page=./compositions/aberration

Figure 2: Compositional sketch of MacCallum’s aberration.

that time, no intuitive sense of how they would be enactedin performance. The piece was constructed as follows:

1. τi(t) were created using a random process designedto produce curves that generally oscillate between amaximum and minimum tempo and change direc-tion with some average frequency.

2. The time of every beat and subdivision (triplets, six-teenth notes, and quintuplets in this case) was com-puted for every voice from τi(t) and written to a file.

3. A simulated annealing algorithm was run to find Risuch that ⊕3i=1(Ri, τi(t)) ≈ (RT , τT (t)).

4. Matlab was used to create a template showing theposition of every bar, beat, and subdivision for allvoices, along with lines overlayed to show differentoutputs of step 3.

5. Steps 2–4 were repeated until the simulated anneal-ing algorithm produced output with a high degreeof voice exchange without too many instances ofpolyrhythms containing more than two subdivisionsin half a beat.

6. Simple compositional sketches would be made to in-vestigate the features of (Ri, τi(t)).

7. Steps 1–6 were repeated until a suitably interestingoutput was produced.

8. The composition was then done directly on top ofthe template in pencil (Figure 2), and the results tran-scribed using Sibelius for the parts and OmniGrafflefor the score (Figure 3 – see also Section 4.4).

There are a number of difficulties inherent in the stepslisted above:

• The distance of the approximation⊕3i=1(Ri, τi(t)) ≈(RT , τT (t)) is directly dependent on τi(t) which werechosen a priori using a random process rather thanbeing treated as free variables. This is not necessar-ily a problem, indeed, in the case of aberration, thiswas a feature of the work and a point of composi-tional investigation.

• Steps 1–7 offer little in the way of compositional in-tervention. When the simulated annealing algorithmproduced output that was deemed unusable for onereason or another, it was difficult to apply constraintsto have it avoid similar conditions during future ex-ecution.

• Step 8 is extremely cumbersome, error-prone, andforces the composer to commit to a given outputonce the composition of material has begun. If changesto any of the τi(t) need to be made at a later time,any material created must be discarded.

• Step 8 must be completed with little or no audition ofmaterial during the compositional process, prevent-ing the composer from experimenting with material.

Delicate Texture of Time was produced in a manner sim-ilar to the method listed above, with the exception that thetools had become easier to use and more robust, and AdobeIllustrator was used in place of OmniGraffle, however theproblems listed above remained present in the process.

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4. COMPUTER-AIDED COMPOSITIONALPROCESS

In this section we present an implementation of the pro-cedure described previously aided by the timewarp∼ ex-ternal for Max/MSP [8] and computer-aided compositionprocesses implemented in the OpenMusic environment [9].

The compositional objective driving our discussion is thedetermination of n rhythmic lines (corresponding to n in-strumental performers), each following a given tempo τi(t)(i = 1...n) and merging to produce a target rhythm (RT , τT ). 4

The target rhythm can come from previous compositionalprocesses or material, or it can be arbitrarily decided andspecified by the composer. It can be expressed with rhyth-mic notation (RT , τT ) or (equivalently) directly as a se-quence of durations (rT ).

Our problem is therefore to find Ri (i ∈ {1...n}) givenrT and τi(t), such that:

⊕ni=1(Ri, τi(t)) ≡ rT

The combination of n lines exponentially increases the searchspace of this problem, which makes it slightly more com-plex than the equivalence issues mentioned in Section 2.4.As a first step, we will consider that n = 1 (and eventuallydrop the subscripts i to simplify the notation). As we willsee, the presented methods easily scale to greater numbersof rhythmic lines.

4.1 Resolution with One Voice (n = 1)

As in Step 2 (Section 3.3), a first simplification we make tothe problem is to preliminarily choose a number of possible

4 We suppose — especially in the case of complex tempo variations —that each performer will be assisted, typically by a click-track.

pulse subdivisions for each voiceRi. Each subdivision (S)yields a regular rhythmic pattern (notated RSi ) which willbe used to compose a “quantification grid”. Given thesepatterns RSj and the tempo curve τ(t) a sequence of du-ration rSj = RSj ⊗ τ(t) can be computed for each sub-division (see Figure 4). Currently this part of the processis performed in Max/MSP using the timewarp∼ externalas described in [8]. The results (rSj ) are communicated toOpenMusic through a simple file export/import protocol.

Note that at this point, if the subdivision is known andthe tempo curve τi(t) does not change, the conversion ofrS back into R is relatively straightforward.

Figure 4: Generating a sequence of durations rSi startingfrom a beat subdivision S and a tempo curve τi(t) (S = 2).

The same procedure is applied for different values of S(S1, S2, ...) yielding a superimposition of lines (rSj ) fol-lowing the same tempo curve τ(t) (Figure 5).

Figure 5: Superimposition of rSj (Sj = {1, 4, 5, 6, 7}).

Figure 6: Finding elements of rT in rSj .

Figure 7: Reconstitution of RSj for each subdivision S according to the selection in Figure 6. (Note that RS7 = ∅.)

The search procedure then consists in finding and mark-ing an element in one of the different rSj which best matcheseach of the elements in the target rT (see Figure 6). Con-straints that govern acceptable combinations of subdivi-sions can also be applied to this search procedure to offer adegree of control over the general polyrhythmic complex-ity of the output.

From these marks it is easy to reconstitute a simple rhythmRSj for each value of S, containing one single rhythmicfigure or subdivision (Sj) and considering every markedelement in rSj as a played note, and every non-marked el-ement as a silence (see Figure 7). An “abstract” rhythm Ris then created regardless of the tempo curve τ(t), whichwill be equivalent to r if played back following τ(t).

A delicate part in the process is the merging of the dif-ferent lines RSj back into a single voice R. As the tempocurve has been abstracted (it is the same for every RSj ),some OpenMusic tools such as the merger function can beused for this operation [10]. Another solution is to per-form a “local” quantification of the sequence r obtainedfrom the combination of rSj = RSj ⊗ τα, where τα isan arbitrary value of the tempo [6]. This quantificationprocess is generally straightforward and reliable using ex-isting tools (e.g. omquantify 5 ), given the known tempoτα and the limited set of allowed subdivisions correspond-ing to the different Sj . Figure 8 shows the rhythm R0 =RS1 ⊕ RS4 ⊕ RS5 ⊕ RS6 ⊕ RS7 merging the lines RSjfrom Figure 7. This rhythm corresponds to the target se-quence rT from Figure 6, if played following the initialtempo curve τ(t): R0 ⊗ τ(t) ≡ rT .

Figure 8: Rhythm merging the RSj from Figure 7.

5 http://support.ircam.fr/docs/om/om6-manual/co/Quantification.html

4.2 Resolution with n Voices

The previous procedure is easily adapted to more than onevoice. Considering our initial problem of finding Ri (i ∈{1...n}) such that ⊕ni=1(Ri, τi(t)) ≡ rT , we just need toreproduce n times the process of generating the lines rSjias in Figure 5.

The search is then extended so as to look up in the n dif-ferent voices for an element of rSji matching each elementof rT . According to the selection, separate sets of rhythmsRSji will be generated for each voice, merged into Ri and

gathered in a polyphony as a result of the overall process.

Here as well, the graphical representation and alignmentof polyphonic rhythmic structures with different, time-varying tempos is a tricky aspect of the polyphonic exten-sion, but stands out of the scope of our present discussion.The OpenMusic poly editor allows for the assignment oftempo changes approximating τi(t) at every pulse of then different voices, and to visualize/play these voices as asingle score (see Figure 9).

Figure 9: Aligned representation of 3 voices (R1, R2, R3)in the OpenMusic poly editor, with tempo changes approx-imating τ1(t), τ2(t) and τ3(t).

http://support.ircam.fr/docs/om/om6-manual/co/Quantification.htmlhttp://support.ircam.fr/docs/om/om6-manual/co/Quantification.html

Figure 10: OpenMusic visual program implementing the rhythm matching process. The rhythm matching interface at theright allows to visualize/edit the matching between the target rhythm rT and the various lines’ tempo-varying grids r

Sji .

4.3 User Interface and Supervision of the Process

The process presented in the previous sections in principlecould be completely automated and packaged in a blackbox. The main choices (inputs) for the user are the tempocurves τi(t), and the allowed rhythmic subdivisions Sj inthe different voices. Still, visualizing the different stepsis crucial to understand the process and eventually tweakthese parameters in order to obtain relevant results, hencethe choice and importance of a visual programming envi-ronment like OpenMusic where each of these steps is ma-terialized by a module, and where all intermediate resultscan be inspected and edited (see Figure 10).

More importantly, composers’ choices can be taken intoaccount in the search part of the process where elements ofthe different rSji are selected to compose the rhythms Ri.The algorithm may make unfortunate choices in the caseof equivalent matches, and sometimes the “best” choice interms of distance may not be the best in terms of read-ability, playability of the result, or because of any othercompositional reason (e.g. controlling voice exchange ordensity, see for instance Step 5 in Section 3.3).

The main module of the visual program in Figure 10 there-fore comes with a specific user interface (visible at the righton the figure) which extends the traditional multi-seq ob-ject of OpenMusic and allows the composer to visualizeand make the choices of the elements in rSji according tovisual judgements or other arbitrary motivations. Compu-tation can therefore temporarily stop here to leave spacefor manual edition and operations, before proceeding todownstream parts of the data processing.

4.4 A Note on Score Notation

In the works presented above, the parts that the musiciansread from are typeset according to standard notational con-ventions in which spacing between notes is set in order tominimize page turns without sacrificing readability. Thescore, however, is prepared in such a way that the hori-zontal distance on the page between two notes Ni and Njis proportional to the duration of Ni (see for instance onFigure 3). This redundant representation of time (rhythmicand proportional) allows one to see clearly the intendedtemporal relationships between the individual parts and toeasily correlate moments in the score with the notation asseen by the performers.

Proportional notation that allows for complete and accu-rate control over the space between notes is impossible inmost environments necessitating the use of graphic designsoftware such as OmniGraffle for aberration or Adobe Il-lustrator for MacCallum’s more recent works. The useof two separate environments for the score and parts canlead to differences between the two causing confusion inrehearsal.

Currently, OpenMusic scores represent time proportion-ally (chord-seq) or notationally (voice), but not both simul-taneously. Recent work has been done to extend the nota-tion objects and provide a hybrid (redundant) representa-tion of time.

5. FROM COMPSOSITION TO PERFORMANCE

One of the goals of the compositions described in Section 3is the representation of the inherently plesiochronous na-ture of human musical performance. It is the musical ma-terial itself, however, that carries this representation; thosepieces do nothing to elicit a performance that would fore-ground this feature the way, for example, the distribution ofthe musicians across a large distance in space would. Wepresent in this section two recent projects designed withthis performative aspect in mind, and which also prob-lematize the relationship between music notation and itsrealization.

5.1 Windows of Musical Time

If the “musical now” is a small window of continuouslyvarying width, what would we find if we could construct aperformance context that would increase the scale of thatwindow to the point that its contents become musical ma-terial and even form? MacCallum’s recent work Hyphosfor alto flute, bass clarinet, viola, and electronics is a com-positional study meant to explore this idea. As in aber-ration and Delicate Texture of Time, the performers listento click-tracks to aid them as they perform long nonlinearaccelerandi and decelerandi, however, in Hyphos, they aremeant to only use the click-tracks in rehearsal and dispensewith them in performance. The use of different slow, grad-ual, continuous changes in tempo for the three performersis designed to defamiliarize the performance context by re-moving the fictional shared tempo. As the performers at-tempt to follow their individual temporal trajectories, theirvertical relationships vary over time with respect to thoseprescribed by the score, and the “window of the now”,bounded by the two musicians leading and trailing the oth-ers, is brought to the foreground.

The compositional challenge here is to construct materialthat satisfies musical and æsthetic goals despite potentiallyextreme variation in performance. To this end, a set oftools that reconfigure the score to represent the temporalrelationships of the individual parts during a given perfor-mance is essential for composers looking to gain a deeperunderstanding of the nature of the performative variabilityand develop compositional strategies and intuition that relyon it.

This work highlights the latent dualistic role of the scoreas providing a set of prescriptive instructions for perfor-mance, as well as being a representation of that whichwas performed. In the case of a score for two musicians,one may be able to follow the prescribed score and men-tally reconcile the visual and auditory representations ofthe composition, however, as the number of parts increases,the complexity becomes unmanageable and the score, as anotational representation of the performance, is no longerof any value. Without a visual aid describing what actuallyhappened, constructive communication with and betweenperformers is hindered.

5.2 External Sources of Temporal Control

Hyphos, described in Section 5.1, was a study in prepa-ration for a collaborative project between MacCallum andchoreographer Teoma Naccarato that remains ongoing atthe time of this writing. In Choreography and Compositionof Internal Time, 6 pulses extracted from wireless electro-cardiogram (ECG) units worn by dancers serve as click-tracks for musicians in real-time. The musicians render ascore, but as in Hyphos, the temporal relationships betweenthe different parts is in constant flux as the dancers per-form choreography designed to affect change in their car-diac function that approximates the general contour of theprecomposed τi(t). The use of biosensors here is intendedto foreground the limits and variation of human bodies inperformance, as well as to intervene in the compositionaland choreographic processes.

6. CONCLUSION

We presented formal and compositional approaches for deal-ing with poly-temporal rhythmic structures in computer-aided composition. These formalisms and general work-flow emphasize both computational and interactive con-siderations at manipulating musical time and rhythmic no-tations. Computer-aided composition provides interactivemusical representations at the different steps of the processand allows for the combination of systematic/automatedprocedures with compositional interventions.

The presented framework is suitably general to be usedfor the generation and manipulation of rhythmic structures.It can, for example, be seen as a supervised rhythmic quan-tification tool, enabling the production of notated rhythmicapproximations of a given sequence of linear onsets, us-ing variable tempo tracks and/or polyrhythmic scores. Wehave also emphasized, in the discussion of recent composi-tional projects, how it is likely to be used in more interac-tive situations such as when the tempo information, for ex-ample, becomes a reactive input causing the different stepsand views of the corresponding musical representations toupdate on the fly.

Acknowledgments

This work is part of the French National Research Agencyproject with reference ANR-13-JS02-0004.

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[4] P. Desain and H. Honing, “Tempo Curves ConsideredHarmful,” in Time in contemporary musical thought,ser. Contemporary Music Review, J. D. Kramer, Ed.,1993, vol. 2, no. 7.

[5] F. Jacquemard, P. Donat-Bouillud, and J. Bresson, “AStructural Theory of Rhythm Notation based on TreeRepresentations and Term Rewriting,” in Proc. Mathe-matics and Computation in Music, London, 2015.

[6] C. Agon, G. Assayag, J. Fineberg, and C. Rueda,“Kant: a Critique of Pure Quantification,” in Proc. In-ternational Computer Music Conference, Aarhus, Den-mark, 1994.

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[8] J. MacCallum and A. Schmeder, “Timewarp: A Graph-ical Tool for the Control of Polyphonic Smoothly Vary-ing Tempos.” in Proc. International Computer MusicConference, New York/Stony Brook, USA, 2010.

[9] J. Bresson, C. Agon, and G. Assayag, “OpenMusic.Visual Programming Environment for Music Compo-sition, Analysis and Research,” in ACM MultiMedia2011 (OpenSource Software Competition), Scottsdale,AZ, USA, 2011.

[10] O. Delerue, G. Assayag, and C. Agon, “Etude etréalisation d’opérateurs rythmiques dans OpenMu-sic, un environnement de programmation appliquéà la composition musicale,” in Actes des Journéesd’Informatique Musicales (JIM), La Londe, les Mau-res, France, 1998.

1. Introduction 2. Preliminary Definitions2.1 Rhythmic Figures / Durations / Tempo2.2 Equivalence2.3 Polyphony2.4 Varying Tempo

3. Compositional Context3.1 Motivation3.2 Virtual and Emergent Tempos3.3 Methods

4. Computer-Aided Compositional Process4.1 Resolution with One Voice (n=1)4.2 Resolution with n Voices4.3 User Interface and Supervision of the Process4.4 A Note on Score Notation

5. From compsosition to performance5.1 Windows of Musical Time5.2 External Sources of Temporal Control

6. Conclusion 7. References