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TEMPO CURVING AS A FRAMEWORK FOR INTERACTIVECOMPUTER-AIDED
COMPOSITION
Jean BressonUMR STMS: IRCAM-CNRS-UPMC, Paris
[email protected]
John MacCallumCNMAT - University of California, Berkeley
[email protected]
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:[email protected]:[email protected]://creativecommons.org/licenses/by/3.0/
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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
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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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Figure 3: Score of MacCallum’s aberration.
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}).
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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
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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.
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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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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