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Empirical Musicology Review Vol. 7, No. 1-2, 2012
49
What is Entrainment?
Definition and applications in musical research
MARTIN CLAYTON
Durham University, Department of Music
ABSTRACT: Entrainment theory describes the process of interaction between
independent rhythmical processes. This paper defines entrainment in this general
sense, then briefly explores its significance for human behaviour, and for music-
making in particular. The final section outlines a research method suitable for studies
of entrainment in inter-personal coordination, and with reference to published studies
suggests that the study of musical entrainment can be a source of rich insight also for
the study of human social interactions and their meanings.
Submitted: 2012 January 6; accepted 2012 July 13
KEYWORDS: dynamic attending, neuronal oscillators, recursiveness, interaction
THE term ‘entrainment’ refers to the process by which independent rhythmical systems interact with
each other. ‘Independent rhythmical systems’ can be of many types: what they have in common is
some form of oscillatory activity (usually periodic or quasi-periodic in nature); they must be
independent in the sense of ‘self-sustaining’, i.e. able to be sustained whether or not they are entrained
to other rhythmical systems (thus sympathetic vibration, as when a violin’s soundboard vibrates at the
same frequency as one of its strings, is not an example of entrainment). In order for interaction to take
place some form of coupling must exist between the rhythmical systems, and this too can take many
forms. This process of interaction may result in those systems synchronising, in the most common
sense of aligning in both phase and period, but in fact entrainment can lead to a wide variety of
behaviours.
The classic example of entrainment is that of pendulum clocks, which were observed by the
Dutch physicist Christiaan Huygens to synchronise when placed on or suspended from a common
support (see Pikovsky et al. 2001, 357ff). Numerous other mechanical instances of this phenomenon
have been identified, but the phenomenon also extends to the biological world, where examples include
those of synchronising fireflies, and the resetting of body clocks by sunlight (circadian entrainment).
Even these few examples reveal some of the many variables encountered in the study of entrainment.
Huygens’s original example was of two clocks mutually influencing each other (symmetrical
entrainment); the fireflies are many, with each individual potentially both influencing and being
influenced by several others; circadian entrainment features one rhythmic process – the cycle of day
and night – influencing the internal body clocks of many individual organisms, while those individuals
are not able to influence the time of the sun’s rising and setting (asymmetrical entrainment).
Entrainment between rhythmic processes may thus be one to one, one to many, or many to many, and
may be symmetrical or asymmetrical. This paper will focus on examples of entrainment that are
particularly relevant to human behaviour, and more narrowly still to musical behaviour. Even within
this relatively narrow brief, entrainment comes in many forms, in terms of the number of rhythms
involved, their timescales (periods), (a)symmetry, and so forth.
Entrainment is not a single phenomenon that occurs only in human musical behaviour: it is an
abstraction describing a process common to many different phenomena occurring at different scales of
time and space, in both biological and mechanical systems. As such it has been studied with the aid not
only of observation of natural phenomena, but also with the help of mathematical models developed
within the broader framework of dynamical systems theory. It is possible to study musical entrainment
productively with reference to only a tiny fraction of this substantial research field, but nonetheless a
certain part of this wider body of theory is indispensible. One essential abstraction is the notion of
phase in periodic or quasi-periodic processes. Many observable rhythmical processes are by their
nature continuous, but the more or less arbitrary definition of a reference point (such as the audible tick
or a clock or the moment at which an insect begins to emit light) allows us to study the phase
relationships between them. If we are concerned with entrainment between musicians, we might
identify the moment an individual strikes a drum-head or taps a foot as the focal point of a quasi-
periodic movement, and based on this choice we may study the relative phase of a particular pairing. If
Empirical Musicology Review Vol. 7, No. 1-2, 2012
50
two such events occur at precisely the same time then the rhythms are in phase (relative phase 0°), if
one occurs precisely midway between the other they are in anti phase (relative phase 180°), and so on.
If two rhythms are entrained, then, they do not necessarily fall precisely into phase with one another.
Rather, the evidence for entrainment will be (a) a stabilisation of the relative phase relationship, and (b)
the reassertion of this stability following a perturbation. In other words, when two rhythms are
entrained the relationship between them (be it in-phase, anti-phase, or somewhere in between) will
stabilise, and will be sufficiently robust to reassert itself; in the example of the clocks the test of
entrainment is not simply that they tick in synchrony (which could happen by chance), but that if we
upset that synchrony, for instance by momentarily stopping one of the pendula, they will re-
synchronise.
ENTRAINMENT IN HUMAN BEHAVIOUR
An example of human entrainment has already been noted above – the resetting of internal body clocks
by sunlight. In fact, many instances of entrainment can be observed both within and between human
individuals. Examples on a very small scale include entrainment between different neuronal oscillators,
or between ‘pacemaker’ cells found in the heart. Still within the individual but on a somewhat larger
scale, many common actions involve the synchronisation of movements between different body parts –
for example walking. At the inter-individual level, many forms of manual labour clearly involve the
synchronisation of actions between individuals – the cooperative pounding of roots or grains, for
example, or a cooperative sawing motion. High-performance athletes may benefit from types of
entrainment not experienced by most individuals: for instance, rowers and cross-country skiers tend to
entrain their breathing to their limb movements (Steinacker, Both et al., 1993; Fabre, Perrey et al.,
2007). In short, entrainment is displayed in many different forms of human behaviour, with periods
ranging from a few milliseconds to a day (and possibly longer). That observed in inter-personal
entrainment tends to fall within a somewhat narrower temporal range.
Looking at this from a somewhat different perspective, entrainment affects our behaviour in
many ways: if our body clock falls out of synchrony with the cycle of day and night we suffer from jet-
lag, which can result in unpredictable patterns of tiredness and wakefulness, irritability and so on. We
will suffer in very different ways if our heart pacemaker cells do not function properly. The broader
implications of entrainment for human behaviour have been discussed across a range of disciplines
including social psychology and even history. Historian William McNeill, for instance, begins his 1995
book with reflections on the visceral effects of military drills and what he calls ‘muscular bonding’,
extending his discussion to topics including dance, religious ceremonies and social cohesion in small
communities. In the process he tackles many questions which have interested social scientists at least
since Durkheim, who argued that:
“Probably because a collective emotion cannot be expressed collectively without some
order that permits harmony and unison of movement… gestures and cries tend to fall
into rhythm and regularity, and from there into songs and dances.” (Durkheim 1995
[1912]: 218)
One of the important aspects of human entrainment to have been studied to date is inter-limb
coordination: a long tradition of tapping coordination studies has shed considerable light on this
phenomenon. The basic dynamics of two independent but coupled body parts (published studies focus
most frequently, but not exclusively, on index fingers) can be modelled with the help of a simple
mathematical equation known as the Haken-Kelso-Bunz (HKB) model, which simply assumes the
existence of two rhythmical systems and some form of coupling between them (it is not therefore
dependent on any specific biomechanical information): this model predicts two stable relationships, in-
phase and anti-phase, with the anti-phase relationship becoming progressively less stable as the
frequency of oscillation increases (Kelso, 1995: 54ff). This is exactly what empirical studies of finger-
tapping show, whether the two index fingers belong to one individual or two.
A related group of studies explore unintended entrainment between two individuals
performing actions such as swinging pendula from the wrist or rocking in rocking chairs (Schmidt &
O’Brien, 1997; Richardson et al., 2005; Richardson et al., 2007). These experiments clearly show that
when individuals interact socially, for example in conversation, the rhythms of their actions tend to
become entrained. A key factor here is mutual attention in social interaction, because simply being in
the same room is not a sufficient condition for entrainment to occur.
Indeed, according to Mari Riess Jones’ Dynamic Attending Theory, attention is a key factor in
human social entrainment. Jones proposes that attentional behaviour is quasi-periodic, and that our
attentional rhythms may become entrained to regularities in our environment, which may include
rhythmical behaviours of other individuals (Jones & Boltz, 1989; Large & Jones, 1999). The
entrainment of attentional rhythms can be understood as underpinning a number of human social
behaviours, including speech and music: if I can entrain to your behaviour and you to mine then we are
able to coordinate our actions. Numerous examples of human behaviours – notably musical ensemble –
Empirical Musicology Review Vol. 7, No. 1-2, 2012
51
suggest that some mechanism of this kind must exist, while for others – such as turn-taking in
conversation and other aspects of linguistic behaviour – it may also be a convincing explanation.
A particular phenomenon which seems to be distinctive of entrainment in and between
humans, which is related to the periodicity of attention, is our ability to coordinate actions with an
external periodic auditory stimulus. The notion that this is exclusive to humans has been challenged by
Patel: exploring his hypothesis that beat perception is linked to vocal learning (2006), studies appear to
show that a cockatoo is able to synchronise its movements with a musical stimulus, if intermittently
(Patel et al., 2009). Whether or not this proves to be a widespread ability in other vocal-learning
species, it certainly seems to be fundamental to many behaviours that we think of as essentially human,
such as the use of language and music. Dynamic attending, especially when employing auditory
information, permits a wide range of temporally coordinated behaviours in humans. Timing
coordination can be observed in many different species, but the range of applications of inter-personal
entrainment seems to be uniquely broad in humans, precisely because we can attend rhythmically to a
range of senses including hearing.
ENTRAINMENT IN MUSIC – DISTINCTIVE FEATURES
As noted above, entrainment theory is not domain specific, but is rather an abstraction that can be used
to make sense of many different phenomena. In studying entrainment in music too we can find many
different phenomena sharing some common features. In fact, I argue that it is important to distinguish
between different manifestations of the phenomenon (see Clayton, in press), and that it is convenient to
do so at three different levels, viz:
1. Intra-individual entrainment takes place within a particular human being. An important
phenomenon at this level is the entrainment of networks of neuronal oscillators, which appear
to be responsible for metrical perception (Large & Kolen, 1994; Large, 2000, 2010; London,
2004). Another aspect of intra-individual entrainment, as noted above, is the coordination
between individual body parts (e.g. the limbs of a drummer).
2. Inter-individual/ intra-group entrainment concerns co-ordination between the actions of
individuals in a group, which is essential for ensemble playing in any musical tradition. This is
largely facilitated by the entrainment of attentional rhythms to auditory information, although
other sense modalities – vision in particular – often also play a part.
3. Inter-group entrainment concerns the coordination between different groups. This is less
widely recognised, being a rare phenomenon in Western art music, but in fact it is a
widespread phenomenon (see Lucas et al., 2011).
These different levels of musical entrainment are interdependent, most obviously in the sense that each
builds on the previous level: intra-personal entrainment allows us to perceive metrical structures in
musical stimuli and to coordinate our actions to those structures; without this it would not be possible
for individuals to play in time with each other; and without musicians playing together in groups we
could not have multiple groups interacting with each other. The interdependence may not all be in one
direction, however. Clayton (2007) demonstrates how hierarchical timing relationships can come into
being without prior planning or explicit recognition, as an emergent behaviour of a group of people
performing quasi-periodic actions: this raises the possibility of metrical patterns emerging directly from
joint action, rather than necessarily coming into existence first at the neuronal level and then being
expressed behaviourally.
As noted above, entrainment does not necessarily result in synchronisation in phase between
rhythms of matching periods. Musical behaviour offers many other manifestations of entrainment,
some of which may even be unique to human music-making.
A. Different rhythms can entrain not only in unison, but also in hierarchical or polyrhythmical
relationships. Examples of coordination between musical parts which are relatively slow and
those which are relatively fast (in relationships such as 2:1. 4:2:1, or 6:3:1) are so common in
music as to be trivial. Less common but still very widespread are polyrhythmic relationships
between parts (3:2, 4:3); Clayton (2007) introduces an example of a 3:2 relationship which
emerges unintentionally from musical interaction.
B. Hierarchical relationships can not only be observed behaviourally, as in the case of parts
which move at different speeds but are mutually coordinated. They also account for metrical
percepts: computer models which aim to illustrate this process in a simplified form, show how
hierarchical percepts can emerge spontaneously in response to relatively simple stimuli (Large
2010).
C. Just as common as musical parts in hierarchical temporal relationships are parts with matching
periods which are synchronised out of phase – for example, a snare drum that falls in an anti-
phase relationship with a bass drum. While so many commonly-cited examples of entrainment
seem to involve in-phase relationships, it is important to keep in mind that in music,
entrainment can involve a wide range of phase relationships.
Empirical Musicology Review Vol. 7, No. 1-2, 2012
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D. Entrainment can be symmetrical (as in Huygens’s clocks) or asymmetrical (as in circadian
rhythms). In the case of music it can be either: symmetrical in an ensemble made up of peers,
asymmetrical when people play or dance along with pre-recorded music they cannot
influence. It can also be relatively symmetrical: in most musical ensembles any individual can
influence any other, but in practice some people are more likely to have influence than others
(e.g. conductors, section leaders, soloists, senior musicians). Music may then be a particularly
good forum for investigating the interdependence between timing coordination and social
power relationships.
In summary, musical entrainment is recursive: individuals perceive and generate hierarchical temporal
structures; they coordinate their actions within groups; and groups of people coordinate to form larger
groups. It is also diverse: it can involve matching periods as well as hierarchical and polyrhythmic
relationships, it is out of phase as often as it is in phase; and it can fall almost anywhere on the
symmetrical-asymmetrical continuum. Musical entrainment is observed with periods in a range of
roughly 100-2000 msec (corresponding to frequencies of 0.5-10 Hz), from the fastest beat to a typical
measure (metrical and hypermetrical structures can however be considerably longer than 2000 msec,
see e.g. Clayton, 2000, p. 87).
Given the diverse examples of entrainment in musical performance, it follows that a wide
range of methods may be applied in studying these phenomena. The particular focus of the work
summarised below has been the role of entrainment in interpersonal and inter-group interactions, as
manifested both in the sounds produced and in patterns of movement. The analyses are therefore
sensitive both to musical knowledge and to personal and social factors – in other words they take
account of the fact that entrainment is being observed in real-life, meaningful human activities, and
assume that both the features of the music that people aim to produce, and the range of meanings that
the activity holds for them, will be significant in relation to the entrainment dynamics per se.
INVESTIGATING MUSICAL ENTRAINMENT: LESSONS FROM CASE STUDIES
This final section briefly introduces an approach to the study of musical entrainment in natural settings,
which employs a ‘stroboscopic’ method designed for the investigation of entrainment between quasi-
periodic rhythms (Pikovsky et al., 2001, Clayton et al., 2005). This method is straightforward in
principle, involving the following steps:
a) Identification of quasi-periodic rhythms and extraction of time series data
b) Calculation of relative phase relationships from pairs of time series
c) Investigation of entrainment using this relative phase data, employing visual inspection and
statistical measures
The case studies referred to here all concentrate on either inter-individual or inter-group entrainment.
The rhythmic processes at stake can be either sound-producing or silent movements: timings can be
taken from the onsets of particular sounds (in an audio file), or identifiable points in a periodic motion
such as the moment a drumstick strikes a drum head, or the highest or lowest point in a foot-tapping or
head-nodding movement (from video recordings – motion capture technology would be another source
of this data). Whichever rhythm is at stake, and whatever the data source, series of time data points
need to be derived.
The next phase in this process involves the calculation of the relative phase of each point in
one series with respect to the other series. If we are looking at the location of bass onsets with respect
to a ride cymbal part in a jazz ensemble, for example, the two cymbal onsets closest to a given bass
onset define a period, and the location of the bass onset with respect to that period is calculated. The
relationship of the bass part to the cymbal part is thus expressed in a single series of phase angles.
When these phase angles are plotted against time, it is possible to get an immediate
impression of the relationship between the two rhythms. If they are unentrained, that is uncoupled, then
the relative phase plot will generally proceed as a gradual drift, showing up on the chart as a series of
diagonal lines. If they are entrained then the relative phase will be stable, and the plot will form a (more
or less) horizontal line.
Another way of plotting the same data is to simply plot the distribution of phase angles on a circle,
in effect removing the time dimension. The grouping of data points will indicate the phase relationship
to which the rhythms tend. If the distribution is unimodal, calculation of a mean vector will enable a
quantification of this mean phase angle [µ] and the degree of entrainment (r, on a scale from 0 to 1).
Multimodal distributions may indicate hierarchical or polyrhythmic relationships. Three stages of this
process are illustrated in Figure 1: (a) a time series plot, (b) a plot of relative phase against time (in
seconds) and (c) a phase distribution plot with mean vector.
Empirical Musicology Review Vol. 7, No. 1-2, 2012
53
a. #
b. #
c. #
Fig. 1. Investigating the relative phase of two musicians’ silent hand gestures (data selected from the
study reported in Clayton, 2007, for clarity of illustration). a. Time series data. b. Relative phase of the
harmonium player’s finger taps calculated with respect to the singer’s hand taps. c. Distribution plot of
the data from chart b with mean vector (r = 0.961, µ = -8°). (Chart c and mean vector calculation
produced in Oriana).
Three studies cited here apply this method in different ways (further studies using similar
methods can be found in Clayton et al., 2005). Clayton (2007), from which the illustrative data in
Figure 1 are derived, studies the relationships between musicians in an Indian classical ensemble. This
paper considers two different aspects of an ostensibly unmetred performance: a) entrainment between
the silent hand gestures of soloist and harmonium accompanist, and b) that between different players of
the drone lute tanpura, who are not supposed to coordinate with each other or with the music they hear.
Doffman (2008) uses similar methods to study the timing relationships between the members of jazz
trios, and correlates these findings with interview material from the same musicians talking about
Empirical Musicology Review Vol. 7, No. 1-2, 2012
54
timing relationships and their socio-musical significance. Lucas et al. (2011) applies the same methods
to the study of relationships between different groups in the Afro-Brazilian Congado ritual, again
seeking to make sense of the results in terms of participants’ musical and ritual relationships. All of
these studies employ audiovisual recordings of real-life performances in natural settings. Of these three
studies Clayton uses data derived from video observation, Doffman’s timing data were derived through
a process of onset detection using audio files, while Lucas et al. derived timing data by tapping along to
audio files (which were also contextualised with the help of video recordings). The different sources of
the data affect the timing resolution to some extent, but otherwise are treated in very similar ways.
In the case of Clayton (2007), this approach demonstrated that in the case of the silent hand
gestures, the two musicians were clearly coordinated – but loosely so, with r = 0.67 a much lower index
of the strength of coupling than those reported in other studies. (As Figure 1 illustrates however, at
times the coupling between the two is nonetheless much stronger than this, with r = 0.961 for this ten
second extract). The study of the relationships between tanpura players reveal a mixed picture:
sometimes they show no phase stabilisation, but in one pairing the two players show a (clearly
unintentional) stabilisation, which occurs in a 3:2 relationship (the periods of the two plucking patterns
are roughly 3 seconds and 2 seconds). In this case the stabilisation occurs between the soloist Veena
Sahasrabuddhe and her student sitting behind her, and occurs when the student fixes her visual
attention on the soloist’s back or shoulder: visual information seems to be important, and to this extent
the entrainment must be asymmetrical as the soloist cannot see her accompanist. Less surprisingly
perhaps, a study of the relationship between the soloist’s hand tapping and her own tanpura playing
showed entrainment in a 3:1 relationship. As noted above, these findings illustrate not only that
interpersonal musical entrainment can occur unintentionally, but that it can do so in hierarchical or
polyrhythmic relationships, and can result in temporal hierarchies at least as complex as those
intentionally reproduced as musical metre.
While the previous study was concerned largely with exploring the possibility of unintentional
entrainment, Doffman (2008) looks at the musical and social coordinates of timing relationships
between jazz players. Not surprisingly, the pairs of musicians (drummer, bassist, and guitar or piano
soloist) are tightly entrained, with r typically >0.9. More interestingly, small nuances and shifts
between relatively tight and loose coordination, or between a particular musician being slightly ahead
or slightly behind another, can be intensely meaningful for these musicians, and these phenomena are
tightly interwoven with musicians’ estimates of their own and others’ capabilities and characteristics as
musicians, and with their understanding of the ideals to which jazz performers should aspire. Doffman
studies the dynamic shifts between tighter and looser coordination, and concludes that timing
relationships aimed for in jazz groove cannot be reduced to a single ideal (e.g. in phase relationship and
degree of entrainment); rather, the ideal relationship is inherently dynamic and playing jazz involves
meaningful variations within the permissible range of looseness and out-of-phaseness.
Lucas et al. (2011) employ the same ‘stroboscopic’ method to investigate entrainment
between groups in a form of ritual processional music. Here ethnographic study strongly suggests that
the groups are invested in the notion that playing in time together is an index of ritual unity;
concomitantly it is important not to fall into time with groups from other communities, which would
indicate a breaking down of necessary ritual bonds and barriers. Study of several occasions on which
different groups play in close proximity allowed a number of different factors to be distinguished.
Different groups belonging to the same community entrain, and fall into synchrony, relatively easily –
at least when their tempi are fairly close together and they are in close proximity. When the groups
belong to different communities this is not so: sometimes they manage to retain their mutual
independence, using strategies such as exaggerating tempo differences and looking away from each
other; at other times one or both groups will simply stop playing to avert the possibility of falling into
time with the other. On one occasion, however, two groups performed a mutual greeting ceremony
while attempting to avoid playing in time, despite the fact that they were playing similar rhythms at
similar tempi. The result was that they actually fell into a tightly entrained relationship for over two
and a half minutes (r = 0.988), but they managed to do so out of phase by 223°, which meant that
although they were tightly entrained they did not perceive the relationship as such. Again, the complex
interrelationship between entrainment dynamics, intentions and meanings is apparent.
These three studies, as varied as they are, all address a particular type of musical entrainment
phenomenon – they are concerned with studying the interactions between people while making music
in real-life situations. The various findings demonstrate not only that it is possible to shed light on
entrainment dynamics in natural musical performances, but also that it is possible to relate these
findings to information about the intentions, experiences and discourses of the people involved. This is
a particular approach to a range of entrainment phenomena in music, which is interdisciplinary and
committed to rigour in both quantitative and qualitative research methods and to a principled
investigation of their interrelationship (see Clayton, in press). There can be little doubt that much more
can be learned about human music-making, indeed about human interactions in general, through
studies of this nature.
Empirical Musicology Review Vol. 7, No. 1-2, 2012
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!
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