hypotheses on the choreographic roots of the …

ACTAS DEL X ENCUENTRO DE CIENCIAS COGNITIVAS DE LA MÚSICA

Alejandro Pereira Ghiena, Paz Jacquier, Mónica Valles y Mauricio Martínez (Editores) Musicalidad Humana: Debates actuales en evolución, desarrollo y cognición e implicancias socio-culturales. Actas del X Encuentro de Ciencias Cognitivas de la Música, pp. 477-495. © 2011 - Sociedad Argentina para las Ciencias Cognitivas de la Música (SACCoM) - ISBN 978-987-27082-0-7

HYPOTHESES ON THE CHOREOGRAPHIC ROOTS OF THE MUSICAL METER

A case study on Afro-Brazilian dance and music LUIZ NAVEDA AND MARC LEMAN

GHENT UNIVERSITY

Abstract A great part of the musical engagement is dependent on the human capacity to perceive

and produce of temporal regularities in music. In Western music theory, these regularities are related to the concept of musical meter. A brief overview of the theories of musical meter reveals that a number of approaches are built upon abstract concepts of musical structure, axiomatic rules of preference or hypothetical models. In this paper, we argue that dance provides reliable explanations for a number of mechanisms behind the emergence of meter. We demonstrate our hypothesis by means of a case study on the analysis of dance and music recordings of Afro-Brazilian samba. The movement and audio data was processed by using TGA analysis (topological gesture analysis). The results show that the structure of musical meter is not only more elaborated in the dance gestures but also provides richer elements to support the concept of meter in music. We demonstrate that the structure of meter is readily available in the choreographic forms and delineated by biomechanical constraints of the dancer's body. We suggest that the structure of musical meter might have been formed as an entangled musical-choreographic form and was further fragmented in a disembodied view of musical knowledge.

Resumen Gran parte del trabajo musical depende de la capacidad humana para percibir y producir

regularidades temporales en la música. En la teoría de la música occidental, este aspecto se relaciona con el concepto de métrica musical. Un breve resumen de las teorías de la métrica musical revela que una serie de enfoques se basan en conceptos abstractos de la estructura musical, en reglas axiomáticas de preferencia o modelos hipotéticos. En este trabajo, sostenemos que la danza proporciona explicaciones fiables para una serie de mecanismos detrás del surgimiento de la métrica musical. Demostramos nuestra hipótesis mediante un estudio de caso sobre el análisis de las grabaciones de la danza y la música del samba Afro-brasileño. Los datos de movimiento y audio se procesaron mediante el análisis de TGA (análisis topológico del gesto). Los resultados muestran que la estructura del compás musical no sólo es más elaborada en los gestos de la danza sino que también proporciona elementos más ricos para apoyar el concepto de métrica en la música. Demostramos que la estructura de metro está disponible en las formas coreográficas y está delimitada por las limitaciones biomecánicas del cuerpo del bailarín. Sugerimos que la estructura del compás musical podría haber sido formada como una forma enredada de música y coreografía y se fragmentó en una visión descorporeizada de conocimientos musicales.

Our frequent admonition - stop thinking and dance - isn't to say that the motion is unthinkable. It's to say that the

body is capable of understanding more things at once than can be articulated in language. One has no choice

but to think with the body. Browning (1995)

Introduction The human capacity to perceive and engage in temporal regularities in music is a key

musical ability without which the vast majority of music forms, traditional dances, rituals and other forms of musical engagement would not have evolved in the human culture. The concept of musical meter captures the structural properties formed by temporal regularities perceived in the auditory stream, from the viewpoint of the Western theory of music. However, scholars have long been

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struggling with gaps in representation, modeling and conceptualization of meter. From simple time prescriptions – such as the ones seen in a musical score – to computer algorithms for beat and meter detection – such as the ones used in computer processing in MIR applications1 – the formalization behind models of meter still requires more developments. Common sources of inconsistencies in these models include the inherent metrical ambiguity of music styles (McKinney & Moelants 2006), lack of isochrony in the compound metric levels (e.g., Blom & Kvifte 1986), cultural or personal preferences for certain tempos or metric structures (e.g., London 2009), and the perception/performance of several parallel metric structures (e.g., Parncutt 1994), among others. The attempts to find proper generalizations for the mechanisms that are behind the emergence of musical meter seem to have produced abstract2

The application of theories of embodiment (e.g., Merleau-Ponty 1962; Varela, Thompson, & Rosch 1991) to musicology (e.g., Leman 2007; Toiviainen, Luck, & Thompson 2010; Van Noorden & Moelants 1999) and the contributions inherited from ethnomusicology (e.g., Blom & Kvifte 1986; Chernoff 1991; Clayton, Sager, & Will 2004) and cognitive sciences (e.g.,Trainor & Trehub 1992) demonstrated that some of the problems in explaining the emergence of meter could be solved by looking at the moving body and how it shapes musical behavior (in special, oscillatory movements). This shift towards a broader epistemological framework where musical engagement takes place and interacts with the human body (and not only a body that follows a mind) showed that certain aspects of rhythm and meter could be explained by the inherent biomechanics (see, for example, London, Gritten, & King 2006; Van Noorden & Moelants 1999) and morphology (Naveda & Leman 2009) of the musical movement

models or parameterizations. These theoretical artifacts are then used to fill gaps between the theory and the diverse forms of the phenomenon of meter. Examples of these ‘abstract’ mechanisms and structures include representations such as metrical grids, metaphors of periodicity, mental models, oscillatory predispositions or simple axioms (some of these examples will be discussed in Section ‘Symmetry, periodicity and abstractions’).

3

In this paper, we argue that the structure of musical meter may have inherited more than phenomenological parallelisms with dance forms. We suggest that musical meter may have originally developed from interactions between music and dance, whose primary characteristics have been imprinted in both modalities. And intrinsically dependent music-dance structures. In the next sections, we outline a non-exhaustive review of the main theories of meter, our hypotheses and an experiment that aims at investigate how the musical meter is reflected on dance gestures. The experiment is realized in the context of Afro-Brazilian samba music and dance.

and its cultural and social forms such as dance, social choreographies and spontaneous movement. Indeed, if one looks at the sociocultural displays in which music appears in most of the societies, it is difficult to ignore the recurrence of dance and music phenomena at the same time and spaces.

Outline of the study In this study, we investigate how dance gestures inform about the structure musical meter, in

the attempt to provide better insights on the theory of meter. In the Section ‘Background on meter’, we will delineate the questions and theories that form the background of this study. In the experimental part, described in the Section ‘Methodology’, we delineate our methods and explain how music and movement analysis were merged in the analysis of musical meter in the dance space. The methods were applied to a data set of 13 movement recordings realized professional dancers, specialized in Afro-Brazilian samba. Section ‘Results’ displays the results for all dancers, which focuses on the movement of hands, feet and how musical meter is represented in space. Section ’Discussion’ discusses the impact of the results against the theory of musical meter, which are summarized in the Section ‘Conclusion’.

Background on meter Meter is a concept that seem not being easily detached from the surface of the sound in the

auditory domain or, at least in the available symbolic and sensory descriptors of music. Symbolic information such as pitch and duration of musical events do not contain metrical structures per se. In fact, most of the metrical information in traditional Western notation is prescribed a priori and simplified as a system of isochronous bar and beat levels. Metrical structure is certainly more complex. State-of-the-art of the models for beat detection based on low-level features of musical audio are struggling to

1 Music information retrieval. 2 The use of the word abstract in this context means that the models or set of rules proposed in theories of musical meter that are weakly supported by causal explanations. 3 The terminology musical movement refers to any movement driven or accompanied by music. This definition includes spontaneous movement to music, dance, ancillary and expressive movements of musicians, among others.


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achieve acceptable recognition levels for beat and bar levels, even in an universe of Western popular songs4

Symmetry, periodicity and abstractions

. Meter could then be seen as a ‘higher-level’ concept. This means that the structure of meter seem to emerge from more complex interactions between perceptual cues, time, memory and reasoning. These contextual levels of information include not only low- and mid-level sensory and perceptual cues such as the perception of intensity in sound, the ability to discriminate musical events, rhythm figures, tonal and temporal contexts but also the entangled cultural context that surrounds musical phenomenon. In these contexts, multimodal and multidimensional information might take part in definition of the sensation of metrical engagement. Images, movements, dances, words, among others will affect and shape (conflict, reinforce and complement) certain aspects of perception of temporal regularities. These extrinsic musical influences could also be recollected from memory and be enacted as sensorimotor references in the act of listening, as seen from a perspective of ecological perception (Gibson 1979; Parncutt 2007).

The concept of musical meter is represented in different ways in musical theory, pedagogy and performance. Representations of meter range from simple musical notation to geometric schemas and sets of proposed rules organized as theories. Possibly, the most disseminated concept of musical meter resembles a hierarchical structure of beats and metric levels depicted as a grid, as seen in figure 1. In this representation, the establishment of temporal regularities caused by past events (expectations) reinforce or conflict with the metric structure, which is seen as something that resides in the mind of the listener (Temperley 2000). This view was strongly disseminated through the work of Lerdahl and Jackendoff (1996) and similar contributions such as Longuet-Higgins and Lee (1984).

Lerdahl et al.’s concept of meter – expressed in the structure seen in figure 1 – displays the hierarchies between metric levels as layers (vertical) distributed along time (horizontal). The idea of periodicity and infinite temporal structure (circularity) is implicitly denoted by the symmetry and layered distribution as a grid. The sensation of temporal regularity or periodicity in music is induced by the recurrence of metrical accents, here represented by equally spaced dots on each layer. It can be said that metric accents are musical events that reinforce the metric structure by redundancy or periodicity in the expected metrical grid (Sethares 2007). However, the nature of these metrical accents is still object of discussion among researchers. Lerdahl and Jackendoff (1996) defines metric accents as “any event at the musical surface that gives emphasis or stress to the musical flow”. Such definition shows that metric accents are not only limited by changes in intensity of the musical events but also duration (Longuet-Higgins & Lee 1984), phrasing, grouping or melodic content. Although Lerdahl and Jackendoff (1996)’s definition of accents fulfills most of the possible variations of accents, it has been accepted that acoustic cues are not sufficient to explain the emergence of musical meter (Large and Palmer 2002).

Periodicity is another important characteristic of meter and the vast majority of representations stress this aspect in one way or another and often by visual metaphors. For example, in the examples of necklace notation displayed in the left side of figure 2 (see Grant 2009; London 2004; Sethares 2007), the visual metaphors rely on simple regular polygons or circles that suggest the circularity in the structure of metrical hierarchies. Other examples such as notation of poetic meter (Sethares 2007, p. 59), grid (Temperley 2004) or block (McAuley 2010) representations at the right side of the figure suggest periodicity by means of symmetry and flow. The importance of periodicity is also stressed by Large and Palmer (2002) and Palmer and Krumhansl (1990), whose approaches assume that the relationships between the perception of rhythm and metrical periodicities are based on ‘internal self-sustained oscillations’. The idea of an internal model that produces periodicities is also presented in studies on African rhythm and meter. Waterman (1948) suggested the existence of a feeling of ‘metronome sense’ while Jones (1959) mentioned the existence of ‘an underlying beat’ in the concept of meter in African music. Agawu (1992) speaks about a ‘metronomic framework’ and Nketia (1963) uses the term ‘regulative beat’. Together with these hypothetical assumptions on the existence of internal periodic clocks, there is a tendency to understand the representation of beat or metric structures as isochronous temporal regularities. It demonstrates the importance of another aspect of metric representations: the symmetry of metric structures.

Symmetry is widely present in the representations of musical meter. The frequent aid of geometric representations as a form of grasping the structural properties of meter seem to denote the apparent ambiguity and complexity of the ‘deep structure’ of musical meter. The examples seen in figure 2 depict several aspects of meter such as structural symmetry, intricate relations between hierarchical levels, well-formedness or preferential forms. These characteristics were wisely organized as a set of metrical preferences and a set of preference rules in Lerdahl and Jackendoff (1996), whose structure is also represented by equally spaced dots in a circle, seen in figure 2. London (2004) for

4 See, for example, thelast results of the contests on meter and beat detection at Mirex: http://www.music-ir.org/mirex/.

http://www.music-ir.org/mirex/�

http://www.music-ir.org/mirex/�

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example, makes use of geometric symmetry formed by lines between dots to demonstrate how different rhythm structures are discriminated and fit well-formedness rules.

Figure 1. Hierarchical representation of the structure of meter (based on Lerdahl & Jackendoff 1996) . The third

row displays beat level. The beat can be subdivided in other sub-levels, which are often realized through binary or ternary divisions (here exemplified as binary subdivisions). The same process is propagated up to higher levels in

binary or ternary multiplications.

By looking at representations of musical meter, it is possible to observe three elements that are prevalent in the concepts of meter: (i) figurative geometric representations that are linked with periodicity or temporal regularity, (ii) symmetrical representations that provide economic forms to communicate organization, intricate structural relationships and complexity and (iii) abstract models that fulfill predispositions for certain metric organizations, tempo distributions or temporal/periodic behaviors. Although these three characteristics provide useful categorizations of the temporal structure of musical meter they do not reveal the causality behind the phenomenon of meter: geometrical representations do not explain the relationship between meter and periodicity nor the need or cause of the symmetrical organization. Why metrical engagement engenders certain preferences for metric organizations and periodic behaviors? Where these preferences come from? What are the contexts that form or drive the formation of temporal regularities and musical structure of meter?

Hypothesis Dance is one of the phenomena that often accompany music practices. Our hypotheses try

to respond to some of the abstractions behind the theories of meter by exploring the music-dance relationships. If musical meter depends on dance, then important characteristics of the musical meter will have developed in both modalities. Moreover, if the music-dance interdependence is fundamental to the concept of musical meter, then abstract components of meter could be found in the dance structure.

We assume that the concept of meter could have been generated from entangled forms of music and dance. This ‘scenario’ is not limited to ritualistic and old forms of music and dance traditions that were communal/social practices at some stage (e.g., dances such as the Gigue, Allemande). A great part of the contemporary musical cultures such as the dance music, music from Africa and African diasporas are still connected with lively dance practices. From the broad perspective on meter described in the last sections, it is now plausible to suggest that dance traditions may contain repositories of sensorimotor relations with music that were shaped by cultural choices in society. Although a large diversity of music-dance relationships are reported (Hanna 1987)5, dance forms were probably more entangled with musical engagement on the stages where musical (and choreographic) knowledge remained tacit and merged in community6

“… dancing is essentially a termination, through action, of certain kind of symbolic transformation of experience... a dance is visually apprehended, kinesthetically felt, rhythmically ordered, spatially organized phenomenon which exists in three dimensions of space and at least one in time.” (Williams 1997, quoted in Hanna 1987)

. This relationship is still dominant in a great part of musical contexts, which includes classical European music forms and the indubitable relevance of dance in the pop culture. At this point, we might need a working definition of dance. The following definition proposed in Hanna (1987) illustrates insightful concepts that are present in several definitions of dance:

5 Including dances without music, dances in the darkness, dances without metrical or musical interdependence as described in Hanna (1987). 6 Hoerburger (1968), dealing with dance ethnography, call this first emergence of music/dance forms as the ‘first existence’ of dance, as seen as a a form of tacit knowledge shared in social displays. When dances are transformed into explicit knowledge and detached from music, it enters in what he defined as ‘second existence’.



Figure 2. Examples of representations of meter. See description in the text.

As in several definitions of dance (see, for example, other definitions in Buckland 1983; Carroll 2001 ; Hanna 1987) dance appears to enact a sort of symbolic transformation. In the dance-music contexts that populate a great part of popular music and dances, dance may be strongly characterized by ‘rhythmic order’, which unfolds in spatial organization in time and diverse modes of synchronization and entrainment. Within the context of popular dance-music contexts, lets suppose a qualitative change from ordinary to choreographic gesture were realized through engagement with music, in special the engagement with rhythm. This transformation would imply that music and dance share spatiotemporal cues and that musical qualities are synchronized with body movements. Not only the perception of movement in dance would be qualified by musical qualities but also the sensation of musical qualities would be influenced by choreographic/kinesthetic actions. In this scenario, temporal regularity and metrical structure could emerge in both kinesthetic and auditory domains as a shared structure. In some cases, auditory cues will sufficiently mirror all fundamental characteristics of meter;

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in other cases, auditory cues will inform only part of the metric structure. In this hypothetical scenario, it seems logical that a theory of meter solely based on music would require abstract concepts to fulfill the gaps provoked by the epistemological division between dance and music disciplines. Likewise, it seems natural that a methodology that aims at observing dance-music relationships should somewhat invert the symbolic transformation: to make use of sound as an action-perception carrier of kinesthetic sensations, or to make use of movement cues as an action-perception carrier of musical elements. In this study, we focus on the second method.

Meter and dance The latent relationship between meter and movement or dance is profusely documented in

studies on non-Western cultures and much less explicit in Western musical theory. The key ethnographical literature in dance and music offers a number of music-dance relationships in which studies on African music or music from the African diaspora have been specially relevant. Chernoff (1991) argues that the participative model of musical performance in African music ensures that there is always a metrical or phraseological anchor in the movement of dancers, which was also confirmed by Temperley (2000). In the analysis of drum patterns in Afro-Brazilian Batuque, Kubic (1990) found that the only available cue for beat annotations was the movement of dancers. Blom and Kvifte (1986) investigated the metrical ambivalence of Hardingfele music in Norway. They demonstrated that the gangar dance is essential for the coordination of meter and tempo among musicians. Similar contributions were made by Martin (1979) and Felfoldi (2001) in relation of East-European dances. Grau (1983) reported on the yoi traditions in Bathurst Islands (North Australia). Yoi traditions make use the word ‘dance’ to refer to dance, music, singing and rhythm characterization (if seen from a Western division of practices).

More recently Leman and Naveda (2010) studied how musical meter is connected with the morphology of the ‘basic gestures’ and Naveda and Leman (2010) demonstrated how meter is imprinted in the topological configuration of gestures. Both studies concentrated on samba and Charleston dances. Naveda and Leman (2010) suggested that the representation of meter in dance seem to reflect a structure composed of layers and phases, hereafter called layered-phasic model. This model of metric layers differs from the layered models described in the introduction because it subdivides the metric levels in sub-levels differentiated by phase. For example, there are two metrical accents on the half-beat positions in a 2/4 bar, which are considered by the Western music theory as pertained to a single metric-level (metric level half-beat). It was shown that, in some dances, these two metrical accents are performed in different locations in relation to the dancer’s body and exhibit different characteristics. Such characteristics differentiate each phase of the half-beat levels, which gives origin to the subdivision shown in figure 3. This figure illustrates the traditional representation as layers (left side) and the layered-phasic model (bottom side) that displays both layer and phase information of the metric level. Although several ‘necklace’ representations (London 2004; Sethares 2007) implicitly suggest the phase structure of metric levels, to the best of our knowledge, the issue was not clearly demonstrated through dance gestures before Naveda and Leman (2010).

Figure 3. Two different viewpoints on the structure of meter. First (left side) the traditional representation as

layers. Second (bottom side), the layered-phasic model that displays both layer and phase of the metric level.



Universe of study The samba culture offers a optimum case study to investigate relationships between musical

meter and dance. It has an entangled dance and music culture that has a profound impact on Brazilian identity. Its exhibits complex rhythmic and polymetric texture and paradigmatic dance styles (such as the samba-no-pé that is focused in this study) accompanied by a several musical styles described in the literature.

Three characteristics are widely attributed to samba music in both academic and instructional literature: samba exhibits (i) a binary meter (e.g., 2/4), (ii) accentuated in the second beat and (iii) a rhythmic texture characterized by syncopated rhythms (Chasteen 1996; Mariani 1986; Moura 2004, among others; Salazar 1991; Sandroni 2001). The harmonic and melodic structure of samba music are based on a Western tonal model but the notion of rhythmic priority dominates the musical texture in all sub-styles. Sub-styles of samba are differentiated by a large range of variations such as musical structure, lyrics, geographical distribution, choreography, instrumentation and sometimes only differentiated by its terminology (Fryer 2000). Examples of these sub-styles include the samba-carioca, samba-de-roda, pagode, partido-alto, samba-canção, and samba-enredo, which are accompany by samba dances, such as the samba-no-pé choreography.

Although the binary prescription of meter (e.g., 2/4) in samba music is a common point among musicians and scholars, samba is frequently described as a polymetric form (e.g., Browning 1995; Fryer 2000): a musical texture built up different layers of metric structures. These metrical structures display a diverse combination of rhythms, improvisative passages, periodicities and accent groups that provide a weak support to a binary meter, from a musical viewpoint. Authors like Browning (1995), Naveda and Leman (2010) and Sodré (1979) claim that this rhythmic signature was intentionally designed to provoke rhythmic and metrical ambiguity and, by consequence, provoke movements.

Very few scholars have attempted to describe the movement in samba dances. hand, it also demonstrates the self-sufficiency of the transmission of dance traditions. Although the literature of samba music does not rarely allude to the importance of dance, the contrast with the profuse documentation about samba music seems to indicate the effect that Desmond (1994) described as ‘nearly unnoticed symbolic system’. It refers to the idea that the body is so present in the life of a person (including scholars) that it becomes difficult to perceive it as a concrete text or carrier of information.

So far, text-based descriptions of dance are the most common reports on samba dances found in the bibliography. Such descriptions tend to disarticulate inherent complexity of the body engaged in dance (Calvert et al. 2005; Desmond 1994; Hanna et al. 1979; Ungvary et al. 1992). Browning (1995) and Mariani (1986) undertake two rare attempts to represent and analyze the ‘text’ that lies underneath the gestural content in samba. While Browning adopted an immersive perspective on samba culture by studying samba dances, Mariani tried to describe the gestures in samba by means of Laban’s analysis and notation (Laban 1928; Laban & Lawrence 1947). In a series of studies on samba dances and music, Naveda and Leman (Leman & Naveda 2010; Naveda & Leman 2009, 2010) analyzed dance gestures from a computational perspective. The studies reveal cross-modal elements of the gestural forms, however, an extensive analysis of large collections of dances are still needed.

In the experiment that follows we use categories of musical meter to qualify the dance gestures. This process sheds light to ordinary trajectories provided by movement descriptors using descriptors of the music.

Methodology

Subjects Thirteen Brazilian subjects participated in the experiment. All participants were professional

samba dancers with more than 5 years of experience of performance in dance. Each subject performed two dance sequences (13 recordings in total) realized in Brazil and Belgium. We recorded 7 female and 6 male subjects. Although we realized recordings in different tempi, in this study we will only report on the results of the recordings realized at 80 BPM.

Procedure

Task The dancers were asked to perform simple sequences of basic samba dances in samba-no-

pé style within a delimited area (diameter = 4 m). They were asked to avoid improvisational turns in the body orientation (frontal performance, in relation to a single direction). All subjects were asked to

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perform two sequences of dance at 80 BPM and 120 BPM. Only the sequences in 80 BPM are analyzed in this study.

Motion recordings We recorded the dance performances using a motion capture system (Optitrack/Natural

Point) that consisted of 12 cameras positioned around a squared aluminum structure (4 x 4 meters) and a computer workstation. Each session was 60 s in length and was recorded at a frame rate of 60 Hz, interpolated to 100 Hz in the editing phase. The dancers wore a dance suit with 34 reflective markers attached to it; the markers provided the point-set representation of body morphology. The markers were placed on the head (3), upper arms (3 + 3), upper back (3), hips (4), hands (3 + 3), thighs/knee (2 + 2), shins (1 + 1), and feet/toe (3 + 3). The recording sessions were edited in the ARENA software (Natural Point) and exported as C3D files. The files were imported into Matlab by using the MoCap toolbox (Toiviainen & Burger, 2008). The transformation of the set of 34 marker points into a stick figure containing 20 joint segments was also computed using MoCap Toolbox7

Stimuli

. The trajectories of each joint of the dancer were normalized with respect to one reference point and orientation of the dancer’s body (the point is defined as the centroid of the body across markers and the orientation as a frontal view with respect to the left and right hips). This procedure subtracts the influence of whole-body rotation and translation from the raw trajectories.

The stimuli used to perform the samba dances were composed of looped samples of a samba percussion ensemble (surdo, tamborim, and caxixi). The samples were recorded from professional samba musicians in Brazil, using a multitrack recorder. Two stimuli were used to accompany the dancers: one sequence performed in 80 BPM and one sequence performed in 120 BPM. The beat and 1/4 beat of each stimulus were manually annotated in order to provide the metric cues necessary to implement the methods (see Section ‘Universe of Study’).

Selection of dance sequences We manually selected a set of 32-beat homogeneous sequences from the original motion

capture recordings, audio sequences and audio annotations. The categories of metrical accents were categorized from the annotated beat and 1/4 beat points using the models of metric hierarchies displayed in see figure 3, which generated a set of time stamps and categories o meter, as exemplified in figure 4.

Data analysis

Topological gesture analysis (TGA) The Topological Gesture analysis (TGA) is a technique introduced in Naveda and Leman

(2010) that permits the observation of relationships between movement and musical cues in space. In short, it consists in the projection musical cues onto the gesture’s trajectories and the interpretation of the gestural space as a topological structure. Originally applied to dance, this technique maps the topological regions that form of the space around the dancer according to musical cues. There are several reasons use TGA technique for this study. TGA analysis is able to map the strength of how dancers use the space in relation to the musical meter. The focus on topological structures and the transformation from the Euclidean space of the capturing system to topological spaces makes the results from the TGA technique more likely to be compared with models of meter proposed in the bibliography.

The extraction of features using Topological Gesture Analysis (TGA) involves (i) projection of cues, (ii) discrimination of point clouds and the (iii) generalization of topological abstractions in convex hull representations.

Projection of musical cues The projection of musical cues onto dance trajectories is the key to qualify dance trajectories

with musical information. Musical cues consist in any temporal descriptor of the musical texture. In this study, we used the categories of metrical accents annotated in the time domain. The categories used in this study convey the layered-phasic model of metric levels, illustrated in figure 3, which includes both metrical level and the phases of the metrical accents. This model was firstly identified and applied to dance in Naveda and Leman (2010) and prevents several problems in the interpretation of dance gestures, explained in a comparison between different models in the Section ‘Metric Hypotheses’.

7 Joint names: root (centroid of the body), left hip, left knee, left toe, right hip, right knee, right toe, mid-torso, neck, head, left shoulder, left elbow, left finger, right shoulder, right elbow, right finger.



The process of projection is described in figure 4 and figure 5. It involves the transformation of musical cues in topological regions in space, or regions qualified by musical information. Since motion capture and musical stimuli were synchronized in the recordings, the annotation of the categories of metric accents can be projected onto its respective time points in the trajectories of the body joints. This process generates point clouds spread in the space around the dancer.

Discrimination, ratio of recognition and dispersion In the discrimination phase, the point clouds were classified using a quadratic linear classifier

(QDA, see Jain et al. 2000), which was trained with the original categories of metric accents (the process is illustrated in figure 4). Outliers above or below two times the interquartile range of the distribution in each category (all dimensions) were excluded. This results in a set of corrected classified points separated in space by quadratic boundaries (such as ellipsoids or paraboloids) and a set of incorrectly classified points (error). The proportion of points that were correctly classified by the quadratic boundary was indicated by the ratio of recognition (elsewhere represented by rr). The ratio of recognition is the ratio between correctly predicted and incorrectly predicted (including outliers). It is intuitively linked to the quality and strength of the link between musical cue and point cloud in space. If the dancer uses a discriminated region in space to perform gestures synchronized to a musical structure (characterized by musical cues) this region becomes linked or qualified by this musical structure.

The level of dispersion of the resulted point-clouds was computed as the generalized variance of the set of points (Wilks 1932; Wilks & Olkin 1960). The generalized variance indicates the dispersion of the point clouds and indicates if this relation of movement with a space is compact or loose. The generalized variance is calculated as the determinant of the covariance matrix of the positions for the points of the point cloud region.

In summary, the ratio of recognition indicates how often the dancer uses a region of space in synchronization to a metric level; it informs how consistent is the relation between musical cue and use of space. The generalized variance indicates how this region is dispersed or compact in space.

Figure 4. Process of projection of the point clouds. In the top graph, the representation of metrical hierarchy of

the music is illustrated along the time-domain. The categories are projected on the spatiotemporal domain using the time position of each category, which results in point clouds distributed in space. Further discrimination of these clouds is realized using a quadratic discriminant analysis. The discrimination process results in spatially

discriminated point clouds that carry the quality of the projected cue.

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Figure 5. Processes of (a) projection, (b) accumulation of cues into point clouds and (c) discrimination of the space as topologies. For illustrative purposes, only the beat levels (first and second beat) are represented.

Topological abstractions Topological regions are represented by geometric figures plotted against a stick figure. They

delimit regions in space in which musical that are significantly synchronized with the dance gestures. The abstractions are automatically generated from the point clouds, which are transformed in 3-dimensional polygons. The transformation of point clouds into polygons basically implies on defining a form in which point cloud can be generalized in space (see Carlsson 2009, for an overview of the theory behind it). In this study we assume that the points that define the boundary of the point clouds form a polygon that represents the topological region. We define this polygon as a convex hull, which is the minimal convex set containing the point cloud. Figure 5 displays the processes from projection of cues to the topological abstractions. In this example, only two categories of metric accents are projected onto movements: first and second beat.

Results The results provide representations for virtually any joint trajectory on a stick figure.

However, it is not feasible to display results for all body parts. In the following sections we will concentrate on the movements of one hand (Section ‘Hand Gestures’) and one foot (Section ‘Foot gestures’) and in summaries of descriptors for all body parts (Section ‘Strength of metrical representations’). Before these results, in the next section, we will demonstrate the response of the TGA representation to different models of meter.

Metric hypotheses Figure 6 displays the results of different models of metrical cues applied to the trajectories of

the right hand. In the first row, we applied the layered-phasic model (see figure 3 in Section 3, which includes layers and phases along 2 beats. The application of this model results in eight discriminated regions that reflect the metric levels in the gestural space of the right hand. The sequence of labels on the topological regions indicate that the gesture starts on the first beat in front of the body of the dancer and is projected towards the second beat, far from the dancer’s body, in clockwise direction. For each phase and metric level there is a determined region in the space, whose consistency is described by the high ratio of recognition – rr (ratio above bars). Note that metric level 1/4(4) seem to be not so consistent (rr=.56) in comparison with other metric levels. It is also possible to infer the strength and compactness (or dispersion) of the beat levels in the dance gesture: they exhibit high rr (rr=.94) but are concentrated in a small region, denoted by small generalized variances (bars).

The application of a more simplistic concept of layered metric levels such as the one displayed in the second row (right) shows that the lack of phases does not fit in the topology of the gesture. Note that in this case, the phases of the metric levels 1/2(1) and 1/2(2) are merged in a single category. This results in a topological region that ‘bridges’ two regions in space that were used at different phases. This region then trespasses the ‘ring’ topology of the hand’s choreography and disrupts the ratio of recognition and real values of dispersion (generalized variance).



Figure 6. Results of different models of metrical cues applied to the trajectories of the right hand. Topologies are displayed in the left graphs while the models and respective generalized variance (bars) and ratio of recognition (ratios above bars) are displayed in the right graphs. In the first row, it is displayed the model used in this study, with 2 beats and metric levels with phases. In the second row, a simple model of metric levels without phases is

displayed (see text for the discussion). On the third row, a model of 4-beats projects an hypothetical 4-beat meter on the gestures (see text).

For another side, if a model with four beats is applied (third row) to the same gesture, the consistent metric topologies that occupied one region in the layered-phasic model start to collapse in two regions (see metric levels 1 and 3, 2 and 4) or disappear (see metric levels 1/2(2) and others). In this case, the discriminant analysis could not find specific and consistent spaces where the dancer is synchronized with the meter, which results in very contrasting ratios of recognition. One metric level takes over another because they basically occupy the same region in space, hence they should not be differentiated.

These three examples demonstrate how dance gestures convey a very sensitive and complex structure that goes beyond traditional models of meter.

Hands gestures Figure 7 shows the results for the left hand of 13 dancers. These results show a diverse

profile of gestural morphologies (shapes and trajectories) and relationships with meter but very similar topological relations. Topological relations are very simple relationships between objects. For example, topological relations of one object to another may indicate if these objects are close, inside, apart or contain each other. They are insensitive to distances and orientations (in the Euclidean space) and rely in relationships with other landmarks. In our case, the main topological relationships can be traced by the relations of the topologies to the dancer’s body. The results indicate that except dancer 6 and 10, all dancers exhibit a symmetrical disposition of metric levels in a form of a ring Beat

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levels (first and second beat) take opposite sides in the gestural structure and are often more compacted than the other levels although the position in relation to the body of the dancer is not fixed. It is interesting to note that metric levels 1/4-beat are often more dispersed, most of the shapes exhibit a ring topology and are performed in anti-clockwise direction (while right hand will show clockwise movements).

Foot gestures Figure 8 shows the topologies for the right foot of 13 dancers. As indicated by the arrow

below the stick figure, the viewpoint of the results shows a top-view of the right-foot trajectories with the dancer facing the top of each window.

Figure 7. Topologies of dance gestures for 13 dancers, left hand. Each geometric figure (convex-hull) defines a

discriminated space where the dance gesture is synchronized with the metric cues (labels on the geometric figures). Thin lines indicate trajectories of the left hand.



Figure 8. Topologies of dance gestures for 13 dancers, right foot. Each geometric figure (convex-hull) defines a discriminated the space where the dance gesture is synchronized with the metric cues (labels on the geometric

figures). Thin lines indicate trajectories of the right foot.

The configuration of topologies for the right foot appears to be more complex than the structure of the hands. Topological regions are inevitably more interviewed due to the biomechanical limitations of the feet, which do not own the same freedom of movements that the hand articulations permit. However, only the dancers 6, 10 and 12 show more complex and asymmetrical configurations, which seem to be aligned with the level of symmetry observed in the results of the right-hand. Beat levels are symmetrically distributed in extremities in the movement of dancers 1, 2, 3, 8, 9, 11 and 13. Dancers 1, 4, 5, 8 and 13 seem to mark beat levels using lateral positions while the other dancers use forward-backward movements. As observed in the case of hand topologies, the beat levels are generally more compact while 1/2 and 1/4 metric levels are more dispersed, especially at the frontal extremities of the foot’s gesture.

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Strength of metrical representations The generalized variance and the ratio of recognition (explained in ‘Topological

abstractions’) are fundamental indices for the evaluation of the strength of the representation of meter in the trajectories of dance gestures. Figure 9 displays the global result of these two indices for all 20 gesture trajectories.

The results of the ratio of the normalized generalized variance displayed in figure 9 show few significant differences. The dispersion of the space used by the metric level 1/4(4) seem to be marginally higher than the other metric levels, followed by metric level 1/4(2). These two metric levels are positioned just before beat levels (metric levels 1 and 2, as seen in figure 3), which suggest that dispersion may be a result of a preparatory movement towards the return of the beat level. All other metric levels, including beat levels, exhibit a high variance, which may hide internal differences between gestures (not discussed in this study).

Figure 9. a) Mean of the normalized generalized variance for all joints and all metric levels. The generalized variance was normalized by the maximum result among the metric levels of each joint. b) Mean of the ratio of

recognition for all joints and all metric levels.

Figure 10. Number of results of ratio of recognition above thresholds, rr > 0.3, rr > 0.6 and rr > 0.9 per body joint

(for all subjects and metric levels).



The results of the ratio of recognition displayed in figure 9b do not exhibit a clear prevalence of any metric level but indicates a rather consistent representation of all metric levels in the dance gestures. Mean rr values above 0.6 indicate that an average of at least 60% of the gesture trajectories were synchronized with specific metric levels in delimited regions of the space. Further exploration of these results are shown in figure 10. The graphs show the number of topological regions by body segment that exhibit a ratio of recognition above three thresholds (rr > 0.3, rr > 0.6 and rr > 0.9). They indicate that above rr > 0.6 and 0.9, lower limbs (down to the ankles) arms and hands show more results with high rr while torso, neck and head are less subjected to high rr. Higher results of ratio of recognition in these graphs indicate that the most redundant gestures of the most periodic dance performances are concentrated on the extremities of the dancer’s body.

Discussion The results shown in the last sections offer a limited view on the topology of selected

gestures and a broad summary of how meter is reflected in the dance gestures. We use the analysis of dance gestures to support a discussion on musical meter and our proposed hypotheses, which will not be easily validated or refuted from a statistical point of view.

Response to different models of meter The form that dancers enact metrical characteristics in space reveals more insights on the

structural complexity of meter than the models discussed in the introduction. One of the novel perspectives offered by these results is related with the layered-phasic structure of the metric levels in dance, discussed in the Methodology. Metric engagement in dance is revealed by means of action in the space, which shows a richer set of properties in relation to musical properties. These regions of space are spatially different from each other and reflect a structure that is periodic, non-isochronous and organized in phases. This concept differs from the necklace notations presented in figure 2 because it shows causal relationships with the choreographical forms, bio-mechanical constraints of the body and expose finer aspects of the engagement with meter. For example, the up-beat of the first beat (1/2(1)) has different properties and occupy spatial regions than the up-beat of the second beat (1/2(2)). Different phases of the same metric level probably recall different psychophysical sensations. They involve different kinesthetic properties, different impacts on the proprioperception system (Phillips-Silver & Trainor 2007) and intentional gestural behavior (Gallagher 1995). From the perspective of ecological perception (Gibson 1979), it is quite feasible to imagine that these sensations impregnated in some extent the concept of meter that was projected onto music theory and practice. For example, recent studies on the analysis of microtiming deviations in samba also found non-isochronous structures on different phases of microtiming deviations and interactions with meter and dynamics (Gouyon 2007; Naveda et al. 2011).

Symmetry, biomechanics and periodicity Symmetry of dance gestures is another very important aspect that dialogs with our

hypotheses. Our results demonstrate that gestures are organized in structures that are dependent on the morphology of the human body. The organization of metric levels in space often takes a form of opposed levels, mirrored or inverted to the opposite limb (left-right opposition, not visualized here) whose biomechanics are likely to create a prevalence of binary symmetries such as binary sub-divisions, multiplications or oppositions. It is known that the morphology of the body interacts with the action-perception processes (Pfeifer & Iida, 2005) involved in human cognition. Since body movements must keep minimum equilibrium profiles to support and give equilibrium to a limited body, it is likely that metric structure in the dance gestures is somewhat affected by this these constraints. The similar profiles of rr of metric levels shown in the results (Section ‘Strength of metrical representations’) suggest that there is an equal representation of the metric levels in the dance gestures, which is periodic in time and redundant in space. The high rr found in the extremities of the body indicate that symmetry is equally and symmetrically distributed to the pair-wise limbs (left-right).

The same bio-mechanical constraints that affect the morphology of dance forms and metrical organization seem to affect preferences for tempo and metrical distributions. It has been claimed that the biomechanics of the body influence preferences for tempo (Styns et al. 2007; Van Noorden & Moelants 1999), rhythmic disambiguation (Brown et al. 2006; Phillips-Silver & Trainor 2007), spontaneous movement to music (Toiviainen et al. 2010) and entrainment (Clayton et al. 2004). One may also suggest that the topologies of dance gestures are able to convey the metric structure as a preference for certain distributions of metric levels, as seen in the results (Section ‘Metric hypotheses’) and that metrical configurations of music styles are represented in the dance choreographies. These hypotheses confirm results found in the analysis of Charleston choreographies in (Naveda & Leman 2010). In this study, it was found a configuration of 4-beat metric levels for the movement of feet and

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2-beat levels for the movement of hands, which agrees with the two possible metrical signatures of Charleston music, 2/4 and 4/4.

General discussion The trend between symmetry and periodicity is reflected in the results of ratio of recognition.

High rr levels indicate redundancy in the use of space, and thus indicate the performance of periodic gestures. One may say that the task proposed to subjects involved instructions that lead to periodic gestures or that dance is not only structured in one single style/step performed in loop. However, in the proposition of the task, we oriented the subjects to avoid improvisational styles and turns, which did not excluded improvisative and random choices within the style (as observed in some of the dancers). It is also certain that this limited spectrum of dances do not represent a significant sample of the possible dance displays within a dance style nor a significant diversity of dance styles in human culture. But it is certain that our universe of shares similar characteristics with other dance-music traditions.

Our results show how periodicity is reflected in redundancy in the use of space. A focus on kinetic properties of the movements or in the organization of the dance displays in large phrases could lead to other viewpoints and insights on the role of periodicity in dance. Attempts to investigate periodicity in the shapes and topologies of dance gestures are aligned to morphokinetic and topokinetic elements of the relation of the humans with space (Mullis 2008; Paillard 1991), but more research is needed to approach its relation to meter, music and dance.

Homogeneous dance displays do not encompass the variability of a dance style. However they are common elements in dances and there must be important reasons for such as movements being selected from others and prevailing against other possible random, out-of-synch or out-of-phase movements (Hanna et al. 1979). If these reasons were solely supported by biomechanics, we would expect a low degree of diversity of dance forms since humans share a rather common body apparatus. If these reasons were only based on environmental conditions, we would expect low degrees of diversity of dance forms in societies that evolved in the same environment. We cannot ignore the influence of symbolic or ancillary movements or traces of ordinary movements (movements of work activities, walking, etc), which are difficult to detect imply in a degree of uncertainty. Dance gestures certainly share metric properties with the musical matter when dance and music are connected in a dance-music ecology, but diversity in dance displays seem to be affected by a myriad of extraneous influences.

Finally, the symmetry, periodicity and consistency of the layered-phasic 2-beat model of meter demonstrated to be deeply entrained in the dance gestures of samba. Results from the metrical hypotheses test and the profile of topological regions in the Section Results confirm the presence of binary meter in samba dances. The weak support of the musical texture for a 2-beat metric signature in samba music indicates a relation of mutualism between music and dance forms as proposed in Sodré (1979) and Browning (1995). However, it would be an over-simplification to reduce meter in samba culture as a binary signature. Recent studies on timing (Gerischer 2006; Gouyon 2007; Naveda et al. 2011), dance (Leman & Naveda 2010; Naveda & Leman 2009, 2010), entrainment (Lucas 2009) in Afro-Brazilian cultures such as the samba report on intricate forms rhythmic engagement, which are beyond simplistic concepts of Western prescription of meter. The very same complexity may pose unbearable challenges to the analysis of African music and music from the African diaspora, from a Western theory based a concept of music without dance.

Conclusion In this study we tried to demonstrate that abstract concepts behind the theories of meter are

readily available in the dance gestures. The universe of samba dances and music was used to demonstrate how meter unfolds from dance engagement. More specifically, we have proposed that characteristics of models of musical meter such as symmetry, periodicity, preference rules for tempo and distribution of metric levels may simply reflect properties perceived in dance and in the morphology of the human body, but attributed to music.

The results found in the case of samba dances show that dance forms enrich the concept of musical meter by formulating constraints to the music forms. Although one cannot fully claim that dances are ultimately responsible for the musical meter, it is plausible that these two forms still exchange and have exchanged mutual influences. The application of dissociative and anthropocentric viewpoints on culture and science may have left an incomplete object defined as music detached from the original structure and its original connections with dance. Reports on such music-dance ecologies in the literature confirm that the division between dance and music is culturally specific, which could lead to a narrowed theory of ‘musical meter’ (if this concept exists as such).



There is a need for more approaches on dance and music subjects that are able to uncover the mutualism that connects music and dance in the societies. It is especially important to focus on comparative data sets but also to understand how other societies look at music and dance practices. New perspective from diverse cultural backgrounds may help to find better solutions for the modeling and computation of musical meter in combination with more knowledge about the psychophysics of human movement.

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