Rhythm histograms and musical meter: A corpus study of Malian percussion music
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Studies of musical corpora have given empirical grounding to the various features that characterize particular musical styles and genres. Palmer & Krumhansl (1990) found that in Western classical music the likeliest places for a note to occur are the most strongly accented beats in a measure, and this was also found in subsequent studies using both Western classical and folk music corpora (Huron & Ommen, 2006; Temperley, 2010). We present a rhythmic analysis of a corpus of 15 performances of percussion music from Bamako, Mali. In our corpus, the relative frequency of note onsets in a given metrical position does not correspond to patterns of metrical accent, though there is a stable relationship between onset frequency and metrical position. The implications of this non-congruence between simple statistical likelihood and metrical structure for the ways in which meter and metrical accent may be learned and understood are discussed, along with importance of cross-cultural studies for psychological research.
KeywordsMusical rhythm Cross-cultural study Situated cognition
Introduction and background
It is unlikely that mental representations for meter are learned through formal training; very few of the listeners in these experiments had had exposure to music theory. Rather, the rhythmic organization of temporal patterns appears to be instantiated in the frequency with which events occur in various metrical positions in Western tonal music and affects [their] perceived relatedness and memory strength (Palmer & Krumhansl, 1990, p. 739).
Two important points are evident in Fig. 3. The first is that shapes of these histograms do not correspond to patterns of metrical accent as per Turkish music-theoretic descriptions of each usul (Holzapfel 2015, p. 7). Second, while there is a strong correspondence between note probability and the usul stroke patterns, in some pieces other onset locations are also prominent (e.g., Sofyan, Türk Aksaği). The correspondences between usul and onset position are thus weaker than those found between metric position and onset position in corpuses of Western music, which led Holzapfel to conclude that in comparison to Western music, meter in Turkish music is less stratified (Holzapfel, 2015, p. 24).
Africanist musicology has long noted that rhythms in African music can be anchored to the underlying meter in less predictable, more contingent ways, particularly by affording anti-phase (off-beat) and other, more complex types of non-congruent rhythm-meter-relationships (Kolinski, 1973; Locke, 1998, 2010; Kubik, 1988; Arom, 1991; Agawu, 2006; Burns, 2010) Indeed, a repertoire in which a significant portion of note onsets tend to be non-congruent with the metrical framework is Kolinski's (1973) very notion of a "contrametric" style of rhythmic organization. Contrametric rhythm is particularly characteristic of much sub-Saharan rhythm (see Locke, 1982; Arom, 1991). Yet the underlying meters to which sub-Saharan rhythms are anchored are, in most cases, the same meters found as in Western music. The most common meters involve a cycle of four even beats that are subdivided into two, three, or four more rapid subdivisions.1
Here we present data from a corpus of Malian drum ensemble performances; like Holzapfel, our corpus is characterized by contrametric rhythms. In addition to expanding the cultural and stylistic range of rhythmic analysis, our study furthers the aims of corpus studies of musical rhythm and meter in two ways. First, the statistical analyses we present are derived from the timings of complete actual performances, rather than from score-based symbolic representations. Second, our data include all parts of the musical texture, rather than the melody alone. This gives us a richer and more ecologically valid picture of the context in which listeners might acquire rhythmic knowledge. Moreover, by including the complete musical texture and by giving a broader situated account of how these pieces are heard, we suggest how mental representations for meter in characteristically contrametic styles may be developed.
Ensemble and repertoire
Malian jembe drum ensemble music is the musical component to larger, often participatory social events that involve dancing, eulogy, masquerade or puppetry, and other modes of social interactions (Polak, 2007). The percussion ensembles typically involve three distinct musical roles: a variative lead drum, a timeline or "hook" drum that plays the characteristic asymmetrical rhythmic figure that identifies each particular piece, and one or more ostinato accompaniment parts that maintain the basic beat. These roles are assigned to specific instruments in the ensemble (Polak, 2010). The lead part is played on the first jembe, a goblet shaped drum played with the percussionist's bare hands. The hook part is played on the dundun, a cylindrical drum beaten with a stick. The accompaniment part is played by a second jembe. Metrical structure in Malian drum ensemble music most often involves an isochronous 4-beat cycle, or multiples of it such as an 8-beat cycle. In our corpus, the subdivision is always ternary; in Western music it would be analogous to 12/8 meter. In addition, many pieces of Malian drum music, including all of the pieces analyzed here, involve a gradual increase in speed over the course of an entire performance.
Our corpus is comprised of three different pieces, "Manjanin," "Maraka," and "Woloso," which are part of the core repertoire of standard pieces in the urban tradition of Bamako, the Malian capital (Polak, 2012).
In 2006/2007, R.P. collected a set of multi-track and video recordings of 15 complete live drum performances in Bamako, Mali. Unidirectional microphones (AKG C-419) clipped to the drum rims recorded uncompressed audio to a mobile digital four-track studio (Edirol R4). Recording sessions took place open-air, where there was little acoustical crosstalk of instruments and reverberation from walls. Sony Vegas Pro versions 11 and 12 (http://www.sonycreativesoftware.com) was used for audio synchronizing, archiving, and editing; Sony Soundforge Pro (ver. 10) and Steinberg Wavelab (ver. 7; http://www.steinberg.net) were used for audio editing and automatic onset (attack point) detection. Onset markers were checked by eye and then onset timings were converted into a text file and imported into Microsoft Excel (Microsoft, Redmond, WA) for formatting and cleanup.
Over the course of the entire metric cycle the positions which articulate the hook rhythms (marked with asterisks) are among the most frequently articulated positions in each histogram. Apart from the downbeat (1.0 = start/end-point of metric cycle), onbeat positions (2.0, 3.0, or 4.0) are articulated less frequently than the upbeat metrical positions that immediately precede them, save for the 3rd and 4th beat positions in Maraka. Mid-beat positions are articulated much less frequently than the surrounding positions. Even more significant is the fact that the most frequent onset locations in each histogram are not periodically spaced (e.g., if one uses the four or five most frequent onset locations as points of reference, Manjanin and Woloso exhibit a 2 + 3 + 3 + 4 spacing, while Maraka exhibits a 2 + 4 + 3 + 2 + 1 pattern), while the underlying metric beats are periodic. Thus, while at least three levels of rhythmic hierarchy seem clearly evident based on event frequency, the histograms in Fig. 5 are inherently equivocal as to the location and relative accent of on-beat versus off-beat (mid- and up-beat) positions, as they cannot align their aperiodic onset frequencies with the periodically occurring beats of the 12/8 measure.
Corpus studies of Western musical rhythm (Palmer & Krumhansl, 1990; Huron & Ommen, 2006; Sadakata, Desain & Honing, 2006) have presumed that the frequency of a note onset's occurrence within a measure is correlated with its metric status, and, based on this presumption, these studies have made inferences how these frequencies may support a statistical learning process for musical meter. These inferences are in accord with various generative theories of musical rhythm (Lerdahl & Jackendoff, 1983; Longuet-Higgins & Lee, 1992; Temperley, 2010), in which surface rhythms are derived from operations on an underlying metrical structure. This approach also underlies many beat and meter finding algorithms used in music information retrieval contexts (Jehan, 2005; Ellis, 2007; Tomic & Janata, 2008; Mauch & Dixon, 2012). Oscillator models of musical meter (Large & Palmer, 2002; Large, 2008) derive metrical accent and stability from stimulus-driven resonances that are dependent upon event frequency and regularity, and most EEG and MEG studies of rhythm perception presume that salient musical events are more likely to occur on the beat (Brochard et al., 2003; Abecasis et al., 2005; Snyder & Larger, 2005; Vuust et al., 2009). However, the shared presumption that onset frequency is correlated with metrical accent holds only contingently, that is, for the corpora of Western classical and popular music that were used in these studies, and for which these models were developed.
We would therefore argue that it is the recurrence and stability of rhythmic figures in a given style or repertoire (and the metrical predictability this recurrence entails) that is required for learning characteristic rhythm-meter relationships. While some styles may involve a strong correlation between note frequency and metrical accent, this is not required for the developing mental representations for meter in informal listening contexts (cf. Palmer & Krumhansl quote above). As Large, Herrera & Velasco (2015) have shown, cyclically recurring rhythms, even if they do not contain explicit beat-level periodicities, can still give rise to a sense of beat. To this we would add that situated encounters with characteristic rhythmic patterns can engender not only a sense of beat, but a sense of the higher and lower levels of metrical organization as well.
A listener's metrical knowledge comes from hearing rhythms and melodies in their normative, real-world situations (Keller 1999). In those contexts, performance timing information supplements the learning of rhythmic patterns, as we have suggested above. Moreover, we are able to orient characteristic usul or hook rhythms—the rhythmic layers that have the greatest influence on event frequencies within the measure—relative to other parts of the musical texture that project metrically unambiguous accompaniments (e.g., the Jembe2 part in Fig. 4). Once the relation of the hook to the basic meter has been grasped in the context of the full ensemble, one can "fill in" missing metrical information when one encounters it in isolation. Indeed, one may learn to associate particular timbres, frequency ranges, and articulations with various rhythmic and metric cues: the sound of the dundun is indicative of the hook part, and signifies that it is likely to mark an alternation of "onbeat" and "offbeat" locations, for example. As our Malian percussion pieces typically accompany dancers in social settings, there is also a rich visual and kinesthetic context in which these rhythms are experienced. The dancers' movements may be observed, or better yet, one might learn to dance, and thus know how to move with the music, whether or not one is actually dancing (Agawu 1995).
We therefore posit that while the frequency of onset occurrence of events doubtless plays a role in our acquisition of rhythmic and metric knowledge, those frequencies occur in holistic contexts that include timing, timbre and other auditory, visual, and sensorimotor channels of perception. Combinations of these cues forge associations between statistically common rhythms and their characteristic metrical orientations. In so far as the rhythms in a corpus have a stable and distinctive set of features that afford a clear affiliation with a particular meter, their statistical attributes may serve as cues for the appropriate metrical templates. While our Malian corpus is illustrative, it is not that different from most other corpuses of music marked by recurring characteristic rhythms (e.g., a "Mazurka" corpus or a "Reggae" corpus).
The study of African rhythm shows that the perceptual and cognitive relationships between rhythmic patterns and their interpretive metric frameworks are more complex and the potential for their independence is larger than has been assumed. It is also a reminder to psychological researchers that the behaviors we observe, and the models we create to explain them, are both often contingent facts of particular cultures, and that cross-cultural research can usefully inform our psychological theories in important ways.
Meter in African music, and especially how to notate/represent it, has been a contentious topic in the ethnomusicological literature. However, a strong consensus has emerged that while the rhythms found in various West African musical styles are characteristically different from those found in many Western musical styles, their basic metric frameworks are not; this is based both on the testimony of African musicians and dancers as well as music analysis (Locke, 1982, 1992, 1998; Nzewi, 1997; Anku, 2000; Agawu, 2006; Burns, 2010; Polak, 2010).