Strange Attractors: Thoughts on rhythm and meter in polyrhythmic space.

(DURHAM) In his explanatory notes to Mode de valeurs et d’intensités, Olivier Messiaen highlights the fact that the notated meter of 2/4 is merely for ease of performance, thereby discouraging the performer from attempting to fit the music into any accustomed a priori metric framings. When creating music in polyrhythmic space we may find ourselves thinking about notated meters in a similar way, simply selecting them by virtue of either pure convenience or correspondence with particular format unit durations. This is nothing new, as the bar line has always had a tenuous relationship with musical meter as it is performed and perceived. As composers have sought ever-new approaches to rhythmic organization and temporal flow, any conceptions of meter as a locally repetitive, cyclical force of forward musical motion which is somehow directly reflected in the time signature have often been redundant, or at least of minimal consideration.

In my own work using polyrhythmic space I have gradually become aware of how the very act of composing with two or three layers of discrete pulse streams – streams which can often affect one another locally in, at times, quite unexpected ways – can produce a different sense of forward motion, one that is naturally reflected in the more complex local and global temporal relationships between the elements of the polyrhythmic limbs. Speaking from experience, these time points (the individual elements in each limb) can naturally exert strong, attractive motional forces across the time skein, creating a complex web of continual temporal flux. One feels oneself composing toward and away from time points in the underlying structural layers, using them as rhythmic poles around which the push and pull of musical time can be organized. Can these irregular motional forces in polyrhythmic space be potentially understood as some kind of complex meter? Or, taking a step back: when composing stratified rhythms in polyrhythmic space, what happens to the idea of meter?

Recent research has greatly increased our understanding of the cognition of musical meter as well as its function and manifestation in the temporal structure of music, although much remains unclear. David Epstein (1995) and Jonathan Kramer (1988) both view meter as part of a bipartite overall temporal structure, with meter on the side of what they both term ‘chronometric time’ and rhythm on the side of ‘integral time’. Epstein highlights the importance of a perceptible periodicity, but for Kramer this is less important, as he focuses more on its manifestation through metric accent, also asserting that we can even perceive ‘deep meter’ at the highest levels of temporal structure, even up to the duration of an entire movement. Justin London (2012: 15-18) takes these ideas further in his study of the cognitive facets of meter, developing the idea of metric entrainment, whereby meter functions in the perceptual middleground as a framework for motional continuity and listener expectation; the cognitive entrainment to a metric pulse. Viktor Zuckerkandl was one of the first theoreticians to conceive of meter as a wave, beating at multiple levels of periodicity across the temporal skein. Applying this concept to our question of meter in polyrhythmic space, it would seem logical to infer that any polyrhythmic structure has the potential to create multiple simultaneous metric waves which must be somehow collated and internalized by the listener. The wave conception of meter has been adopted and developed by many researchers since Zuckerkandl, some of whom have also begun to explore how listeners group local rhythmic patterns into multiple high-level patterns of meter, and how specialist and non-specialist listeners rely in different ways on regular and irregular pulse configurations. Based on recent research into dynamic attending theory (DAT), Mari Riess Jones has hypothesized the existence of what she terms metric binding:

Entrainment is a biological process that realizes adaptive synchrony of internal attending oscillations with an external event. Different event timescales correspond to marked (i.e., accented) metric levels. Time spans within a metric level can elicit a corresponding neural oscillation, which has a persisting internal periodicity, manifest as a temporal expectancy. It ‘tunes into’ recurrent time spans at a given level by adjusting its phase in response to temporal expectancy violations at that level.

Building on four assumptions regarding neural oscillations which she claims are shared by several current DAT models (that they are self-sustaining, stable, adaptive, and that multiple related oscillations can be triggered by multiple time levels), she goes on to propose that:

Whenever two or more neural oscillations are simultaneously active, over time their internal entrainments lead to binding and formation of a metric cluster. A metric cluster comprises sets of co-occuring oscillations with interrelationships that persist due to acquired internal bindings. Entrainments among internal oscillations promote binding, which strengthens as a function of:

1. Duration of co-occurring oscillatory activity.
2. Phase coincidences, and
3. Resonance (i.e., relatedness) among oscillator periods.

For composers experienced with working in polyrhythmic space, this hypothesis may ring true on several levels, as our compositional process is often centrally concerned with the construction of just these types of metric clusters (to use Jones’s term); i.e., multiple musical (thus, neural) oscillations of pulse streams which may have varying degrees of duration, phase coincidence, and resonance, as outlined in Jones’s three points above. Speaking in general terms, we understand that whilst music based upon what is necessarily a more irregular network of attractive forces may create practical problems for performers (in terms of ensemble coordination – specifically, the coordination of different simultaneously-operative pulses and, possibly, meters) as well as listeners (some of whom may lack the listening experience or even the ability to effectively entrain with more complex rhythmic/metric structures), that it can nevertheless be a vehicle for rich, deeply affecting artistic experiences.

In conclusion, let us briefly return to the question of gestural rhythm and metric continuity/discontinuity as it relates to larger formal constructs. From the standpoint of practical composition, the widespread use of gestural rhythms and phrases can frequently create additional problems of formal development, the main one being: how do these more-or-less isolated rhythmic gestures help to move the music forward? I use the word ‘isolated’ here to describe a gesture that is composed to be in some way (rhythmically or harmonically) self-sufficient (i.e., it has no directly-connected rhythmic antecedent, and no immediately obvious consequent). Conceptually, it comes more or less from nothing (silence/stilness), returning to nothing. Such gestures may (while still exhibiting their properties of self-sufficiency) be composed within a sequence of similar gestures (which may or may not overlap), forming part a larger gestural whole. Magnus Lindberg highlights this very issue in his program note to Corrente II (1992):

After having written a Piano Concerto in 1991 preceded by three works for different orchestral effectives (Kinetics, Marea and Joy) I felt that I had come to an end with a certain musical expression and also compositional technique. All these works were based upon an extended chaconne principle with chord chains cycling around, undergoing constant transformation and being articulated in a very gestural way. The musical paradox and evidently also the challenge was the discrepancy between a brick-like method expressed in a world of gestures (with all difficulties involved in conceiving music out of phrases) aiming at a continuity in terms of progression and development.

What Lindberg means specifically by “aiming at a continuity in terms of progression and development” is not completely clear, but from the context it is logical to intuit that Lindberg was (at least in part) searching for a more through-going sense of temporal continuity (as opposed to music composed of gesture/phrase islands, i.e., the ‘brick-like method’ described above); one that could perhaps make more use of the periodicity of meter, with its in-built continuity of forward motion? It is also interesting that he mentions his so-called ‘chaconne principle’ of harmonic organization in this context as well, as it is an idea not unrelated to our discussion of pc set permutations in Section 1.1. As described by Ilkka Oramo, Lindberg’s ‘chaconne principle’ abstracts the idea of the historical chaconne by creating chains of pc sets with specific pitch/register mappings which are more or less fixed throughout the work. In practice, he typically composes through these reservoirs of pitches in the same order, creating strong harmonic relationships based on global repetitions, but with a huge amount of local rhythmic control and flexibility. As Oramo points out in his analysis of Lindberg’s Corrente II, the composer did not, in fact, abandon the chaconne principle for this piece as stated in the program note (indeed, Oramo’s analysis shows that the piece is actually built around the composer’s most rigorous working out of the chaconne principle up to that time), but rather has gone on to use it in the majority of his subsequent work.

To summarize the main points in our discussion of rhythm and meter in polyrhythmic space, we have seen that:

1. the potentially attractive motional forces operative between polyrhythmic elements can very naturally be made to function as both foreground rhythmic ‘signposts’, as well as the foci of middleground metric pulses and background structural supports,

2. the individual elements can be as strongly- or weakly-emphasized as the composer wishes, and can be operative at various levels of magnification within the temporal structure, providing as much fine or coarse rhythmic detail as the needs of the music dictate, and

3. from a contrapuntal standpoint, the composer has enormous freedom in determining the nature of the relationships between polyrhythmic layers (the spectrum between stratification and integration).

If working with a high-integer polyrhythm over the course of a longer section of music, she is able to exploit the (by definition) absolute stratification of the polyrhythmic elements to create music which is made up of perceptibly discrete layers. At the opposite end of the spectrum, she may also compose a more integrated music, weaving a unified structure through the layers, exploring the latent possibilities of microrhythmic coordination and development through the composite rhythm.

A final word: as we will see in my own compositional practice, it is critical to note that the polyrhythmic canvas as we have defined it thus far can be viewed as the temporal analogue to the permuted ‘harmonic fields’ which we discussed in Section 2.0, in that it creates complex patterns of durational interrelationships through a section of music; intricate networks of temporal tension and release which can then be used as raw material for composition. Again, it is the composer’s task to uncover the interesting ‘hidden’ potentialities latent in these parallel harmonic and temporal structures.

This material has been excerpted from Chapter 2 of my PhD portfolio commentaries at Durham University. A complete, fully referenced pdf copy of the entire thesis is available here.

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