Dualisms of Time

Abstract

What is the nature of time in music? This paper considers a number of dualisms in the way we conceive of and describe time, and how these dualisms function in the relationship between music analysis and music perception. Using the opening of the second movement from Webern's Symphony as a case study, McTaggart's tensed/un-tensed distinction will be considered, as will the static/dynamic conceptions of Newtonian/Einsteinian time. Various analytical methods will be placed relative to these dualisms.

Keywords:

• Time
• Analysis
• Perception
• McTaggart
• Einstein
• Webern

Music is a chronologic art, as painting is a spatial art. (Stravinsky, 1947)

Recently, I was teaching a session on serial technique and was asked by a student, ‘Can you hear these transformations?’ The question is deceptively simple, its answer rather complex. As a starting point, one might assert that the ability to perceive any musical pattern is a function of familiarity and complexity: the more familiar with a piece of music one is, and/or the more straightforward it is, the more able one is to recognize pattern (see Berlyne, 1960; Konečni, 1982). But if we conceive of familiarity as resulting from previous experience, and complexity as a measure in part of the amount of musical information within a given duration (let's say a bar), then both are contingent upon a rather more fundamental, though nebulous, variable: time.

The piece of music in question was the opening of the second movement from Webern's Symphony, Op. 21. As shown in Figure 1, this is a relatively simple example of serial technique: a row is presented linearly in the clarinet, whilst its retrograde appears simultaneously, split across three supporting instruments.

Figure 1 Webern's Symphony, Op. 21 (ii)—Opening Theme.

To ask ‘can you hear this transformation?’ is actually to ask whether it is possible to detect aurally—and then ascribe significance to—a pattern present in the score. But, in the case of this extract, various features of the music seem designed to throw the listener off the scent: the possible grouping of the surface into pairs of notes, the duplicated pitches, the splitting of the retrograde between parts, the use of horizontal (melodic) and vertical (harmonic) presentations. But there also exists a more fundamental hurdle: to hear the horns and harp as presenting a retrograde of the clarinet's row necessarily requires first a complete hearing of the clarinet, for how otherwise would the listener know, for instance, that the harp's initial B♮ is related to the last note in the clarinet? So, on a first listening, definitively no: it is not possible to hear the retrograde transformation. Detection requires multiple listenings, which is to say that memory—the temporal past—becomes crucial. And if one is sufficiently able to recall on subsequent listenings how the clarinet material ends, and to begin to make associations with the opening of the horn and harp material during the act of re-listening, then expectation—the temporal future—also comes into play, as predictions about how the horn and harp might end begin to form.

It takes time to hear music, and it takes interactions between the temporal domains of past, present and future to detect patterns in, and ascribe significance to, music. And yet, in the context of musicology as a whole, surprisingly little research has been undertaken on the nature of time in music, despite the vast array of musical dimensions with which it intersects (see Kramer, 1988). That fact perhaps mirrors the greater imprecision we have in our more general definitions of time, for there remains in the scientific and philosophical communities great debate over what time actually is. This article considers two dualisms in the way we conceive of and describe time—the tensed/un-tensed distinction, and the Newtonian/Einsteinian distinction—and considers ways in which different analytical approaches to music adopt different positions relative to those dualisms.

Tensed and Un-Tensed Time

Taking Varèse's famous description of music as ‘organized sound’ (Varèse & Chou, 1966, p. 18), whilst the entities the composer organizes are pulses of sound with characteristics of pitch, volume, duration, and so on, the background against which that organization takes place is time. Indeed the word ‘composition’ derives from the Latin compositio, meaning ‘putting together’, which is to say in the case of music the placing—or ordering—of sounds upon a blank temporal canvas.

As such, one might reasonably conceive of time in music as a coordinate system akin to a ruler, the units of which might be the tactus or the bar, the second or the minute. Relative to that coordinate system, the composer positions sonic events, and a score might be conceived of as a set of instructions as to when, relative to one another, performers must produce certain sorts of sounds. Such spatial metaphors are actually rather common in the way we process our understanding of time more generally: analogue clocks offer a visual-spatial representation of the structure and passing of time, as do timelines and calendars, and we use language such as a duration being long or short, of looking forward, or of the amount of time (see Gentner, 2001; Guck, 1981).

In any case, in this spatial, coordinate-based model, the present moment has no special significance: a score exists in its entirety irrespective of how far through a given performance the listener is. An analogy is to consider the score as functioning like a roll of film used for cinematic projection: the complete set of images on the film exists in its entirety despite the fact that only one image is brought into view at any given moment. And yet when watching that film, the viewer only ever has access to a single image at a time: he or she must imagine what has come before and what is yet to come. Likewise, the music listener continually navigates the temporal domains of past (memory), present (perception) and future (expectation), as musical information unfolds over time, and as mental representations of what is being heard are constructed.

A key philosophical paper in this area is John McTaggart's 1908 work, ‘The Unreality of Time’, in which the author defines two series of terms that we use (and confuse) to conceive of and describe time: the un-tensed series, and the tensed series (1908, pp. 457ff).¹ In the former, time coordinates such as ‘23rd June 1981' and ‘23rd June 1982' are used to pinpoint events. These are organized from earlier to later, but that organization does not rely on when the present moment lies: 23rd June 1982 is later than 23rd June 1981, irrespective of the date today. As such, we are able to conceive of time as an abstract set of coordinates with definite relative position, as in a timeline, a calendar, or, in the case of music, a series of bars. Conversely, in McTaggart's tensed series, time coordinates are conceived of relative to the present moment, through terms such as ‘two weeks ago’ or ‘tomorrow’. So, where one is a series from earlier to later, the other is a series from past to future.

Returning to the Webern extract, whilst it is possible to observe the (un-tensed) temporal coordinates of the row and its retrograde within the score, if it is possible to hear that relationship without a score, then memory—the past—must play a role: as described above, the listener must hear the full clarinet line before it is possible to hear the horns and harp material as being related. So whilst the set of coordinates present in a composition—and indeed the language used to describe musical patterns—often correlates with an un-tensed conceptualization of time, the act of listening to and drawing significance from music correlates more with a tensed view of time.

In fact, past experience plays a much more comprehensive role in music perception than this simple example expresses. As Robert Snyder has written:

One of the major discoveries of the last one hundred and fifty or so years about the human mind is that much of the activity of the brain and the mind is not available to awareness but remains implicit, evidenced particularly in the acquisition of skills. Many kinds of physical skills, such as knowing how to ride a bicycle or produce a clear tone on a musical instrument, require memory for their execution, but those memories are not available to consciousness, and cannot be described verbally. More recently, another form of implicit long-term memory has been proposed—implicit perceptual representations. This form of memory is involved in unconscious statistical learning that keeps a record of regularities in the environment, and structures unconscious expectations about environmental events. Schemas are describable in terms of this kind of implicit memory, being involved in, e.g., the unconscious generation of expectations about musical events as a piece unfolds. (2009, pp. 108ff)

The process of recognising pattern in music therefore requires several steps, all of which are intimately linked to a tensed version of time. First, the listener divides up perceived information into chunks: ‘a memory chunk is a group of 3–4 items related by association; a musical grouping consisting of 3–5 notes would be a chunk, and a phrase consisting of several of these groupings would be a higher-level chunk’ (Snyder 2009, p. 108). These chunks are then compared by the mind with previously learned schemas (from earlier in the piece, or from earlier in one's life) in order to bring about pattern recognition, and expectations for the likely content of future chunks. The ‘perception’ of musical patterns might therefore be better described as ‘apperception’, in that new experiences are interpreted with respect to mental representations of previous, related experiences.

In the case of the Webern extract, whether or not the retrograde relationship is aurally recognizable is therefore contingent on whether the listener has chunked the clarinet line as a self-contained unit, and grouped the combined horns and harp material into a second, self-contained unit. Only then might the mind recognize the relatedness of the two chunks. However, two problems present themselves. First, as Snyder notes:

Memorability appears related to how ‘chunkable’ a sequence is, which will depend on the amount of repetition of items, and to boundaries formed by discontinuities or changes in a sequence; the more clearly a sequence can be subdivided in units of chunkable size, the more likely it is to be recalled. (2009, p. 108)

To chunk the clarinet as separate from horns-plus-harp seems unlikely given that there exists a much more straightforward way of dividing the material. As shown in Figure 2, quite apart from all the local note pairs, the quaver silences in bars 5 and 7 offer easily perceptible points of subdivision, particularly given that, if divided in this way, the third chunk is rhythmically similar to the first.

Figure 2 Webern's Symphony, Op. 21 (ii)—Perceptual Chunks.

This chunking obeys a number of Lerdahl and Jackendoff's Grouping Preference Rules: GPRs 2, 3, 5 and 6 (see Lerdahl & Jackendoff, 1983, pp. 36–67). And whilst it is possible that the listener might add the dotted-crotchet of bar 5 to the first chunk, and begin the second chunk in bar 6, either way, bar 7 would likely form the start of the third chunk. Interestingly, the chunking shown in Figure 2 does suggest the possibility of hearing another pattern in the score: the mirror symmetry in rhythm, and in pitch at the interval of a tritone, about the centre of bar 6. As such, the central of the three chunks might most easily be heard as palindromic, and that recognition might serve as a schema for hearing the palindrome extend into chunks one and three.

In any case, to perceive this music as being formed of two layers (horizontal subdivision) when a much more straightforward chunking system of three subsections (vertical subdivision) exists seems unlikely. In order to test that hypothesis, a small-scale, informal experiment was run in which eight participants (all experienced musicians) were asked, after two listenings, to describe notable musical features in the extract. In their free text responses, seven participants grouped the music into the subsections shown in Figure 2, whilst only three described the layering of instruments; no one grouped the horns and harp into a single chunk.

Further to the issue of chunking, a second and more fundamental problem with hearing the full clarinet line as a single unit exists here. As Snyder states, ‘Because of the limitations of working memory, music can be immediately comprehended only on the time-scale of 5–8 seconds' (2009, p. 114). At the indicated pulse of  = c54, the 11 bars take just over 12 s, and indeed many performances take longer still: Boulez's version on the Complete Webern collection, for instance, takes a little over 16 s. In other words, in the presence of the possible chunking system shown in Figure 2, it seems even less likely that, without prompting specifically to do so, the listener would divide the musical material in the necessary way so as to be able, later, to recognize the retrograde relationship. It may be for reasons such as this—and indeed the absence of a familiar tonal syntax—that Robert Francès has concluded of transposition, inversion, and retrograde transformations that ‘serial unity lies more on the conceptual than on the perceptual level', and that ‘when thwarted by melodic motion, rhythm, and the harmonic grouping of tones, it remains very difficult to hear’ (1988/1958, pp. 126ff). More specifically regarding the Webern extract, no one in the aforementioned experiment described in their free text comments any recognition of the retrograde relationship; likewise Nicholas Cook has described in his own test of the first movement that, again, no listeners reported the serial structure (1987, p. 296).

There may be a difference, then, between the units created in the act of composition—bars, note rows, etc—and the units created in the act of listening—perceptual chunks. This is not a problem per se, but becomes one when analysts of music aim to comment on perceptual experience (unity, direction, closure, energy etc.), but do so using the units of composition.² Importantly in the present context, that disparity is related at a deeper level to a dualistic view of time, or, more specifically still, in the mixed applications of tensed and un-tensed conceptualizations of time: a coordinate-based analysis does not take account of how the mind forms mental representations through interactions between memory, perception and expectation.

Newtonian and Einsteinian Time

Music analysis can objectify the score or subjectify the listening experience. The tensed/un-tensed dualism, in the case of music at least, is primarily a case of subject/object focus: time itself does not behave differently between these models. However, other dualisms in the conceptualization of time do indeed comment on the behaviour of time, rather than on its mere description.

One important example is the extent to which there is consistency in the rate at which time is perceived to pass. On the one hand, we measure time in periodic intervals of uniform duration: seconds, days, etc. Indeed, many aspects of our lives are acutely attuned to these periodicities: we wake up, become hungry, and later tired, at similar times of day, sometimes so attuned to the rhythms of time that we awaken moments before the alarm clock rings. However, on the other hand, it seems that the human mind sometimes experiences duration as a function of information content: how many of us have experienced the perceived dragging of time during a slow day at work, for instance? Indeed, time can sometimes feel as if it has come to a near standstill when one is waiting, travelling, bored, and so on.

In one model, then, time passes at uniform speed. In the other, the rate at which time is perceived to pass is determined by the content within a given duration. As such, the variation comes not in time itself, but in the mind's measurement of a duration. However, that distinction echoes a deeper reality in the nature of time. The branch of physics known as classical mechanics, and, in particular, the work of Isaac Newton, has as a premise the concept of absolute time: it is possible to measure an interval of time absolutely, irrespective of any movement by the observer. However, in one of the most significant paradigm shifts in modern physics, Albert Einstein proposed in the special theory of relativity that, when moving at particularly high speeds, the rate at which time passes slows down. And according to the general theory of relativity, time and space are interwoven into the broader continuum of spacetime, meaning that space and time are actually affected by what happens in the universe (e.g. spacetime is warped in the presence of a large planet). These theories have been proven experimentally beyond question (see Dainton, 2010, pp. 194–212, 313–27 and 343–67; Lockwood, 2005, pp. 23–89).

As such, time is not the stable and regular force we imagine, and the Newtonian/Einsteinian distinction stands as a useful metaphor to the perception of time in music. As Thomas Clifton has written, there is often a difference between ‘the time a piece takes' as opposed to ‘the time a piece presents or evokes' (1983, p. 81). Studies have demonstrated a wide range of factors that affect the perception of duration, from the amount and type of information/activity contained in a timespan (and, consequently, the time taken by the brain to code that data), to familiarity with the type of information/activity under consideration, to the state of mind of the listener (see Clarke & Krumhansl, 1990, p. 200; Kramer, 1988, pp. 322–374). More recent music-specific research has highlighted the additional effects of the degree of ‘dynamism’ or ‘activity’ in the music, the pattern of tonal modulation, the clarity and regularity of the metre, and the order in which materials are presented (Bailey & Areni, 2006; Firmino, Bueno, & Bigand, 2009; Marvin & Brinkman, 1999).

Such factors were intuited by Stockhausen, sometime before the emergence of empirical evidence, in his article ‘Structure and Experiential Time’. Stockhausen observes that,

 … Experiential time is in a state of flux, constantly and unexpectedly altering. An apparent paradox is immediately explained: the greater the temporal density of unexpected alterations—the information content—the more time we need to grasp events, and the less time we have for reflection, the quicker time passes; the lower the effective density of alteration (not reduced by recollection or the fact that the alterations coincide with our expectations), the less time the senses need to react, so that greater intervals of experiential time lie between the processes, and the slower time passes. (1958, p. 64)

Stockhausen goes on to analyse the opening of the second movement from Webern's String Quartet, Op. 28, through this conceptual lens. In particular, he focuses on the idea of ‘alteration'—the way in which one sound transforms into another—concluding that the flow of time is a function of the way in which the composer controls change within and between various dimensions of the music, in order to manipulate expectation and surprise within the listener.³

However, that Einsteinian conception of time is rare in music analysis: the object of analysis is more often the score, rather than the listener's perception of that score, and the units used to describe patterns in music are most often the (Newtonianly stable) bar, tactus or second. As per the tensed/un-tensed distinction above, that in itself is not a problem, except when attempts are made to describe the perceptual experience using a Newtonian analytical framework that does not take account of Einsteinian warping. To exemplify the potential problem, Figure 3 reproduces the opening material of the second movement from Webern's Symphony, and continues by showing the first variation.

Figure 3 Webern Symphony Op. 21 (ii)—Theme and First Variation.

As can be seen, there is, according to the score, an important temporal equivalence between the theme and its variation: both are set over the same number of bars (11, albeit with anacrusis and overspill in the variation), and as such the sections might reasonably be described as being ‘the same length’, or, further still, ‘temporally balanced’. Such an observation would also have justification from a perceptual point of view: as Snyder has written:

Current theories of pulse suggest that it arises from patterns of oscillation set up in the brain that entrain to the pulse of a piece of music, forming a dynamic memory within which unfolding temporal events are organized, and yielding a framework that guides immediate temporal expectations about when events are experienced as most likely to occur. (2009, pp. 110ff)

The maintenance of the minim-level tactus and 2/4 time signature across the two sections in Figure 3 potentially brings about a perceptual connection.

However, as above, Webern again throws the listener off the scent. First, the rhythmic surface of the variation consists largely of two- and three-note groups, and these groups often phase across bar-lines, breaking down the sense of regular pulse and thereby making the underlying 11-bar structure somewhat hard to perceive. Second, the tempo changes from  = 54 to 66: a 22% increase in the rate at which each tactus beat is superseded by the next. As such, where, according to the score, the theme lasts for just over 12 s, the variation, excluding the anacrusis and overspill, lasts for 10 s: a 16% decrease in the overall duration. Third, there is a significant increase in the complexity of content in the variation. This factor is rather more difficult to model, but if complexity is taken simply to mean the number of attack points within a given duration, then whilst the theme contains 38 individual note attacks, the first variation contains 109: a 187% increase. Indeed the palindromic structure set up in the theme is expanded in the variation such that the row and its transformations are now presented twice, the second time in full retrograde. Given such complicating factors, is it reasonable to expect score-less listeners to be able to hear the 11-bar equivalence without prompting?

In order to test this question, a small-scale, informal experiment was again carried out with eight listeners, all experienced musicians. Participants were asked to listen to the extract in Figure 4 without a score, and were told they would be asked a question on what they had heard.⁴ Additionally, they were told that the extract consisted of two sections, and that the entry of the violin marked the rough point of division between those parts. After a single listening, participants were asked to describe as a proportion the relative duration of the sections. Answers varied widely, from 35:65 at one extreme, to 70:30 at the other. No participants heard the sections as 50:50.

Figure 4  Analytical Intersections.

Whilst such an experiment in no way proves that the tempo and/or complexity changes are themselves warping the perception of time—as described above, a multitude of factors have been shown to affect the perception of duration—what is clear is that the bar-based equivalence of the score was far from straightforwardly communicated to these listeners: the two parts were not perceived to be ‘the same length’. Whether the direct result of content changes, or the effect of other variables, time warping is clearly taking place in the minds of listeners to a significant degree. So again we see how a dualism in the way we conceive of time plays a role in the way we conceive of music: the Newtonian model provides a useful means of analysing the score; an Einsteinian model may prove more useful when analysing the listening experience.⁵

Dualisms of Time

In her article, ‘The metaphor of musical motion: Is there an alternative?’, Judith Lochhead undertakes a detailed comparison of music analyses by Allen Forte, Christopher Hasty and David Lewin, noting a predominance of either tensed or un-tensed language in their work (1990, pp. 83–103). Likewise, there are many analyses of the Webern Symphony that demonstrate greater propensities within the tensed/un-tensed dualism and also the Newtonian/Einsteinian dualism. For instance, brief comparison of analyses by Nelson (1969) and Rahn (1987) reveals a significant difference: Nelson begins by asking the reader to observe intervallic and temporal coordinates in his transcription of the row; Rahn begins by asking the reader to listen multiple times to the opening, and to build up a musical memory of its intervallic content.

However, it is possible to place not just individual analyses, but broader analytical methods, relative to the dualisms discussed above. A simple example is the distinction between the type of proportional analysis espoused by theorists such as Ernő Lendvai, and the phenomenological analysis undertaken amongst others by Lawrence Ferrara (see Lendvai, 2000/1971; Ferrara, 1984). In the former method, focus is placed on the score, such that durations are calculated largely based on a Newtonian conception of time (primarily by means of bar-or tactus-counts); durational proportions are identified based on un-tensed relationships rather than on individual perceptions of those relationships. In the latter, the focus is placed on the listener, with particular significance given to content-time (space-time) connections, and on the way in which perceptions of those connections unfold over repeated listenings as mental representations are refined. Figure 4 places these analytical methods relative to an intersection of the two dualisms discussed above, viewing them as parts of a broader whole in the way we conceive of and describe time.

Differences between Schenkerian theory and Generative theory offer a more subtle example (see Schenker, 1969/1933; Lerdahl & Jackendoff, 1983), and one that breaks the parallelism described thus far between un-tensed and Newtonian time, and between tensed and Einsteinian time. Both analytical theories place emphasis on the listening experience, but the rule systems on which significance is attributed differs greatly between the two (though sometimes they offer similar findings). As such, Generative theory might be placed further along the tensed axis of Figure 4 than Schenkerian theory, inasmuch as its well-formedness and preference rules are more explicitly based on the way in which listeners chunk the perceptual surface during the act of listening. However, another difference lies in the analytical graphs themselves: where those of Generative theory tend to be laid out in a fairly grid-like, Newtonian pattern (with equal horizontal distance between tactus pulses, for instance), Schenkerian graphs often freely warp the horizontal, temporal axis, giving greater space (and time!) to moments of greater structural significance.

Such examples demonstrate the way in which conceptualizations of music are based on—or at least betray, at a deeper level—differing understandings of time. In fact, the tensed/un-tensed and Newtonian/Einsteinian dualisms discussed thus far represent two of many such disparities in the way we conceive more generally of time. As shown in Figure 5, there are various additional dualisms that have immediate relevance to music.

Figure 5 Dualisms of Time.

Taking the discrete/continuous dualism as an example, as Eric Clarke and Carol Krumhansl have noted,

The notion of ‘item’ in music is somewhat difficult: music consists of a continuous flow of information in which the idea of individual, discrete items must be treated with great care. It is too easy for musical notation, which indicates clearly distinguishable discrete events (notes), to be uncritically assumed to be directly equivalent to the way the music sounds. (1990, p. 219)

Indeed analytical methods such as Schenkerian theory, through a focus on voice leading and the consequent connections between notes, have as a crucial underlying principle the continuous nature of time. Others, such as Set theory, tend to focus instead on the transformation of discrete musical units (sets) in order then to identify broader organizational patterns (see Forte, 1973). And taking one further dualism as an example, metre in music involves a careful balance between linearity and cyclicity, inasmuch as a succession of pulses are grouped into periodically recurring patterns of weak and strong; likewise formal structures that involve repetition entail a balance between linear change and periodic recurrence.

As Stravinsky wrote in his Poetics of Music, ‘music is a chronologic art, as painting is a spatial art’ (1947, p. 29). But the nature of time is much less straightforward to conceptualize than the nature of space, with its three dimensions that are, by definition, easy to visualize. As shown in Figure 5, there are potentially many subtle dualisms in the way we conceive of time. Of course, in reality, we are constantly and unconsciously moving around within these dualisms, synthesizing different positions within and between multiple axes to suit our needs in any given situation. In music too, many different combinations are possible: Set theory, for instance, is largely un-tensed, Newtonian, discrete, directional, cyclic and relational. As such, the two-part interaction shown in Figure 4 can be extended to include the multiple dimensions of Figure 5, with each dualism on its own axis. But whilst no one coordinate-combination is per se the most appropriate when considering music, each offers subtly different perspectives: the rule systems used by the conscious mind to analyse music may well differ significantly from the rule systems used by the unconscious mind to listen to music, and this difference reflects, at a deeper level, the varying ways we conceive of and describe time.

Notes

1 McTaggart uses the terms Series A for tensed terminology, and Series B for un-tensed terminology. He goes on to claim that because the meaning of the un-tensed is reliant on the relationships of the tensed, time could not be said to exist without the tensed. However, the tensed view, he claims, is metaphysically unsound because it is based upon an infinite regression. His argument for the consequent ‘unreality of time’ has been refuted on various grounds (see Danton, 2010, pp. 13–43, for the full argument and subsequent refutations), though need not concern the present paper, which relies simply on the two-part distinction.

2 Analytical methods such as Lerdahl and Jackendoff's Generative Theory of Tonal Music (1983), and Adam Ockelford's Zygonic Theory (2013), are notable exceptions, in their placing of perceptual chunks at the heart of their approaches.

3 See Clarke and Krumhansl (1990, pp. 221–222), for a critique of Stockhausen's method.

4 The Boulez Complete Webern was used, for despite the fact that Boulez adopts slightly slower tempos than those indicated in the score, his realization preserves the 16% reduction in duration in the variation.

5 In fact, whilst music scores might be conceived of as being the archetype manifestation of Newtonian time, the physical amount of space taken on the page by each bar is a function of the amount of material contained therein: an interesting Einsteinian warping of the musical timeline.