XuThe Musical Arrow of Time - The Role of Temporal Asymmetry in Music and Its Organicist Implications
- Prelude
- A discursive quest
- Interlude
- Musical consequence
- Postlude
- Appendices
The Musical Arrow of Time
The Role of Temporal Asymmetry in Music and Its Organicist
Implications
Qi Xu
Submitted in partial fulfillment of the requirements for the Doctor of
Musical Arts degree, The Juilliard School
May 2022
2022 Qi Xu
ALL RIGHTS RESERVED
Revision: 9b437739
Abstract
Adopting a performer-centric perspective, we frequently encounter two statements: “music flows”, and “music is life-like”. In this dissertation, our discussion builds on top of the two statements above, resulting in an exploration of the role of temporal asymmetry in music (generalizing “music flows”) and its relation to the idea of organicism (generalizing “music is life-like”). In particular, by the term “temporal asymmetry”, we focus on its two aspects. The first aspect concerns the vastly different epistemic mechanisms with which we obtain knowledge of the past and the future. Following a discussion on epistemology, we focus on a particular musical consequence: recurrence. The epistemic difference between the past and the future shapes our experience and interpretation of recurring events in music. The second aspect concerns the arrow of time: the unambiguous ordering imposed on temporal events gives rise to the a priori pointedness of time, rendering time asymmetrical and irreversible. A discussion on thermodynamics informs us musically: the arrow of time effectuates itself in musical forms by delaying the placement of the climax.
Organicism is invited to the discussion of temporal asymmetry. Organicism functions as a mediating topic as it engages with the concept of life as in organisms. Therefore, on the one hand, organicism is related to temporal asymmetry in science via a thermodynamical interpretation of life as entropy-reducing entities. On the other hand, organicism is a topic native to music via the universally acknowledged artistic idea that music should be interpreted as a vital force possessing volitional power. With organicism as a mediator, we may better understand the role of temporal asymmetry in music. In particular, we view musical form as a process of expansion and elaboration analogous to organic growth. Finally, we present an organicist interpretation of delaying the climax: viewing musical form as the result of organic growth, the arrow of time translates to a preference for prepending structure over appending structure.
Autobiographical Note
Qi Xu, from Shenzhen, China, is currently a DMA candidate at the Juilliard school studying piano performance. During his undergraduate years at Columbia University, Qi explored other disciplines with the intention of facilitating his musical understanding. In particular, Qi was selected in 2014 as a Columbia-Oxbridge Scholar to study abroad for one year at Cambridge University where he read mathematics. With exposure to other disciplines, Qi specializes in approaching piano performance with reason and sentiment combined. As such, during his recital tour in Germany 2015 for example, he was hailed by the German press as “a storyteller and wild riders” (Rheinische Post) who stands in “between drama and charm” (Derwesten).
To my parents
Acknowledgements
I would like to express my gratitude to many people, whose help has made this dissertation as it stands now possible in the first place.
First and foremost, I would like to thank my advisor Dr. Lasser for his guidance throughout the dissertation preparation process. His advice, feedback, and encouragement have been inspiring me since the brainstorming stage of the dissertation.
I would also like to thank my reviewers Dr. Dawe and Dr. Reynolds for their time and effort invested in reviewing the dissertation. Without their thorough and constructive feedback for revision, the dissertation would not be as polished as it is today.
Moreover, I would like to thank the entire DMA committee chaired by Prof. Gottlieb for giving me the opportunity of being part of this lovely community and program. Through classes, seminars, performances, and conversations, I learned and grew not only as a musician and a scholar, but also as an artist, a citizen, and ultimately a well-rounded human being.
I would like to thank Dr. Kaplinsky for her mentoring since 2009. Ever since studying with Veda, she has been more than a piano professor to me. She is my life mentor and a source of inspiration. Through her teachings in piano and life, the years of studying with her have influenced and shaped my artistic vision and worldview.
I would also like to thank Dr. Raekallio for his piano instruction during my DMA years. The conversations with him have strengthened my appreciation of the beauty of piano playing resulting purely from the machinery of piano technique.
Finally, I want to thank my family and friends for their continuous support throughout my life. Because of them, I am constantly humbled, motivated, and uplifted to devote myself to building a better world: a world, as indicated later in this dissertation, where darkness, destruction, and chaos normally prevail as the arrow of time favors thermodynamic equilibrium.
Prelude
Peculiar properties of time
The universe is metamorphosed into a story about running brooks, poetry, and music
Time is a topic that fascinates many for its ability to inspire awe and curiosity. Due to its omnipresence, time is discussed from various perspectives, under different contexts, and across many disciplines. On the one hand, we observe and conceive time in everyday life for practical purposes. Musicians improve their crafts by carefully engineering time-related musical structures, thereby manipulating audience’s anticipation and expectation in order to achieve desirable performance effects. For example, performers take advantage of time by using devices such as rubato to create the so-called “magical moments”, a term listeners frequently use to describe expressive performances. On the other hand, philosophers, scientists, and theologians whose areas of study are less relevant to everyday life, inquire into the issue of time as one of the fundamental questions of our universe. Augustine, in his “Confessions”, makes a remark concerning the puzzling and paradoxical quality of time:
What is time? [...] We surely know what we mean when we speak of it. We also know what is meant when we hear someone else talking about it. What then is time? provided that no one asks me, I know. If I want to explain it to an inquirer, I do not know. (Augustine 2009, 230)
His remark suggests that time seems to possess the magical property that, upon conscious inspection, resists rational understanding and explanation. Similar phenomenon can be found in the field of art. For example, in music, listeners are astounded by outstanding performances. However, upon reflection, listeners are likely unable to explain, in technical terms, how exactly the performance they experienced is expressive. Metaphorically, musicians are like magicians who inspire awe from the audience, yet must deliberately hide the method from which the awe is derived. We attribute the term “genius” to performers who have the ability to present successful performances whose inner workings remain inexplicable to the general public. In his “Critique of Judgment”, Kant even goes as far as to claim that the inner working remains ultimately unknown to the author himself/herself:
[If] an author owes a product to his[/her] genius, he himself[/she herself] does not know how he[/she] came by the ideas for it; nor is it in his[/her] power to devise such products at his[/her] pleasure, or by following a plan, and to communicate [his/her procedure] to others in precepts that would enable them to bring about like products.(Kant 1987, 175)
It is through the resistance to rational understanding that genius is valued for its uniqueness, originality, and singularity. Similarly, time is puzzling and fascinating, as it easily can be felt, but hardly understood.
After recognizing the peculiarity of time, we may indulge our curiosity further by asking: what exactly are the outstanding properties of time? What specifically makes time a fascinating topic worth discussing?
Explanatory power of time
By answering the questions above, we become capable of acquiring a better understanding of musical topics, as music necessarily takes place in time. The explanatory power of time is essential to understanding music. Metaphorically, as suggested by Schlesinger in his book “Aspects of Time”, time can be interpreted as a container, such that “every event occur[s] at some point in time”.(Schlesinger 1980, 3) We can further extend the metaphor of the container by saying that a better understanding of the container (i.e. time) pertaining to its characteristics will also benefit our understanding of its contained object (i.e. music). Using the dichotomy of form and content, we may claim that our perception of the content is shaped by the form through whose medium the content is presented. Moreover, even for the sake of a musical discourse, some conclusions about music are possible to be derived only if we look at a bigger picture: time and its general properties. For example, without having the idea of temporal asymmetry in mind, we might not recognize that, as will be discussed in later sections, some microscopic musical structures such as phrase model (see section 2.2.3.3) and macroscopic structures such as musical form (see section 2.2.3.2) bear close resemblance. Once we use time as an overarching topic, a “point of intersection”, we are then empowered to explain and organize various issues of interest that are commonly encountered in music. Time as a topic, has the organizational power to group commonality in music that is difficult to notice otherwise. In particular, structures occurring at different organizational levels can be explained by a common root cause: attributes of temporal processes at different scales that permeate musical phenomena.
The organizational power found in the topic of time is akin to that of music history. If we study each piece in isolation without considering the overall historical trend and stylistic characteristics as a organizing mechanism, then the repertoire of musical works would look disparate in the sense that all pieces are unrelated. One of the practical consequences is that, for performers, preparing a piece for performance becomes less effective. In such a case, one has to treat the piece as completely new, without referring to knowledge, from prior experience, of pieces similar in style (e.g. those from the same composer or from the same time period).
Time as a universal theme across topics
Thus, to embark on a journey inquiring the nature of time and its relations to music, amounts to crosscutting the universe (or the topoi) of musical discourse. The term “crosscutting” here is borrowed from the field of computer science, as appeared in the term “cross-cutting concerns”. For a software application, we can often find ways to decompose it into different logical units bearing various possible names: functions, components, modules, features or concerns. For example, a typical music streaming application consists of several components. It should have a component that provides searching functionality, so that users can search for specific songs using a set of keywords. Meanwhile, for users without a specific search target in mind, who simply would like to explore new songs, the application should have a component that displays a collection of recommended songs to the users. Finally, the application should provide a component that plays the selected song, i.e. a music player component. The above examples of components are, using terminology from computer science again, encapsulations of the application’s functionality. Metaphorically, we can think of the components as being encapsulated into separate “capsules”, with each capsule attaining a clearly-defined boundary. Musically, the metaphor of capsules can be illustrated using the example of sonata form: a piece of music exhibiting the formal structure of sonata form is divided into components named exposition, development, and recapitulation. Each component in sonata form can be viewed as a capsule with clearly-defined boundary. For example, the boundary between the exposition and development is often notated visually by a repeat sign. The advantage of encapsulation is that the architecture of the application’s design is well structured. Practically, the team of developers can adopt a strategy called the division of labor to assign tasks concerning well-defined components to specific team members.
With each component clearly defined and encapsulated, however, there are some “aspects” that span across multiple components. Continuing the musical metaphor using sonata form above, certain topics are found across exposition, development, and recapitulation. For example, the topic of tension-release is an aspect that spans across exposition, development, and recapitulation. For the example of the music streaming application above, one aspect that is found in all three components is that the user must be logged-in in order to use the streaming service provided by the application. Therefore, user authentication in this case, is a cross-cutting concern of the application, because it metaphorically “crosscuts” three components. In computer science, there is a specific design paradigm named “aspect oriented programming” (abbreviated as “AOP”) that primarily deals with cross-cutting concerns:
AOP is often defined as a technique that promotes separation of concerns in a software system. Systems are composed of several components, each responsible for a specific piece of functionality. But often these components also carry additional responsibilities beyond their core functionality. System services such as logging, transaction management, and security often find their way into components whose core responsibilities is something else. These system services are commonly referred to as cross-cutting concerns because they tend to cut across multiple components in a system.(Walls 2011, 10)
The concept of cross-cutting concerns can be applied to music via analogy. In addition to the specific metaphor using sonata form illustrated above, we approach music from a multitude of perspectives depending on topics of interest. As such, the field of music is partitioned into various subdisciplines. Given an arbitrary piece of music, if the interest is on the socio-historical context under which the piece is composed, then the approach takes the form of music history. If the interest is on the musical aspect of the piece, treating it as an ahistorical and autonomous object of art, then we can pick applicable analytical devices to approach the piece as we see fit. Among examples of analytical devices are harmony, counterpoint, formal analysis. As a practical consequence, these partition schemes give rise practically to standard courses commonly found in music conservatories: music theory (which can be further divided into harmony and counterpoint), music history, performance practice, etc. Within the broad categories above, there are specific theories providing unique insight and interpretations to the piece. To name a few, we encounter theories such as species counterpoint, sonata theory, Schenkerian analysis, music set theory, serialism, neo-Riemannian theory, etc. These theories may have their unique merits, weakness, and areas of focus. For example, Schenkerian analysis is effective in treating tonal music (or more specifically, a certain subset of tonal music), and its explanatory power becomes questionable once we take into account non-tonal music. However, regardless of the specific contents proposed by various theories, they necessarily share a common feature that, by definition, they are studies of music, which invariably involves arrangement of events in time. Now, we can extend the analogy of cross-cutting concerns as follows: different subdisciplines and theories about music act as encapsulated components designated to explain musical phenomena. Meanwhile, the issue of time serves as a cross-cutting concern, because it tends to “cut across multiple components” of musical research.
Unique feature of time: directionality
Among properties of time, the most distinguishable one concerns directionality of time. We seem to have a priori knowledge of the directionality of time that is independent from empirical observations. In particular, the directionality of time entails that time necessarily flows from the past to the future. A piece of music can then be figuratively described as a process of unfolding, whose direction points from the past (that is already unfolded and visible), through the present, into the future (that is still folded and invisible).
Space and time: a comparison
The directionality of time becomes even more evident once we contrast it with that of space. In space, there is no absolute direction, in the sense that directions can be named arbitrarily: there is no intrinsic difference between left and right, at least for the macroscopic world we experience daily. It is up to our conventions that we define directions the way they are. For example, some countries adopt left-hand traffic while others adopt right-hand traffic. Meanwhile, both conventions are equally justified. In other words, spatial directions amount to arbitrary choice, rather than necessity. In mathematics, in order to determine the direction of cross product, the orientation of the vector space must be determined in advance. However, the choice of orientation is arbitrary. It is due to convention, not necessity, that we often pick the orientation such that the cross product abide by the right-hand rule, as opposed to the equally valid left-hand rule. Furthermore, we can consider a thought experiment as follows. A possible universe in which right-handed population is majority is not substantially different from the one in which left-handed population is majority. The arbitrariness of choice precisely refers to the fact that outcomes resulting from different choices are indistinguishable from one another.
Physicist Feynman, in his publication “The Feynman Lectures on Physics”, delivers a vivid parable illustrating the problem of distinguishing spatial orientations:
[Imagine] that we were talking to a Martian, or someone very far away, by telephone. We are not allowed to send him any actual samples to inspect; for instance, if we could send light, we could send him right-hand circularly polarized light and say, “That is right-hand light—just watch the way it is going.” But we cannot give him anything, we can only talk to him. He is far away, or in some strange location, and he cannot see anything we can see. For instance, we cannot say, “Look at Ursa major; now see how those stars are arranged. What we mean by ‘right’ is …” We are only allowed to telephone him.
Now we want to tell him all about us. Of course, first we start defining numbers, and say, “Tick, tick, two, tick, tick, tick, three, …,” so that gradually he can understand a couple of words, and so on. After a while we may become very familiar with this fellow, and he says, “What do you guys look like?” We start to describe ourselves, and say, “Well, we are six feet tall.” He says, “Wait a minute, what is six feet?” Is it possible to tell him what six feet is? Certainly! We say, “You know about the diameter of hydrogen atoms—we are 17,000,000,000 hydrogen atoms high!” That is possible because physical laws are not invariant under change of scale, and therefore we can define an absolute length. […] we start to describe the various organs on the inside, and we come to the heart, and we carefully describe the shape of it, and say, “Now put the heart on the left side.” He says, “Duhhh—the left side?” Now our problem is to describe to him which side the heart goes on without his ever seeing anything that we see, and without our ever sending any sample to him of what we mean by “right”—no standard right-handed object. Can we do it? (Feynman, Leighton, and Sands 2011)
The task involves teaching the concept of left and right in a purely verbal manner, without referring to any potentially shared experience. The solution, according to Feynman, is a convoluted yes, involving substantial use of atomic physics:
In short, we can tell a Martian where to put the heart: we say, “Listen, build yourself a magnet, and put the coils in, and put the current on, and then take some cobalt and lower the temperature. Arrange the experiment so the electrons go from the foot to the head, then the direction in which the current goes through the coils is the direction that goes in on what we call the right and comes out on the left.” So it is possible to define right and left, now, by doing an experiment of this kind.(Feynman, Leighton, and Sands 2011)
Later on, the author proceeds to the notion of antimatter which further complicates the issue, as it makes the choice of handedness arbitrary again. Without going into further details that are beyond the scope of this paper, the sheer difficulty of answering the seemingly trivial question of distinguishing left from right demonstrates how absolute direction in space can be a tricky issue to tackle.
In contrast to that of space, orientation of time seems to be straightforward. We consider again the thought experiment of teaching the Martians. However, this time we would like to teach them about distinguishing the past from the future. Then the task is simple. By definition, every process takes place in time. Therefore, we may take any process, and mark its beginning and ending. Then the relationship between the marked beginning and the ending corresponds exactly to that between what we call the past and the future. The remaining task is about finding a strategy to mark the beginning and ending, as well as distinguishing between them. The instruction for the curious Martians is straightforward. If they are enthusiastic about music, then we may teach them the following: have a notebook ready, and start playing a piece of music. Then write down notes upon detecting change of states. Without loss of generality, we may assume that there are only two possibilities for change of states: from silence to sounding music, and from sounding music to silence. The rules of writing notes are the following: 1) if a change of state is encountered and the notebook is empty, write “beginning”, and 2) if a change of state is encountered and the notebook already has the word “beginning” on it, then write “ending”. The two rules uniquely determine the way through which the beginning and ending are marked. From the example above, we can easily notice the crux of the issue. Orientation of time is straightforward because we have ways to remember the past (by the act of note-taking in the example), whereas the idea of remembering the future leads to absurdity. In other words, we are able to potentially recall the past with the help of objects named “records”. On the other hand, the only way we can inquire into the future is through prediction. Acts of recalling and predicting involve procedures and mechanisms that are fundamentally different. Knowing the difference between the two, as we will discuss in section 2.1.2, is crucial for a better understanding of music, since music perception heavily depends on recall (of heard events) and anticipation (of upcoming events).
Meanwhile, orientation of time is non-arbitrary. More specifically, the choice of the direction in which time flows is not a play of words or convention in the following sense. Let us consider again the above thought experiment of imagining a possible world in which left-handed population is majority. This time, instead of substituting right-handed population for left-handed one, we choose to imagine a possible world in which time flows backwards, in the sense that cause follows effect, instead of preceding it. Consequently, such a possible world would be saliently different to our current one. In fact, it might be incomprehensible to us at all. How could we possibly understand a world in which effect precedes cause? For instance, how could we enjoy a concert whose finale comes before the opening movement? The reason for such incomprehensibility is partly that the temporal order of events is embedded in the definition of causality. Therefore, cause, by definition, comes before effect. Analogously, the opening movement, by definition, comes before the finale. To claim otherwise amounts to stating a logically contradictory statement: the finale which by definition follows the opening movement is at the same time preceding the opening movement, constituting a logical conjunction of mutually exclusive propositions.
Artistic implications
The difficulty of imagining the backward flow of time shows how the directionality of time is so ingrained and hard-coded in our mindset, that if the directionality is modified, then time is rendered incomprehensible and perplexing. In fact, it is precisely due to our ingrained intuition of the flow of time, that artists often employ strategies to counter such intuition in order to achieve surprising dramatic effects. For example, in his magnum opus “Faust”, Goethe describes an unrealistically strange scene:
Show me the fruit that rots before it’s plucked and trees that grow their greenery anew each day! (Goethe 1985, 1:131)
Spoken by Faust during his confrontation with Mephistopheles, the quoted passage addresses Faust’s wish to experience the impossible and the transcendental. It is under such circumstance that the wager between Faust and Mephistopheles unfolds, becoming the direct cause for plots that follow. It is worth noting how the author conveys the idea of experiencing the impossible and the transcendental: through portraying time poetically, achieving the poetic effect by means of distorting our common sense about time. Our common sense indicates that fruits rot after they have grown mature, and trees turn green annually. In the quoted passage, however, the sequence of their life cycle is substantially altered by means of temporal disorientation and acceleration, respectively. While tampering with our common temporal perception may render time incomprehensible as discussed previously, which is undesirable for many practical purposes, it may be desirable for the sake of artistic effects. Besides the literary example of Faust illustrated above, manipulating temporal perception (in particular, temporal disorientation) is in general a powerful expressive tool across many artistic disciplines. Consider additionally the following examples: a broken glass is restored spontaneously from the floor onto the table; a dead person rises from the tomb. As discussed previously for similar examples, we are perplexed upon encountering such examples because they are incomprehensible. However, if we now try to adopt an artistic perspective, we may experience the emotion of wonderment, making these examples awe-inspiring. In fact, the emotion of awe in this case has the same origin as the previously mentioned quality of incomprehensibility. It is exactly due to our lack of ability to comprehend temporal disorientation, that we feel awe as an emotional consequence. Such an awe-inspiring device is commonly found in the arts as well as religions. For example, rising from the dead (i.e. resurrection) is one of the cornerstones of Christianity, directly resulting in Easter being the most important day in the liturgical calendar. The same concept finds its importance in music as we take into consideration religious and religion-inspired repertoire. In section 1.1, we will examine Bach’s St Matthew passion and the Christmas oratorio as a musical case study of the concept of resurrection, which is itself a powerful musical tool that takes full advantage of temporal disorientation.
Further decomposition
We need to take a step back, and realize that when we state that “time flows from the past to the future”, the statement can be further decomposed into two substatements. The first states that, when viewed from the present, the future looks vastly different from the past. The sheer distinguishability between the past and the future gives rise to the notion of temporal asymmetry. Metaphorically, we may imagine an old style weighing balance, with the past and the future respectively placed at both ends. Then temporal asymmetry amounts to assigning the past and the future different weights. The second substatement further adds a condition that assigns a specific arrow of time relating temporal events in a particular order, that time flows definitively from the past to the future, never the other way around. Extending the weighing balance metaphor, the second substatement specifies that the future weighs more than the past, such that if we were to place a drop of water on top of the balance, it would naturally flow toward the future end of the balance.
In the following section, we will use the two substatements previously mentioned as a point of departure. The goal is to discuss a few general remarks relating to the notion of temporal asymmetry from a philosophical and scientific perspective, thereby providing necessary prerequisites and insights for a musical discourse that follows.
A discursive quest
Necessity of extramusical discussion
Before discussing and exploring how temporal asymmetry is at work in music, it is deemed necessary to get a sense of how commonly we encounter the topic of temporal asymmetry across various fields. By expanding the universe of discourse so that the scope of discussion is no longer limited to music, we may better understand how we are situated in discussing the topic of temporal asymmetry. Poetically speaking, we may view the collection of all fields of study as comprising an atlas. Then our objective is to have a birdview of the entire landscape of such an atlas. Consequently, when the discussion eventually returns to music, we are equipped with a holistic understanding of the topic.
We would like to inquire how various disciplines observe and explain the phenomenon of temporal asymmetry. In particular, the focus is on the key concepts that frequently emerge during the inquiry. By focusing on these key concepts, we discover essential topics that a fruitful discussion must presuppose. For example, one important aspect of temporal asymmetry involves the idea of trace: records of the past that make our inference of the past different from prediction of the future. Once we know the role of trace in a scientific context, the same topic can be used to guide our musical inquiry: how the role of memory shapes the way we listen to music and anticipate upcoming musical events. For example, we may ask about how musical repetitions take advantage of our memory capacity. That is, after knowing the key concepts relevant to temporal asymmetry in non-music subjects, we obtain the knowledge of what to look for in music. One may challenge such indirect strategy by claiming that it is entirely feasible to conduct an inquiry into the issue of temporal asymmetry in music without consulting other fields of study, treating it as a purely musical quest. While it is true that confining the topic exclusively to music is feasible, it may be nevertheless limiting. The limiting factor can be stated in two ways: the topic of time is often ignored in music, and the topic of time is actively discussed in other fields.
Temporality as an ignored topic in music
We start with a counterintuitive observation: we often overlook the role of temporal structure in music. It might sound unexpected at first. After all, in instrumental lessons, one of the most frequently discussed topics is timing. Additionally, meter, rhythm and formal analysis, being temporal structures, are indeed essential components of music theory and analysis. Finally, music is, by definition, dependent on time as its essential medium of expression. However, there are two justifications backing up the claim that temporal structure is an overlooked topic, corresponding to two perspectives:
Temporality in comparison to other musical topics
Temporality in music compared to other disciplines
Temporality in comparison to other musical topics
Within the realm of musical discourse, if we consider conventional music theory and analysis, an important rule of thumb states that pitch is more important than rhythm. In other words, the majority of theories, as far as Western art music is concerned, focus on pitch contents and pitch relations more than their rhythmic counterparts. For example, theoretical constructs central to Schenkerian analysis often omit rhythmic structures in the sense that a typical Schenkerian graph reveals more insights about pitch relations (i.e. Ursatz and prolongational spans) while discarding most of the durational information. Because the durational information is filtered out, we are typically unable to recover the temporal proportion of the original piece it represents. In other words, Schenkerian graph as a graphical representation of analytical reduction, filters out durational information because the theory considers rhythmic structures as constructs subordinate to pitch contents and pitch relations, hence are subject to omission in analysis. In fact, we may even find abuse of notation in a typical Schenkerian graph: note values become time-irrelevant. For example, Salzer puts a footnote in his Schenkerian treatise “Structural hearing”:
The use of half-notes for chords of the structural progression in the graph is not intended to indicate time-values, but to differentiate structural points from chords having a different function. (Salzer 1952a, 1:12)
Therefore, rhythmic structure may be described as, borrowing ideas from Schenkerian analysis itself, foreground elaborations that can be omitted due to its lesser importance.
We obtain a similar conclusion if we consider serialism as a case study. One of its central constructs is the tone row along with rules for manipulating the tone row. However, the tone row, evident from its name, is an exclusively pitch-based construct. It is possible that the development of total serialism is a response to the limitations of the pitch-based serialism, by extending the same technique to musical parameters other than pitch, in particular, rhythm. However, simply seen from the historical development itself, we can observe the implied subordination: serialism was developed with pitch manipulation in mind, and it was later on extended and ported to cover rhythm and other parameters such as timbre and dynamic. Therefore, rhythm is a subordinate to (or alternatively, a derivative of) pitch. More explicitly, when we apply serialist techniques to rhythmic parameters, we necessarily ignore temporal structures because serial rules were not originally developed with the concept of temporal structures in mind.
Additionally, if we inspect the overall historical development of music theory and analysis, then we also discover that most of the elaborate theories focus on pitch relations. To name a few examples, species counterpoint is appropriate for pre-tonal music; tonal harmony is, by definition, suited to tonal music; serialism and pitch-class set theory are designed for post-tonal music. Moreover, as illustrated in the case of total serialism, theories of temporality are often extensions to pitch-centric theories, i.e. as a by-product. In particular, the procedure of theory-building can be described as follows: we start with formulating symbolic representation of pitch entities (e.g. tone row as representation of the twelve pitch classes) and rules for its symbolic manipulation (e.g. inversion, retrograde, and retrograde-inversion). Then we try to apply the same theory, as the collection of symbolic representation and rules for symbolic manipulation, to non-pitch parameters. For example, we try to see what happens if the tone row is populated by parameters other than pitch classes. We may establish an arbitrary mapping in which pitch class 0 is substituted by durational value of eigth-note, pitch class 1 by quarter-note, and so on. At this point, we may already see a potential issue: because the theory is not developed originally with rhythmic parameters in mind, many constructs in the theory are questionable. For example, for a rhythmic “tone” row, does it make sense to employ a collection of twelve elements? For pitches the collection of twelve is justified because of tuning constraints in equal-temperament. However, the choice of twelve becomes arbitrary once rhythmic parameters are in question.
We may argue that, it is exactly for the reason of subordination that David Lewin, a proponent of neo-Riemannian theory, writes in his treatise “Generalized Musical Intervals and Transformations” the following statement:
This chapter takes as its point of departure a figure showing two points s and t in a symbolic musical space. The arrow marked i symbolizes a characteristic directed measurement, distance or motion from s to t. It intuits such situations in many musical spaces, and i is called “the interval from s to t” when the symbolic points are pitches or pitch classes. (Lewin 2007)
In order to present the central theme of the book, i.e. the mathematical model named “Generalized Interval System” (GIS), the author clearly understands that it is the most natural to base the model on pitch relations first, hence providing an intuitive motivation for the GIS model. Evident in its naming, the GIS model is built upon the notion of interval, which has very specific connotations: in music, the term interval unequivocally refers to intervallic distance between pitches. If, instead, we would like to refer to temporal intervals, we need to further specify the term by prefixing it with additional qualifiers, e.g. time-span interval. After introducing GIS using its originating idea of the interval, the model naturally extends, or using its own terminology, generalizes to rhythmic parameters.
Many authors also acknowledge the general neglect of temporal structure in the scholarly field. As the Grove article on “Theory, theorists” points out, the 1985 (which is considered a recent publication in the field of music academia) issue of the journal “Music Theory Spectrum” dedicates its entirety under the title “Time and Rhythm in Music”. (Palisca and Bent 2001) The editor of the issue, in the opening words, states that the temporal dimension of music remains “the less explored of the music’s two major domains”.(Rowell 1985) Within the same issue, Kramer suggested in his article “Studies of Time and Music: A Bibliography”, that musical “time has not been widely recognized as an independent field of study”.(Kramer 1985) Then, he enumerates specific observations as evidence:
The New Grove has no article on time; RILM has no separate category for time; The Music Index has only recently begun to list articles under the heading “Time.”(Kramer 1985)
Additionally, in the preface of the 1960 book “The Rhythmic Structure of Music”, the authors outrightly point out that the study of the temporal aspect of music “has been almost totally neglected in the formal training of musicians since the Renaissance. There are many textbooks on harmony and counterpoint but none on rhythm”. (Cooper and Meyer 1960, v) It is true that their statements might turn out to be outdated by the current century as the development of music scholarly research has been constantly ongoing. Furthermore, ethnomusicologists are justified in arguing that the neglect of rhythm might be peculiar to Western art music only. In other words, the neglect is not universal, but applicable to only a particular subset of the entire human society. However, the fact that temporality (i.e. rhythmic structures) had been a relatively overlooked topic within the realm of mainstream Western music theory and analysis for several centuries is itself noteworthy.
Temporality in music compared to other disciplines
If we compare music externally to other fields, i.e. when comparing it to science and philosophy, we observe that musical discourse often avoids explicitly discussing time in itself. To make the statement clear, consider the very term of temporal asymmetry. In the context of music, many readers would consider it a borrowed term in the sense that it appears foreign to us. Upon encountering the term, we find it more natural to interpret it as a term borrowed from physics: in particular, the thermodynamic temporal asymmetry. In other words, time is a built-in topic that is native to physics, whereas it is foreign to music. Upon contemplation, it is worth emphasizing again how ironic and peculiar the case is. On the one hand, time as a topic is overlooked both within the field of music and in comparison to other fields of study, to the extent that discussing time in itself would seem out of place in a musical discourse. On the other hand, the role of time is in fact, more crucial in music than in physics. Firstly, the physical reality of music, when interpreted as a physical process, is temporal in a strictly physical sense. Therefore, since musical events are themselves physical phenomena, what applies to general physical phenomena must equally apply to musical events, if not more. Secondly, in addition to being physical, music also attains its psychological reality, such that we must take into consideration our subjective experience of time. It is exactly for this reason that music theory is considered a completely different discipline from acoustics, even though both are ultimately studies of sound.
Therefore, time plays another role in music: it governs the mental representations of music, in addition to the physical reality of music. One may argue that the same statement is true in physics where mental representations are necessary if we were to understand anything at all: we need mental representations of the universe in order to conduct any study of it. However, the crux of the statement resides in that physical theories often intentionally omit subjective experience of time altogether. Suppose that for reasons analogous to optical illusions, we happen to believe that time flows more slowly today. Meanwhile, we have the same degree of belief that a time-measuring device is credible. However, we find ourselves in a conflicting situation since our belief of slower time contradicts the evidence given by the time-measuring device. In such a case, omitting subjective experience of time is equivalent to saying that we necessarily reject the belief of slower time in favor of the evidence given by the time-measuring device.
The above thought experiment may sound unsurprising in a physical context. However, if the same situation happens in music, the outcome may turn out to be vastly different. Suppose that we encounter a similar situation, with the time-measuring device being a metronome. The situation is given that when listening to a performance of a piece, most of the listeners in the room believe that the performer is unable to maintain a steady tempo. The performer responds by showing evidence from an accurate metronome that the tempo is steady in the metronomic sense. However, in this case, musicians would still keep the conclusion that the tempo is not steady with the following justification: it is true that the tempo is steady in the physical sense, yet the performer still fails to maintain a steady tempo musically as the harmonic content, dynamic, register and texture may require some nuanced timing for the listeners to experience a musically steady tempo. Notice that we are not discussing timing nuances such as rubatos that listeners can detect. Instead, we are talking about the counterintuitive performance technique that, sometimes in order to achieve the perception of a steady pulse, performers need to do the exact opposite by avoiding a metronomically accurate playing.
For example, consider variation 22 from Rachmaninoff’s Paganini rhapsody (see figure 1.1). The entire variation can be interpreted as a huge crescendo in many senses of the word: the dynamic is increasing, the texture (i.e. number of simultaneous notes played) is thickening, and the register is ascending. Meanwhile, the steady quarter-note beat is the rhythmic pulse throughout the entire passage (see figure 1.2).
Because so many musical factors contribute to the feeling of musical accumulation, if we keep the metronomic steady tempo throughout, we may have the auditory illusion (analogous to optical illusion) that the performance is speeding up, as the crescendo in dynamics, thickening in texture and ascending pattern suggest accelerando without actually playing one. As a result, with this expansive musical passage, the performer may have to judiciously make the interpretive decision of stretching the tempo if the goal is to convey a sense of a steady march-like progression, in order to cancel out the effect of speeding up implied by the non-temporal parameters mentioned above.
The rhythmic nuances which performers frequently exercise show that listener’s perception is prioritized over scientific measurement. For the exact reason, we have the controversy of the metronome: a metronome that measures physical time gives equality of durations, yet it is not strictly equivalent to the notion of steady pulse musicians have in mind. This is why, as commonly observed in instrumental lessons, one of the main challenges in rhythmic training involves reconciling the musical notion and the metronomic notion of rhythmic pulse. On the one hand, instructors emphasize the importance of metronomes as an indispensable aid to understanding rhythmic pulse. However, on the other hand, the emphasis on metronomes almost always accompanies an equally urgent reminder concerning the potential misuse that, by excessive use of the metronome, students may lose the organic quality of rhythm that is vital to true musicianship. Therefore, students are repeatedly told that a musically steady pulse does not equal mechanically metronomic playing (the same way the slogan “correlation does not imply causation” is repeatedly spelled out in science classrooms).
In his book “Sound and symbol”, Zuckerkandl makes a clear distinction between physical time and musical time. According to his schematic comparison, the primary distinction between the two is that physical time is the “form of experience” whereas musical time is the “content of experience”, (Zuckerkandl 1973, 202) invoking the dichotomy between form and content. As such, physical time serves as mere measurement of events while musical time acts as “an active force”(Zuckerkandl 1973, 247) that produces musical events. One of the specific consequences of his claim is that musical time resists measurement attempts. In his own words, time “knows no equality of parts”.(Zuckerkandl 1973, 208) In other words, the statement claims that talking of equal beats is ill-defined for musical time. Speaking of “an equality of times, or of parts of time, has no reasonable meaning in the realm of meter”. (Zuckerkandl 1973, 210) By contrast, in the physical world, equality of time depends on its measurement, so that the “equality of hours is the equality of the distances traveled by clock hands”. (Zuckerkandl 1973, 209) In fact, measurement of moving body can be taken as the very definition of time, as Zuckerkandl points out:
The motion of one body, if it is taken as the measurement of the motion of another body, is called time.(Zuckerkandl 1973, 209)
We can further justify taking measurement of moving body as the definition of time if we consider a common thought experiment in understanding special relativity. The thought experiment is set up in the way given by Feynman in his lectures about “Transformation of time”.(Feynman, Leighton, and Sands 2011) We may consider a railway train. Inside the train, we have a laser pointer placed on the floor of the train, pointing at the ceiling that has a mirror (see figure 1.3).
The laser pointer emits light that returns to its original place after reaching the mirror. Then, we define time by the motion of the laser beam as follows: we record how long it takes to make a roundtrip, and call it one unit of time. Meanwhile, we call the setup of the laser pointer and mirror the clock. Now, as physicists tend to do in their creative thought experiments, the train moves outrageously fast, in fact, at the speed of light. Special relativity tells us that the speed of light is the same for all observers. Additionally, we defined one unit of time using the roundtrip of the laser beam. Consequently, an observer on the ground will find that the clock stops on the train, precisely because the laser beam on the train travels a diagonal path (see figure 1.4). In order to for the laser beam to reach the mirror in the ceiling, it has to travel a distance greater than the horizontal displacement of the train (because the diagonal path corresponds to the hypotenuse of a right triangle, which is the longest side of a triangle). However, because the train moves horizontally at the speed of light, the laser beam can never reach the mirror in the ceiling by completing the diagonal path, let alone making a roundtrip.
This thought experiment shows the slogan that “moving clocks run slower”.(Feynman, Leighton, and Sands 2011) Meanwhile, it also shows that physically, time is defined by its measurement. In other words, it is not that physical time favors measurement, but rather, measurement is the very definition of time. In Feynman’s words:
Now if all moving clocks run slower, if no way of measuring time gives anything but a slower rate, we shall just have to say, in a certain sense, that time itself appears to be slower in a space ship. All the phenomena there—the man’s pulse rate, his thought processes, the time he takes to light a cigar, how long it takes to grow up and get old—all these things must be slowed down in the same proportion, because he cannot tell he is moving. The biologists and medical men sometimes say it is not quite certain that the time it takes for a cancer to develop will be longer in a space ship, but from the viewpoint of a modern physicist it is nearly certain; otherwise one could use the rate of cancer development to determine the speed of the ship! (Feynman, Leighton, and Sands 2011)
On the other hand, music rejects measurement. The mechanical pendulum-based metronome is an example of devices that achieve durational equality by using the motion of the pendulum. In fact, Zuckerkandl might have exactly the example of the metronome in mind when writing, so in later passages he invokes arguments similar to the example of steady tempo given above, in order to demonstrate his schematic comparison:
What do we mean, then, when we demand that musicians play in time; demand, that is, that they preserve equality of measures and beats? The poor performer who takes all sorts of liberties with time is censured for the capricious inequality of his measures and beats. By what concept of equality do we measure this inequality? Certainly not by the concept of measurably equal lengths. […] There is no such thing as a musician whose performance does not depart from mathematical equality within certain limits; accurate experiments have given amazing proof of how great such departures can be without even being noticed by the listener. (Zuckerkandl 1973, 210)
In comparison, the discussion of time gains its ontological status in philosophy and natural science in the way time is often explicitly recognized and spelled out. Without loss of generality, we consider a few selected examples in the field of philosophy and physics to illustrate how time is explicitly spelled out.
Philosophy: transcendental time determination
As a philosophical figure, Kant is accredited for his formulation of transcendental idealism, which is considered by many to be an important milestone in the development of epistemology. One of the central concerns in transcendental idealism is about construing the adjective “transcendental” that prefixes a considerable portion of his terminology. For example, by observing the chapter outline of his magnum opus “Critique of pure reason”, we readily find that the work is divided into two halves bearing the titles “Transcendental Doctrine of Elements” and “Transcendental Doctrine of Method”, respectively. The first half is subsequently divided into two parts: “Transcendental Aesthetic” and “Transcendental Logic”. Finally, the part of “Transcendental Logic” is further divided into “Transcendental Analytic” and “Transcendental Dialectic”. Without regarding the specific meaning of each term, the sheer prevalence of the adjective prefix “transcendental” indicates its importance to understanding the theory as a whole. Therefore, it is necessary to discuss the term “transcendental” in the context of a Kantian framework, if one is to fully understand and appreciate its philosophical endeavor.
Whenever attaching the prefix “transcendental” in its Kantian sense to some terms, we invariably deal with the necessary and universal conditions through which the very existence of our cognitive experience is possible. In the introduction to the “Critique of Pure Reason”, Kitcher interprets two terms that are transcendental-prefixed. Firstly, for transcendental philosophy, its goal is to
investigate the necessary conditions for knowledge with a view to showing that some of those necessary conditions are a priori, universal and necessary features of our knowledge, that derive from the mind’s own ways of dealing with the data of the senses. (Kant 1996, xxxi)
Similarly, for the specific chapter on transcendental deduction, its goal is to
show that certain concepts that are a priori, in the sense that they cannot be derived from sensory data, are necessary for all cognition, and so are a priori in the sense that they describe universal and necessary features of all the objects of which we can ever have any knowledge. (Kant 1996, xliii)
Finally, in Kant’s own words, he begins the introduction to first edition with a section on “The Idea of Transcendental Philosophy”:
even among our experiences there is an admixture of cognitions that must originate a priori, and that serve perhaps only to give coherence to our presentations of the senses. For even if we remove from our experiences everything belonging to the senses, there still remain certain original concepts, and judgments generated from these, that must have arisen entirely a priori, independently of experience. These concepts and judgments must have arisen in this way because through them we can […] say more about the objects that appear to the senses than mere experience would teach us; and through them do assertions involve true universality and strict necessity, such as merely empirical cognition cannot supply. (Kant 1996, 44)
By comparing the three excerpts above, one can summarize a few common qualities that a topic must enjoy in order to be considered transcendental. Firstly, it must deal with a priori concepts, i.e. concepts that are independent from empirical observations. In order to make his claims specific, he stresses that they are “not those that occur independently of this or that experience, but those that occur absolutely independently of all experience”. (Kant 1996, 45) Secondly, those concepts must be necessary (as opposed to contingent) and universal. Finally, they must be concepts through which our cognitive experience is made possible, i.e. being necessary conditions for the very existence of cognitive experience.
Therefore, Kant’s philosophical ambition is to lay the epistemological foundation upon which we can have any cognitive experience at all. How transcendental idealism achieves such foundational work is to examine the mechanism through which our cognitive faculties enable and shape our experience. Consequently, all experiences (whether internal or external) as sensory data must necessarily and universally conform to the structure and form imposed by our cognitive faculties. In fact, the shift in focus is precisely what justifies Kant to compare “his revolution in epistemology to the Copernican revolution in astronomy”. (Kant 1996, xxxi) It follows that the epistemological foundation laid by transcendental idealism is also the foundation of essentially all sciences: it is through the structure and form of our internal cognition that we can possibly make claims about the external world.
Now, given the importance and ambition of transcendental idealism, it is noteworthy to observe the role of time within the framework of the theory. One may be surprised at how the issue of time has been elevated to a status of unrivaled prominence. To see how time is a topic in focus, we consider two aspects offered by the Critique.
The first aspect identifies time as “the formal condition of inner sense”. (Kant 1996, 153) The very first part of the Critique, titled “Transcendental Aesthetic”, consists solely of explicit discussion of space and time. In other words, Kant immediately brings into discussion the topic of space and time, thus marking them for consciousness. We may safely speculate that dedicating the entire opening portion of the book proves how space and time are highly esteemed topics. In fact, he makes the claim that “transcendental aesthetic cannot contain more than these two elements, i.e., space and time”. (Kant 1996, 93) Being fundamental topics in his philosophical construction, he presents space and time as the two “pure forms of sensible intuition”. (Kant 1996, 75) He claims that space and time are transcendental in the sense that they are non-empirical: they are forms without contents, and they necessarily and universally give rise to the possibility of empirical contents. For a better understanding, we can refer to the container metaphor mentioned on page once more: space and time are containers in which everything is arranged, related, organized and cognized. However, space and time themselves do not contribute to the content of experience. Therefore, the metaphor enables us to cognize the world in a bottom-up manner: the process starts with space and time without content that are analogous to an empty container. Then we provide space and time with empirical content to make it perceptible to us. What Kant does involves a similar thought experiment, except that it proceeds in a top-down manner. Our task is to ask: what are we left with, if we try to remove all empirical content from our experience? He continues that
if from the presentation of a body I separate what the understanding thinks in it, such as substance, force, divisibility, etc., and if I similarly separate from it what belongs to sensation in it, such as impenetrability, hardness, color, etc., I am still left with something from this empirical intuition, namely, extension and shape. These belong to pure intuition, which, even if there is no actual object of the senses or of sensation, has its place in the mind a priori, as a mere form of sensibility. (Kant 1996, 73)
By comparing the way Kant argues for space and time, we may arrive at the conclusion that time is a pure form of sensible intuition that is more fundamental than space. At first glance, space and time appear complementary: space is “nothing but the mere form of all appearances of outer senses”, (Kant 1996, 81) whereas time is “nothing but the form of inner sense”. (Kant 1996, 88) Note that the parallelism in wording suggests that space and time are responsible for outer senses and inner sense, respectively. However, he then immediately clarifies his point by asserting that time is (indirectly) responsible for all outer senses as well:
Time is the formal a priori condition of all appearances generally. Space is the pure form of all outer appearances; as such it is limited, as a priori condition, to just outer appearances. But all presentations, whether or not they have outer things as their objects, do yet in themselves, as determinations of the mind, belong to our inner state; and this inner state is subject to the formal condition of inner intuition, and hence to the condition of time. Therefore time is an a priori condition of all appearance generally: it is the direct condition of inner appearances (of our souls), and precisely thereby also, indirectly, a condition of outer appearances. (Kant 1996, 88)
This passage vividly suggests that time is more fundamental than space, as it is the formal condition of both inner and outer appearances, whereas space is limited to “just outer appearances”. (Kant 1996, 88)
The second aspect identifies time as “the transcendental time determination”. Later in the Critique, Kant embarks on an investigation of a puzzling concept: schematism. Schematism involves the study of the “transcendental schema”, (Kant 1996, 211) which mediates between the “category” and the “appearance”. (Kant 1996, 210) In particular, he examines how “an object is subsumed under a concept”. (Kant 1996, 209) For example, in music analysis, we often subsume musical objects (which are technically sensory data, i.e. appearance) that bear rondo-like properties under the concept (or category) of rondo. Kant sees a potential issue here: while subsuming object under concept produces no problem in most cases, it may produce problems in cases involving a special kind of concept, that is, pure concept of understanding. According to him, in subsuming an object “under a concept, the presentation of the object must be homogeneous with the concept”. (Kant 1996, 209) However, pure concepts of understanding are special because they “are quite heterogeneous from empirical intuitions”. (Kant 1996, 210) Therefore, the problem reads:
How, then, can an intuition be subsumed under a category, and hence how can a category be applied to appearances[…]? (Kant 1996, 210)
In order to solve the problem, the idea of a transcendental schema is devised. Metaphorically, it serves as a third-party broker, such that it is “homogeneous with the category, on the one hand, and with the appearance, on the other hand”. (Kant 1996, 210) Without going into technical details of terminology, we readily see that transcendental schema is devised as a solution to an urgent philosophical problem: reconciling the experiential content of the world and our cognitive faculties. It is then natural to ask what the candidates of a transcendental schema are.
Now comes the surprising part: transcendental schema is nothing but transcendental time determination. It is surprising because the statement explicitly identifies time as the basis of all cognition and understanding, including those that are considered non-temporal. He argues:
Time, as the formal condition for the manifold of inner sense and hence for the connection of all presentations, contains an a priori manifold in pure intuition. Now, a transcendental time determination is homogeneous with the category (in which its unity consists) insofar as the time determination is universal and rests on an a priori rule. But it is homogeneous with appearance, on the other hand, insofar as every empirical presentation of the manifold contains time. Hence it will be possible for the category to be applied to appearances by means of the transcendental time determination, which, as the schema of the concepts of understanding, mediates the subsumption of appearances under the category. (Kant 1996, 211)
Note that this is a strong claim: transcendental time determination is not a candidate (among others) for transcendental schema, but the “schema of concepts of understanding”. (Kant 1996, 211) In short, the term transcendental schema is nothing but an alias for transcendental time determination. It follows that in his subsequent enumeration of transcendental schemata (i.e. transcendental time determinations), time cries out for attention. For example, the schema of actuality “is existence within a determinate time”, and the schema of necessity “is the existence of an object at all time”. (Kant 1996, 217) Moreover,
the schema of magnitude, the production (synthesis) of time itself in the successive apprehension of an object; the schema of quality, the synthesis of sensation (perception) with the presentation of time–or, i.e., the filling of time; the schema of relation, the relation of perceptions among one another at all time (i.e., according to a rule of time determination); finally, the schema of modality and of its categories, time itself as the correlate of the determination of an object as to whether and how it belongs to time. Hence the schemata are nothing but a priori time determinations according to rules; and these rules, according to the order of the categories, deal with the time series, the time content, the time order, and finally the time sum total in regard to all possible objects.(Kant 1996, 217)
Ubiquitous time parameter: a brief survey of physics with respect to the role of time
Among disciplines of natural science, physics stands as one of the major fields of study. In fact, one might go as far as to claim that physics is the most fundamental discipline of natural science. For example, physicist Feynman once made a remark that might cause elevated debate due to its alleged implication of condescension:
Physics is the most fundamental and all-inclusive of the sciences, and has had a profound effect on all scientific development. In fact, physics is the present-day equivalent of what used to be called natural philosophy, from which most of our modern sciences arose. Students of many fields find themselves studying physics because of the basic role it plays in all phenomena.(Feynman, Leighton, and Sands 2011)
Within the taxonomy of physics, one major branch is mechanics. In the narrow and etymological sense of the word, mechanics concerns the “motion of material bodies”,(Goldstein, Poole, and Safko 2002, 1) i.e. change in spatial position of material bodies in time. However, in its general sense, it concerns time evolution: change in states of physical systems in time. One should note that time evolution is a generalization of motion (or inversely, motion is a special case of time evolution), as motion of material bodies is precisely change in positional states in time. However, we may be interested in issues other than positional state. For example, in quantum mechanics, we are interested in the time evolution of wavefunctions. Therefore, taking time evolution as the general definition of mechanics enables us to subsume under the term a variety of theories, as what constitute “states” and “physical systems” is open to interpretation.
Now comes the crucial observation: the role of time remains prominent throughout the development of mechanics. The claim runs parallel to the previous section, that time is explicitly spelled out in the field of physics. In particular, it has very specific meaning in physics: the time parameter is, in the most literal sense, spelled out in writing down equations of motion.
Poetically speaking, the concept of time remains ubiquitous in mechanics. Ubiquity of time implies two statements: 1) time remains to be a fundamental concept in all formalisms, surviving through theoretical developments; 2) within debates on interpreting physical formalisms, time is a topic that often gains consensus among different parties. We will show what the two statements mean in light of the three examples of formalisms: Newtonian mechanics, Lagrangian formulation, and quantum mechanics. 1
Throughout the history of mechanics, people argue and debate about representations of the world, and formulate various formalisms that try to capture different aspects of it. In particular, they often argue about ways of encoding states of physical systems: how should a theory represent and specify the instantaneous state of a physical system? Newtonian mechanics and Lagrangian formulation (i.e. classical mechanics) use position (hence velocity as the time derivative of position) as the conceptual basis, while quantum mechanics uses wavefunctions. Now, one must realize that no matter how the concepts of position and wavefunctions differ, they are nothing but mathematical functions parametrized by time. Therefore, people seem to reach consensus on the interpretation of time. It is true that one might argue about the rate at which time flows, but everyone must agree that time evolution of physical state depends explicitly on time. The dependency on time is best illustrated in notation, where the parameter t representing time is visible in all mathematical representations of physical state. Similarly, the equations in their general form governing the time evolution (e.g. equations of motion) of physical state are all differential equations containing time-derivatives. Now, even without knowing the semantics of the following equations or the meaning of the term “time-derivative”, one can readily see the presence of letter t:
| Newton’s second law | Euler-Lagrange equation | Schrodinger’s equation |
| $F = m\frac{dv}{dt}$ | $\frac{\partial L}{\partial x} = \frac{d}{dt}\left( \frac{\partial L}{\partial\dot{x}} \right)$ | $i\hslash\frac{\partial\Psi}{\partial t} = H\Psi$ |
Simply from the syntactical appearance of these equations, one can observe that symbols differ as fundamental concepts of different theories vary. However, the symbol of letter t that is reserved for time is persistent across all equations. Time stands the test of time (no pun intended).
One can be provocative and claim that time is a completely imaginary concept. Nevertheless, in the end of the day, one has to deal with the letter t in practice. The ubiquity of time manifests itself in the painstaking process of solving the differential equations. In contrast, we should note how easily (or even necessary) musicians can bypass time altogether. Instead of thinking about issues of timing directly, performers are encouraged to tackle them using techniques of breathing, bodily gestures and pictorial imaginations. In instrumental lessons, we are unlikely to hear instructors say: “don’t rush at the rest! In fact, the timing of the rest should be linearly proportional to the magnitude of your musical intensity just now. Also note that the musical intensity is also a function depending implicitly on time”. It is more common to hear the following hypnotic-style argument: “don’t rush at the rest! Now, close your eyes and take a deep breath. You should imagine that you are on a field trip, and you see the expansive landscape. You can even hear the birds sing! Do you want this quiet moment to last longer, or you just want to go home and sleep? Of course you want to take time here!”
A justification for ignoring temporality: musical time
One potential explanation for avoiding explicit discussion of time in music is that, instead of discussing time as a concept in itself, we often focus on its derivatives: constructs built on top of time that describe experienced time, as opposed to objective time. Music, as essentially a study of human expressions which just happens to use the medium of time, prioritizes time in its wrapped form: experienced time. The statement has two layers of meaning.
The first layer pertains to the idea that musicians differentiate between physical time and musical time. For example, rhythm and meter are constructs built on top of the metronomic notion of physical time. However, they are not interchangeable, as the metronome controversy discussed on page indicates. To this end, Zuckerkandl dedicates a new term “metric wave” to describe musical rhythm. He then used this newly coined term to explain what it means to have musical equality of time, given that durational equality of time is an ill-defined idea in music:
To play in time musically does not mean to play tones that fill equal lengths of time, but tones that give rise to the metric wave. (Zuckerkandl 1973, 210)
The exact meaning of the metric wave is open to interpretation. It is possible that the metric wave is more of a poetic metaphor than a term to be taken literally. However, the sheer fact that the author designates a unique term shows the necessity of describing rhythm in a way that is exclusive and specialized to music.
The second layer pertains to the idea that musicians differentiate between temporality and atemporality in music, which eventually abstracts away the topic of psychological time altogether. More specifically, given the musical notions of rhythm and meter, we can construct derivative concepts: harmonic rhythm and hypermeter (more generally, large-scale rhythmic organization). At this point, we observe an interesting shift of focus. Harmonic rhythm primarily focuses on harmonic organization. However, harmony (in particular, tonality) itself is atemporal because it prioritizes pitch relations. By the same token, hypermeter focuses on grouping structures. The question remains to discover what the grouping criteria are. In particular, the question addresses ways in which we determine hypermetrical boundaries. As we move from meter to hypermeter, i.e. from microscopic metrical structures such as a phrase, to macroscopic hypermetrical structures, the discussion inevitably incorporates additional considerations such as harmonic progression and melodic contour due to the increased complexity. For example, establishing a metrical pattern may simply require durational information (i.e. knowing the onset and duration of each note-producing event) as well as description of dynamics using strong and weak beats, while disregarding other information such as pitch content, let alone harmonic progression and melodic contour. However, determining a hypermetrical pattern (or more generally, large-scale musical pattern spanning an extended period of time) requires much more information. For example, how do we decide if a moment is a so-called “structural downbeat”? What does it mean to have a hypermetrical strong beat? Durational and dynamic information alone will not suffice to explain. To arrive at a decision, one must necessarily address additional concerns such as the tonal and motivic scheme of the piece. It is for this cause, the very term of hypermeter stirs up debates and confusions in the theoretical literature. Yust, in his book “Organized time", points out that “the concept of hypermeter seems to change its colors depending on the analytical situation, coming to mean different things to different people”. (Yust 2018, 123) For example, many authors treat hypermeter and phrase structure interchangeably. As indicated by Krebs, Lester “appears to equate hypermeasures and phrases”. (Krebs 1992, 84) More interestingly, as observed by Smyth, some authors arrive at self-contradictory conclusions about hypermeter:
Schachter blurs the distinction between a “group of measures” (grouped by virtue of similar surface rhythms and accentual patterning) and a “phrase” (a musical segment ending with a cadence). Failing to retain the crucial distinction he drew in his first article between durational and tonal rhythm, he (like Berry) effectively turns the phrase into a hypermeasure. (Smyth 1992, 82)
One should not dismiss the blurring boundary between phrase and hypermeasure as failures authors commit. Instead, to think positively, the blurring boundary reveals a praiseworthy quality of musical organization: variety. As we move toward large-scale musical structures, i.e. structures spanning an extended period of time, we necessarily have to consider more musical factors contributing to their analysis and interpretation. The increased complexity of music’s internal organization corresponds exactly to the notion of musical variety that is praised and valued throughout music history. For example, Zarlino brought up the idea of variety as early as 1558, noting the importance of “variety in the movement of the parts and in the harmony; for harmony is nothing other than diversity of moving parts and consonances, brought together with variety”. (Zarlino 1976, 52) Later on, Niedt spelled out the importance of musical variety by dedicating an entire chapter titled “On the Necessity and Grace of Variation in General”, echoing how variety was cherished during the Baroque period:
nothing in human life can be more pleasant and necessary than variety, in artistic as well as in natural things. Were it not for summer and winter, sowing and harvesting, frost and heat, day and night, and so forth, what creature would be able to endure this mortal life? […] Indeed, the ear knows of no greater pleasure than in the variety of many tones, songs, and melodies. […] the greatest charm rests in Variation, whether it be performed by the human voice or by various instruments. (Niedt 1989, 73)
In order to see how the importance of variety is style-agnostic, consider a more recent figure. Schoenberg treated the dichotomy between variation and repetition as the determinant for effective composition, thereby highlighting the role of variety in music:
A motive appears constantly throughout a piece: it is repeated. Repetition alone often gives rise to monotony. Monotony can only be overcome by variation. […] Variation means change. But changing every feature produces something foreign, incoherent, illogical. It destroys the basic shape of the motive. […] Accordingly, variation requires changing some of the less-important features and preserving some of the more-important ones. (Schoenberg 1970, 8)
The term “increased complexity” above refers to the increasing number of possible interactions between musical events. Simply put, large-scale musical structures provide more ways in which musical events can be defined. Increasing number of well-defined musical events then enables analysts to examine a richer collection of musical interactions. For example, in discussion of form, we inevitably examine large-scale musical events bearing different names. Sometimes formal sections bear generic letter names such as “AAB” for bar form, while on other occasions formal sections acquire dedicated names indicating their functions such as recapitulation in sonata form. Then, we are entitled to speak about the “recapitulating event”, referring to either the moment where recapitulation section begins (i.e. timepoint) or the recapitulation section itself (i.e. timespan). With either interpretation, we may now relate the recapitulating event to other musical events in the piece. For example, we can ask conventional questions about the relationship between recapitulation and development: how does the retransition at the end of the development lead the music into the recapitulation, creating a strong sense of return? Furthermore, we can be less conventional by asking questions that are more creative. It is a perfectly valid question, for example, to ask: how does the development section of the piece as a whole relate to the opening three-note motive? It is a creative question precisely because we are now inspecting relationship between musical events across different organizational levels. Using the language of Schenkerian graphs, we are connecting elements between a foreground graph and a background graph.
A musical case study: Variations and Fugue on a Theme by Handel, Op. 24
To show how the creative question above is at work in music, consider the following example. The opening theme of Brahm’s “Variations and Fugue on a Theme by Handel, Op. 24” can be characterized locally (i.e. focusing solely on the melodic segment) by its three-note ascending scale figure: B-flat, C, and D (see figure 1.5).
Our task is to see what this local structure of the three-note ascending scale reveals about musical events occurring at a larger level. Therefore, we look at the distant future: the last variation before the final fugue (i.e. variation 25). The reason for choosing this specific excerpt is not arbitrary. We make the choice due to its musical significance. In fact, one may even claim that variation 25, in the context of its large-scale formal organization, is more significant than the prolonged fugue that follows. The justification resides in the preceding two variations (variation 23 and 24). From a performer’s perspective, variation 25 sounds like a major arrival point, precisely because it is well prepared and anticipated by the preceding two variations. Variation 23 and 24 are unique among other variations, in that they can be treated as a single variation, forming a two-variation long build-up leading to variation 25. In fact, for performers, the two variations as a whole make up a giant creascendo: variation 24 is the intensified version of variation 23 in the sense that it is equipped with a written-in accelerando (i.e. replacing eighth notes by sixteenth notes) as well as an expansion in register (see figure 1.6). It is for this reason, it would be absolutely inappropriate to take time in between variation 23 and 24, as one normally would not interrupt a crescendo.
Given the importance of variation 25, it follows that one should examine its internal structure. For performers, the task is vital, since one must find justified ways to bring out the climax and convince the audience musically (as opposed to analysts who can rely on verbal means). The variation, consisting of eight measures, has three structural moments. One might ask for the precise definition of structural moments. For now, we may safely take a performer’s perspective and define it operationally: moments are structural if they are less fault-tolerant. Such an operational definition of structural moments is self-evident if we apply it to daily life. Moments in life are important (i.e. structural) if they are events that we, using everyday language, cannot afford to screw up. As a result, a DMA entrance exam at Juilliard is a structural moment when compared to, say, practicing the 86th measure of some Beethoven sonata in room 481f.
Musically, performers can then evaluate the following question: in variation 25, what are the moments in which we are least willing to make a mistake? Interpreting the question positively, what are the moments we must bring out perfectly in order to be satisfied with the performance? The three candidates for such moments are: the beginning of the first four-measure phrase, the beginning of the second four-measure phrase, and the downbeat of the penultimate measure (see figure 1.7). The first two choices are straightforward because they correspond to phrase boundaries, performers need to clearly present them in order to make the audience aware of groupings that are vital to musical understanding. Now, the third is a structural moment because, just like variation 25 itself, it is well-prepared by the measure before: a virtuoso measure that is characterized by ascending scales and outrageous octave leaps, forming a bursting crescendo that finally arrives at the downbeat of the penultimate measure.
Note now that the three structural moments of variation melodically delineate a three-note ascending scale figure: B-flat, C and D. The figure is identical to the opening three-note figure aforementioned. Poetically speaking, the opening figure organically grows through time, and eventually becomes the Leviathan who is capable of causing the climax of the piece, the same way it stirs up gigantic waves in the sea (taking its biblical definition) or in society (in the Hobbesian sense).
The projection of the opening three-note figure into variation 25 is reinforced if we incorporate a larger context by considering preceding variations. A common argument is that one does not have to find variation that far from the opening theme to establish parallelism. For example, the theme (the first eight measures of the piece) itself also possesses three structural moments identical to variation 25. However, notice that the statement above is not true: for the theme, we are not justified to assign the same structural moments as those of variation 25, namely, downbeats of measure 1, 4 and 7. In particular, the downbeat to the penultimate measure in the theme is not a structural moment. In fact, it is intentionally masked if we examine the preceding measure: its melody ascends linearly to F. Then the voice is abruptly and haphazardly cut off by the minor-third leap, in contrary motion, downward to D (see figure 1.8).
Because of the leap, we get a sense of interruption and discontinuity. Lasser’s theory of contrapuntal voices may help explain why the leap conveys a sense of interruption. In his “The spiraling tapestry”, Lasser proposes a musical structure termed “contrapuntal voice”. A seemingly monophonic voice of melody may be decomposed into possibly many “contrapuntal voices”,(Lasser 2008, 8) thereby creating “ ‘single-line’ counterpoint”(Lasser 2008, 8) within a monophonic texture. In Lasser’s words, a melody “which we normally consider to be a single voice, is in fact made up of a multiplicity of Contrapuntal Voices engaged in counterpoint with each other within the melody”.(Lasser 2008, 8) By proposing the idea of contrapuntal voices, we are equipped with better analytical tools in analysis of melodic contours. Specifically, we are able to understand monophonic melody from a more contrapuntal and polyphonic perspective. In practice, some instruments have limited capability in performing polyphonic texture. For example, it is physically impossible for a string instrument to play five moving voices simultaneously (whereas such task is simple for keyboard instruments such as piano and organ). However, it would be musically untrue to claim that compositions for string instruments are incapable of carrying out counterpoint. From a listener’s perspective, we vividly recognize, through our musical instinct, that in a piece for solo string instrument, there are multiple events happening concurrently. Figuratively, we may imagine that a monophonic voice played by the string instrument is capable of conveying a theatrical sense of conversation between voices, personified as characters. For example, we may consider Bach’s third cello suite (see figure 1.9).
For the example, we are justified to assume that Bach invariably carries over his polyphonic compositional style and method into this work for solo cello. However, due to the physical constraints of the cello as an instrument, he has to flatten the polyphonic texture into a monophonic one. Therefore, our task is to restore polyphony (because the piece is by a polyphonic composer) from monophony (because cello is inherently limited). For such a task, Lasser’s formulation of contrapuntal voices provides insights into identifying and interpreting contrapuntal voices necessary for restoring polyphonic understanding of a monophonic voice.
For identification of contrapuntal voices, Lasser points out that the crux lies in the distinction between leap and step in a monophonic texture:
When discussing melodic contour, we habitually treat conjunct [i.e. stepwise] and disjunct [i.e. leap] motion as two distinct but nonetheless equal ways in which adjacent pitches can relate in a melody.[…] Though visually, we see notes move by step or by leap, to the ear, notes connect only by step, not by leap.(Lasser 2008, 7)
Thus, the life cycle of contrapuntal voices is controlled by the melodic motion classified into step and leap motions. In particular, leap motion creates a new contrapuntal voice:
When a “leap” appears on the musical surface, it is aurally understood as the cessation of one Contrapuntal Voice and the beginning of another Contrapuntal Voice. Defined in this way, leaps or disjunct motion, disappear from the actual experience of a melody […](Lasser 2008, 8)
Applying the idea of contrapuntal voices to our analysis of figure 1.8, by moving the melody in linear motion up to F, listeners are following closely on the contrapuntal voice. However, the leap effectively breaks the contrapuntal voice and creates a new contrapuntal voice. Therefore, the D on the downbeat, being the beginning of the new contrapuntal voice, cannot serve the role of arrival point since no event comes before it. The leap causes the D on the downbeat to behave drastically differently from that in variation 25. In variation 25, the D is prepared using a leap-free linear motion, which gives it the role of inevitable arrival. The minor-third leap in this case, thus breaks the three-note figure that is the basis for establishing parallelism.
By the same token, we find that in variations leading to variation 25, the downbeat on the penultimate measure is often evaded. It is then possible to speculate that the evasion is intentional, with purposes similar to evaded cadence: by leaving the tension hanging and unresolved, the music can then delay its resolution, which effectively strengthens the resolution when it finally arrives. To see how evasion is achieved, we start our analysis from variation 21. In variation 21, the downbeat D in the penultimate measure is evaded by the trick of grace notes: D is technically the beginning of the measure, yet turning it into grace note successfully shifts the real downbeat to F. Similar technique happens with variation 22. On the penultimate measure, the music experiences a sudden pullback in the sense that the music is being reset to the beginning of the variation in the literal sense: the first half of the penultimate measure is identical to that of the first measure of variation 22. Now for variations 23 and 24, D finally becomes the downbeat of the penultimate measure. However, it is evaded again due to figuration: the scale with crescendo on the first and third beat causes the motion to be directed to the second and fourth beat of each measure (see figure 1.10).
The musical case study above illustrates how musical structures across different organizational levels may interact, contributing to the increased complexity as musical time climbs the ladder of abstraction. In fact, it also answers a potential objection. One may object by claiming that concepts of musical time at a higher organizational level does not lead to increased complexity. On the contrary, it leads to simplification. The claim is that large-scale musical structures serve as reductions that in fact simplify analysis. Consider the example of Schenkerian graphs: as we traverse from foreground graphs to background graphs, we are essentially in a process of moving from a low organizational level (i.e. surface level that includes all musical events) to a high organizational level (i.e. level that includes only major events). In other words, we are climbing the “ladder of abstraction” precisely in its originating and etymological sense: the linguistic concept, introduced by Hayakawa, that words subsume ideas of “greater generality and applicability” (Hayakawa 1990, 159) as we metaphorically climb the abstraction ladder. One may then argue that such process is reductional, as it terminates with a background graph consisting of relatively few major events (i.e. the Ursatz) because many of the details (i.e. elaborations) are being omitted during the process. What we eventually obtain from the process is a simplified representation consisting of number of notes (for example, six for a typical Ursatz) that is orders of magnitudes lower than its foreground representation (by literally counting number of notes that appear on the printed score). Similarly, from a formal perspective, we observe that in analysis, a piece is often reduced to its formal components. For example, analysts subsume a variety of pieces into the catalog of rounded-binary form, ignoring pitch contents specific to each piece. How can a simplified representation reconcile with the idea that large-scale structures introduce additional complexity?
To counter such potential objection, one must recognize how multiple organizational levels introduce more possibilities for cross-level interactions. It is true that the background level of a Schenkerian graph is simple, which mostly contains just the Ursatz itself. However, the essential part of the analysis is to see how different levels interact. In particular, in Schenkerian analysis, we are entitled to say that its value resides in comparing graphs side by side, showing how events across different organizational levels inform each other. It is for this reason that by convention, Schenkerian graphs are often notationally presented in a way that they are aligned vertically, in order to highlight the reductional process, rather than the individual graphs (e.g. musical illustrations found in the second volume of Salzer’s “Structural Hearing”(Salzer 1952b), see figure 2.2 on page ). In other words, it is through the relationship between Schenkerian graphs representing different organizational levels, that the theory attains its explanatory power. Similarly, knowing the three formal divisions of the sonata-allegro form alone is pointless, because the real musical value concerns how these formal divisions are brought to life through composers’ crafts, which requires us to closely examine how musical details relate to the formal divisions.
In order to illustrate how interactions between multiple organizational levels introduce more musical possibilities, we present a simple combinatorial argument here. Consider a 16-measure metrical model that is recursively divided into groups of two, giving us a collection of the following musical groupings (see figure 1.11): a 16-measure phrase group, two eight-measure phrases, four four-measure subphrases, eight two-measure phrase segments, and 16 one-measure measures. The collection consists of five levels of organization: 16-measure level, eight-measure level, etc. We can do analysis with this metrical model in three different ways. For this toy example, what constitutes analysis here is simply counting pairwise relations within the collection of musical groupings enumerated above. For this example, we focus on unordered pairs, in the sense that the relationship between phrase A and phrase B is the same as that between phrase B and phrase A. Combinatorics gives us useful mathematical tools for counting, making this example a combinatorial argument.
The first approach is what can be called a forgetful simplification. With this approach, we are forgetful such that when moving to a higher level of organization, we forget about lower levels of organization. Applying to Schenkerian analysis, this is to say that we keep only the background Schenkerian graph, and feed all other graphs (i.e. foreground and middle ground graphs) into the paper shredder. Therefore, we start with the 16-measure metrical model, and through the analysis, are left with only one thing, namely, the 16-measure phrase group. Everything at a lower level of organization (e.g. eight-measure phrase) is lost. With this approach, the analysis is not really informative because the number of possible pairwise relations is zero (assuming that we disregard self-relating pairs).
The second approach is more telling. The analysis is no longer forgetful, because we realize that every organizational level is born equal: a four-measure subphrase is not inferior to a 16-measure phrase group. Therefore, by principle of indifference, we should not forget about four-measure subphrases in favor of the 16-measure phrase group. However, at this stage, we are still reluctant to consider pairwise relations across levels of organization because we think such relations cause categorical error: we are comparing apples and oranges. In this case, how many pairwise relations can we produce? Introducing tools from combinatorics, we have a function $\left( \frac{n}{r} \right)$ that gives us the number of possible ways to select r items from a collection of n items where the order of selection does not matter. With the introduced tool, we can count now:
$$\left( \frac{16}{2} \right) + \left( \frac{8}{2} \right) + \left( \frac{4}{2} \right) + \left( \frac{2}{2} \right) = 155$$
The third approach takes into consideration pairwise relations across levels of organization. Suppose now, after reading the musical case study presented in section 1.2.1, we realize that comparing a four-measure subphrase to an 16-measure phrase group is not comparing apples and oranges. Instead, such comparison might be musically insightful. In such a case, how many pairwise relations can we produce? The answer is given by:
$$\left( \frac{16 + 8 + 4 + 2 + 1}{2} \right) = 465$$
By comparing the numbers, we can readily see what it means to have more possibilities. It literally refers to more number of pairwise relations through counting.
Therefore, we arrive at a seemingly counterintuitive conclusion: reduction of musical time is not about simplifying analysis. Rather, it makes us better appreciate the complexity of music as an inherently “hierarchical and temporal" (Yust 2018, 6) construct. In fact, Salzer arrived at the same conclusion by stressing the interdependence and symbiosis between structure and prolongation. On the one hand, he points out that the central concern of Schenkerian analysis is the “distinction between structure and prolongation”. (Salzer 1952a, 1:13) Meanwhile, he warned the reader with a beautifully written passage:
It is wrong to assume, however, that finding the structural framework constitutes the sole purpose of this [Schenkerian] approach. On the contrary, structural hearing implies much more. It enables us to listen to a work musically, because by grasping the structural outline of a piece we feel the full impact of its prolongation which are the flesh and blood of a composition. Thus the structural outline or framework represents the fundamental motion to the goal; it shows the direct, the shortest way to this goal. The whole interest and tension of a piece consists in the expansions, modifications, detours and elaborations of this basic direction, and these we call the prolongations. (Salzer 1952a, 1:14)
Performers are in a position to resonate with the above statement. As one studies a piece of music in order to present it for on-stage performance, the symbiosis is particularly evident. While the knowledge of formal structure is necessary for a better musical understanding, it is through elaborations in the Schenkerian sense that music manifests its manifold of expressive variety aforementioned (see page ), bringing into life its “flesh and blood”. (Salzer 1952a, 1:14) Knowing that a piece conforms to the formal schema of exposition-development-recapitulation is informative, yet the true musical interest rests in how it carries out the schema, making it stand as a unique work of art among the oeuvre of all works. In short, realization and (more importantly) deviation of formal schema in practice (in terms of both composition and performance) are the force behind the expressive power listeners readily feel. Poetically speaking, if one is to say that the value of music analysis by reduction is not the end result, but the very process itself, then for performers, the process of preparing a piece for performance becomes a narrative itself. In fact, it is a metanarrative whereby performers produce a series of analyses (e.g. Schenkerian graphs) in order to become intimately familiar with the piece. Most importantly, the significance of such metanarrative is precisely the very personal experience of living with the music, along with all the joy and frustration associated that take place off-stage. When compared to the totality of this metanarrative, its end result, i.e. a successful performance of the piece, becomes less significant.
Summary
Throughout this chapter, we focused on a particular statement: time is often overlooked in music. The way we elucidated the statement was to interpret it using multiples perspectives. With the first perspective, we explored how within the field of music, time receives relatively little attention. With the second perspective, we explored how explicit discussion of time is often deliberately avoided in music. To highlight the deliberate avoidance, we contrast the treatment of time in music to that in other disciplines where time as an essential concept is often spelled out. Finally, with the third perspective, we attempt to justify the observation that time is overlooked in music by proposing that the explicit notion of time has been abstracted away as we focus on increasingly abstract conceptions of music that are based on notions such as musical phrase and musical variety. Therefore, the focus of musical discourse has shifted away from time.
With the statement that time is overlooked in music, the natural necessity of looking elsewhere arises. After all, we may argue that if a topic in a field is overlooked, then it is also likely that the field would lack terminologies and methodologies required to address the topic. On the other hand, disciplines with explicit focus on the topic of time are likely to inform and inspire us on the topic. One natural place to continue the current discussion is the disciplines where the notion of temporal asymmetry acquires its etymological origin, to which we should refer even for the sake of faithfully introducing the concept. Therefore, we examined the topic of temporal asymmetry as explicated in the fields of science and philosophy, which will be our focus of chapter 2.
The arrow of time in science and philosophy
Recall that in section 3.3, the notion that time flows implies two separate statements. The first states that the past and the future are distinguishable. The second states that they are distinguishable in a unique way such that we are entitled to say that time necessarily flows from the past to the future. We should note that the first statement is a necessary condition for the second statement, but the converse is not true: it is possible to have distinguishability of the past and the future, yet time flows in an opposite direction (if the term “opposite” means anything at all).
This chapter is dedicated to the discussion of these two statements from the perspective of philosophy and science. In particular, in section 2.1, we focus on the philosophical perspective of the distinguishability (i.e. the first statement) between the past and the future, with emphasis on epistemology. In section 2.2, we focus on the scientific perspective of the arrow of time (i.e. the second statement), with emphasis on thermodynamics.
Record and trace
We begin by considering a simple musical question: how is the beginning distinguished from the ending? The question may appear silly at first because everyone knows its answer, the same way everyone knows what time is according to Augustine (see page ). But if we continue along the way Augustine argues, we then ask: do we really know their distinction? We should acknowledge that this simple musical question is real as it is what musicians must seriously consider in practice. The beginning and the ending of a piece of music must be constructed differently and under different set of premises. For example, more specifically, how is an introduction different from a coda? Knowing the difference between the two directly affects one’s interpretive decisions. Musicians may give various creative answers. For example, one may argue that the introduction is more anticipatory in function, whereas coda is more conclusive. Meanwhile, others may argue that a coda can also be anticipatory if we take into consideration that it may serve as the transition to the next movement.
A motivational musical case study: introduction and coda in the first movement of Beethoven’s piano sonata, Op. 111
The first movement of Beethoven’s piano sonata Op. 111 serves the purpose of illustration. Conveniently, the movement simultaneously contains both an introduction and a coda, enabling us to compare them for differences. The role of the introduction in this example is non-arbitrary: it introduces, prepares and anticipates the theme on measure 19 (see figure 2.1). Knowing its anticipatory function, we can then claim a provocative statement: the theme is motivically captured and represented by a single note, the note C representing its downbeat as well as the tonic of the key of the whole movement. One can further notice that for this sonata, the single pitch C plays a more crucial role than C minor as a key by observing its two movements: the first movement (C minor) and the second movement (C major) are tonally related by parallel relationship. For parallel keys, the commonality primarily rests in the shared tonic note instead of scale. Motivically speaking, the introduction then represents a process of affirming the note of C, thereby achieving its anticipatory function, i.e. it helps listeners to anticipate the arrival of C as the downbeat of the exposition.
The musical strategies to achieve the anticipatory function are the following. Our goal is to prepare the theme represented by the note C. Firstly, we adopt a performer’s perspective and examine a crucial section within the introduction: measures 6-10 (see figure 2.2). The section is crucial because for performers, it is tricky both technically and musically. Technically, the section is challenging as it requires precise sound control (voicing of six simultaneous notes with pianissimo) and rhythmic control (i.e. the double dotted rhythm, again with pianissimo). More important, the section is musically challenging because one easily produces a monotonic performance primarily due to the double dotted rhythmic pattern throughout. In response, listeners might be disoriented and wonder what this section is musically about. Ironically, a convincing performance of this section represents a musical sense of disorientation: a process of searching, as if the music is poetically personified, and wanders aimlessly in the dark. The distinction between a musical sense of disorientation and disorientation as expressing a listener’s confusion is in the same manner we have good and bad surprises: musical surprises and disturbing surprises due to suboptimal performance. Through depiction of musical disorientation, the anticipatory function is achieved in this section by a successful representation of searching: we anticipate the theme because the theme is previously unclear, hidden in the musical mist. As a result, the anticipatory function of the introduction is nothing but the urge to clear the musical mist as one musically searches for C: a clearly presentation of the note C at the downbeat by itself may sound abrupt and haphazard. However, if the note is prepared by the introduction such that it results naturally as the inevitable consequence of the introduction, listeners are satisfied in the same way one finds satisfaction in solving puzzles.
Secondly, in addition to the section highlighted by measures 6-10, the entire introduction prepares the arrival of note C by deliberately avoiding it until the onset of measure 19. We observe the avoidance by enumerating all musical events involving note C in the introduction (see figure 2.3). The trill in measure 1 is the first melodic appearance of C, however, it is de-emphasized and destabilized by two musical devices: the trill representing melodic instability and unstable harmonic support (i.e. diminished seventh). Then in measure 2, C occurs in the second quarter beat. However, notice how it is musically passing: melodically it acts as a passing tone between B on the downbeat and D on the strong third beat (not only in terms of metrical accents if we notice the dynamic marking); dynamically it is explicitly marked by p; metrically it occurs on the weakest quarter beat (note that the fourth beat is empty, making the second beat the weakest beat that listeners can perceive). After measure 2, surprisingly (unsurprising to our analysis), note C is completely absent until measure 10. However, C is again de-emphasized in measure 10, albeit having a dynamic mark of f: the sole purpose of the three last eighth notes of the measure is to lead into the downbeat of measure 11. Between measure 11 and measure 18, the interpretation of C is more evident by noticing its role as the upper neighbor of B. Despite the harmonic evidence that B is supported by a dominant harmony throughout the introduction (in which case B is the leading tone to C), locally (between measure 11 and measure 18) B is harmonically stabilized by repeating localized harmonic resolutions: B on the downbeat of measure 11 is the resolution of an augmented sixth, and the two occurrences of B on the downbeat of measure 13 and measure 15 are the resolutions of diminished fifth between the outer voices. As a result of B appearing as a stable pitch locally in the section between measure 11 and measure 18, C can be locally interpreted as the upper neighbor of B.
The section consisting of measure 11-18 is one of the best examples illustrating how musical interpretation is entirely contextual: the local musical evidence in the section indicates that C is the upper neighbor of B. Only through retrospection after hearing the downbeat of measure 19, one can realize the reversal of role: B is in fact, the leading tone to C, restoring our understanding that B is supported by a dominant pedal throughout the introduction.
The coda that begins on the third beat of measure 146, on the other hand, is more controversial concerning its function within the piece. For this example, the coda is less definitive in function because we can argue that its function may be interpreted in two distinctive ways: conclusive or anticipatory.
A word is needed to justify the identification of the coda. We can identify coda using musical parallelism (or more generally, pattern matching) between the end of the exposition section and the end of the recapitulation section. In sonatas without a coda, by definition, the end of the recapitulation is also the end of the entire work. Conversely, if there is musical material after the end of the recapitulation section, then we identify it as the coda. Now the question is reduced to the identification of the end of the recapitulation section. Empirically, the match between the end of the exposition and that of the recapitulation is exact, in the sense that the last phrase of the exposition is the same to that of the recapitulation, up to a difference in keys (i.e. the exposition often ends with a “wrong” key that is later corrected by the recapitulation). For example, consider the first movement of Beethoven’s piano sonata Op. 2, No. 3. The final phrase of the movement (see figure 2.5) is the transposed version (disregarding minor difference in musical details) of the final phrase of the exposition (see figure 2.4).
In both places, the ending is characterized by four measures of stormy sixteenth notes consisting of broken octaves, followed by two measures of cadential closure. Such parallelism between the ending of the exposition (see figure 2.6) and that of the recapitulation (see figure 2.7) is also found in the first movement of Op. 111.
