the metaphor of musical motion is there an alternative.pdf

The Metaphor of Musical Motion: Is There An AlternativeAuthor(s): Judy LochheadSource: Theory and Practice, Vol. 14/15 (1989/1990), pp. 83-103Published by: Music Theory Society of New York StateStable URL: http://www.jstor.org/stable/41054225 .Accessed: 25/05/2013 16:14

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The Metaphor of Musical Motion: Is There An Alternative*

Judy Lochhead

Almost thirty years have passed since Peter Westergaard wrote: "The problem of rhythm in contemporary music lies not in the difficulties of extending traditional analytic concepts to handle increasing complexity in new music, but in the inadequacy of traditional analytic concepts to handle any music/'1 Much has been written about the temporal structures of music since Westergaard's article, and much has been gained. But little attention has been directed toward the kinds of temporal concepts that underlie analytic and theoretic constructs. Since music is a "temporal art/'2 we might well consider the nature of temporal conception in thought about music as a means toward clarifying some of the problems associated with it.

Difficulties surrounding temporality are not unique to music; a satisfactory way of conceiving the nature of time has been a problem since the beginnings of philosophy. One issue has particular relevance to thought about music: the concept of temporal

*An earlier version of this paper was given at the annual meeting of the Society for Music Theory, November 9, 1985, Vancouver.

Westergaard, "Some Problems in Rhythmic Theory and Analysis/' Perspectives of New Music, 1/2 (1962); reprinted in Perspectives on Contemporary Music Theory, ed. Benjamin Boretz and Edward T. Cone (New York: Norton, 1972), 226-37.

2Music has been considered a "temporal art/' by a great variety of writers: philosophers and aestheticians as well as musicians. For instance, Stravinsky in his Poetics of Music in the Form of Six Lessons (trans. Arthur Knodel and Ingolf Dahl, New York, 1947, 29), writes that "...music is based on temporal succession and requires alertness of memory. Consequently music is a chronologic art, as painting is a spatial art." And from a somewhat different perspective, Susanne K. Langer writes in Feeling and Form: "Music makes time audible, and its form and continuity sensible." (New York: Charles Scribner's Sons, 1953), 110.

Recently some writers have been exploring spatial features of musical sound. For instance, see Robert Morgan "Musical Time/Musical Space," Critical Inquiry, 6/3 (1980), 527-38; Thomas Clifton, Music as Heard: A Study in Applied Phenomenology (New Haven: Yale University Press, 1979); and Jonathan Bernard, "Inaudible Structures, Audible Music: Ligeti's Problem, and His Solution," Musk Analysis, 6/3 (1987), 207-36 and The Music of Edgard Varse (New Haven: Yale University Press, 1987). These writers are careful to point out the interpntration of temporal and spatial features. While recognizing the validity and usefulness of exploring the spatial metaphor, I have chosen to focus on temporality here because this is the essential dimension of musical phenomena and because it poses so many conceptual difficulties.

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flow or passage bears directly on the idea of musical motion. An investigation of a provocative philosophical controversy surrounding the notion of temporal flow will be our entry into this issue and into more general questions of temporality. Before jumping into the philosophical discussion of temporal flow, let us briefly consider the idea of musical motion itself.

"Motion" is often attributed to the sounding successions we know as music. Such attributions may be found in virtually all sorts of writing about music. A few examples demonstrate some of the various ways that motion enters into our discourse. In his "Remarks on the Recent Stravinsky" Milton Babbitt comments on The Symphony of Psalms: "The final sound of the [first] movement is a G-major root position triad, defining - by structural parallelism - a motion [my emphasis] of a minor third from E to G."3 Maury Yeston writes: "Taken as continuous flow, [equidurational] pulses suggest no necessary internal groupings but merely a tempo of recurrence, and to create rhythmic or metrical motion of any consequence, it would require the placement of some interpretive accents within the flow."4 And, Charles Wuorinen writes: "Disjunct linear motion, therefore, has often been a signal to critics that 'melody' is absent."5 In such discussions of musical motion, one cannot point to any thing that actually moves.6 Insubstantial sounds have temporal extension and are succeeded or preceded by other sounds or silence. In the excerpt above, Babbitt refers to two successive harmonies, built on E and G, between which he ascribes a motion. Yeston, positing a sort of background flow of pulses, notes a motion between certain temporal places of the flow that are marked by accent. For Wuorinen, motion occurs between successive notes. For each writer, motion describes a relationship between entities - harmonies, accents, notes - in a temporal succession. Musical motion is an attribute of succession whose meaning is metaphorical, and the immediate source of the metaphor is temporal flow or passage.7

The metaphors of both temporal and musical motion assume meaning from the motion of objects in space which itself is relative: objects move both through space and

3Milton Babbitt, "Remarks on the Recent Stravinsky/' Perspectives of New Music (1964); reprinted in

Perspectives on Schoenberg and Stravinsky, ed. Benjamin Boretz and Edward T. Cone (New York: Norton, 1972), 168.

4Maury Yeston, "Uninterpreted Rhythmic Structures," The Stratification of Musical Rhythm (New Haven: Yale University Press, 1976), 35; my emphasis. 5Charles Wuorinen, "Melody," Simple Composition (New York: Longman, 1979), 53; my emphasis.

6One could argue that sound itself is the consequence of moving air which in turn sets in motion various structures of the inner ear, but most discussions of musical motion are not directed to this aspect of sound. And further, most (if not all) authors do not argue that a higher-level concept of musical motion rests on a lower-level concept of the generation and perception of sound.

7Note that the metaphor of "flow" is also present in the Yeston quote above.

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The Metaphor of Musical Motion 85

with respect to other objects.8 The temporal metaphor is similarly relational; it requires the assertion of a first-order time that moves or flows with respect to a second-order time. The musical metaphor involves a relative motion between sounds - usually as notes, intervals, or chords - and time, or time and tonal space.

The validity of the temporal metaphor was vigorously debated by a variety of philosophers and scientists in the early years of this century, but the musical metaphor has been employed quite uncritically. The issues arising from the debate illuminate some of the difficulties surrounding music's temporal structure, to which Westergaard alluded, and suggest some new avenues for thought about music. In particular, I will be concerned with the relevance of the issues for thought about contemporary music. Changes in structure that characterize twentieth-century music have posed significant problems for those who think about that music, and the problem of temporal structure in contemporary music has been one of the most perplexing.

The discussion will proceed as follows: Part I summarizes the central issues of the philosophical and scientific debate; Part II presents a critique of three recent articles on the temporal structures of twentieth-century music in light of these issues; Part III draws together the two prior parts with general remarks on the significance of the philosophical issues for musical concepts.

I

J. M. E. McTaggart sparked debate about temporal motion in an article appearing in 1908 called "The Unreality of Time/'9 He demonstrated that "time is unreal" by arguing that the two ordinary ways of conceptualizing time are logically impossible. One of these ordinary conceptions of time involves the metaphor of motion and the other explicitly denies it.

These two notions of time are commonly referred to as dynamic and static time. McTaggart called them the A-series and the B-series. Dynamic time, or the A-series, is characterized by tensed language that corresponds to the temporal determinations of future, present, and past. For example, imagine you live in Montana and that it is New Year's Day. On this January day you read about and look at pictures of the previous

8The question of how metaphorical meaning arises is complex and beyond the scope of this paper. By asserting that the meaning of temporal motion derives from spatial motion I imply not that there is no real sense in which temporal motion, in a broader meaning, exists but only that there are no substantial entities which change spatial location (see also note 16 below).

Marion Guck explores how metaphor may function for analysis in "Musical Images as Musical Thoughts: The Contribution of Metaphor to Analysis/' In Theory Only, 5/5 (June 1981), 29-42.

9J. M. E. McTaggart, 'The Unreality of Time/' Mind, 17 (1908), 457-74.

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year that has occurred, you watch the football games that are occurring now, and you make plans to turn over a new leaf during the year that will occur. Underlying this tensed language is a notion of change. For instance, on that January day in Montana, an event you expect to occur - say the first taste of a summer tomato - changes from being a pleasant expectation, to being a flavorful experience in July, to being a fond memory in November. It is this concept of change that incorporates the metaphor of temporal motion.10

Static time, the B-series, is characterized by non-tensed language such as before/after and earlier than /later than. Temporal designations such as these determine permanent order relations. For instance, if an event X at rYl is earlier than an event Y at T2, then X is always earlier than Y. Or in other terms, the first day of 1989 is

always earlier than the last day of 1989.

Descriptions of time according to the A- or B-series account for one kind of temporal structure: the relation between successive events in a temporal series or, in other words, order relations. Comprehensive considerations of time in philosophical and scientific literature deal with structures of order and of duration or extension. The question of temporal extension is a separate but not unrelated issue; it will not, however, be considered here.

Both the dynamic and static formulations of time are conceptually available in

ordinary language (and, as we will see later, both occur in music theoretic and analytic language). It is just as common for us to refer to dynamic processes such as growth, decay, evolution, and development as to static relations such as before /after, earlier than/ later than. In ordinary usage the two concepts are equally viable. McTaggart, however, pointed out a contradiction in these two conceptions which led to his assessment of time's unreality.

McTaggart's argument has two parts. The first follows from the assumption that time involves change. Since the order of events is permanent in the B-series, McTaggart maintains that static time cannot accommodate change. Consequently, the meaning of the earlier than /later than designations of the B-series depends upon the future-

present-past relations of the A-series. In other words, the meaning of the B-series

depends on the temporal determinations of the A-series. The second part of the

argument demonstrates that the A-series is contradictory. Every event in an A-series has, simultaneously, all three mutually incompatible temporal determinations; for

McTaggart, it is contradictory to say that an event that is present now, will have been in

10The direction of this motion itself changes according to usage. If we say that an event changes from

being future, to being present, to being past, there is a motion from future toward the past. But if we say that an object becomes something - for instance, an apple becomes rotten - then there is a motion from the past toward the future.

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the future, and will be in the past. The contradiction is not resolved by the assertion that the event has these temporal determinations "successively." This assertion requires the introduction of a second-order time series. The moments of this second-order series would also have, simultaneously, the three temporal determinations, requiring again the assertion of a third-order series, and so the argument goes in an infinite regress. Since the A-series remains contradictory and since the B-series depends on the A, time is unreal.

This proof incited proponents of both the A- and B-series to develop arguments countering McTaggart' s thesis. The issues that were central to the debate between the advocates of the A- and B-series arose either as responses to McTaggart's proof or to the competing theory.11 Issues pertinent to the musical considerations below are presented as "arguments." Four arguments that characterize the B-theory are presented first, followed by four that characterize the A-theory.

Arguments of the B-theory

First argument: The B-series can accommodate change. This argument rests on the distinction between temporal becoming and temporal change, an important distinction since McTaggart defines change as a fundamental feature of time. B-theorists contend that McTaggart confused becoming with change. They show that becoming reflects the temporal relation of an observer to a series of events and that becoming is not intrinsic to the events themselves. In the B-series change is defined by different states of an object at different times. Change may be demonstrated when it may be stated, for instance, that at an earlier time the apple is ripe and at a later time the apple is rotten.

Second argument: The concept of static time best models time in the objective world. Underlying the B-theorisfs position is the assumption that time is an existent in the

11 A limited bibliography about the debate over the A-and B-series is given below. For a more complete list of references see Richard Gale, The Philosophy of Time (London: Macmillan, 1968), 503-06. Works by B- theorists include: Bertrand Russell, Introduction to Mathematical Philosophy (London: George Allen & Unwin, 1919) and An Inquiry into Meaning & Truth (New York: Norton, 1940); Nelson Goodman, The Structure of Appearance (Cambridge, Mass.: Harvard University Press, 1951); W. V. Quine, "Mr. Strawson on Logical Theory/' Mind 62 (1953); A. J. Ayer, The Problem of Knowledge (London: Macmillan, 1956); and J. J. C. Smart, Philosophy and Scientific Realism (London: Routledge & Kegan Paul, 1963). Works by A- theorists: C. D. Broad, Scientific Thought (London: Kegan Paul, Trench, Trubner, 1923); P. Marhenke, "McTaggarf s Analysis of Time/' The Problem of Time, University of California Publications in Philosophy 18 (1935); W. S. Sellars, "Time and the World Order/' Minnesota Studies in the Philosophy of Science III, H. Feigl, G. Maxwell, and M. Scriven, eds., (Minneapolis: University of Minnesota Press, 1962); M. Capek, 'The Inclusion of Becoming in the Physical World," The Concepts of Space and Time, M. Capek ed., Boston Studies in the Philosophy of Science, XXII (Dordrecht: D. Reidel, 1976); and J. N. Findlay, "An Examination of Tenses," Contemporary British Philosophy III, H. D. Lewis, ed. (London: George Allen & Unwin, 1956).

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physical world and that a concept of time that models this objective existent is prior to any other concepts. The third argument rests on this assumption.

Third argument: Temporal becoming is mind-dependent. The change that characterizes a future event when it becomes present or when it becomes past exists only in relation to an observer; in other words, becoming depends on the "now awareness" of an observer. B-theorists claim that while some events in the natural world may be described as a uni-directional, asymmetric, and transitive series, they give no evidence of temporal becoming apart from the orientation of an observer.12

Fourth argument: Temporal designations in terms of the B-series are prior to designations in terms of the A-series. B-theorists contend that the statement "Schoenberg died in the past" means "Schoenberg died at a time that is earlier than another time."

Arguments of the A-theory

First argument: The tensed determinations of future, present, past, and the related notion of becoming, are structures of the objective world. A-theorists argue that we have certain knowledge of future, present, and past events, and of the becoming of events. They assert that experience reveals the objective structures of the world, and further, that the time of an observer is not distinguishable from that of the objective world. The determination of an object or event as past, present, or future by an observer involves a cognitive act, and "if the cognitive acts of an observer are intrinsically either past or present, then the objects of these acts, which are contemporaneous with them, must likewise be intrinsically either past or present."13 Since time of the objective world

12The temporal properties of uni-directionality, asymmetry, and transitivity characterize the "classical"

concept of time. With respect to the succession A B C D these properties may be formulated as follows:

A is followed by B, B by C, and C by D Uni-Directionality

If A is before B, then B is not before A. Asymmetry

If A is before B, and B is before C, then A is before C. Transitivity

The reader interested in further investigation of temporal properties and concepts may find the following books useful: P. C. W. Davies, Space and Time in the Modern Universe (New York: Cambridge University Press, 1977), and Bas. C. van Fraassen, An Introduction to the Philosophy of Time and Space (New York: Random House, 1970). 13 P. Marhenke, //McTaggart/s Analysis of Time/' University of California Publications in Philosophy, 18 (1934), (Berkeley: University of California Press, 1935), 162.

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does not differ from the time of experience, becoming and the tensed determinations of future, present, and past are objective structures of time.

Second argument: An important ontological difference exists between past and future events, and this difference is linked to the objectivity of "now" awareness. As proof, the A-theorists demonstrate that the four-dimensional representation of space and time gives evidence of the ontological difference of the past and future. They argue that, according to relativity theory, no event in the causal past of any single conceivably real observer can ever be contained in the causal past of any other conceivably real observer. In other words, past events in the physical world differ from future events, and the "now" of subjective awareness is that moment that separates the future from the past.

Third argument: A-theorists argue that temporal determinations of the B-series are dependent on temporal determinations of the A-series. They show that the statement "Schoenberg's birth is earlier than Schoenberg's death" means that if Schoenberg's death is in the present, Schoenberg's birth is in the past. Further, A-theorists claim that the translation of a tensed statement into an untensed one results in a loss of meaning. For instance, the statement "Schoenberg's death in 1951 is later than his birth in 1874" does not tell us that Schoenberg is dead.

Fourth argument: McTaggart' s claim that the A-series is contradictory involves a confusion between A- and B-relations. For McTaggart, the B-series entails permanent relations of order between permanent events. The A-series entails impermanent relations between permanent events. A-theorists argue that McTaggart conceives of the A-series as a sort of "chorus-line" of events across which the "spotlight of the present" moves, giving each event its momentary presentness. Such a conception confuses the defining features of A- and B-time. According to the A-theorists, the A-series involves a change of time, that is, it involves the changing of events, not just a change in the relations between events.14

An investigation of the philosophical assumptions that underlie arguments for the A- and B-series sheds some light on the differences between the two notions of time.15

14The A-theorists used this reasoning to counter the argument that the B-series can accommodate change. The argument of the B-theorists rests on the distinction between an event and an object: an event does not change in the B-series but an object may exist in different states at different times. A-theorists demonstrate that change is not implied by statements tracing the history of an object - by such statements as the apple is ripe ab one time and rotten at another. They argue that the ripe state of the apple at the earlier time never changes, and thus it is an event in the B-series. Since these are permanent states of the object, the B-series cannot accommodate change. 15A question that parallels the debate over dynamic and static time concerns the role of metaphor in the constitution of meaning. Proponents of a literalistic theory of metaphor would make the following claims: (1) motion in space has a significance specifically linked to that context, and (2) the transfer of that significance from one context to another, in a metaphor, results in a loss of meaning. Proponents of a constitutive theory would claim that metaphorical language constitutes unique meanings, in fact, by

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While the questions about time that characterize the debate arise from analytic philosophy in the twentieth-century, the two sides reflect different concerns within this philosophical tradition. Arguments for the B-series are based on the assumptions of Philosophical Realism. B-theorists assume that language can describe an independently existing and objective reality. Historically, such a notion of reality has been defined by existence and existence has been associated with universais and the transcendance of time, in other words, with permanence. The permanence of order relations in the B- series gives time a reality that has proved difficult to define otherwise. In the A-series, no part of time can be said to ''exist7': the past is no more, the future is not yet, and the present is constantly becoming the past. Thus, with no existing or permanent parts, dynamic time is not a "real" structure of the world.

Arguments for the A-series come out of philosophies that develop from experience and ordinary uses of language. A-theorists argue from the position that reality involves a reciprocal relation between the world and the language-user. They contend that language reflects this relation and thus reveals the reality of the world.

The debate over static and dynamic time generated commentary demonstrating the validity not only of one conception over another but also of both so long as they are not confused. One writer, J. N. Findlay, claims that statements about the world characterized by both A- and B-relations are meaningful but that the non-tensed statements of the B-series are preferable since they are always true or false.16 He shows that McTaggart's paradox results from a combination of the two kinds of relations: to say that a single event is simultaneously past, present, and future is to ascribe the permanent relations of the B-series to the A-series.17

The arguments offered by A- and B-theorists above are formulated to support the priority of one conception over the other. It is not my purpose to advocate either position as preferable; rather, it is to use the issues arising from the debate as a basis for a critique of theories about music's temporal structure. The critique will be concerned not only with what conception (or conceptions) of time operate in a theoretical context

applying a concept from some specific context to a different context. In the instance of motion that is our concern here, the application of a concept that derives from a visual and tactile confrontation with space to the less comprehensible domain of time constitutes not only a new meaning but, further, a terminology where none existed.

16J.N. Findlay, 'Time: A Treatment/' Australian Journal of Philosophy, 19 (1941); reprinted in Gale, The Philosophy of Time, 143-62.

17Phenomenologists would approach the distinction between dynamic and static time quite differently. Martin Heidegger characterizes time as a fundamental framework of experience - a temporal spread that encompasses future, present, and past. This temporal framework is prior to and makes possible other conceptions of time. The static and dynamic conceptions are two such possibilities.

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but also with the larger philosophical assumptions that underlie the conception. Further, using a philosophical basis for the critique allows for an interdisciplinary comparison between temporal concepts that provides a broader base for understanding the problem of music's temporal structures.

II

Part II presents critiques of articles by three authors: Allen Forte, Christopher Hasty, and David Lewin.18 Each article offers a theory about temporal structure based on examples drawn from the twentieth-century repertory. I have chosen to focus on theories about music from the present century precisely because much of this music challenges the applicability of well-established temporal concepts about music and brings to a head the general problem of formalizing the temporal structures of music.

Allen Forte bases his model of temporal structure on the premise that "duration is the most important aspect of rhythm" (90-91). This premise seems to put Forte's theory outside the boundaries of this critique since the debate over static and dynamic time concerns the order structures - not the durational (or metric) structures - of time. The two structures are not so clearly separable, however, and the order relations implicit in Forte's theory deserve attention here.

Forte models temporal structure by translating durations (as defined by a score) into numerical values. This translation makes comparison of durational quantities simpler and allows one to determine proportional relations between durations. In Forte's words, this translation "presentisi rhythmic structure in the clearest possible way, detached from ordinary notation, with its bias toward traditional interpretation" (91). Example la cites the first six bars of the fifth movement from Webern's Bagatelles for String Quartet, Op. 9. Example lb shows Forte's "proportional graph" of those bars, which I have annotated.

In Example lb the numbers associated with either solid horizontal lines or broken diagonal lines represent the values of the durational units that can be determined from the score. A durational unit in Forte's model can refer to either of two different kinds of temporal extension: what I call note duration and resultant duration. The precise extent of either kind of durational unit is figured between the beginning (attack) and end

18Allen Forte, "Aspects of Rhythm in Webern's Atonal Music/' Music Theory Spectrum, 2 (1980), 90-109; Christopher Hasty, "Rhythm in Post-Tonal Music: Preliminary Questions of Duration and Motion/' Journal of Music Theory, 25/2 (Fall 1981), 183-216; and David Lewin, "Some Investigations into Foreground Rhythmic and Metric Patterning," Music Theory: Special Topics, ed. Richmond Browne (New York: Academic Press, 1981), 101-36. My critiques are not comprehensive; they address only those aspects of the three articles that are pertinent to a consideration of underlying temporal concepts.

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Example la: Webern, Six Bagatelles for String Quartet, Op. 9, V, bars 1-6

uerst langsam (J^ = ca. 40)

l ' 'LJJLlL ' ' VJLJJ F^ rrr an Sieg amSttg

tmitDampftr Z^= il

" - ,4 - ,4 y K = i = ^ uh J

/w- = = - ^^ ^^^^ iw> ==- - Copyright 1924 by Universal Edition A.G., Wien. Copyright renewed; All Rights Reserved; Used by permission of European American Music; Distributors Corporation, sole U.S. and Canadian agent for Universal Edition

Example lb: Forte's "Proportional Graph" of Webern, Op. 9, V, bars 1-6 [Author's annotations in bold]

Palindrome 1 : 2 6 6 18 6 6

i 24 2 : 3 48 4 j i

5 j j j 1 2 ' i2 12 12 | i : 3o :

(12:24) [i2]^**r; ^ . : 1 ! i

K &

t i2 12 .-# ri2^w.^ [12] 36 ' J : 30 ; ; N

18 t

24 ^ / [6] 36 24

| I 12 : 18 2 : 3

(release) of a note, but a durational unit is not necessarily equivalent to the actual

sounding length of a note. In Forte's terms, a durational unit "is determined by the

following pair of nodes [i.e., for our purposes, dots on the proportional graph]: attack to attack, attack to release, release to attack, and release to release ." (94)

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A note duration is simply the temporal extent of a note as defined by its attack and release. In Example lb, the non-bracketed numbers and solid horizontal lines represent note durations. Bracketed numbers and broken diagonal lines represent resultant durations. These durations "result from" the beginnings and/or ends of two different notes. For instance, the quarter rest of violin I, in bar 2 in the score is represented by the durational unit of 12 in the proportional graph. This unit results from the release of violin II's, viola's, and cello's notes in bar 1 and the beginning of violin I's note in bar 2. The numerical values of durational units are determined by assigning the integer value 1 to a duration that will allow one to assign whole integers to the longer durations in the music being considered. In Example lb the value 1 is assigned to the thirty-second note of a dodecatuplet: thus, a half -note equals 24, a quarter-note equals 12, an eighth equals 6, and so forth.

With a numerical translation of the score in place, one may consider the proportional relations between the durational units. Forte points out that in bar 1 two durational units of 12 articulate the longer duration of 24. The comparative relation between the unit of 24 and the articulating unit of 12 forms a "proportio dupla." (95) Forte continues his discussion of the movement by showing the proportions formed by various durational units. An instance of the "proportio sesquiltera" in bars 4-5 is annotated on Example lb. He also identifies a palindrome in bars 4-6 involving the durational values 6-6-18-6-6. My annotations above and below the graph of Example lb make explicit the proportional relations.

The comparative relations between durational units in Forte's model depend not on the uni-directionality of the dynamic series but on the permanent relations of the static series. Forte's comparative structures are isotropie; that is, they imply temporal directionality but not uni-directionality. For instance, Forte's "proportio dupla" does not distinguish between 2:1 and 1:2.19 Directionality of any sort is not a defining property of a palindromic sequence. A visual representation will clarify: it is impossible to distinguish a palindromic sequence - 8 8 16 8 8 - from its retrograde - 8 8 16 8 8. However, within the constraints of the B-series, the ordering of two durations of 8 before a duration of 16 and two after it clearly defines the palindrome.

For the most part, Forte's model of temporal structure rests on a concept of static time. The model, as such, does not address certain kinds of temporal issues. For instance, the first bar is described as a "proportio dupla," but that concept does not concern the problem of how the proportion manifests itself when the durational unit 24

19While the terms duple proportion and sesquiltera proportion do mean, strictly speaking, 2:1 and 3:2, Forte does not use them this way. In his article the term "proportio dupla" refers to both 2:1 and 1:2 and "proportio sesquiltera" to both 3:2 and 2:3.

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(the "1" of the proportion) cannot be said to "exist" until the end of bar 1, until after the first unit of 12 has already occurred. The model deals with formed, not forming, structures.

Further, the model does not address the changing status of a structure. For instance, it does not deal with the change from an incomplete to a completed structure over the time of bar 1 through the downbeat of bar 2. While B-theorists claim that the B-series can accommodate change, the situation is somewhat more complicated in Forte's model since it involves durational proportions. The model compares durational events which sometimes are and sometimes are not discrete. It does not follow the changes over time of a single object. Thus even though the durational events require time to manifest themselves, the model is not concerned with the actual "unrolling" of the duration.

Forte's model clearly rests upon assumptions of the B-series. It describes formed or permanent structures characteristic of that series. Durations have an objective existence and comparative relations between them are fixed. The model posits not structures which manifest themselves over time but a definite web of comparative relations.

Hast/s article specifically addresses motion as an aspect of music and links it to duration. The first part of his work considers motion and duration from a philosophical perspective, and the second part presents a temporal model for post-tonal music based on the ideas of the first part.

The philosophical discussion rests on the assumption that motion is a structure of temporal phenomena (185). Hasty cites the work of the philosopher Errol Harris to clarify the idea that temporal continuity and change are at the root of musical duration and motion.20 Harris points out that temporal succession on the one hand requires a notion of change or impermanence between events and, on the other, a continuity binding events together. Duration itself is inconceivable without a notion of continuity between diverse events in succession. Harris's comments describe the dual nature of temporal relations and are reminiscent of the distinctions characterizing the debate over dynamic and static time.21

Hasty develops a concept of musical motion from this dual nature of temporal relations. He states that between the successive events of a musical presentation there is a qualitative relation. If, between two or more events, a unity obtains by means of a continuity between them, then the qualitative relation may be described as musical

20See Errol E. Harris, "Time and Eternity/' Review of Metaphysics, 29 (1976), 464-82.

21Both the static and dynamic formulations of time require some notion of change and continuity. Philosophical treatments of time have always pointed out this duality, but in either formulation, one of these two features is emphasized.

Theory and Practice




motion. It is the formation of a unity, what Hasty calls a structure, that is the basis of musical motion. In his words: "What is required for motion to take place is the formation of a structure or whole between or among events" (191).

Hast/s discussion of the philosophical basis of musical motion clearly rests on a

dynamic concept of time that feeds his interest in developing a temporal model applicable to the structures of post-tonal music. He posits the "mind-dependence" of motion, contending that "it is not the material world itself but our mode of cognition which creates temporal relations" (191). His definition of motion by the "formation of a structure" rests on a uni-directional temporal series, the tensed series of dynamic time. While born of the continuity of a structure, musical motion depends on the impermanence of temporal relations as defined by a dynamic series.

The model of temporal structure Hasty presents in Part II of his article proceeds from his definition of musical motion. To illustrate his model, Hasty considers the first 12 bars of Webern's Op. 27 Piano Variations, third movement.22 The present discussion will focus on bars 1-2 only; Example 2a cites these bars. Hasty begins with the assertion that the "continuous event of bars 1 and 2 [is] a unit," what he also refers to as a "structural component" (197). The discussion of this unit demonstrates its structural coherence by showing various ways in which the different elements of the unit are associated. Hasty calls such associations segmentations, and conceives of them as different interpretations, different ways in which a "group of notes may be heard" (197). While Hasty identifies four segmentations of the unit in bars 1-2, Example 2b cites only the first.

In Segmentation 1/1 (shown in Example 2b) the first and last notes are united by an equivalent duration and are called element A; by the same criterion, the second and third notes are united and called element B. These two elements are themselves united as a "higher-level structure" by interval class similarity: both are instances of ic 1. Other factors associating Elements A and B are contour - a retrograde (or inversional) relation - and pattern of metrical accent - again a retrograde relation (weak, strong/strong, weak).

Hasty identifies another unifying factor of the first segmentation: interval-class association. In Segmentation 1/1, the ic content of the four possible three-note subsets is shown in brackets. These ic formations, the bracketed ic numbers, are the basis of a unifying association. Hasty writes that in Element A there is "a motion from...[l, 4, 5] to [1, 3, 4] while in B this order is reversed" (199). This retrograde relation between the ic associations unites A and B as a structure.

In his philosophical discussion, Hasty asserts that musical motion and continuity result from the formation of structures. Such a theoretical concern rests firmly on a

^Hasty also considers bars 1-5 from Stefan Wolpe's Form for Piano (1959).

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Example 2a: Webern, Variations for Piano, Op. 27, Movement III, bars 1-2. Hasty' s analysis of Unit 1.

Ruhig fliessend J=ca. 80

fli i-Unitl 2 nr

1 P f

in 1 - I pf 1 - * ?

f

i Copyright 1937 by Universal Edition. Copyright renewed; All Rights Reserved; Used by permission of European American Music; Distributors Corporation, sole U.S. and Canadian agent for Universal Edition

Example 2b: Hasty7 s "Segmentation 1/1/' Webern, Op. 27, Movement III, bars 1-2

j[ y^ ^^^^T^S; ' >v + contour (retrograde) ' r ' bow + order of ic association (retrograde) [1,4,5] '*y [ 1 , 3, 4] + pattern of metrical accent (retrograde;

(i.e., A - weak, strong ^-^^ B

- strong, weak)

[1,3,4] V^Xl^S] A B i

+ duration ( o ) + duration ( 4 )

- duration A/B

dynamic concept of time. While his intention is to provide an understanding of the

dynamic relations we conceive as musical motion, his demonstrations of such motion sometimes bear traces of static relations. For instance, Hasty's assessment that there is a motion from the ic association [1, 4, 5] to that of [1, 3, 4] does not account for the fact that the specific ic associations of El? are realized one-by-one (they "come into being") while the ic associations of D occur "all at once." The structure imbuing motion here

may be said to "click into place" or, as Hasty suggested in earlier comments, may

Theory and Practice




"spread through" the events as a "mutual conditioning or relationship" (191). Such a movement would be experienced at the cognitive moment of [1, 3, 4], the downbeat of bar 2. Further, we might note that the retrograde relation of accentual and contour structures between Hasty's A and B elements of Segmentation 1/1 occur in bar 2; the structures similarly occur "all at once."23 In all of these instances, the motion resulting from the completion of a structure occurs in a cognitive instant with the movement "spreading backward," so to speak, over the time of the structure. To suppose cognition of a "forwardly directed" motion would require expectation and thus some prior knowledge of the structure which is being complded. While the dynamic model of time is not entirely precluded in these instances, the question remains whether the conceptual ascription of motion need attend to the cognitive nature of its apprehension.

In another segmentation involving bars 1-2, a static model operates for the ic association for the El? of bar 1 and the D of bar 2. The order in which ics occur for the El? is 4, 5, 1 and the order for D either a) 4, 3, 1; b) 1, 3, 4; or c) * depending on how one decides to figure that order: according to a) a chronological presentation of the music, b) temporal proximity to the D, or c) all at once. Hasty's decision to show ic association by the normal form, as [1, 4, 5] and [1, 3, 4] reflects a concern for the "formed" relations of the static model, not the "forming" relations of dynamic time. Hasty bases his model of temporal structure on dynamic concepts of time and concerns himself directly with the question of motion in post-tonal music. However, by not attending to temporal factors of cognition and by relying on some static relations, the model presents more a mixture of static and dynamic time. While such a mixture is not necessarily a failure of the model, clarification of the types of information generated by differing concepts is theoretically valuable.

Lewin's article presents a theory that models our experience of metric structure and in particular our experience of a metric downbeat. While Lewin presents several successively more complex versions of the theory, my discussion considers only the first and simplest version.

Lewin's theory departs from a perceptual problem presented by Jeanne Bamberger.24 Bamberger expected listeners to respond to a series of identical pulses separated by successive durations of 2, 3, 4, and 5 units as an ametric phenomenon. The 23In the case of the retrograde contours, one might argue that the structure "is becoming" during the entire duration of the D of measure 2, since the durational similarity and thus the contour structure is not manifest until the D completes itself. 24The problem was presented during the panel "Cognitive Approaches to Music Composition and Perception," First International Conference on Computer Music, Massachusetts Institute of Technology, October 29, 1976.

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problem was that listeners heard the pulses as a metric phenomenon. Both the expected ametric response and the reported metric response are shown in Example 3a. Lewin's theory addresses the question of why listeners might respond in such a metric way. The following presentation of Lewin's theory will focus on the downbeat occurring with the fifth pulse, or in other words, the downbeat occurring after the second bar line of the "metric response" in Example 3a. Lewin argues that this fifth pulse has a unique prominence.

Lewin represents the attack points of the pulses on a time line and labels each attack point according to elapsed units of time; the units are reckoned according to Bamberger's original experiment. This time-point series is shown in Example 3b. The downbeat of present concern is annotated at time-point 14 in the example. Lewin's discussion proceeds through four chronological stages of the listening process that correspond to the time-points 2, 5, 9, and 14. Each stage represents a moment when certain durational units are perceived.

The first stage of the listening process occurs at time-point 2, at which time a durational unit of 2 has elapsed. The second stage occurs at time-point 5. At this stage Lewin claims that a listener perceives a duration of 3 units (that duration between time

Example 3a: Two responses to a series of identical pulses separated by durations of 1, 2, 3, 4, and 5 units of time

Ametric I I I I h Response m * s j

(rit.) MU Metric 4 I I I I Response 4 J J J J

Example 3b: Lewin's "Tune-point series" of the successive pulses

Metric Downbeat

1.1..1...I , , , i t = 0 2 5 9 14

Theory and Practice




points 2 and 5) and a duration of 5 units (that duration between time points 0 and 5). Further, he daims that the duration 2 of Stage 1 is retained.

For each stage of the listening process, Lewin constructs a table indicating perceived and retained durations. Example 3c shows the table for Stage 2. The uppermost row of the table indicates durational units that are possible in this instance. The row underneath tabulates the number of those durational units perceived and retained at Stage 2. At this stage, t equals 5, there are entries of 1 under the durations of 2, 3, and 5.

What kind of information the model yields at the "downbeat" of time-point 14 is crucial for Lewin's theory. This is Stage 4 of the listening process, and its durational table is shown in Example 3d. The durational entries of Stage 4 are unique with respect to prior stages because two durations, 5 and 9, have been "experienced" twice while no duration was experienced more than once in Stages 1-3. Lewin attaches special significance to the fact that the durational entries at Stage 4 are unique as such. He calls this numerical fact the "peaking" at this stage and correlates this peaking with the perception of a downbeat at time-point 14.

Lewin takes the theory out of the laboratory and uses it to consider metric structure in the sixth piece of Schoenberg's Op. 19 miniatures. The score of bars 1-6 is given in Example 4a and is annotated to show time-points. Lewin is interested in whether his model of durational peaking confirms the notated downbeat of the piano's left-hand chord in bar 5; this is the only notated downbeat of the first 6 bars. Example 4b is an annotated version of Lewin's durational table for bars 1-6. The column on the far left

Example 3c: Stage 2 of Lewin's listening process

d= 234579 12 14 number of d 110 10 0 0 0 experienced at t = 5

Example 3d: Stage 4 of Lewin's listening process

d= 234579 12 14 numberofd 1112 12 11 experienced at t = 14

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100 LOCHHEAD

shows time-points that correspond to attacks in the score, and the uppermost, horizontal row shows possible durations for this excerpt.

In Example 4b I have underlined the row of duration entries for time-point 17, the time-point corresponding to the notated downbeat of bar 5. I have also circle four entries in this row; these entries are Lewin's numerical peaking values. Time-point 17, unique with its four peaking entries, bears what Lewin calls a "maximal functional ictus" (112), and the durational peaking at time-point 17 confirms the notated downbeat of bar 5.

Lewin's model rests on a dynamic concept of time. It formulates a notion of uni- directional change by showing the transformation of perceived and retained durations over the chronological stages of a listening process. While the concept of motion is not

explicitly incorporated, it underlies Lewin's model. The durational transformations, which are charted in the duration tables, formulate the "becoming" of a structural event, in this case an ictus or downbeat. And, the chronological stages of the listening process stand in a tensed (past and present) relation to one another.

The model posits a listener and a listening process taking into account the

apperception of temporal units. The approach does not tackle the problems of listener

competence and individual differences between listeners. Nor does it consider what kinds of durational discriminations are perceptually possible. For instance, in the discussion of the metrical response in the Bamberger sequence of durations, Lewin does not consider why a listener perceives a durational unit of 2 rather than a durational unit of 1 at time-point 2; since a single unit has been heard, it seems likely that a listener would interpret it as a duration of 1 and successive units would be reckoned according to it as a standard.25 The listener in Lewin's model is not a real or ideal listener but rather a function of temporal change.26

Lewin's model does not posit the permanent relations of static time; it allows for

multiple temporal designations of a single event. For instance, a durational event may be perceived, retained at Stage 1, retained at Stage 2, and so on.27 In terms of the A- theorists, Lewin's model incorporates a changing of events.

^One might posit a sophisticated cognitive process that would compare a succession of durations and arrive at a largest common unit. Lewin does not make such an argument here but rather provides for such a process theoretically. 26 As a side issue here, it is interesting to note that Lewin's model of time resembles Husserl's model as it is articulated in The Phenomenology of Internal Time-Consciousness, trans. James Churchill (Bloomington: Indiana University Press, 1964), especially 48-52. For an analysis of Husserl's ideas on temporality, see Izchak Miller, Husserl, Perception, and Temporal Awareness (Cambridge, Mass.: MIT Press, 1984).

27In the article "Phenomenology, Music Theory, and Modes of Perception/' Music Perception, 3 (1986), 327-392, Lewin employs a similar model in which the interpretations of events change over time with the addition of new perceptual information.

Theory and Practice



The Metaphor of Musical Motion 1 01

Example 4a: Schoenberg, Six Little Piano Pieces, Op. 19, VI, bars 1-6

Time-point: 0 3 7 10 11 12.5

Sehr langsam ( J) 12 3

ii 4 iffy* i f^t^^1 i,- ^

pppp |J. ^,

13.5 16 17 19 20 20.5 22

JL^I ww - ^t'F 5>' h*. E~^j fj- v^/

%h

Used by permission of Belmont Music Publishers, Los Angeles, California 90049

Example 4b: Lewin's Durational Table of Schoenberg, Op. 19, VI [Author's annotations in bold]

a= 0.5 1 15 2 2.5 3 3.5 4 45 5 5.5 6 6.5 7 75 8

f= 3 1 7 11 1

10 2 1 2 11 1 2 2 2 1 12.5 1112 2 1 2 1 13.5 2 1 2 2 12 1 12 1

Downbeat nrtlA7. . 16 21 3222 11112 1 Downbeat nrtlA7. . ^ } 3 2 g> 2 1 1

11112 1 Q 1 C> 1

1

ot Dar 5 19 3 113 3 3 2 112 2 2 3 2 20 411343311223312 20.5 1421344321223413 22 1432354322233413

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102 LOHEAD

IH

In a famous passage from Book XI of the Confessions, Augustine laments, "What, then, is time? I know well enough what it is, provided that nobody asks me; but if I am asked what it is and try to explain, I am baffled."28 Such an observation serves us well: We know full well what rhythm and musical motion are, but our attempts to comprehend them in a formal sense, while not unsuccessful, often result in paradoxical questions about the very nature of these temporal structures.

Some of the enigmas surrounding temporality flow from the visual bias of our language and hence our conceptual models. Language equips us better to describe

phenomena visually rather than aurally apprehended.29 Sounds, like time, are

ephemeral and cannot be said to exist tangibly like a book or clarinet. Our inability to make time and sounds "stand still" so that they may be "grasped" conceptually, so that we may "see" what they are all about, may be correlated with a less developed descriptive language and conceptual framework applicable to temporal and sounding phenomena.

The discourse of theories about music's temporal structure works within similar constraints of language, constraints that bear on the problem of music's temporal structure in music theory and analysis. For example, the theoretical question of what constitutes a musical whole (a group or segment) and the analytical determination of such a whole are temporal issues related to the problem of how temporal unity itself is

possible. The problem arises from two ways of conceptualizing time that underlie the distinctions between static and dynamic time. So on one hand, time is conceived as a succession of discrete nows or moments in which distinct differences of before and after obtain. On the other hand, time is conceived as constant flow in which the continuous distinctions of past, present, and future obtain. These two views must be reconciled in order to formulate the concept of a temporal whole or unity; that is, distinct moments of succession must be unified by continuous flow.

The problem of temporal grouping and segmentation engages both these

requirements. The basis for continuity and an articulation of the events constituting a

temporal unit require both a theoretical and an analytical determination. These determinations, while not simply made and resting on issues of criteria, are essentially temporal, engaging questions of order, duration, unity (among others).

28St. Augustine, Confessions, Book XI, trans. R. S. Pine-Coffin (New York, 1977), 264.

29The philosopher Don Ihde discusses this bias in Listening and Voice: A Phenomenology of Sound (Athens, Ohio, 1976), see Chapter 1, "In Praise of Sound/' 3-16.

Theory and Practice



The Metaphor of Musical Motion 1 03

The distinction between static and dynamic time that has been our focus here touches on only a part of the larger topic of temporality, but it raises issues of significance for music. The title of this essay, while ostensibly asking if alternatives to the metaphor of musical motion exist, asks another question: What are the temporal concepts underlying thought about music and how do they bear on theoretical and analytical problems concerning temporal structure?

The discussion above showed a static model underlying Forte's theory. The structures he formulates are tunelessly true and not mind-dependent. Hasty's theory addresses the question of musical motion directly, but his structures bear traces of static relations, and Lewin's theory formulates structure according to dynamic principles. Mere identification of the type of model underlying a theory is not the goal here, but rather a means by which to comprehend what that theory can and cannot assert about musical phenomena.

Difficulties surrounding each model are various and often raise ontological questions about music itself and our observations of it. For instance, about theories based on static time, one might ask if structure can exist apart from its apprehension and whether a listener is implicit in the theory itself. One would also have to inquire after the role of the score in determinations of structure: specifically, does the visual information of the score provide a satisfactory representation of the sounding phenomena, or does visually apprehended information correspond to that which is aurally apprehended?

About theories based on dynamic time, one might ask whether motion is said to occur in the transformation of a single event or musical object, or is it a quality of relation between different events? Questions about the score arise here too: is motion attributable to the eye scanning the page in analysis or performance? How does its visual information correspond to the sounding experience of the listener? Questions about the role and capabilities of the listener arise for dynamic time: Can listeners make the kinds of temporal discriminations proposed by the theory? How does one investigate structures that either implicitly or explicitly involve a listener? These questions often border on larger philosophical issues but they are fundamental to any theoretical or analytical approach, and as such they deserve attention.

It has been my purpose here not to endorse any particular mode of temporal conception but to clarify the assumptions underlying theories about the temporal structure of music. Such a clarification allows us to better understand what a theory may or may not achieve and thus to foster progress toward a more satisfactory understanding of music's temporal structure.

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Article Contentsp. [83]p. 84p. 85p. 86p. 87p. 88p. 89p. 90p. 91p. 92p. 93p. 94p. 95p. 96p. 97p. 98p. 99p. 100p. 101p. 102p. 103

Issue Table of ContentsTheory and Practice, Vol. 14/15 (1989/1990), pp. 1-215Front MatterIntervallic Process and Autonomy in the First Movement of Debussy's Sonata for Cello and Piano [pp. 1-12]Intervallic Transformation and Closure in the Music of Stravinsky [pp. 13-34]Schoenberg's Op. 14 Songs: Textual Sources and Analytical Perception [pp. 35-58]Pitch Centricity as an Organizing Principle in "Speculum Speculi" of Charles Wuorinen [pp. 59-82]The Metaphor of Musical Motion: Is There An Alternative [pp. 83-103]The Clock Diagram: An Effective Visual Tool in Set Theory Pedagogy [pp. 105-121]The Potential and the Actual: Process Philosophy and Arnold Schoenberg's Violin Concerto, Op. 36 [pp. 123-137]Interval Cycles in Alban Berg's String Quartet Opus 3 [pp. 139-177]Function and Pitch Hierarchy in Movement II of Bartk's Fifth Quartet [pp. 179-186]ARTICLE-REVIEWThe Context of Composition [pp. 187-201]

REVIEWSReview: untitled [pp. 203-209]Review: untitled [pp. 209-215]

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