Precedents for a Theory of Projection

This chapter evaluates the work of two theorists who have discussed in detail the phenomenon of “projection.” Friedrich Neumann's concept of the rhythmic pair differs from this book's account of projection most obviously in its isolation of the pair as an autonomous “whole,” its separation of rhythm and meter, and its invocation of time point for the determination of an event's boundaries. Although Neumann describes the process through which a rhythmic pair might become unified as a “higher order” discrimination, he does not consider the process through which equality is produced and removes projection from meter in order to characterize an exclusively rhythmic order that in many respects resembles Hugo Reimann's dynamic, “organic” model. By contrast, Moritz Hauptmann is concerned with the process whereby determinate duration and equality are created and proposes a theory in which meter, quite apart from rhythm, is regarded as a dynamic, organic phenomenon arising from an innate human disposition for equal measure. The chapter then considers Hauptmann's analysis of the formation of duple meter.

General Character of the Opposition

This introductory chapter presents the opposition of rhythm and meter as a contrast of (1) rhythm as variegated pattern and meter as periodic repetition; and (2) rhythmic and metric accent or, more broadly, rhythm as event and meter as a measurement of the duration of that event. These oppositions are sufficiently commonplace that it has not been necessary to invoke the work of specific theorists except to document the most radical interpretation of metrical accent as durationless. In one form or another, these two interpretations can be found in most current writing on rhythm and meter. Although these interpretations need not be seen as incompatible, there has been a tendency in more systematic treatments to posit one or the other as fundamental. Ultimately, it is in the opposition of meter and rhythm that one encounters most poignantly the opposition of law versus freedom, mechanical versus organic, general versus particular, or constant repetition of the same versus spontaneous creation of the ever new.

Toward a Music of Durational Indeterminacy

This chapter argues that the intense reduction of projective potential observed in Webern's op. 22 and carried even further by the brokenness of the projective field in Babbitt's Du invites comparison with efforts in the years following the Second World War to eliminate meter's hold on the attention and its involvement in the formation of those more or less determinate sonic durations called “phrases.” Boulez's le marteau sans maître's nine movements offers a great variety of approaches to musical continuity. Excerpts from this work thus lead to the end of the present study—a consideration of rhythm in music that has attempted to renounce meter's efficacy in the formation of phrase. An examination of these excerpts allows for a consideration of some more general questions of rhythm and some of the novel experiences offered by “the New Music.” The chapter also discusses the distinction of “constituent” and “phrase” in detail.

Problems of Meter in Early-Seventeenth-Century and Twentieth-Century Music

This chapter assesses meter in early-seventeenth-century and twentieth-century music. Specifically, it analyzes compositions by Monteverdi, Schütz, Webern, and Babbitt. Monteverdi's “Ohimè, se tanto amate” from the fourth book of madrigals presents a metrical subtlety rarely encountered in eighteenth- and nineteenth-century music. Here the projective field is very mobile, and mensural determinacy is restricted to relatively small measures. Meanwhile, Schütz's concertato motet “Adjoro vos, filiae Jerusalem” from the Symphoniae sacrae, Book I (1629), demonstrates extremely subtle rhythmic detail and great projective contrast used in the service of a compelling larger gesture. Here the repetition of small melodic figures is used for the creation of complex projective fields that serve the continuity of phrases and sections. The chapter then looks at the much smaller measures and much greater ambiguity in some music of the twentieth century.

Obstacles to a View of Meter as Process

This chapter addresses the obstacles to viewing meter as a process in which the determinacy of the past is molded to the demands of the emerging novelty of the present. “Novelty” here implies freedom—the freedom to make decisions and to create experiences that are not mere repetitions and that are not, therefore, predetermined. Viewed in this way, meter is not distinct from rhythm as “general” is from “particular” or as “law” is from “freedom.” There are, however, two customary interpretations of meter that do support such a distinction and therefore demand refutation. The first is the reduction of meter to the mechanical repetition of equal durations conceived as habit. The second is the identification of meter with measures conceived as time spans that contain rhythmic events and which can be joined as contents of yet larger containers.

The Limits of Meter

This chapter examines the durational limits of meter. If meter and projection are synonymous, the problem is simply that of determining the durational limits of projection. If, nonetheless, “hypermeter” is intended to name a measuring of duration that is distinct from meter and yet meter-like, one might also ask how this measuring is carried out and how it is related to meter proper. To seek the maximum length of projection is to ask how far mensural determinacy can be stretched. Since mensural determinacy is gradually attenuated, evidence of projection or projective potential will, as a rule, become progressively weaker as duration increases. But since mensural determinacy can be enhanced or reduced by any factors that contribute to the particularity of the event, there can be no general or context-free limit to determinacy.

Meter as Projection

This chapter presents the term “projection” in discussing the process in which a mensurally determinate duration provides a definite durational potential for the beginning of an immediately successive event. Projective potential is the potential for a present event's duration to be “reproduced” for a successor. This potential is realized if and when there is a new beginning whose durational potential is determined by the now past first event. Projective potential is not the potential that there will be a successor, but rather the potential of a past and completed durational quantity being taken as especially relevant for the becoming of a present event. The chapter then argues that projection is nothing other than meter—that projection and meter are one.

Two Eighteenth-Century Views

This chapter discusses two viewpoints on rhythm and meter from the eighteenth century. In Der vollkommene Capellmeister (1739), Johann Mattheson systematically develops the concept of die Rhythmik as a means of uniting meter as the division or measuring of time and rhythm as the particular course this measuring takes in melody. To understand this concept, it will be necessary to explore Mattheson’s subtle and, from a twentieth-century perspective, quite unfamiliar terminological distinctions. In his attempt to reconcile what one might think of as rhythm and meter, Mattheson is not concerned with the distinction between measure and “rhythmic pattern,” but rather with the distinction between mensuration and movement and their mysterious union in Rhythmik. The chapter then considers the modern pulse theory that emerged midcentury and was most systematically formulated by Heinrich Christoph Koch in 1787. Koch does not explicitly oppose meter and rhythm, in part because he conceives of metrical grouping as a creative and spontaneous process of one's imaginative faculties. Nevertheless, what one would call “meter” and “rhythm” are opposed implicitly in Koch's theory as an opposition of unity and multiplicity or as the givenness and fixity of pulse against the creative activity of grouping.

The Spatialization of Time and the Eternal “Now Moment”

This concluding chapter assesses the attribution of spatiality and timelessness to musical events. The novel experiences offered by the New and post-New Music have been the subject of considerable speculation concerning the temporality of postwar compositions and people's experience of “time” in general. These speculations have centered on two characteristics that distinguish the new music from the old: the spatialization of time and the experience of the moment as an autonomous, timeless, or eternal present. These notions already appeared in earlier discussions of structure and of meter conceived as cyclic return. There it was argued that the spatialization of time and the autonomy of a present freed from becoming are products of conceptualization. However, in postwar avant-garde aesthetics, these categories are adamantly applied to perceptual acts.

Overlapping, End as Aim, Projective Types

This chapter looks at several passages from a single composition in which relatively large measures are formed. In the Allegro from the first movement of Beethoven's First Symphony, two large phrases or periods compose the exposition. The first of these is a virtually unbroken gesture. The second period, though more broken, is also a highly continuous gesture. Although there is a sharp break between the two periods, there is little discontinuity between the second period and the repetition of the first. The chapter then considers the concept of “overlapping,” conceived most generally as the joining of ending and beginning. Since tonal continuities and discontinuities play an essential role in overlappings, one needs to study more closely the interaction of tonal and projective potential and the notion of end as goal. The chapter also presents a rudimentary theory of “projective types” or durational patterns that present more or less specific possibilities for overlapping.