Perceiving meter in romantic, post-minimal, and electro-pop repertoires

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Fall 2015 Perceiving meter in romantic, post-minimal, and electro-pop repertoires James Edward Skretta University of Iowa Copyright 2015 James Edward Skretta This dissertation is available at Iowa Research Online: Recommended Citation Skretta, James Edward. "Perceiving meter in romantic, post-minimal, and electro-pop repertoires." DMA (Doctor of Musical Arts) thesis, University of Iowa, Follow this and additional works at: Part of the Music Commons

2 PERCEIVING METER IN ROMANTIC, POST-MINIMAL, AND ELECTRO-POP REPERTORIES by James Edward Skretta A thesis submitted in partial fulfillment of the requirements for the Doctor of Musical Arts degree in the Graduate College of The University of Iowa December 2015 Thesis Supervisor: Assistant Professor Jennifer Iverson

3 Copyright by JAMES EDWARD SKRETTA 2015 All Rights Reserved

4 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL This is to certify that the D.M.A. thesis of D.M.A. THESIS James Edward Skretta has been approved by the Examining Committee for the thesis requirement for the Doctor of Musical Arts degree at the December 2015 graduation. Thesis Committee: Jennifer Iverson, Thesis Supervisor Benjamin Coelho Robert Cook Nicole Esposito Kenneth Tse

5 To everyone and their own unique, phenomenal experience of life. ii

6 ACKNOWLEDGEMENTS There are many people to thank for helping bring this thesis to life, but here I want to direct attention to just a few. Thank you to Dr. Jennifer Iverson for your countless hours of consulting and suggesting revisions. Your mentorship has helped me grow tremendously as an academic and, perhaps more importantly, as a compassionate human being. Thank you to Dr. Kenneth Tse, who has helped me to become the saxophonist and musician I have always wanted to be. Your critical ears will remain with me for the rest of my life. Lastly, thank you to my friends and family, without whose support I would surely not have the courage to purse the things I want. Thank you for helping me grow into the person I have come to be. I wish to also express gratitude to all of humanity for the astounding achievement that is music itself. Music fills in the unknown empty spaces in my life, ultimately gluing everything together. iii

7 PUBLIC ABSTRACT Musical meter is a framework that listeners and performers use to organize music as it occurs in time. You might imagine counting along to a song on the radio or moving to the beat played by the DJ at a club. Meter is a type of cognitive synchronization anticipation mechanism: the music we perceive informs our ability to predict the music we are about to hear. How is this possible? When hearing music, listeners synchronize to the music, aligning their mental pattern of counting with the most strongly perceived musical events. Though music itself is an external stimulus, counting along is internal and occurs cognitively in the mind of the listener. What happens when the music presents events the listener does not anticipate? The listener may choose to count through the unpredictable moments, or the listener might choose to give up their old pattern of counting and adopt a new pattern. This thesis analyzes these types of situations in a variety of musical styles. By drawing on what is known about both music cognition and metric theory, I offer a means to predict how a listener may react in these unpredictable situations. These reactions depend on two factors: the emphasis or accent perceived in the audible music itself and each individual s conditioned musical memories. Historically, music theorists have characterized meter according to a strict set of relationships between rhythmic durations. This thesis shows, however, that musical meter is dynamic entity that resides within the mind of each individual listener. iv

8 TABLE OF CONTENTS LIST OF FIGURES, TABLES AND EXAMPLES vi INTRODUCTION xiii CHAPTER ONE..1 Building a Metric Hierarchy in Thom Yorke s The Eraser...16 Shifting the Pulse in CHVRCHES s The Mother We Share..25 Shifting Attention and Entrainment in Thom Yorke s A Skip Divided.38 Flexibility of Projection in Thom Yorke s The Mother Lode 59 CHAPTER TWO..71 Perceiving Meter in John Adams s Hallelujah Junction...72 Accent as Metrical Informer and Perpetuator 82 Accent and Metrical Dissonance...95 Metrical Dissonances and Changing Textures in Hallelujah Junction 108 Formal Structure as Related to Metrical Dissonance CHAPTER THREE.120 Metrical Dissonance in Schumann s Symphony in C Major, Op Main Theme and Codetta. 124 Transition to Subordinate Theme.133 Subordinate Theme..137 Perceiving Changes in Phenomenal Meter Related to the Tactus Representing Phenomenal Meter.148 CHAPTER FOUR 153 Phenomenal Meter in Rodney Roger s Lessons of the Sky.153 Perception of the Tactus Perceptual Boundaries in the Context; Projection and Perception at 100 bpm: I 161 Projection and Perception at 100 bpm: II 164 Projection and Perception at 100 bpm: III Projection and Perception at 100 bpm: IV Tactus Disrupted by Phenomenal Accent 176 Hypermetrical Organization above the Tactus Level..178 CODA..189 BIBLIOGRAPHY v

9 LIST OF FIGURES 1.1. Hauptmann, Die Natur der Harmonik und der Metrik, p Neumann, Die Zeitgestalt, vol. 2, p Projection as represented by Hasty Strata as presented by Yeston Lerdahl and Jackendoff s dot-map A representation of meter as presented by London Hierarchical periodicities for a 100 bpm quarter note LIST OF TABLES 1.1. Summary of Lerdahl and Jackendoff s Metrical Preference Rules (MPRs) Possible hierarchic configurations as tempo changes..161 LIST OF EXAMPLES 1.1a. Yorke, The Eraser, introduction in unmeasured notation b. Yorke, The Eraser, first three pulses c. Yorke, The Eraser, failure of four-pulse hierarchy d. Yorke, The Eraser, failure of two-pulse projection e. Yorke, The Eraser, three-pulse projection vi

10 1.1f. Yorke, The Eraser, three-pulse projection, two-pulse projection failure g. Yorke, The Eraser, three-pulse projection extended h. Yorke, The Eraser, introduction with waveform analysis i. 1.2a. Yorke, The Eraser, mm. 1-8 as conceived. 24 Chvrches, The Mother We Share, mm. 1-9, meter as heard b. Chvrches, The Mother We Share, mm. 1-9, meter as composed c. Chvrches, The Mother We Share, mm. 1-3, half-note projection as heard d. Chvrches, The Mother We Share, mm. 1-3, quarter-note projection as heard e. Chvrches, The Mother We Share, mm. 1-2, meter as heard with stereophonic effect f. Chvrches, The Mother We Share, mm. 1-3, meter as heard, failure of dottedsixteenth-note projection g. Chvrches, The Mother We Share, mm. 1-3, meter as heard, quarter-note projection coordinated with sixteenth-note groupings h. Chvrches, The Mother We Share, mm. 2-3, meter as heard with alignment of strong accents i. 1.2j. Chvrches, The Mother We Share, mm. 2-3, meter as reinterpreted with alignment of strong accents.34 Chvrches, The Mother We Share, mm. 1-9, meter with conservative hearing k. Chvrches, The Mother We Share, m. 6, alignment of voice patch after drum and bass entrance a. Yorke, Skip Divided, introduction, possible quarter-note pulse projection b. Yorke, Skip Divided, alternative possible quarter-note pulse projection beginning with first entry c. Yorke, Skip Divided, mm. 1-3, beginning of metric hierarchy as heard d. Yorke, Skip Divided, mm. 1-28, meter as heard e. Yorke, Skip Divided, mm , lyrics with poetic accent as heard vii

11 1.3f. Yorke, Skip Divided, mm , lyrics with poetic accent as heard g. Yorke, Skip Divided, mm , lyrics with poetic accent as heard h. Yorke, Skip Divided, mm , lyrics with poetic accent as heard i. 1.3j. Yorke, Skip Divided, mm , lyrics with poetic accent as conceived Yorke, Skip Divided, mm , lyrics with poetic accent as conceived k. Yorke, Skip Divided, demonstration of metrically malleable drum set l. Yorke, Skip Divided, mm. 1-27; with immediate entrainment m. Yorke, Skip Divided, mm ; lyrics heard as syncopated n. Yorke, Skip Divided, mm ; strong v. weak projection ascending through the metric hierarchy a. Yorke, The Mother Lode, mm. 1-4, meter as heard b. Yorke, The Mother Lode, mm. 5-25, rhythm of lyrics in meter as heard c. Yorke, The Mother Lode, mm , meter as heard d. Yorke, The Mother Lode, mm e. Yorke, The Mother Lode, mm , meter as heard Adams, Hallelujah Junction, mm Adams, Hallelujah Junction, opening motive with possible metric interpretations a. Adams, Hallelujah Junction, mm b. Adams, Hallelujah Junction, mm. 1-6, composite of both pianos Adams, Hallelujah Junction, mm. 1-15, composite score with interruptions of rhythmic motive a. Adams, Hallelujah Junction, mm. 1-15, composite score with unchanging projection schema b. Adams, Hallelujah Junction, mm. 1-15, composite score with adjusting projection schema..81 viii

12 2.6. Adams, Hallelujah Junction, mm , composite score a. Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart b. Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart inducing a three-pulse projection schema c. 2.8a. Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart inducing a three-pulse projection schema and stable metric hierarchy 89 Adams, Hallelujah Junction, mm. 1-14, composite score with accent chart and no adjustment to dotted-quarter-note entrainment b. Adams, Hallelujah Junction, mm. 1-14, composite score with accent chart; entrainment adjusting to phenomenal surface Adams, Hallelujah Junction, mm , composite score with accent chart and cardinal numbers indicating distance between strong accents Adams, Hallelujah Junction, mm , composite score with accent chart showing potential for eighth-note projection Adams, Hallelujah Junction, mm , composite score with accent chart showing potential for eighth-note projection Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart showing cardinality of various phenomenal layers a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers b. Adams, Hallelujah Junction, mm , composite score c. Adams, Hallelujah Junction, mm , composite score with accent chart showing strong accents and numerical labeling of visual layers from 2.13a a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers b. Adams, Hallelujah Junction, mm , composite score with accent chart showing strong accents and numerical labeling of visual layers from 2.14a a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers..109 ix

13 2.15b. Adams, Hallelujah Junction, mm , composite score with numerical labeling of visual layers from 2.15a c. Adams, Hallelujah Junction, mm , composite score with numerical labeling of eighth-note onset interval a. Adams, Hallelujah Junction, mm , original score with cardinalities of interpretive layers b. Adams, Hallelujah Junction, mm , onset points in a MOD-12 space (per quarter-note) Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition main theme a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , melodic rhythm giving rise to quarter-note projection b. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , melodic rhythm giving rise to quarter-note projection with three-pulse hierarchy a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition codetta b. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , first violin melody Schumann, Symphony in C Major, Op. 61, first mvmt., mm. 102 to 53, end of exposition and beginning of second statement Schumann, Symphony in C Major, Op. 61, first mvmt., mm , melodic rhythm Schumann, Symphony in C Major, Op. 61, fourth mvmt., mm Schubert, Symphony in B minor, D. 759, first mvmt., mm Schumann, Symphony in C Major, Op. 61, first mvmt., mm , transition to subordinate theme with change from three- to two-pulse hierarchy a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition subordinate theme b. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , interruption of two-pulse projection x

14 3.9c. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , change from two- to three-pulse hierarchy d. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , stable metric hierarchy e. 3.9f. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , phenomenal conflict in a continued two-pulse projection schema Schumann, Symphony in C Major, Op. 61, first mvmt., mm , adjusting entrainment to match phenomenal nadir accent g. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , adjusting entrainment to match phenomenal climax accent a. Three-pulse metric schema with a quarter-note tactus requiring adjustment at the quarter-note level b. Three-pulse metric schema with a quarter-note tactus requiring adjustment at the eighth-note level c. Three-pulse metric schema with a quarter-note tactus requiring adjustment at the sixteenth-note level Schumann, Symphony in C Major, Op. 61, first mvmt., mm , hypermetric reinterpretation with meter signatures reflecting phenomenal meter a. Rogers, Lessons of the Sky, mm b. Rogers, Lessons of the Sky, mm. 1-9 renotated to reflect quarter-note pulse a. Rogers, Lessons of the Sky, mm b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note pulse a. Rogers, Lessons of the Sky, mm b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note tactus a. Rogers, Lessons of the Sky, mm b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note pulse a. Rogers, Lessons of the Sky, mm b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note tactus xi

15 4.5c. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal tactus d. Rogers, Lessons of the Sky, mm , showing tactus shifted e. 4.4c. Rogers, Lessons of the Sky, mm , renotated to reflect dotted-eighthnote tactus 175 Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal tactus d. Rogers, Lessons of the Sky, mm , strong/weak tactus representation f. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter g. Rogers, Lessons of the Sky, mm , renotated to reflect parallelism c. 4.1c. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter Rogers, Lessons of the Sky, mm. 1-9, renotated to reflect phenomenal meter d. Rogers, Lessons of the Sky, mm. 1-9, renotated to reflect phenomenal meter locally e. 4.4e. 4.4f. Rogers, Lessons of the Sky, mm. 1-9, renotated to reflect phenomenal meter with strong/weak pulse relationship.185 Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter with strong/weak pulse relationship.186 xii

16 INTRODUCTION Here are three confounding experiences relating to the perception of pulse and meter: 1) upon listening to a composition and cognitively determining a metric framework, you look at the composition s score and discover that your aural interpretation of meter disagrees with what has been notated; 2) when listening to a piece of music, you simply cannot latch onto a steady pulse, despite the score showing unchanging meter signatures; 3) while listening to a performance or recording, you and a compatriot disagree about the location of the downbeat in a metric pattern. These three situations describe listening experiences that are incongruent with metric schemas, whether represented in notation or garnered (without notation) from experience. These situations further imply that meter arises within each individual listener and is resultant from sensing music as heard. This thesis will explore these situations from multiple perspectives and examine their theoretical and phenomenological underpinnings. Work on the perception and interpretation of meter falls under two generally distinct fields: 1) music perception and cognition and 2) analytical and theoretical approaches to metric theory. This thesis aims to present a dynamic, cognitively-derived model of metric theory that draws on both the phenomenal experience of perceiving and cognitively processing music and the theoretical work of the music theory community. In the pages to come, I present analyses which show that metric perception hinges upon both the bottom-up concerns and insights of recent metric cognition research, as well as the top-down perspective of more traditional, hierarchical metric theory. Work in perception and cognition tends to focus on the microscopic functioning of pulse perception and how those pulses give rise to a metric hierarchy, while traditional metric theory tends focus on how this hierarchy xiii

17 influences a more static organization of the perceived pulses. My analyses focus on both the micro-metric functioning of pulse perception and macro-metric functioning of hypermetric organization and ultimately shows how these different levels of metric meaning are mutually supportive. These analyses examine musical repertoire spanning a variety of western styles, including electro-pop compositions by Thom Yorke and Chvrches, a romantic-era symphony by Robert Schumann, and contemporary minimal works by John Adams and Rodney Rogers. Chapter One introduces central concepts of a dynamic, cognitive model of metric theory through an analysis of contemporary electro-pop songs by Thom Yorke and Chvrches. The songs all exhibit metric ambiguity; that is, they contain passages of irregular pulse and meter. Underlying these analyses are two fundamental concepts: metric projection and metric hierarchy. The phenomenon of metric projection historically reaches back as far as Moritz Hauptmann (1873) and Friedrich Neumann (1959). More recently, Christopher Hasty (1997) explores projection from a phenomenological perspective while examining the philosophical conflict between the static representation of musical sounds in notation with the dynamic becoming of those sounds in temporal space. Hasty s principal aim is to deconstruct the historical view of meter serving as a mere container for rhythm by arguing, rather, that it is a listener s perception and understanding of emerging rhythmic durations that give rise to a type of cognitive anticipatory schema, which I will refer to frequently as a metric schema. Metric projection is by nature hierarchical, and thus, musical meter is characterized by the coordination of at least two levels of pulse projection: a tactus level and a multiplication- or division-of-the-tactus level (Cooper & Meyer 1960, Cone 1968, Yeston 1976, Lehrdahl & Jackendoff 1983, Lester 1986, London 2012, et. al.). xiv

18 The remainder of the Chapter applies the concepts of metric projection and metric hierarchy to passages from a number of recent electro-pop compositions in which it may be challenging to discern a clear metric schema. These challenges, similar to those described in the opening paragraph, become manifest as situations in which it is difficult to perceive a stable sense of meter, in which listeners may need to adjust their sense of meter due to a changing patterns of accent perceived from musical surface, or in which a listener may discover a latent meter that emerges as the musical landscape changes. This music can be said to exhibit metric ambiguity; that is, these passages may present challenges to readily perceiving pulse and meter. Chapter Two more deeply examines the relationship between the phenomenal musical surface and the cognitive metric schema through an analysis of John Adams s Hallelujah Junction. Adams s composition poses a number of challenges to perceiving meter, as patterns of accent (phenomenal accent) frequently defy the listener s expectations (metric schema). I employ John Roeder s (2003) accent taxonomy to calculate the prominence and regularity of phenomenal accent types and use the results to suggest which phenomenal cues are most likely to induce a metric schema. The second half of Chapter Two takes the accent analysis and relates the uncovered patterns of phenomenal accent to what Harald Krebs (1999) calls metrical dissonance. According to Krebs, metrical dissonance occurs when patterns of phenomenal accent conflict with either one another or the notated meter signature. Krebs precisely defines metrical dissonances by labeling recurring patterns of accent interpretive layers according to a common duration. For example, an accent pattern of three eighth notes conflicting with a pattern of four quarter notes is designated as a dissonance of three eighth notes against eight eighth notes. Using this method, I draw interpretive layers directly from the Roeder-style accent analysis of Hallelujah Junction. Especially significant for Krebs is that an interpretive layer may xv

19 be metrically dissonant with the notated meter signature. Krebs calls these dissonances subliminal dissonances. Dissonances of this variety may be perceptually indistinguishable, for the notated meter signature is not on its own an audible phenomenon. Chapter Three shows that formal sections are often characterized by recurring types of metrical dissonance. A close analysis of the exposition to Robert Schumann s Symphony in C Major, Op. 61, shows that both the main and subordinate theme areas exhibit their own metrical characters. In this way, formal functions are conveyed not only by their melodic and harmonic characteristics, but also by the phenomenal metric qualities that cue a listeners cognitive metric schemas. Much of the exposition exhibits subliminal dissonances, for rarely do the most salient patterns of phenomenal accent align with the unchanging notated 3/4 meter signature. If a meter signature does not accurately reflect phenomenal meter, what use is it? A performer may find it particularly useful to learn a musical passage against an unchanging metric framework, especially when the contents of that passage are metrically irregular but must be performed with rhythmic precision. This then suggests the possibility for two distinct musical experiences: 1) a learned metrical understanding dictated primarily by the cognitive metrical accent and 2) a phenomenal metrical understanding dictated primarily by the phenomenal musical surface. To show how these two experiences differ, I reinterpret passages of the Op. 61 Symphony s exposition to convey the phenomenal metric experience in a way that the original, unchanging metric notation cannot. Because meter occurs in time, we must also consider the relationship between the perception of phenomenal accent and the rate at which that accent seems to occur. Chapter Four explores this relationship by drawing heavily on the work of Justin London (2012), who presents fascinating research regarding the human capacity for pulse perception. London establishes xvi

20 limits to temporal perception which point to a psychological preference for moderate tempos. Through some elegant analysis, London argues that the adult human is cognitively biased to perceive pulses at rates between approximately 86 and 100 beats per minute. I relate how this tempo preference influences the perception of phenomenal meter in Rodney Rogers s Lessons of the Sky. Rogers indicates an eighth-note tempo between 192 and 208 beats per minute, which suggests for a quarter-note tempo near 100 beats per minute. Why is this important? The vast majority of Rogers s composition is notated in meter signatures that do not suggest the performer to adopt a quarter-note pulse (e.g. 3/8, 5/8, 6/8, 7/8, 5/16, etc.). Despite this, it is only reasonable that large sections of the composition may exhibit a comfortably- perceived quarter-note pulse that defies the notated meter signatures. Using the available patterns of phenomenal accent, I renotate large portions of Lessons of the Sky to convey more clearly a phenomenal metric experience. Initial renotations strictly represent the tactus level, but by relating parallel features of phrases defined by their patterns of phenomenal accent, I derive musically intelligible reinterpretations that reflect phrase structure on a hypermetric level. These analysis ultimately show that hypermeter is not merely a theoretical derivative of quasi-symmetrical projections above the tactus but, rather, a phenomenal gestalt. In summary, meter is constructed in the listener's mind through the confluence of three factors: the hierarchical nature of metric projection, the perception of phenomenal accent on the musical surface, and the rate at which that accent occurs. At the center of this model lies the conflict between the dynamic nature of perception and cognition and the static tendencies of the metric hierarchy. My analyses suggest a symbiotic relationship between these two constructs, while respecting the limited cognitive capacity of the human brain. The final analyses ultimately represent meter as heard at all levels of the metric hierarchy. xvii

21 CHAPTER ONE In this opening Chapter I will explore examples from a number of recent popular and electronic compositions that may present challenges to readily perceiving pulse and meter. These challenges may manifest as situations where it is difficult to perceive a stable sense of meter, where listeners may need to adjust their sense of meter due to a changing musical surface, or where a listener may discover a latent meter that emerges as the musical landscape changes. Such scenarios exhibit metric ambiguity. Through these compositions I will introduce central concepts of a cognitively-based dynamic model of metric theory that are fundamental to the analytical methods I will later employ. The idea that meter is a cognitively-derived gestalt is certainly not new. In fact, support for such an idea exists in the writing of at least two authors from the late eighteenth century. Johann Philipp Kirnberger s discussion of meter in Die Kunst des reinen Satzes ( ) suggests that, If one hears a succession of equal pulses that are repeated at the same time interval experience teaches us that we immediately divide them metrically in our minds by arranging them in groups containing an equal number of pulses; and we do this in such a way that we put an accent on the first pulse of each group or imagine hearing it stronger than the others. 1 Heinrich Christoph Koch (1787) seems to agree with Kirnberger, suggesting in Versuch einer Anleitung zur Composition that the ability to differentiate and organize a series of successive pulses lies in the nature of our sense and our powers of imagination. 2 Although these isolated 1 Kirnberger 1982, 383 [1776, ]. I have added emphasis in this citation by italicizing sections which point to the cognitive functioning of musical meter. This citation is drawn from Mirka 2009, 5. 2 In der Natur unserer Sinnen und unserer Vorstellungskraft Koch 1787,

22 statements are significant insights, the writings by eighteenth- and early- to mid-nineteenth- century authors are grounded in the needs of musical composition. The metric theory therein reflects such motivations, focusing on the division and hierarchical organization of pulses into twos, threes, and fours, and the most suitable meter signature for a composition s most prevalent pulse and subdivision levels. 3 Later writings, dating from the late-nineteenth century to present day, tend to characterize and reconcile musical meter in two generally opposing views: a dynamic unfolding of rhythms as they occur across novel moments of existence and a recurring pattern of generally fixed points which serve as reference for the unfolding rhythms. Christopher Hasty eloquently summarizes the opposition of these two views in his exceptional book Meter as Rhythm (1997): Any discussion of rhythm and meter in music will involve decisions concerning the nature of time, succession, duration, and continuity--topics that are usually conceived in classical scientific terms. Moreover, an analysis of meter in which meter is conceived as cyclic repetition will explicitly invoke the discontinuity of number and will result in the representation of rhythm as a systematic whole of coordinated periodicities in which all the parts are ultimately fixed in a scheme of changeless relationships. 4 How can the experience of meter and rhythm be both continuous and recursively cyclical? Friedrich Neumann (1959) sees these as not entirely opposed, but rather functioning in tandem. He proposes this by presenting the concept of inner time and outer time: inner time is characterized by spontaneity and is inherently rhythmic, while outer time is characterized by the mathematical precision one might use when counting along with music. Neumann suggests that 3 For further reading, see Mattheson 1981 [1739], Koch 1787, and Kirnberger 1982 [1776]. Houle 1987 also presents an analysis of eighteenth and early-nineteenth century meter signatures and their uses in Meter in Music, Hasty 1997, 10. 2

23 inner time can be placed into outer time, uniting the indefinite possibilities of unfolding rhythm with the definite structure of metric hierarchy. 5 Modern metric theory has evolved to present this opposition in two essential concepts: metric projection and metric hierarchy. Metric projection is the psychological phenomenon that describes the cognitive ability to use the judgement of prior events to predict or anticipate future events. Projection has been alluded to in its most nascent form by a number of theorists in the past century and a half, chiefly Moritz Hauptmann and Friedrich Neumann. Hauptmann and Neumann both describe how the projection of a pulse requires the perceiving of at least two prior metrical events. In Die Natur der Harmonik und der Metrik (1873) Hauptmann describes how metrical determination requires two event beginnings: For the beginning of metrical determination we must take an interval of time that at first is still undivided. Two successive audible beats, supposed one second of time apart, may be the sensible image of such an interval of time. These two beats enclose only one space of time. But with the two beats we have, not one, but two [beats] determined. With the second beat, marking the end of the enclosed space of time, there is given the beginning of a second space equal in duration to the first. At the end of this second space we may expect a new beat, which, however, cannot happen earlier than at that point of time without causing an interruption, a curtailment of the time determined for us by the two beats. A single beat then cannot determine a space or magnitude of time. Rather it denotes only a beginning without an end. But with two beats following one another we obtain a whole determined in time, of which the space of time enclosed by the two beats is the half. The first metrical determination is not of a simple interval of time, but of a twofold or repeated one. A simple [impulse] is not a metrical unit, and cannot stand as a metrical whole. A single thing in metrical determination has its meaning only as part of the whole, as first or second. For the metrical whole, from its first determination onwards, is an undivided double, a twin unity. 5 Neumann 1959,

24 Figure 1.1. Hauptmann, Die Natur der Harmonik und der Metrik, p The arcs represent time intervals. Numbers 1 and 2 represent event beginnings. 6 What Hauptmann calls an undivided double, Friedrich Neumann dubs as a rhythmic pair. In Die Zeitgestalt (1959), Neumann draws conclusions similar to Hauptmann: We turn now to the inclusiveness of discriminations and begin with the simplest case in which two discriminations are contained in a discrimination of higher order. To this end we place two real or imagined, temporally adjacent events delimited by points at a certain easily comprehensible distance from one another (ex. 1). This interval might amount to about one second, but within certain limits a larger or smaller interval could be chosen, depending on the rhythmic capability of the reader. Given the two events A and B, two discriminations are defined, one from A to B and one from B to a concomitant, unknown potential limit (S), such that the intervals A-B and B- (S) are, in fact, equal (ex. 2). The existence of this potential limit is immediately known to us when a third event C enters. We are then easily, and with great accuracy, able to say whether C coincides with (S) (ex. 3a), or if it enters earlier (ex. 3b) or later (ex. 3c). Upon the fact of the potential limit, just explained, and its coming to consciousness is based the ability for time-comparison and consequently all beating of measures, counting of measures--in short, the temporal theory of measurement or Metrik. An uninterrupted whole made up of two discriminations of equal duration and determined by two events and a potential limit we shall call a rhythmic pair or also, simply, a pair. Figure 1.2. Neumann, Die Zeitgestalt, Beispielband, p Hauptmann 1873, Hasty 1997, [Neumann 1959, 18-19]. 4

25 Paul Fraisse seemingly agrees with both Hauptmann and Neumann s take on the phenomenal requirements necessary for pulse projection to begin, stating that pulse projection is acquired from the third sound on. 8 In a recent and exhaustive examination of projection, Hasty (1997) defines metric projection as a process in which the determinacy of the past is molded to the demands of the emerging novelty of the present. 9 Rephrased, metric projection is a cognitive process by which the memory and perception of past phenomenal events informs the expectation of yet-to-berealized phenomenal events. The interval of time spanning the onsets of these events can be perceived to have a measurable durational quantity, which by nature is both divisible and multipliable. Therefore, these projected events can be related hierarchically via a listener s cognition. For example, the perception of two or three consecutive events of relatively short duration may be interpreted to be the derivative of one larger event, and vice versa; in musical terms, we might define these relationships as subdivisions and measures, respectively. Figure 1.3 illustrates Hasty s concept of projection. The diagram blends the visual representations of Hauptmann and Neumann. Both Hauptmann and Neumann discuss that two beats or discriminations are required to create a determinate length. By these terms, the authors more generally mean any perceivable event. In this way, the termination of sound can be just as marked for perception as is the onset of sound. Solid lines numbered 1-4 represent sound events and their duration, while arced lines represent the duration between the onset 8 Fraisse 1982, Hasty 1997,

26 and/or termination of events, both the onset and termination being perceivable metric events. 10 Notice how Event 1 is characterized by two perceptible points: an onset and conclusion. Figure 1.3: Projection as represented by Hasty. Of these two points, the conclusion is most significant as its function is twofold: determining the duration of Event 1 (represented by Duration A) and defining a point from which we project forward the possibility that the duration of Event 1 will be reproduced. 11 The conclusion of Event 1 creates Duration A, which Hasty defines as a projective that has the potential to be reproduced. This projective Duration A informs the cognition of the projected potential of Potential A, which spans a duration that was perceived to define the duration of Event 1. Potential A is not itself a phenomenal event, but rather a mental representation of the possibility that the phenomenal event measured by Duration A will be reproduced. In this way Potential A is projected forward in time. Whether Potential A is fully realized cannot be determined until the projected duration of A has passed. Therefore, the articulation of Event 2 10 To be thorough, projection can be cued by any stimulus; a blinking light can easily be perceived to cue projection. I will use hearing throughout this document because I am interested in how projection is functioning in relation to audible events. 11 It is important to note the distinction between Event 1 itself and its duration. Event 1 is itself a unique, irreproducible phenomenon, while the duration that determines Event 1 is wholly reproducible and not at all unique. 6

27 confirms the projection. If there are no conflicting events, projection A-A will tend to perpetuate itself, as represented in the diagram by the arcs A2 and A3. The onset of Event 2 also leads to the perception of a longer, hierarchically related Duration B determined by the onsets of Events 1 and 2 and induces its corresponding Potential B. Notice that projection B-B does not interfere with the level of pulsing established by projection A-A. Rather, these relations are mutually supportive. Smaller hierarchical events can also occur without affecting the initial projection A-A. For example, Event 2 (approximately half the duration of Event 1) projects Potential C. This potential is fulfilled by the onset of Event 3. Because the length of Duration A2 is similar to that of Duration A and its Potential A, we do not experience an interruption of pulse projection. How might an event interfere with projection A-A? Note how Event 4a begins before the complete realization of durations A3 and C3 and potential duration B. The denial of fully realized projections is thusly heard as an early entrance (Line X). Inversely, Event 4b begins after the realization of B and C 2 and would be heard as a late entrance (Line Y). Hasty s theory implies that projection may give rise to a feeling of anticipation. Justin London agrees with Hasty, stating in his book Hearing in Time (2012) that musical meter is the anticipatory schema that is the result of our inherent abilities to entrain to periodic stimuli in our environment. 12 Entrainment, or more specifically metric entrainment, describes the involuntarily process by which a listener cognitively synchronizes to a series of regular pulses. This coordination, which happens involuntarily in the brain, allows for the anticipation and subsequent production of a pulse. As we will later see, the involuntary nature of entrainment significant influences one s ability to perceive a pulse and cognitively construct a metric 12 London 2012, 12. 7

28 hierarchy in situations where phenomenal cues (i.e. the perceived musical landscape) are ambiguous with respect to meter. Fraisse provides some interesting details on this generally spontaneous behavior: People fairly easily accompany [a regular succession of sounds with a motor act]. This accompaniment tends to be a synchronization between sound and tap--that is to say, that the stimulus and the response occur simultaneously. This behavior is all the more remarkable, as it constitutes an exception in the field of our behaviors. As a rule, our reactions succeed the stimulus. A similar behavior is possible only if the motor command is anticipated in regard to the moment when the stimulus is produced. More precisely, the signal for the response in not the sound stimulus but the temporal interval between successive sounds. Synchronization is only possible when there is anticipation--that is, when the succession of signals is periodic [regular]. Thus, the most simple rhythm is evidently the isochronal production of identical stimuli. However; synchronization is also possible in cases of more complex rhythms. What is important is not the regularity but the anticipation. The subjects can, for example, synchronize their tapping with some series of accelerated or decelerated sounds, the interval between the successive sounds being modified by a fixed duration (10, 20, 50, 80, or 100 msec). Synchronization, in these cases, remains possible, but its precision diminishes with the gradient of acceleration or of deceleration. The spontaneity of this behavior is attested to by its appearance early in life and also by the fact that the so-called evolved adult has to learn how to inhibit his involuntary movements of accompaniment to music. 13 Thus, when perceiving music in time listeners cognitively project event durations into the yet-to-be perceived future. Hasty s theory suggests that projection gives rise to a heuristic for anticipation. Projection is not, however, limited to precisely anticipated points in time. Rather, Hasty suggests that projection is the anticipation of a range of possibility: Since we do not, in fact, know the future, our anticipation is necessarily provisional and must not be too narrowly circumscribed. Anticipation in this sense is not the projection of a definite outcome but a readiness to interpret emerging novelty in the light of what has gone before. 14 Should the actualization of a musical event concur with a cognitive projection, the listener will continue to project durations of similar intervals, whereupon one may begin to notice a sense of regularity. 13 Friasse 1982, Hasty 1997, 69. 8

29 This sense of regularity is more colloquially termed pulse. If pulses are isochronous, that is, occurring in time at uniform intervals, it can be relatively easy to predict when future events will occur. A recurring pulse that can be easily predicted is not, however, enough for someone to develop a metric schema. Stable meters feature pulse regularity, but, as was noted by Kirnberger and Koch, stable meters also feature an alternation between the relative strength of pulses. The sense of pulse strength is related to both cognitive and phenomenal cues, as pulses are organized in a hierarchical manner which privileges the most strongly perceived pulses. Theories in support of the hierarchical view of meter have become ubiquitous in the late twentieth century. Indeed, an entire volume could be written reviewing such literature, and no modern publication is complete without a brief overview. 15 For example, Maury Yeston (1976) argues from a top-down, Schenkerian perspective that musical meter is the result of the interaction of various pulse strata: In order to create some regular grouping of elements within a simple pulse, there must be some event occurring at regular intervals within it. Such an event may be sounded in the music, or it may be a purely conceptual [cognitive] division of the pulse. Here the conceptual act of considering the pulses of level A in pairs occurs once for every two pulses of level A [Figure 1.4]. This recurrent act of grouping, whether it is conceptual or whether it is represented by something in the music, then becomes a pulse itself (B), having a rate of recurrence that is necessarily slower than the rate of level A. The fundamental logical requirement for meter is therefore that there be a constant rate within a constant rate--at least two rates of events of which one is faster and another is slower. In view of these two necessary rhythmic strata, the question must now be asked: On which level does the meter appear--on level A or on level B? Clearly there is no meter on level A since, by itself, it is ungrouped. Furthermore there is no meter on level B since, 15 Mirka 2009, Hasty 1997, London 2012, Krebs 1999 all spend a significant number of pages in their recent books reviewing the literature of Yeston 1976, Cooper & Meyer 1960, Lerdahl & Jackendoff 1983, and others. 9

30 by itself it is a simple pulse with no slower rate of events (conceptual or otherwise) by which it may be grouped. There is apparently, then, no such thing as a level of meter or a level on which meter may appear; but rather, meter is an outgrowth of the interaction of two levels--two differently-rated strata, the faster of which provides the elements and the slower of which groups them. 16 Figure 1.4. Strata as presented by Yeston. 17 Yeston s view of this hierarchy is elaborated upon by Lerdahl and Jackendoff (1983), who suggest that meter exists as the result of the alternating relationship of strong and weak points in time, or beats, between which results a measureable duration: Fundamental to the idea of meter is the notion of periodic alternation of strong and weak beats. For beats to be strong or weak there must exist a metrical hierarchy two or more levels of beats. The relationship of strong beat to metrical level is simply that, if a beat is felt to be strong at a particular level, it is also a beat at the next larger level. In 4/4 meter, for example, the first and third beats are felt to be stronger than the second and fourth beats, and are beats at the next larger level; the first beat is felt to be stronger than the third beat and is a beat at the next larger level; and so forth. 18 This alternation of perceived strength is famously represented using a dot map, reproduced here in Figure 1.5. If the beats in this example were imagined as quarter-note durations, the alternation of strong and weak at the quarter-note level creates a new level defined by the strong beats (one and three). These beats themselves also alternate relative strength, as indicated by the stress map. Theoretically, this hierarchy can extend to include infinite shorter and longer 16 Yeston, Ibid. 18 Lerdahl and Jackendoff 1983,

31 durations, but in practice such a hierarchy will only include those durations within the bounds of human perception. Figure 1.5. Lerdahl and Jackendoff s dot-map. 19 The authors go on to specify constraints on such a metric hierarchy, whereby all durations are either multipliable or divisible by one another. These constraints also describe the tendency to synchronize to a pulse at an intermediate level of the hierarchy: An important limitation on metrical grids for classical Western tonal music is that the time-spans between beats at any given level must be either two or three times longer than the time-spans between beats at the next smaller level. For example, in 4/4 the lengths of time-spans multiply consistently by 2 from level to level; in 3/4 they multiply by 2 and then by 3; in 6/8 they multiply by 3 and then by 2. Typically there are at least five or six metrical levels in a piece. The notated meter is usually a metrical level intermediate between the smallest and largest levels applicable to the piece. However, not all these levels of metrical structure are heard as equally prominent. The listener tends to focus primarily on one (or two) intermediate level(s) in which the beats pass by at a moderate rate. This is the level at which the conductor waves his baton, the listener taps his foot, and the dancer completes a shift in weight. Adapting the Renaissance term, we call such a level the tactus. The regularities of metrical structure are most stringent at this level. 20 Recent work by Justin London (2012) relates the perception of meter to tempo, and his visual representation of meter displays the hierarchy of relationships suggested by the former authors. London, however, prefers to represent meter in a cyclical manner; Figure 1.6 is a representation of 4/4 meter and its potential hierarchies. Dots represent beats, while the solid 19 Ibid., Ibid.,

32 circle represents the beat or tactus level: four dots are equally spaced along this circle representing four quarter-note beats. Dashed arcs represent potential hierarchical relationships, with dots indicating either duplet or triplet eighth notes and the potential half-note duration. Figure 1.6. A representation of meter as presented by London. 21 While London understands meter to be a fluid and dynamic attentional process, his visual representation, like those by Yeston, Lerdahl and Jackendoff, and others, is limiting. The author addresses this issue: these representations do seem to be akin to traditional metric analyses in that they are static abstractions from real-time processes. This is a fair enough assessment, for any printed two-dimensional picture of musical meter will have to be a kind of snapshot of a dynamic process. I would emphasize, however, that what is important here is not the structure per se, but the set of relationships (and the constraints under which the relationships may occur) that these metric diagrams aim to capture London 2012, Ibid.,

33 London goes on to emphasize the importance of these relationships to developing a sense of meter: Most theories of rhythm and meter have focused on the formation of measures and larger units. But the metric foreground that is to say, the subdivision levels of the beat has an equally significant import on our metrical attending and hence the meaning and motional qualities of a musical gesture. 23 While the authors may disagree on the finer details regarding how these levels interact with a metric structure, it is agreed that a stable sense of musical meter forms as a result of interactions between the levels of a metric hierarchy. Common to all authors is the view that musical meter requires the coordination of at least two levels of projection: a pulse (tactus) and a division or multiplication of the pulse (subdivision or hypermeter). Because any recurring duration within the bounds of human perception (which will be discussed later) can exhibit a pulse, one could likely perceive a half-note duration to be a subdivision of a whole-note pulse. 24 Consider the following situation that gives rise to a stable metric schema: When listening to a series of recurring quarter notes the listener perceives articulations to be occurring at regular intervals without interruption. The regularity with which the quarter notes occur gives rise to a projection schema where one expects to hear continued quarter-note articulations. With more careful listening, the listener hears an accented pitch every third quarter note. Should this accent occur every third quarter note for a sufficient period of time, the listener will come to expect and 23 Ibid., It is important to dissociate the representation of a duration in western notation with the tendency to associate a typically long representation, such as a whole note or dotted half note, with a long duration. Durational representation of modern notation is tempo dependent, meaning that any note value may be perceived as the tactus depending upon the composer s preferences. Durational representation can also be culture dependent. For example, Brazilian styles, such as samba, frevo, and chorro, tend to be notated in 2/4 time, with sixteenth notes representing the most rapid durations, while arrangements of these styles by North American publishers tend to be notated in cut-time with eighth notes representing the most rapid durations. 13

34 be able to predict the interval at which these accented quarter notes will occur. These two interacting levels of projection, the individual quarter-note pulse and the larger, every-thirdquarter accent, will likely lead to the adoption of a cyclical, three-beat metric pattern of one strong pulse and two weak pulses (commonly understood as triple meter). In contrast, consider a different situation that gives rise to metric instability: When hearing a series of repeated quarter notes, the listener perceives a stable pulse defined by recurring articulations of equal duration. However, the interval between pitch articulations is irregular: sometimes pitches are articulated consecutively, while other times one or two silent pulses separate the articulations. A lack of regular projection on multiple levels may prevent the listener from entraining to any predictable or regularized metric schema. Therefore an ambiguous sense of meter will pervade the listening experience. The opening passage of Thom Yorke s composition The Eraser, which I will discuss at length in a moment, introduces this type of metric ambiguity; despite the presence of a clearly-defined quarter-note pulse, the consecutive quarter-note articulations are separated by quarter rests at irregular and unpredictable intervals. This irregularity presents the listener with a challenging perceptual situation, and it may be difficult to decide which metric schema should be adopted. Generally speaking, metric ambiguity can result whenever uniform pulse projection is absent from at least one or more levels of the metric hierarchy. Consider a third hypothetical scenario in which a musical passage will cue in the listener a metric schema that is later deconstructed or challenged; that is, the opening metric schema is revealed as provisional, temporary, or even incorrect once the listener experiences a wider context. Listeners will use the available phenomenal, surface accents for metric projections, even if those accents are later revealed to be "incorrect" or not indicative of the prevailing meter. 14

35 Ethnomusicologist Nathan Hesselink (2014) describes how this particular situation occurs in The Police s tune Bring on the Night. When listening to performances of this tune, audiences tend to naturally entrain to the drum set s accented high-hat. A short while later the bass drum enters, and its articulations are heard to occur on the off-beats. Any listener familiar with western popular music generally accustomed to hearing bass drum articulation fall on beats of a metric schema. Eventually, as the tune plays out, the accented high-hat articulations are, in fact, intended to occur on the off-beats, while the bass drum articulations occur on beats. Because a listener will frame the rhythm of that melody against a metric schema developed from audible cues, it is likely that a listener will experience the composition s melody differently, as if a painting of a still-life were set against a different background; the work truly becomes a new being with the potential to elicit a distinctly different feeling. All three of these hypothetical scenarios reinforce the thesis that pulse projection and hierarchy interact to condition the listener s experience of meter. My analyses in this Chapter focus particularly upon songs that induce entrainment errors or that force the listener to reconsider their pattern of entrainment. Such events may cause listeners to notice a hiccup in their entrainment. By focusing on these ambiguous metric situations, we will gain a deeper appreciation of the cognitive dimensions of meter, based on the dual foundations of metric projection and hierarchy. In the following sections, I analyze electropop tunes by Thom Yorke and Chvrches. Common to these compositions is a regular metric foundation: each composition can be conceived with unchanging, four-beat metric patterns. As just mentioned, however, these examples elicit metric ambiguity when conflicting metric information is introduced, often mid-tune. This conflicting information reflects the emergence of a metric schema that was only latent at the beginning of the tune, pushing aside the faux metric 15

36 projection and entrainment. In this sense, the listener s fluctuating, unstable, or shifting metric experience is much different than the regularity that might be implied by a transcription represented in common time notation. Based on these experiences and through later supporting material, I hope to add gravity to the argument that meter is not a notated phenomenon, but rather a fluid and flexible schema residing in the listener as a cognitive result of the interaction between projection and hierarchy. I believe this assumption is critical to how we experience music and can help explain how we come to understand not only meter, but also phrase structure, hypermetric grouping, and large scale form. Building a Metric Hierarchy in Thom Yorke s The Eraser To demonstrate how metric projection and hierarchy interact in a metrically ambiguous situation let s look at the opening passage of Thom Yorke s composition The Eraser. In this example metric ambiguity arises from incongruences between potential metric hierarchies and the phenomenal accent exhibited by each event. Let s examine this situation in detail by first examining pulse projection and then relating such projection to potential metric hierarchies. Example 1.1a shows the opening passage of The Eraser in unmeasured notation. In the transcription, I have assigned the most salient level of pulse perception (tactus) to the quarter note. Upon first hearing listeners will surely notice a sense of regular pulse. At what point will the listener begin to project such a pulse? While this may be subject to the attentional energy of each individual listener, let s assume optimal attention to the musical surface. As discussed earlier, pulse projection can begin fully after hearing two perceptually relatable event beginnings, whereby the second event onset determines a definite duration to project forward. Without a second beginning, no determinate duration can be assigned to the event which was initiated by 16

37 the first beginning. Recall the earlier discussion of Neumann s rhythmic pair and Hauptmann s undivided double ; the second onset s function is twofold, determining a duration to the first event and beginning a duration to its own event. In Example 1.1b we see that a second onset does not appear until after two quarter-note beats. We can thus fairly assume that pulse projection B B' is not realized until the articulation of the pitches on the third quarter-note beat. Hypothetically, if no further stimuli were presented, projection theory holds that the listener will continue to project half-note durations into the silent future until the listener s attentional energy becomes diverted. The reality, however, is that a new quarter-note articulation succeeds immediately on the fourth quarter-note beat, creating a new rhythmic pair. In Example 1.1c these successive quarter-notes give rise to projection A A'. Despite projection A A' being immediately succeeded by a silent beat, I suspect listeners will similarly continue projecting quarter-note pulses through the silence. And as the music unfolds, projection A A' will be reinforced; in Example 1.1c quarter-notes resume immediately following the silence on the fifth quarter-note beat. Example 1.1a. Yorke, The Eraser, introduction in unmeasured notation. Example 1.1b. Yorke, The Eraser, first three pulses Projection maps adapted to the examples in this document are drawn from Hasty

38 Example 1.1c. Yorke, The Eraser, failure of four-pulse hierarchy. There should be little ambiguity regarding this level of pulse projection; I expect nearly all listeners will effortlessly entrain to a quarter-note tactus. Remember, however, that meter is more than a simple series of isochronous pulses. London describes this problem and reiterates the importance of a hierarchy of pulse relationships: The tactus establishes the continuity of musical motion; without it, no sense of meter is possible. But a tactus, in and of itself, is insufficient for a sense of meter. The tactus establishes a single periodicity, and to be sure, this does give the listener a limited degree of temporal expectancy: something should happen on the next beat. Yet while this is a type of entrainment, if we were to make a representation of this entrainment in metric terms, it would be a series of one-beat measures... At minimum, a metrical pattern requires a tactus coordinated with one other level of organization. 26 Recall how Lerdahl and Jackendoff suggest that meter exhibits an alternation between perceived strength of pulses. If we were to imagine meter cyclically, as represented by London, the greatest perceived emphasis will be assigned to the beginning of the cycle, with emphases of a lesser degree spread equidistantly throughout the cycle. In the introduction of The Eraser in Example 1.1a we see, however, that there is a great degree of rhythmic irregularity, preventing an even distribution of strong and weak pulses. A simple cardinal number analysis reveals this irregularity: the onset of each uninterrupted series of quarter notes is separated by 2, 3, 5, 3, 8, 7, and 2 pulses, respectively. This irregular pattern of rests prevents the listener from discerning a 26 London 2012,

39 regular alternation between strong and weak beats, ultimately conveying an ambiguous sense of metric hierarchy. In this situation, ambiguity arises as a result of the interaction between two perceptual factors: isochronous tactus projection and inconsistent projection at levels above the tactus. Related to these points is the degree of audible prominence (phenomenal accent) which each projection exhibits. I will return to this point shortly, but for now, let s look at how tactus and super-tactus levels affect the development of a metric schema. There are two general possibilities for selecting a metric schema in this passage of The Eraser: a two quarter-note pulse hierarchy and a three quarter-note pulse hierarchy, each of which exhibiting a quarter-note tactus-level projection A A'. In Example 1.1c what happens with projection B B'? Notice that the whole duration of projection A A' is the same durational value as that of the projected duration B', and that the projected durations A' and B' both discharge their potential on the fifth quarter-note beat. This is a prime example of the nascent becoming of metric hierarchy, and were no further music to be presented, it is quite reasonable that a listener could continue to project a relatively stable pattern of either quarter-note pulse durations or halfnote pulse durations reinforced by a sense of hierarchical relation. In Example 1.1d the B-level projection is extended further to show how a two quarter-note pulse may coordinate with the unfolding music. Remember that significant musical cues must be present for projection to be strongly reinforced. While such cues are present in Example 1.1d on the first and third quarternote beats, the silence (*) on the fifth quarter-note beat denies the listener the anticipated audible cue. It is at this point that I believe listeners may question a potential two-pulse level of projection. Is it possible for the listener to project a four quarter-note pulse? While a two-pulse and four-pulse projection are hierarchically related, the unviability of a two-pulse projection does not 19

40 preclude the possibility for a four-pulse pattern to emerge. In Example 1.1c one might reasonably perceive a longer projective potential C that is begun by the initial quarter note. However, the lack of new beginning (second beginning) on the fifth quarter-note beat denies C closure and prevents the emergence of a four-pulse projection. While it is certainly possible for the previously examined half-note projection to continue, further musical cues do not strongly support such a projection schema, leaving open the possibility that another projection schema may better coordinate with the musical surface. Example 1.1d. Yorke, The Eraser, failure of two-pulse projection. In Examples 1.1b and 1.1c the projective potential C provides such an opportunity, and as the music continues to unfold in Example 1.1e, we see a twice-articulated pattern, quarter notequarter rest-quarter note, allow for the becoming of projection D D'. Considering a hierarchy based on three pulses, the listener may reinterpret what has been formerly heard in a new metric context. In Example 1.1f we see such a reinterpretation: whereas the tactus projection A A' continues uninterrupted, the new beginning on the fourth quarter-note beat and the lack of new beginning on the fifth quarter-note beat deny the realization of the potential B'. This denial results in projection B-B' being replaced by D D'. Projection D D' is extended into the future in Example 1.1g. Though initially viable, we come to hear that projection D D' also lacks consistent reinforcement: in Example 1.1g quarter rests occur at the beginning of the fourth and fifth three-pulse projections. 20

41 Example 1.1e. Yorke, The Eraser, three-pulse projection. Example 1.1f. Yorke, The Eraser, three-pulse projection, two-pulse projection failure. Example 1.1g. Yorke, The Eraser, three-pulse projection extended. As I ve shown, the rhythm of the pitch events in the The Eraser s opening passage does not clearly inform the development of either a duple or a triple metric hierarchy. Let s try a different approach. Earlier I mentioned that metric schemas are informed by the prominence of phenomenal accent exhibited by the events perceived on the musical surface. Lerdahl and Jackendoff define phenomenal accent as any event at the musical surface that gives emphasis or stress to a moment in the musical flow. 27 Critically, phenomenal accents are defined by the relative strength or prominence exhibited by an auditory stimulus in its surrounding context. 27 Lerdahl and Jackendoff 1983,

42 How do phenomenal accents relate to one s construction of a metric hierarchy? Lerdahl and Jackendoff relate phenomenal accent to such a hierarchy: Phenomenal accent functions as a perceptual input to metrical accent that is the moments of musical stress in the raw signal serve as "cues" from which the listener attempts to extrapolate a regular pattern of metrical accent. If there is little regularity to these cues, or if they conflict, the sense of metrical accent becomes attenuated or ambiguous. If on the other hand the cues are regular and mutually supporting, the sense of metrical accent becomes definite and multileveled. 28 By metrical accent Lerdahl and Jackendoff refer to the cognitive emphasis that a listener associates with cognitive projections, which thereby gives rise to a metric hierarchy. Metrical accent may be more colloquially understood by the ubiquitous counting along process that takes place in the listener s head. Hauptmann suggested for this type of accent when describing that in the relationship between two determinative events, the determining quality of the second event gives the first event its perceived emphasis: A first as against its second has the energy of beginning, and consequently the metrical accent. In the two-time metre the first member is accented, the second is without. 29 London suggests that that these metrical accents are marked by consciousness. 30 Recall the earlier discussion regarding strong and weak beats. One can fairly assume that an alignment of audible strong and weak beats greatly affects the ability of one to mentally project strong and weak beats into the future. Lerdahl and Jackendoff share further insight: In sum, the listener's cognitive task is to match the given pattern of phenomenal accentuation as closely as possible to a permissible pattern of metrical accentuation; where the two patterns diverge, the result is syncopation, ambiguity, or some other kind of rhythmic complexity. 31 That is to say, while phenomenal and metrical accents are different 28 Ibid. 29 Hauptmann 1873, London 2012, Lerdahl and Jackendoff 1983,

43 phenomena, the musical surface (phenomenal accent) shapes and informs the cognitively emerging metric hierarchy (metrical accent). How does this apply to the example at hand? One might realistically interpret any or all of the pitches that occur in Example 1.1a to be raw signal cues, with each pitch articulation functioning as an accented pulse in a yet-to-be clarified metric context. However, as the sonogram below Example 1.1h clarifies, listeners should notice that each set of pitches is articulated with varying degrees of emphasis. For example, attentive listeners should notice that the pitches on the fourth quarter-note beat are articulated with greater emphasis than those on the third quarter-note beat. Although these accents are subjectively measured by each listener, they may be more objectively measured by analyzing a recording s waveform. In Example 1.1h a sonogram shows the measurement of sound volume associated with each pitch articulation. Generally speaking there are two intensities of accent: a strongly-accented articulation (quarternote beats 1, 4, 6, 9, etc.) and a less-strongly-accented articulation (quarter-note beats 3, 7, 8, etc.). For the purposes of discerning a metric schema, these strongly-accented articulations are more salient, as they will tend to attract the most attention from the listener. Relate the accents in Example 1.1h to the various projection schemas in Examples 1.1b-g. Based on the degree of phenomenal accent exhibited by the passage, it seems the most congruent projection schemas are represented by Examples 1.1e and 1.1f: with strong accents on the first and fourth quarter-note pulses, a three-pulse projection schema is at first most strongly reinforced. However, the congruence between phenomenal accent and a three-pulse projection schema lasts only these first four quarter-note pulses, and I suspect that the feeling of a three-pulse projection will disintegrate when the accented pitches on the sixth quarter-note beat are articulated, thus giving rise to metric ambiguity. 23

44 Example 1.1h. Yorke, The Eraser, introduction with waveform analysis. So, after this ambiguous beginning, what is the real meter in The Eraser? 32 If the audience were to continue listening beyond Example 1a, they would hear the same passage repeated, this time, however, accompanied by rhythm instruments whose patterns recontextualize the passage, making obvious to the listener an undeniable four-pulse metric pattern. Listen once more to the excerpt; Example 1.1i shows the passage barred into four-beat measures. While the silences on the downbeats of mm. 2, 4, and 6 do not reinforce the pattern, the downbeats of mm. 1, 3, 5, 7 and 8 do. Of special significance is the new-found alignment of the harmonic motion at m. 7. I surmise that the listener who deliberately projects a four-pulse metric pattern and listens beyond the phenomenal accent on the musical surface will experience a less ambiguous metric situation. However, unless the listener knows this information beforehand, from a previous hearing, the cues on the musical surface make it virtually impossible to infer a four-pulse metric pattern on the very first hearing. 32 Thom Yorke undoubtedly had a metric schema in mind when he composed the tune. Yet, while some metric patterns are more easily entrainable due to congruencies between phenomenal accent and metric accent, no one metric schema is more real, so to say, than any other. 24

45 Example 1.1i. Yorke, The Eraser, mm. 1-8 as conceived. Shifting the Pulse in CHVRCHES s The Mother We Share Let s examine another type of situation where the musical surface may cue in the listener a change of metric schema. In the opening of Chvrches s The Mother We Share listeners may notice a regular pulse and an unambiguous four-pulse metric schema; in Example 1.2a this pulse is transcribed as the quarter note. 33 I suspect this metric schema is cued by a perceived downbeat 33 Before continuing further, I wish to comment on the inherently imperfect representations of my perceived hearings. While I am arguing for a cognitively-based dynamic model of meter, the only reasonable means to represent metric perception is by using a system that is familiar to musicians. This means presenting analyses using statically notated measures with bar lines. Example 1.2a is perhaps the best example of such imperfection. For example, at m. 5, it is impossible to know during the experienced listening that an incomplete metric cycle (i.e. measure, notated here as a 7/8 measure) has occurred until the phenomenal cues of the bass and drums recontextualize the vocal patch. In actuality it is more accurate to say that in m. 5 the listener hears a 4/4 measure that has been cut short. When listening to an unfamiliar score changes in metric schema can only occur after the perceiving of cues that suggest the schema should be altered, directly contradicting the dictatorial ability that seeing a change in meter (time signature) informs the prediction pattern to follow. Thus, familiarity with the score fundamentally alters the listening experience; having knowledge of an impending variation in metric regularity may allow for the adjusting of one s metric schema immediately prior to or presently during unfolding metric turbulence. Therefore, these representations should be understood to represent a listener s post-experience understanding, as opposed to the process by which such an understanding was achieved. For an excellent demonstration of how one might 25

46 at the first entrance of the female vocal patch and solidified in the same notated measure by the perception of articulations on an anacrustic fourth beat. This entrainment pattern should feel relatively stable and regular through the first three measures. Example 1.2a. Chvrches, The Mother We Share, mm. 1-9, meter as heard. As the excerpt continues to unfold, however, listeners may experience an entrainment hiccup: while entraining to a metric schema whose pattern begins with the initial voice entrance is at first reinforced, the entrances of the drum set and bass in mm. 5 and 6 conflict with this schema. Much like in the previous example by Thom Yorke, the second statement of The Mother We Share s opening melody is recontextualized in mm. 6-9 against the widely used backbeat schema: while the opening vocal melody initially seems to begin on the downbeat (m. demonstrate adjustments in the metric attending process, I strongly encourage seeing Gretchen Horlacher s 2001 article Bartók s Change of Time: Coming Unfixed. Horlacher presents a dynamic representation of Hasty s projection theory as the music unfolds in Bartók s Change of Time from Mikrokosmos. 26

47 2), the reframed melody is seemingly shifted by one eighth-note, such that it now begins on the off-beat of beat one (m. 6). The backbeat schema is so prevalent in contemporary popular music that I suspect in either mm. 6 or 7, the listener will drop their prior entrainment pattern and adopt the new pattern with relative ease, based on the now unambiguous metric orientation provided by the backbeat. Thus, the changing musical surface reveals a latent metric schema. Many listeners may not even realize that they have falsely entrained to the composition s opening passage. This is likely dependent on the quality of attention directed to actively keeping time. It is challenging to precisely know, as each individual s listening experience is a unique, irreproducible phenomenon. The situation encountered by the reframing of the melody in mm. 6-9 may cue what Mark Butler describes as metrical reinterpretation, where the entrance of a new textural layer calls a prior metrical interpretation into question. 34 Such reinterpretation suggests that we reconsider the perceived meter of the opening material, and considering that the remainder of the tune can be comfortably experienced with the entrainment pattern suggested by Example 1.2a, it seems unlikely that Chvrches would compose a solitary measure of asymmetrical meter. Example 1.2b shows this material with metric notation that is parallel to that observed in mm. 6-9: the voice is silent on the downbeat of m. 2 and enters on the off-beat one eighth note later. Butler suggests that such silent downbeats have the potential to create ambiguous beginnings. 35 It is only reasonable that a listener tends to associate the first perceivable event with the beginning of a metric schema, as this first event together with its immediately following event will activate projection. 34 Butler 2003, Ibid.,

48 Example 1.2b. Chvrches, The Mother We Share, mm. 1-9, meter as composed. As I have been discussing, however, I tend to hear no perceivable ambiguity. Despite its incorrectness, the musical surface notated in Example 1.2a cues a relatively clear and unambiguous metric pattern. Why is it that a listener could select this false meter? As a general rule, if there is no rhythmic phenomenal material to which a listener can metrically entrain, a listener will be unable to actively (conscious effort) or passively (unconscious attending) project a pulse coordinated to the musical surface. If the downbeat doesn t sound, we don t hear it. Fair enough. Yet, even with the privileged, retrospective knowledge gleaned through repeated listening and/or analysis that the initial vocal patch entrance occurs on the off-beat, it is in my experience nearly impossible to comfortably and accurately maintain an entrainment pattern that corresponds to the correct or latent metric schema represented in Example 1.2b. With this in 28

49 mind let s more thoroughly examine the opening of The Mother We Share to discover why the entrainment schema in Example 1.2a is so challenging to override. The first sound heard is that of the synthesized keyboard s sustained chord; its entrance is relatively indistinct and lacks apparent rhythmic activity. While this first sound sets the metric attending process in motion, recall that projection cannot begin in earnest until a second beginning is perceived, such as is heard when the voice initially enters. It is possible to argue that the differences between the synthesizer and vocal patch in both timbre and quality of articulation may lead the listener to hear these two initial sounds as separate, metrically unrelated streams, especially when considering the great degree of metric activity which characterizes the ensuing material of the vocal patch. The idea that listeners will understand a musical surface as comprised of separate streams is proposed by Albert Bregman (1990). Thus, my suspicion is that some listeners may not begin the projection process until the vocal patch s second pitch, the first of three A 4s. Nevertheless, the duration between the onset of the synthesizer and the onset of the vocal patch does have a determinate value. This duration is impossible to precisely discern during the first hearing (and still challenging to discern after multiple hearings), but a quantized sonogram suggests this duration to be approximately 1.35 seconds. Considering that a notated measure lasts approximately 2.7 seconds, we might assume the initial duration to be mensurally notated as a half note. Indeed, if one were to listen to the song and entrain to a quarter-note or half-note pulse, then immediately restart the tune and begin consciously projecting a similar pulse at the moment the faintest sound is perceived, it is plausible that one may hear the entrance of the vocal patch coinciding with either a second half-note or third quarter-note pulse projection (Examples 1.2c and 1.2d), and thus helping to solidify the first-vocal-sound-as-downbeat schema. 29

50 Example 1.2c. Chvrches, The Mother We Share, mm. 1-3, half-note projection as heard. Example 1.2d. Chvrches, The Mother We Share, mm. 1-3, quarter-note projection as heard. Is it possible to further extend the model of projection proposed in Example 1.2c? While the half-note projection is confirmed by the dotted-eighth note on the downbeat of m. 1, the tie from the off-beat of beat 2 into the downbeat of beat 3 prevents a confirmation of the half-note pulse. If the listener were to actively project this pulse through this incongruency, they may experience validation when their pattern of projection aligns with the down-beat of m. 2. Even so, I believe this makes for a relatively weak argument to support a half-note projection. More plausible is an extension of the quarter-note projection, demonstrated in Example 1.2d. Though it may be impossible to begin projecting a quarter-note pulse at the onset of the synthesizer entrance, post hoc logic reveals a quarter-note projection schema that seems more reasonable than the previously discussed half-note schema. Example 1.2d shows pulses aligning on m. 1, beats 1, 2 and 4, as well as the down-beat of m. 2. This rudimentary analysis is, however, somewhat naive; we should also consider the phenomenal accent present in the vocal patch. Which audible cues attract the greatest degree of attention? In this instance, the greatest degree of phenomenal accent appears to be associated with pitch changes in the vocal patch, indicated in Example 1.2e by notated accents. Those listening to the song, particularly those 30

51 using headphones, may notice a stereophonic effect that amplifies these accents: pitch change is coordinated with a change in output channel, as indicated in the example. If we were to disregard the accent associated with the pitch change from E 4 to F4 during the third beat, we notice a recurring dotted-eighth note duration. One might then reasonably suggest that, despite the previous discussion of the role of the opening half note s duration, a dotted-eighth note projection schema has the potential to emerge, shown here in Example 1.2f. In this scenario the tenuous half-note potential A' is rapidly replaced by the dotted-eighth note projection B-B'. And this seems a plausible representation, that is until the end of m. 2 where a solitary extra sixteenthnote duration, notated here as a sixteenth rest, prevents the schema from aligning with measure two s down-beat. This syncopated pattern of dotted-eighth notes is a relatively common convention in contemporary rock, pop, and electronic genres. Thus, I suspect that when a listener recognizes this convention, made apparent by the previously mentioned extra sixteenth note, they will adapt to the convention and instead project a quarter-note pulse through the parallel syncopation in m. 3. Upon second hearing, this privileged knowledge now in tow, a listener should be able to project a quarter-note pulse with relative ease, now with the additional reinforcement of accents perceived to be on beats 1 and 4. Broken down to lowest level of subdivision, one might reasonably perceive an underlying sixteenth-note hierarchy represented in Example 1.2g. Example 1.2e. Chvrches, The Mother We Share, mm. 1-2, meter as heard with stereophonic effect 31

52 Example 1.2f. Chvrches, The Mother We Share, mm. 1-3, meter as heard, failure of dottedsixteenth-note projection. Example 1.2g. Chvrches, The Mother We Share, mm. 1-3, meter as heard, quarter-note projection coordinated with sixteenth-note groupings. Significant theoretical work suggested by Lerdahl and Jackendoff supports the plausibility of a listener projecting a quarter-note pulse through the syncopation. In their landmark book A Generative Theory of Tonal Music (1983) the authors put forth a set of Metrical Preference Rules (MPRs) (Table 1.1) that suggest which metric entrainment pattern a listener will prefer when presented with multiple options for a well-formed metric environment (eg. preferring Example 1.2a over 1.2b, despite Example 1.2b s viability). Of particular relevance to The Mother We Share is the third metric preference rule MPR 3, Event : prefer metric schemas where the onset of pitch events coincide with pulses of metric projection, and the fourth metric preference rule MPR 4, Stress : prefer metric schemas in which strongly accented 32

53 events (phenomenal accent) coincide with strong beats of metric projection (metrical accent). 36 When applying these principles to our two potential schemas in Examples 1.2h and 1.2i, we see that when the initial vocal patch entrance is heard as the downbeat (Example 1.2h), phenomenal accent coincides with metrical accent twice per measure. The sequential accent congruence on the anacrustic final beat of m. 2 and the downbeat of m. 3 is especially significant, as it substantially clarifies a sense of both stable quarter-note projection and metric hierarchy. Conversely, when the onset of the vocal patch is heard as the off-beat (Example 1.2i), the solitary accent congruence on beat 3 only weakly reinforces the notated metric schema. In sum, all evidence overwhelmingly favors hearing the initial vocal patch as the downbeat of a fourpulse metric schema. MPR 1 Parallelism Where two or more groups or parts of groups can be construed as parallel, they preferably receive parallel metrical structure. MPR 2 Strong Beat Early Weakly prefer a metrical structure in which the strongest beat in a group appears relatively early in the group. MPR 3 Event Prefer a metrical structure in which beats of level Li that coincide with the inception of pitch-events are strong beats of Li. MPR 4 Stress Prefer a metrical structure in which beats of level Li that are stressed are strong beats of Li. MPR 5 Length Prefer a metrical structure in which a relatively strong beat occurs at the inception of either relatively long: a. pitch-event; b. duration of a dynamic; c. slur; d. pattern of articulation; e. duration of a pitch in the relevant levels of the time-span reduction; f. duration of a harmony in the relevant levels of the time-span reduction (harmonic rhythm). MPR 6 Bass Prefer a metrically stable bass. MPR 7 Cadence Strongly prefer a metrical structure in which cadences are metrically stable; that is, strongly avoid violations of local preference rules within cadences. MPR 8 Suspension Strongly prefer a metrical structure in which a suspension is on a stronger beat than its resolution. Table 1.1. Summary of Lerdahl and Jackendoff s Metrical Preference Rules (MPRs). 36 Lerdahl and Jackendoff 1983,

54 MPR 9 MPR 10 Time-Span Reduction Binary Regularity Prefer a metrical analysis that minimizes conflict in the timespan reduction. Prefer metrical structures in which at each level every other beat is strong. Table 1.1. Continued. Example 1.2h. Chvrches, The Mother We Share, mm. 2-3, meter as heard with alignment of strong accents. Example 1.2i. Chvrches, The Mother We Share, mm. 2-3, meter as reinterpreted with alignment of strong accents. As explained earlier, metric entrainment is by nature a dynamic attending process, and a listener s schema of metric entrainment is directly affected by cues gleaned from a musical surface. Should phenomenal cues cease to correspond with a listener s projection of metrical accent, the listener will adjust their entrainment schema. Let s inspect the remaining audible cues that may potentially affect metric entrainment as the opening of The Mother We Share unfolds. As interpreted in Example 1.2a, the initial attention given to the synthesizer at the song s outset will surely be redirected to the rhythmic vocal patch sending the synthesizer into the 34

55 background. Through mm. 2-3 the listener will likely entrain to the metric schema of Example 1.2h with relatively little conscious effort. The next phenomenal cues with the potential to attract attention and inform meter are the drum set s snare click and high hat, entering on what can be perceived to be the final eighth- and sixteenth-note, respectively, of m. 3. Note that these entrances do not occur on a beat of the entrained metric schema. As just mentioned, we tend to adjust or reconsider our pattern of entrainment only when phenomenal cues significantly conflict with the present entrainment pattern. Here, the snare click continues through mm. 4 and 5 in an uninterrupted series of eighth-notes, every other note coinciding with the pervasive quarter-note pulse notated in Example 1.2a. Thus, it seems likely that the initial entrance at the end of m. 3 will be heard as an anacrusis to m. 4, where we find the first two measures of the vocal patch begin to be reiterated. This perceived anacrusis will likely intensify the already assumed feeling that the vocal patch s initial F4 is the downbeat of a four-pulse metric hierarchy. It is only later, at the end of m. 5, that the listener may hear a phenomenal cue that is significantly incongruent with the initial projection schema, thus requiring a change of projection. The cues in question are the drum set s bass drum and tom-toms. It seems quite unlikely that even a trained musician will be able to maintain a conservative hearing in the initial metric schema, as Example 1.2j hypothetically shows. 37 Notice the significant lack of metric congruence in the bass and drum set, the instruments which generally clarify the metric schemas of most western popular music genres. As it was with selecting between the entrainment patterns suggested by Examples 1.2h and 1.2i, a sense of anacrusis can dramatically affect the clarity of a downbeat. Hence, the drum set s fill at the end of m. 5 makes for a strong anacrusis 37 Conservative listeners may prefer to firmly maintain their pattern of entrainment through conflicting metric information, while radical listeners may prefer to immediately adjust to the changing musical landscape. See Imbrie

56 to the downbeat of m. 6, where the entrance of the male vocal patch, a change of keyboard synthesizer voicing, and entrance of a bass line seem to unquestionably indicate the beginning of a measure. While the drum set appears to be silent on this downbeat, its articulations on the recontextualized beats 2 and 4 indicate the familiar backbeat schema, further intensifying the stability of a new four-pulse metric hierarchy. Lerdahl and Jackendoff also suggest a preference rule supporting entrainment to the backbeat. The tenth metric preference rule MPR 10, Binary Regularity instructs us to prefer metrical structures in which at each level every other beat is strong. 38 The ubiquitous binary backbeat schema thus overwhelms our attention, effectively diminishing the capability of the female vocal patch to influence a separate pattern of metric entrainment. Example 1.2j. Chvrches, The Mother We Share, mm. 1-9, meter with conservative hearing. 38 Lerdahl and Jackendoff 1983,

57 As a final point for dissection, it is my suspicion that, despite the relatively obvious adjusting of meter between mm. 5 and 6 (assuming entrainment as suggested by Example 1.2a), listeners lacking privileged knowledge may be hard pressed to notice the metric reframing of the female vocal patch. If the listener were to entrain to the meter suggested in Example 1.2a, which I have argued to be a strong possibility, a shifting of the entrainment pattern at the bass and drum set s entrance at m. 6 will lead to a shifting of the female vocal patch in relation to this entrainment pattern. Heard originally as beginning on m. 2 s downbeat, the precise rhythmic repetition of this figure now occurs one eighth note later, beginning on the off-beat of m. 6 s downbeat. As previously examined through Examples 1.2h and 1.2i, this eighth-note shift holds gross implications for metric entrainment. The subjective feelings differentiating the rhythmic character of these two presentations are unlikely to be uniform from listener to listener. But it is, nevertheless, the case that former congruencies between phenomenal accent and metrical accent which so strongly reinforced the entrainment pattern suggested in Example 1.2a (eg. the anacrusis of beat 4 and the following downbeat) are now absent from the female vocal patch. Despite the attention demanded by the drum set and bass from mm. 6-9, I still find that the female vocal patch is able to be heard with relative ease due to its high tessitura and distinct timbre. However, it is my experience that such a shift in metric alignment (from Example 1.2h to 1.2i) is rather imperceptible. Only after repeated hearings with attention directed solely to this vocal line have I been able to consciously recognize this shift. What contributes to this difficulty? Consistent with my methodology, the answer lies in the phenomenal cues gleaned from the musical surface. The entrance of the drum set and bass in Example 1.2a at m. 6 is also accompanied by the addition of a new phenomenal cue: a male vocal patch. Interestingly, the rhythm of this patch is strikingly similar to that rhythm which informed our original false 37

58 meter in Example 1.2a. If we directly compare these two rhythms, as done in Example 1.2k, it is easy to discern the similarity between the new rhythm of the male vocal patch at m. 6 and the pattern of accent inferred from the initial female vocal patch in m. 2. It seems that the male vocal patch "stands in" for the female vocal patch, maintaining a stream that reinforces the original "faux" metrical alignment, even after the drum and bass recontextualize and realign the female vocal patch. The end result is that the musical surface maintains ambiguity or multiplicity for a longer time, creating a situation where we are free to toggle back and forth between two possible alignments or two possible interpretations. A useful analogy can be drawn with the perceptual ambiguity seen in the vase/face or Necker cube visual puzzles, which allow for multiple interpretive possibilities as opposed to a definite solution. Example 1.2k. Chvrches, The Mother We Share, m. 6, alignment of voice patch after drum and bass entrance. Shifting Attention and Entrainment in Thom Yorke s A Skip Divided When phenomenal cues on the surface of the music begin to significantly accentuate a previously latent metric schema, as in the excerpt from Chvrches, one may reasonably expect the listener to resolve the conflict by adjusting their entrainment and, ultimately, their metric schema in accordance with the new information. Exactly when a listener abandons the faux entrained schema and invests in the newly emerging latent schema instead remains variable. Christopher 38

59 Hasty says that it is often maintained that a meter, having once been established, will tend to perpetuate itself even against the influence of conflicting rhythmic accent. 39 Andrew Imbrie (1973) suggests the potential for both maintaining an entrained metric pattern and adjusting to the conflicting information: conservative listeners may prefer to firmly maintain their pattern of entrainment through conflicting metric information, while radical listeners may prefer to immediately adjust to the changing musical landscape. 40 Imbrie s conservative and radical labels seem to be related to degree to which a listener s attention is directed inward. Conservative listeners will focus on perpetuating their entrained pattern of counting (metrical accent); on the other hand, radical listeners tend to direct attention outward, aligning with the changing musical surface (phenomenal accent). I am skeptical as to whether a listener is purely conservative or radical in their habit of entrainment, but analyzing the varying approaches in the face of a changing musical surface can shed light on the process of metric selection during the listening experience. Such issues of attention and entrainment dominate the perception of meter in the opening portion of another Thom Yorke composition, Skip Divided. For the moment, let s focus on how the opening phenomenal cues may induce meter. As I discussed with respect to Examples 1.2a-k, a listener will tend to entrain to whichever phenomenal cues occur with the greatest degree of regularity and audible prominence. Skip Divided opens with a sparsely orchestrated soundscape that features a variety of percussive and synthesized sounds: a click succeeded by static, two relatively deep unpitched flams, a shaker-like sound, a strained beep that is pitched at B4, and an 39 Hasty 1997, Imbrie

60 oscillating, intoned buzz. How might a listener begin to derive a metric hierarchy from these events? Examples 1.3a and 1.3b represent the rhythm of these sounds in two differing projection schemas, each having a reasonable potential for entrainment. Here the percussive sounds are presented in unmeasured notation as an undifferentiated series of eighth-note articulations; accents are scored over the notes which correspond to the beep, signifying its phenomenal prominence. While it is impossible in Example 1.3a to perceive a downbeat on the initial eighth rest a silence cannot signal a perceptual beginning the ensuing eighth-note/eighth-rest pattern gives rise to a relatively stable quarter-note pulse projection. The three initial pulse/event congruencies in Example 1.3a firmly establish an entrainment pattern in which the hypothetical pulse projection can be sustained and extended throughout the phrase despite the silences (represented in Example 1.3a with question marks). This hypothetical projection is reinforced by a perception of anacrusis, as the long duration initiated by the accented event is immediately preceded by a short duration. Butler agrees that perceiving short-long rhythms tends to imply anacrusis arrival. 41 The three repeated eighth-note articulations further serve to enhance the stability of this projection schema. Example 1.3a. Yorke, Skip Divided, introduction, possible quarter-note pulse projection. 41 Butler 2003,

61 Example 1.3b. Yorke, Skip Divided, alternative possible quarter-note pulse projection beginning with first entry. Example 1.3b is shown as an alternative to this potential schema. Here we assume a quarter-note pulse projection will be cued by the first heard stimulus two eighth-note articulations. Such a projection schema is seemingly stable for a short while, and even makes more sense in the short term: the beginning of the projection schema coincides with the first stimulus. However, these congruencies are succeeded by three consecutive questionable beats, the first and third being silent and the second coinciding with a vaguely articulated shaker-like sound. It may be possible that a listener projecting the pattern shown in Example 1.3b could maintain their pattern of entrainment through these questionable pulses, where after one would find an alternation between silence and articulation. Yet, three questionable pulses is a significant duration through which a listener must projection metrical accent against conflicting information. Considering this, it seems likely that Example 1.3a is the projection that best informs a pattern of metric hierarchy, despite the initial instability that may be perceived. Preferring the metric schema suggested by Example 1.3a is further supported by the proclivity to hear the prominently articulated B4 beep as a downbeat. Notice how this pitch event, indicated in Example 1.3a by accents, coincides directly with the suggested quarter-note pulse projection. Furthermore, the beep helps to outline the emerging sense of metric hierarchy. Example 1.3c models a potential metric selection process with an enhanced sense of stable 41

62 hierarchy. Although the potential of projection A A, cued by the first two audible events, can be realized (as already hypothesized and discarded in Example 1.3b) I suspect the phenomenal accent associated with the articulation of the beep may suggest a new projective beginning B. The B4 beeps not only give rise to the quarter-note projection B B, but also support the hierarchically related half-note projection C C. These multiple emerging levels above the tactus projection hypothesized in Example 1.3a function to inform a stable metric hierarchy. Example 1.3c. Yorke, Skip Divided, mm. 1-3, beginning of metric hierarchy as heard. In the excerpt from Chvrches I showed that, as the music unfolds, new phenomenal cues emerging on the musical surface may require us to adjust or reconsider our metric schema. In Skip Divided, how might new cues affect our perception of meter? Let s assume the listener has firmly entrained to the meter projected in Example 1.3a. The first instance of new audible material beyond the initial percussive and ethereal tones is the sound of a male voice singing oo. In Example 1.3d this is heard beginning in m. 4 on the off-beat of beat 2, moving shortly after on the off-beat of beat 3. As notated the entrance and movement of this voice obviously conflicts with the suggested metric schema. These phenomenal cues are significantly different in style to the percussive articulations and may draw the listener s attention away from attending to metric schema dominated by the B4 beep, and the long tones heard at this entrance may have the 42

63 potential to change or derail the entrained metric schema. Lerdahl and Jackendoff s fifth metrical preference rule, MPR 5, Length, supports this potentiality: prefer a metrical structure in which relatively strong beats occur at the inception of notes of relatively long duration. 42 To the contrary, I believe that voice s long tones heard at this entrance are unlikely to inspire any change in a listener s metric schema; the potential for the long pitch following the voice s initial quarter-note to inform any sense of meter is dramatically reduced by the percussive rhythmic activity that continues to pervade the musical surface. Furthermore, when perceived as notated in Example 1.3d, the voice s movement on the downbeat of m. 6 directly reinforces the initially suggested metric schema, intensifying its stability. 42 Lerdahl and Jackendoff 1983,

64 Example 1.3d. Yorke, Skip Divided, mm. 1-28, meter as heard. While I surmise a listener will hear the entrance of the voice in m. 4 within the meter defined by the B4 beep that is, integrating the new musical material into the currently entrained metric schema the relatively dramatic shift in the musical landscape at m. 9, cued by the 44

65 entrance of bass tones and more rapidly-articulated percussion sounds, will signal to the listener to adjust the location where the metric hierarchy begins. In my experience, this is passively achieved; the bass s entrance occurs as an eighth-note anacrusis to the new hierarchy beginning, allowing the listener to effortlessly re-entrain to the shifted hierarchy the instant before it is revealed. Notice that, despite the large amount of new audible material, the pattern established and informed by the initial percussive sounds remains relatively undisturbed; when shifted by two beats the B4 beep continues to articulate beats 1 and 3. Recall the metrical preference rules mentioned above. How might these influence the selecting of this shifted meter? Though the articulations of accented pitches generally avoid beats, thereby creating syncopations that violate MPR 3, Event and MPR 4, Stress, choosing to hear the onset of the longest pitches in the bass and newly-shifted voice near the beginning of this metric pattern is seemingly supported by MPR 5, Length. Is it problematic that so few cues beyond the percussion line align with beats? As has been earlier mentioned, once a metric pattern has been firmly established it will tend to perpetuate itself despite the presence of conflicting phenomenal cues. Such overwhelming cues are certainly soon to be found. But in the immediate future it seems plausible for a listener to continue to project the newly shifted meter in Example 1.3d through m. 9; the repetition of mm from mm may help to solidify an emerging sense of stability and regularity. The only remaining phenomenal material in Skip Divided s musical landscape to affect the perception and construal of a metric hierarchy is the vocal lyric line, which, as transcribed in Example 1.3d, enters on an anacrusis to m. 17. Here a listener may notice some initial instability between the currently entrained metric schema and poetic accent suggested by the lyrics I m in a skip divided malfunction. When examined from a poetic perspective in Example 1.3e this lyric can be separated into three syllabic sets, each with a single accent: [I m in a skip] [di-vi- 45

66 ded] [mal-func-tion]. Notice how these accents correspond to the rhythm of the lyrics and how initial instability may arise: the unaccented word I m does not seem well suited to entering on a pulse, nor does the accented word skip seem comfortably placed on the off-beat of beat 1. This period of instability is relatively brief with accent returning to predictable locations congruent with the suggested meter: the vi syllable of divided falling on beat 3 and the func of malfunction on the following measure s beat 2. Consequently, I suspect that a listener who has firmly entrained to the meter suggested by Example 1.3d will with little effort be able to maintain this assumed metric schema through mm s brief instability, based on the lyrics syncopated syllabic accents. Having successfully integrated the new phenomenal material from the musical surface into the currently activated metric schema may perhaps suggest to the listener a degree of future stability. Indeed, the parallel structure between mm and mm should suggest perpetuation of meter, despite the similar initial incongruence between poetic accent and metrical accent in Example 1.3f. The lyric I flap around and dive bomb may even enhance this metric schema s sense of stability with the accented word dive coinciding with the metric accent attributed to the beginning of m. 20. Note Lerdahl and Jackendoff s MPR 1, Parallelism : where two or more groups or parts of groups can be construed as parallel, they preferably receive parallel metrical structure. 43 Example 1.3e. Yorke, Skip Divided, mm , lyrics with poetic accent as heard. 43 Lerdahl and Jackendoff 1983,

67 Example 1.3f. Yorke, Skip Divided, mm , lyrics with poetic accent as heard. As there are no further changes in the pattern of accent in the percussion, bass and upper voice lines, the activity of Yorke s lyrics appear to attract the greatest attention. Assuming no changes to one s pattern of entrainment, the following line, frantically around your light, appears to be littered with conflict between metrical and syllabic accent. 44 In Example 1.3g the accented syllable fran falls on the off-beat of beat 1, as does the syllable round in the following measure. What emphasis do we find between these two stressed syllables? While there is no inherent syllabic accent attributed to ti-cally, the short-long notation of the eighth note and quarter note suggests the long note to carry more accent. Were a listener to consider this shortlong accent they would experience approximately six pulses of contradictory syllabic accent. In fairness, it is impossible to determine a precise quantity to define an overwhelming amount of conflicting phenomenal cues. Who is to say that the six pulses of conflict in this situation are that much more overwhelming than the almost four pulses of conflict discussed in Example 1.3e? Furthermore different listeners will likely have varying tolerances for conservative vs. radical evaluations of new phenomenal cues. Nevertheless, I suspect that a listener may have a challenging time sustaining the previously established metric schema. Were this meter to be maintained, as suggested in Example 1.3h, there would be continued disagreement between 44 It should be mentioned that the pattern of accent exhibited by syllables in spoken language is analytically synonymous with the type of phenomenal accent I have been describing in purely musical contexts. 47

68 syllabic and metrical accent. That is to say, I believe the syllabic accents cease to be heard as syncopations and begin to challenge the entrained metric hierarchy. It is certainly confounding and uncomfortable to perceive meter when presented with such significant and extended incongruence. 45 Considering this perspective requires some necessary amending of Hasty s earlier statement: meter is a flexible and fluid gestalt that, having once been established, will only briefly perpetuate itself against the influence of conflicting rhythmic accent. Should the original meter remain phenomenally unaccented, expect the listener to adjust their pattern of entrainment to conform to a newly-emerging pattern of phenomenal accent. 46 Example 1.3g. Yorke, Skip Divided, mm , lyrics with poetic accent as heard. Example 1.3h. Yorke, Skip Divided, mm , lyrics with poetic accent as heard. 45 How precisely can we discuss a feeling of discomfort that is resultant from cognitive processing? These feelings, or qualia, are entirely subjective phenomenon and likely differ greatly from individual to individual. Such is the general nature of phenomenology. The impossibility of any interpersonal comparison does not, however, detract from the reality that non-well-formed metrical contexts tend to elicit feelings of discomfort. 46 The question of when a listener abandons entrainment to adopt a new metric schema informed by emerging material a radical hearing is endlessly fascinating and deserves more careful empirical study. We might then have a better collective idea of when it is likely that we will hear conflicts merely as syncopations or metrical dissonances (more on this in Chapter Two) versus when we hear conflicts as new patterns for entrainment. 48

69 Therefore, I suspect that listeners who have entrained to the pattern of meter hitherto suggested will naturally adjust their entrainment schema to eliminate discomfort somewhere in mm. 21 or 22. Where precisely might this take place? Recall Butler s earlier suggestion that short-long rhythms tend to imply anacrusis arrival. Perhaps, then, the rhythm attributed to the lyrics frantically around might be more effortlessly perceived as shown in Example 1.3i. With fran beginning on the measure s downbeat, the remaining syllables correspond directly with the suggested metrical accent. Of special significance is the newly aligned accent of round on the downbeat of the ensuing measure. Just previously in Example 1.3f we heard around with the accented syllable round placed firmly on a pulse, what was in this case perceived to be beat 3. It seems apparent to me that, because these two iterations of around are heard in close proximity, how a listener perceives the accent of the first iteration in m. 19 may directly influence the perception of the second in m. 22, conforming accordingly to Lerdahl and Jackendoff s MPR 1, Parallelism. The perceiving of a-bout s accent, in combination with the implied anacrusis of its short-long rhythm, seems, at least in this unique situation, to create enough conflict to overwhelm the listener s prior pattern of entrainment. For validation of shifting the metric schema by one eighth note, one must listen no further than the following line of lyrics, enveloped in a sad distraction. This line is transcribed in Example 1.3j in the newly shifted meter: both syllabic accents on vel and sad now correspond to strong beats of the suggested meter. And while the latter syllables of strac-tion are strongly syncopated against this newly shifted meter, I suspect that their relative brevity is not great enough to affect one s pattern of entrainment. In sum, having heard strac-tion as syncopated is evidence that one has already entrained to the new meter in the ways and at the times described. 49

70 Example 1.3i. Yorke, Skip Divided, mm , lyrics with poetic accent as conceived. Example 1.3j. Yorke, Skip Divided, mm , lyrics with poetic accent as conceived. Let s briefly review my analysis thus far. As transcribed in Example 1.3d, I suggest that listeners will first entrain to a metric schema congruent with the B4 beep, further supported by the short-long, anacrusis-implying pattern and the perceivably pulse-congruent percussion articulations. As the first voice patch enters in m. 4, minimal rhythmic-metric congruencies provide just enough phenomenal accent support to encourage continued projection of the established metric schema. The bass and more active percussion enter on the perceived anacrusis to m. 9 and come to the attentional foreground, encouraging a resetting of the metric hierarchy, while the formerly referential B4 beep may become less present in a listener s attention. A listener who has entrained to the latent meter (suggested by Example 1.3d) may become firmly entrenched in this metric schema during the following eight measures, which repeat two fourmeasure phrases. When the lyrics enter just prior to m. 17, the unchanging percussion and bass cues move to the attentional background and the lyrics now dominate the foreground. Attention to the rhythm and accent of the lyrics ultimately influences the perceived shift in metric schema near mm

71 Upon first hearing, it seems plausible that a listener may not perceive the slightly shifted placement of accent in the percussion instruments. Notice in Example 1.3d how prior to m. 22 accents are notated on the off-beat of beat 1 and precisely on beat 3. After the accent shift, beginning in m. 23, these accents fall on beat 1 and the off-beat of beat 2. It seems even more plausible that a listener who has let the B4 beep subside from their attention entirely may not notice that it is no longer articulated on beats of their newly entrained metric schema. This phenomenon relates closely to the situation discussed in Chvrches s The Mother We Share: as new information is introduced and the changing musical surface draws a listener s attention to new stimuli, it seems reasonable that changes in the musical background may go entirely unnoticed, despite their former influence on construing a metric schema. This is particularly true in the case of a radical listener, for whom new stimuli are especially salient in informing dynamic metric schemas. When the most prevalent phenomenal cues suggest changing one s pattern of entrainment, less dominant phenomenal cues will be ignored or subsumed into the new pattern; meter is flexible and fluid schema that resides in the listener and derived through each listener s dynamic attending. That a listener might not notice the shifted percussion sounds may be telling of this excerpt s metric malleability. London describes metric malleability as the property by which many melodic or rhythmic patterns may be heard in more than one metric context 47 Metric malleability is demonstrated by Examples 1.3a and 1.3b, for instance: multiple patterns of entrainment are plausible. Furthermore, consider the variety of possible interpretations of the percussion activity which underlies Yorke s lyrics in Example 1.3k; each iteration in the example is shifted by one eighth note to present all eight possibilities for metric perception. No matter 47 London 2012,

72 which sounds are privileged as beginning the hierarchy, the constant unfolding of eighth-note articulations and the uneven spacing of phenomenal accent allows for strong accents to be heard to articulate beats under many possible potential interpretations. Example 1.3k. Yorke, Skip Divided, demonstration of metrically malleable drum set. Each rotation features strong accents that align with strong pulses. 52

73 While it may be unfair to generalize the means of composition for all dance music, the pervasive use of looped patches and samples in this style makes it rather challenging to insert single measures of shorter or greater duration, as employed in Example 1.3d, m. 22. Likewise, it may be inaccurate to apply the term measure or bar, as the composition methods used by DJs and mixers are not necessarily amenable to traditional notation. For these reasons, in my experience, it is quite rare to find instances of notated changing meter signatures in electro-pop music. While such instances of changing meter are often characteristic of jazz, progressive rock, and other instrumental genres, electronic dance music does not typically allow for changing or irregular meters. I would be quite remiss if I were to say that EDM is not rich with metric dissonance or ambiguity. 48 I mean to say only that the metric changes that western musicians come to understand through changes in meter signature are generally absent in electronic genres. Thus, I suspect that Yorke has not actually composed an insular measure with one additional eighth-note duration (Example 1.3d, m. 22), nor is it likely that he composed a single measure with either fewer or greater quarter-note durations (Example 1.3d, mm. 7 or 8); this seems illogical given the otherwise pervasive unchanging meter of the composition. It appears most likely that these transcribed changes in meter are merely the result of perception. Let s investigate some retrospective metric projections that reconciles m. 22 s metric ambiguity. Based on the conjecture that Skip Divided features an unchanging latent meter, let s assume that the composition s first audible stimulus occurs on the first beat of the first measure; Example 1.3l transcribes the composition accordingly, as was first attempted to be heard in Example 1.3b. Despite the improbability that a listener can entrain to the meter represented in the 48 For a detailed analysis of rhythm and meter in dance music see Butler 2003, Unlocking the Groove. 53

74 first three measures, the transcription in Example 1.3l shows that many audible cues support the newly suggested meter. The initial voice entrance, formerly perceived to occur on an off-beat, is now shown to be heard as anacrusis arrival, occurring at m. 3, beat 4 and m. 4, beat 1. This prominent arrival is clearly implicated as the strongest pulse of a metric hierarchy. This new arrangement and the congruencies that follow on the downbeats of mm. 6 and 7 better conform to Lerdahl and Jackendoff s MPRs 3, 4 and 5: Event, Stress and Length : the voice enters and moves more frequently on pulses of the suggested meter, these events occur more frequently on strong pulses (downbeats) of the metrical structure, and the longest pitches now occur at the beginnings of measures. Similar preference rules apply to the bass, now entering as a quarternote anacrusis to m. 8, with movement occurring invariably on beats 1 and 4. 54

75 Example 1.3l. Yorke, Skip Divided, mm. 1-27; with immediate entrainment. In Example 1.3l greater congruency between phenomenal accent and metrical accent is not, however, ubiquitous. Notice the large degree of syncopation that is now present in the lyrics shortly after they enter in mm As transcribed in Example 1.3m, initial congruency exists between the phenomenal accent of skip and the metrical accent of measure 16 s downbeat. While 55

76 a corresponding pulse projection succeeds skip (the syllable di), six consecutive pulses that misalign with the rhythm of the lyrics follow. The following line, I flap around and divebomb, exhibits the same incongruence of accent. Despite this large degree of syncopation, I expect that a listener who is firmly entrained to this newly suggested interpretation will be able to effortlessly maintain their pattern of entrainment through the conflict. This seemingly contradicts my earlier suggestion that, in Example 1.3d, mm , six pulses of conflicting accent are enough to overwhelm the established metric schema. Why might a similar number of conflicting pulses be unable to demand a similar change of metric interpretation? Example 1.3m. Yorke, Skip Divided, mm ; lyrics heard as syncopated. Perhaps there is a correlation between hypermetric congruence and maintaining a metric schema in the face of phenomenal conflict. That is to say, stability of the projection of longer durations contributes to the overall stability of the larger metric hierarchy. An arc map in Example 1.3n helps to show where various levels of the hierarchy are strongly reinforced. In this situation each phrase of the lyrics lasts four measures, creating a stable hypermetric grid above the tactus and super-tactus levels. These four-measure-long hypermeters are delineated by the repetition of the initial voice singing oo and the bass line. Each four-measure phrase can be divided into two two-measure sub-phrases, each consisting of an individual line of lyrics. Despite the previously discussed conflict, each two-measure sub-phrase begins with a great degree of congruency: both the oo voice patch and bass articulate an anacrusis on beat 4 and an arrival on beat 1, with each line of lyrics strongly accenting the same downbeats (skip, flap, 56

77 fran, vel, etc.). It seems that this clarity of beginning translates to stability at a hypermetric level and allows a listener to perpetuate this metric schema through the ensuing syncopation. Despite weak projection durations at the quarter-, half-, and whole-note levels, characterized by a lack of congruence between metrical accent and the phenomenal accent of the lyrics, a four-measure hypermetric phrase structure is easily perceptible. Example 1.3n. Yorke, Skip Divided, mm ; strong v. weak projection ascending through the metric hierarchy. I have one final argument for the correctness of the latent meter transcribed in Example 1.3l. Upon entraining to this revised meter, is it possible to return to the original way of entraining suggested in Example 1.3d? When listening to Skip Divided from the beginning I find myself susceptible to entraining to the meter first presented in Example 1.3d. It is my experience, however, that when I become entrained to the revised meter suggested by Example 1.3l, it becomes impossible to return to hearing the meter as was at first experienced. 49 Furthermore, 49 Recall the situation described earlier by Hesselink 2014 when listening to The Police s Here Comes the Night: listeners cannot hear the later passages differently because they have been conditioned by the metric framing of the former material. Inversely, after being conditioned by later material it may be difficult to return to the earlier way of hearing. 57

78 when I begin listening to the recording anytime between the entrance of the bass and the lyrics (mm. 8-16) I tend to entrain to the correct metric schema of Example 1.3l, despite the suggested tendency for entraining to this meter not occurring until several measures later. Thus, I argue that hearing mm as represented in Example 1.3d is dependent upon having also perceived mm. 1-8 in the manner suggested by the same example: a metric schema oriented to the prominent B4 beep establishes a clear metric hierarchy into which the new phenomenal cues are integrated. These multiple understandings are a consequence of the passage s metric malleability. That is, phenomenal accent exhibited by the passage can be interpreted in multiple ways that reasonably lead to the adoption of a few different possible metric schemas (review Example 1.3k). Conversely, the rhythm of the bass and vocal patch, also present at m. 8, are metrically interpretable in only limited ways, ways which are primarily dictated by metrical preference rules. Without having perceived the opening measures as transcribed in Example 1.3a, the music that follows cannot be metrically perceived accordingly. All things considered, the perception of meter throughout Skip Divided is dominated by the changing musical surface throughout the composition s unfolding. As new phenomenal material is presented, attention is drawn away from the previously entrained material. And should this new material sufficiently conform to the prior pattern of entrainment, the new material will be heard as referential to the earlier pattern despite its own implied metric schema. Again, it is impossible to define a precise quantity of cues that would overwhelm an entrained metric schema; this quantity may be different for every listener, or there may even be great swaths of correspondence between listeners. There would be great value in further studying this question empirically. The discussion of meter in Skip Divided shows that not all metric conflict 58

79 may equally influence entrainment adjustments. But it does seem that we ve uncovered a significant factor that affects the perpetuation and possible adjustment of a metric schema: clarity of the metric hierarchy s beginning. While listeners perpetuate a hypothetical metric schema in the face of conflicting phenomenal cues, should these phenomenal cues fail to reinforce the onset or beginning of a metric hierarchy, the listener may be forced to reconsider their pattern of metric attending. Flexibility of Projection in Thom Yorke s The Mother Lode Despite the implied rigidity of musical notation, the human capacity to perceive and compare durations of time is quite flexible. Indeed, it is often the case that musical performances deemed the most expressive are those in which the performer manipulates tempo by performing rubato, ritardando and accelerando to draw attention to particular passages. How might a listener entrain to a pulse through these tempo adjustments? Remember that metric attending is a dynamic process, constantly subject to the changing musical surface. Were our ability to entrain to a pulse dependent upon hearing precisely reproduced durations, our metric attending would derail upon first noticing a fluctuation of any determinate duration. 50 Rather, the ability to perceive a pulse is dependent upon only the potential that a previously perceived duration will be reproduced. And thusly when given that a duration A is followed by a duration B, the listener will be primed for the possibility for duration B to be compared to duration A. Hasty s later comments are consistent with his earlier claim that projection is not the anticipation 50 Before the development of quantization, early MIDI transcribers were unable to recognize the slightest rhythmic and metric deviations. 59

80 of a precisely presupposed outcome, but rather opening the potential to interpret a range of possibilities: Since mensural determinacy is flexible, B can be felt as a reproduction of A s duration without being precisely equal, measured by the clock. However, this flexibility should not be taken to indicate an inability to feel differences among completed durations. It is, rather, a flexibility that accommodates the indeterminacy of present becoming to a definite potential for this becoming. 51 When considering an unfamiliar score, it is impossible to know precisely what the future may portend; it is only as the music unfolds that the listener will know whether their cognitively constructed metric hierarchy will be reinforced. If the judgment of mensural determinacy is flexible, then the metric hierarchy must itself be similarly flexible. To demonstrate, let s look at another Thom Yorke composition, The Mother Lode. Example 1.4a is a transcription of The Mother Lode s opening measures as they are likely to be perceived: twelve consecutive quarter-note articulations of D Major followed by four quarter notes of E minor. The simplest interpretation would suggest an unambiguous metric hierarchy consisting of four measures of four pulses each. The change of harmony is also accompanied by the entrance of a voice patch which provides additional phenomenal weight, in sum conveying a sense of hypermetric anacrusis. Recall how the repetition of the four-measure bass and drum figure in Skip Divided solidifies a sense of metric schema. Here, the four-measure pattern is repeated four times and confirms that The Mother Lode s ostinato piano pulse is congruent with the entrained meter. 51 Hasty 1997, 86. When followed to its logical end, this statement implicates a constantly unfolding metric stream that is somewhat incompatible with the cyclic, hierarchical view of meter that has been central to my analyses in Examples The opposition and reconciliation of these views is a central tenet of Hasty

81 Example 1.4a. Yorke, The Mother Lode, mm. 1-4, meter as heard. This interpretation is not entirely without conflict, however. Notice that on the third repetition, a faint bass drum begins articulating on what are perceived as the off-beats of beats 1 and 3, deviating from both the clearly exhibited quarter-note pulse and also the normative style of western popular genres in which bass drum articulations are typically found on beats 1 and 3 or beats 2 and 4. Lerdahl and Jackendoff may suggest that this particular conflict violates MPR 6, Bass : prefer a metrically stable bass. The authors state, In tonal music, the bass tends to be metrically more stable than the upper parts: when it plays isolated notes, they are usually strong beats; when it plays sustained notes, they are much less likely to be syncopated than an upper part is, and so forth. 52 While Lerdahl and Jackendoff refer more specifically to low pitches in classical repertoire, this preference rule happens to align with the particular style context in which Yorke composes. Despite this preference, it seems likely that no adjusting of one s entrainment pattern will be necessary to accommodate the unstable bass: the aforementioned 52 Lerdahl and Jackendoff 1983,

82 piano cues are simply too pervasive. Rather than creating instability, I suspect listeners will integrate these off-beat articulations into the already entrained metric schema, strengthening the metric hierarchy by adding a level of phenomenal tactus subdivision. Yorke s lyrics, which enter in m. 6, further stabilize the four-measure, four-pulse metric hierarchy; Example 1.4b shows the perceived rhythm of the lyrics. The second and third measure of each four-bar phrase feature somewhat ambiguous rhythms and uncomfortable patterns of accentuation in relation to the tactus. For example, in the second measure of the first four phrases, now and clown, shal and pool, way and goes, and fall and through are all misaligned with the pulse. However, the fourth measure of each phrase distinctly aligns with the change of harmony and piano articulations. Its successive half-note durations further support the metric hierarchy by adding phenomenal accent to the two-beat, super-tactus pulse level. Why might the listener maintain the original metric schema through the uncomfortable second and third measures? As already discussed, the opening sixteen measures (Example 1.4a) established a strong sense of metric hierarchy induced by the piano and (unidiomatic) bass drum. These instruments continue their regular articulation patterns underneath the misaligned lyrics, and the two measures of misalignment do not seem significant enough to overwhelm the regular stream of the piano s and bass drum s phenomenal cues. This slight introduction of discomfort may, albeit, be indicative of a latent meter which is yet to be revealed. 62

83 Example 1.4b. Yorke, The Mother Lode, mm. 5-25, rhythm of lyrics in meter as heard. At m. 29 a significant change occurs on the musical surface. Notated in Example 1.4c, the drum set is expanded from its formerly isolated bass drum articulations to now include a varied high-hat and snare drum. The three quarter notes in the bass drums and the sixteenth-note anacrusis indicate a meter whose hierarchy is coordinated to begin with the bass drum s first quarter note. I suspect that this new phenomenal information will attract a listener s attention for many measures, so much so that it may be quite some time until the listener notices that the 63

84 piano s articulations are no longer coordinated with the pulse of the original entrainment pattern. In Example 1.4c the radical listener now understands these quarter-note piano articulations to occur on the off-beats of each quarter-note pulse. Retrospectively listeners may ask, have I been deceived? Example 1.4d is a transcription of the opening 36 measures adjusted to reflect this latent meter. The bass drum and piano now conform to the suggestions of MPR 6, Bass, as the bass drum appears metrically aligned and the upper voices are syncopated. Example 1.4c. Yorke, The Mother Lode, mm , meter as heard. Example 1.4d. Yorke, The Mother Lode, mm

85 Example 1.4d. Yorke, The Mother Lode, mm. 1-36, contd. 65

86 Example 1.4d. Yorke, The Mother Lode, mm. 1-36, contd. Assuming that there is no durational variation of the piano s quarter-note articulations, an additional eighth note value must be inserted in m. 28 for the bass drum articulations in m. 29 to be perceived as occurring on pulses of the suggested meter. Indeed, when counting strictly a listener may notice that m. 28 seems slightly longer than the measures that both precede and succeed it; Example 1.4e shows the marrying of these two metric interpretations as perceived in real-time. In previous examples the adjusting of one s metric schema to changing cues on the musical surface has been accompanied by an unsettling feeling as the listener s pattern of entrainment is disrupted. In this case, however, m. 28 (as perceived in Example 1.4e) seems to pass with relatively little discomfort. Why might this be? Example 1.4e. Yorke, The Mother Lode, mm , meter as heard. 66

87 I argue that this is related to the flexibility of pulse projection. Recall that both The Mother We Share (Example 1.2a) and Skip Divided (Example 1.3a) exhibit a stream of eighthnote subdivision beneath the more perceptually salient quarter-note tactus. Once having entrained to a quarter-note pulse in these examples, the shifting of phenomenal cues by one eighth note either by addition or deletion was not enough to entirely derail metric attending. Rather, we merely shift the metric hierarchy (cognitive) to align with the musical surface (phenomenal), perceiving slight discomfort in the process. I believe this discomfort is the result of a disruption of the quarter-note pulse projection, which is made obvious due to the constant phenomenal articulation of the eighth-note subdivision, thus creating the sense that the quarternote pulse has been adjusted to 1.5 or.5 pulses. While The Mother Lode similarly features eighth-note subdivisions underneath a prevailing quarter-note tactus, the eighth-note subdivisions are articulated more sporadically, requiring the listener to cognitively project a steady eighth-note pulse. It seems most likely, however, that the listener will focus on the phenomenally articulated quarter-note level rather than give conscious effort to maintaining the rapid eighth-note projection. 53 Notice this in Example 1.4e where we see the brief cessation of bass drum articulations between mm. 27 and 28 attenuating the phenomenal eighth-note subdivision; any sense of subdivision in this short passage must be consciously produced by the listener. This cessation contributes to a feeling of suspended or ambiguous pulse projection, leading the listener to briefly defer projection until clearer phenomenal cues resume at m. 29 when the drum set re-enters. In this way, the precise extended duration of m. 28 is challenging to perceive. 53 London 2012 suggests this to be the result of a cognitive preference for projecting a pulse at a moderate rate. Thus, the very rapid eighth-note projection suggested here requires significant cognitive effort that adult humans prefer to cognitively. Chapter Four contains a detailed discussion of how tempo influences the selection of a metric schema. 67

88 Were the perceived additional duration in m. 28 closer to one quarter note in length, as opposed to the perceived eighth-note duration, it is plausible that the listener might feel a greater sense of pulse suspension, perhaps perceiving m. 28 to have five full beats. Conversely, were the duration of m. 28 one quarter note shorter, the entrance of the drum set at m. 29 may be perceived to be early, likely evoking an even more intense feeling of pulse instability. It is in a similar way that we hear rubato and fluctuations in tempo as expressive devices rather than implicit changes of a metric framework. 54 Regardless, it seems clear that the entrainment process is flexible to the demands of the changing musical landscape, such that slight variances in tempo or note duration can be assimilated into a listener s metric schema with relative ease. Of course, in the case that there is no clear or established schema, phenomenal cues that do not support a sense of pulse projection will merely be interpreted as ambiguous. Nevertheless, the passage from mm is surrounded by two sections which clearly convey metric stability. As the malleable meters and various notational representations of this chapter have shown, musical meter does not exist in arbitrary musical notation of the score. Meter resides in the listener. Metric schemas are the result of cognitively organizing patterns of phenomenal accent perceived on the musical surface into hierarchies. Their perpetuation depends on the degree to which phenomenal accent on the musical surface conforms to the cognitive projection of that metric schema. Although each of the compositions analyzed in this Chapter can be 54 Were the passage to be performed at a significantly slower tempo, thereby increasing the time span of each pitch event, variances of measure duration are likely to be more noticeable. If a quarter-note duration is increased by 200 ms, a half-note duration would be increased by 400 ms and a whole note duration would be increased by 800 ms. 68

89 notated within an unchanging, four-pulse metric grid, their patterns of phenomenal accent do not suggest a static four-pulse metric schema. As new events enter the texture, the listener will adjust their entrained metric schema to coordinate with the changing phenomenal surface accent. Accordingly, as the patterns of phenomenal surface accent shift, the listener may a come to understand a formerly inaccessible, latent meter. This dynamic adjustment in entrainment lends gravity to my argument that meter does not exist in notation. Rather, musical meter is a fluid and flexible schema that resides in the listener and is dependent upon the phenomenal accent exhibited by a musical landscape as related to learned and/or conditioned experiences. Accent s role in discerning a metric schema cannot be understated. In Chapter Two, I will expand upon two mutually supporting accent types, termed by Lerdahl and Jackendoff as phenomenal accent and metrical accent. Phenomenal accent is a physical stimulus that can be perceived with the senses. Recurring patterns of phenomenal accent give rise to metrical accent, the cognitive time-keeping process that Hasty calls projection. Through an analysis of John Adams s Hallelujah Junction, I examine how phenomenal accent and metrical accent interact to affect perception of meter and the stability of the metric hierarchy. Because phenomenal accent is a physical stimulus, I suggest that it can be empirically measured. I employ an accent taxonomy used by John Roeder (2003) to calculate the prominence and regularity of phenomenal accent types and use the results to suggest which phenomenal cues are most likely to induce metrical accent. When phenomenal accent conflicts with cognitively-projected metrical accent, listeners may experience what Harald Krebs (1999) calls metrical dissonance. My analysis reveals that varying degrees of metrical dissonance often define major formal boundaries across the entirety of a composition. In this way, our understanding of formal functions can be directly related to the 69

90 perception of meter. I expand upon this idea in Chapter Three by first analyzing and, then, reinterpreting the exposition of Robert Schumann s Symphony in C Major, Op. 61 to reflect meter as heard. In the Chapter Four I show how perceiving meter is also dependent upon the rate at which phenomenal accent occurs in time. I draw on critical insights from theorist Justin London to present a phenomenal analysis of Rodney Rogers s Lessons of the Sky that displays the close interaction between the perception of meter and tempo. My analysis ultimately shows how unfolding phenomenal cues affect the perception of larger, hypermetric functions. This analysis is supported by the ideas posited in Chapters Two and Three, that the perception of meter at local levels directly informs the highest levels of the metric hierarchy. In sum, my analyses suggest for and support a cognitively-based, dynamic model of metric theory. Metric perception is affected by two mutually supportive, yet, independent mechanisms: the sensing of phenomenal accent and the relation of that accent to a cognitively projected metric hierarchy. Standard thinking assumes that phenomenal accent dominantly informs projection; when attending to an unfamiliar musical landscape, listeners tend to project forward a pulse that is informed by what has been heard. Once established, however, the cognitive projection (metrical accent) associated with that pulse takes on a life of its own, and its hierarchical nature can even overwhelm the inherent power of phenomenal accent to induce that very pulse. Whether a listener conservatively maintains the established pattern of cognitive projection or radically adjusts to the changing musical surface is largely dependent upon the individual experiences of each listener, and these phenomenological differences are endlessly fascinating and worthy of study. 70

91 CHAPTER TWO I spent a fair amount of time in Chapter One discussing how phenomenal accent patterns on the musical surface contribute to perceiving meter. Recall especially the conflict between metrical accent accent associated with the cognitive projection of metric hierarchy and phenomenal accent accent resulting from a change in volume, pitch, timbre, and/or duration. These two distinct yet interrelated accent types interact during the listening experience. Lerdahl and Jackendoff discuss how phenomenal accent serves as a perceptual input to the emergence of metrical accent. The degree to which phenomenal accent occurs in regular or irregular patterns determines the certainty with which a listener can projection respective metrical accent. 55 Not all phenomenal accents, however, equally influence the development of a metric schema. When in the same field of observation, strong energy sources tend to overwhelm the perceptual salience of weaker energy sources. For example, the sun s intense daytime light renders the otherwise visible stars imperceptible. In a musical context, a relatively soft tone is less able to draw attention when surrounded by more prominent stimuli. This phenomenal aspect of the listening significantly impacts metric entrainment. The relative prominence of a stimulus can be a highly subjective phenomenon unique to each listener. 56 However, musical sounds inherently have physically and temporally determinate qualities, perhaps implying that some objective means can be used to measure and classify and compare various phenomenal accents. An analysis of this kind might allow for an objective means to determine the dominant or pervasive metric schema. How might one characterize the 55 Lerdahl and Jackendoff 1983, When approached from a phenomenological perspective, it seems that varying stimuli are measured and related by the degree to which such stimuli have drawn attention. 71

92 varying types of audible stimuli in metrical contexts? In the first half of this Chapter, I will examine the opening passage from John Adams s Hallelujah Junction for two pianos, where phenomenal accent occurs in varying degrees of regularity. Hallelujah Junction s textures are saturated and often ambiguous, but still metrically suggestive and interesting. By adopting a method of accent analysis employed by John Roeder (2003), I intend to make some sense of the composition s often metrically ambiguous texture. My analyses will not always offer a means to hear an entirely clear or unambiguous meter, for the phenomenal cues often conflict with or prevent a listener from projecting metrical accent for long spans of time. In the Chapter s second half, I explore how conflict between phenomenal accent and metrical accent gives rise to what Harald Krebs (1999) calls metrical dissonance. I then relate certain passages and the metrical dissonances they exhibit to larger formal structures that give definition and clarity to Adams s composition. Perceiving Meter in John Adams s Hallelujah Junction Because I am primarily concerned with the phenomenal experience of metric induction, I suggest that the reader first listen to Hallelujah Junction s opening prior to examining the score in Example 2.1. At the composition s outset listeners will hear one piano perform a recurring three-note motive. What metric schema is cued by this motive? I suggest in Examples 2.2a-c that there are limited reasonable ways in which the passage can cue meter. As the motive is three eighth notes in duration, the dotted quarter note appears to be the most likely candidate to be assigned the tactus. Theoretically, this allows for three unique metric interpretations, with each possible interpretation being shifted in the notated measure by one eighth note. Which of these is most likely to be perceived? Recall that the most stable meters feature a great degree of 72

93 congruency between phenomenal accent and metrical accent, whereby the accent of the music reinforces the accent of projection. As we tend to attribute strong phenomenal cues to the beginning of a metric schema, we can safely rule out Example 2.2c, where there is no audible cue on the downbeat. Examples 2.2a and b are more assured alternatives. How might one choose between these two options? Perhaps a listener may prefer to entrain to Example 2.2a, where the downbeat aligns with the E Major chord. Or perhaps the listener may prefer Example 2.2b, where the downbeat aligns with the longest sound duration. 57 Are these equally viable options? Interestingly, the 3/4 time signature notated by Adams reinforces neither option and confuses a simplistic alignment of sound with notated meter; I find it difficult to believe that any listener armed only with the experience of the audible environment would perceive the figure in accordance with Adams s 3/4 notation. Regardless, notation is not the arbiter of whether a listener would prefer to entrain to Example 2.2a or 2.2b. 57 Adams indicates that the performer should depress the damper pedal and allow the sixteenth notes to resonate through the rests. 73

94 Example 2.1. Adams, Hallelujah Junction, mm

95 Example 2.1. Adams, Hallelujah Junction, mm. 1-44, contd. 75

96 Example 2.1. Adams, Hallelujah Junction, mm. 1-44, contd. 76

97 Example 2.2. Adams, Hallelujah Junction, opening motive with possible metric interpretations. As the passage continues in Example 2.3a, the second piano enters performing the same motive in an eventual one-eighth-note cannon. Example 2.3b shows a composite of the two pianos. 58 Here we see that the former metrical preference question resolves itself, as the longest pitch durations (eighth notes) coincide with the greatest number of simultaneously articulated pitches. Consequently, I expect that listeners will tend to coordinate their pattern of entrainment to align with these events. That these eighth-note durations seem to occur at regular, dottedquarter-note intervals serves to enhance the sense of a stable dotted-quarter-note tactus that aligns with the eighth-note durations. Listeners may notice, however, that this dotted-quarternote pattern is not entirely regular. Notice in the composite notation in Example 2.4 that there are multiple occasions, shown with boxes, where the pattern is truncated by one eighth-note: m. 4 beat 1, m. 7 beat 2, m. 12 beat 3, and m. 15, beat For ease of reading I have moved the B, E, and A accidentals into an E Major key signature; the passage is entirely diatonic. 77

98 Example 2.3a. Adams, Hallelujah Junction, mm Example 2.3b. Adams, Hallelujah Junction, mm. 1-6, composite of both pianos. Example 2.4. Adams, Hallelujah Junction, mm. 1-15, composite score with interruptions of rhythmic motive. 78

99 Greater irregularity ensues in the measures that follow (not shown) as the opening material is further developed. It is entirely possible that a conservative listener might maintain the dotted-quarter-note metric projection through the changing pattern upon entraining to the initial projection schema shown in Example 2.5a. A rigid dotted-quarter-note entrainment pattern may merely shift the listener s attention to different parts of the motive, with metrical accent falling first on the root position chord of an eighth-note duration, then the on the pattern s beginning, and finally on the sixteenth-note chord. Should one succeed in maintaining the dotted-quarter-note entrainment pattern, the listener will notice that the phasing of the truncated pattern returns to the initial alignment at m. 13. Example 2.5a. Adams, Hallelujah Junction, mm. 1-15, composite score with unchanging projection schema. It seems possible, however, that the changing rhythmic pattern may cue in the listener a flexible metric schema, as represented in Example 2.5b. The constant eighth-note subdivision may allow the listener to adjust their tactus entrainment to coordinate with the eighth-note 79

100 duration, as was conditioned by the opening measures. In this situation, the metric schema appears to be more strongly tied to phenomenal cues on the surface of the music; although one may hear a recurring pulse, the grouping of those pulses into a hierarchy is highly conditioned by phenomenal accent. Though the dotted-quarter-note tactus in mm is occasionally truncated, these infrequent interruptions do little to disturb the stability of a radical listener s projection at a lower level of the metric hierarchy, namely at the eighth note level. As the passage continues beyond m. 15, however, the newly heard streams of sixteenth notes seem to suggest that a listener might better entrain by conducting a quarter-note tactus; notice in Example 2.6 both the highly irregular occurrence of eighth-note durations and the longer stretches of successive sixteenth-note articulations. In these passages it is my experience that the sense of metric projection becomes greatly attenuated. Despite the great degree of rhythmic activity, the irregularity and oversaturation of phenomenal cues prevents the listener from construing a reliable metric hierarchy above the musical surface. While an abundance of phenomenal cues may seem to indicate the possibility of readily perceiving meter, little or vague differentiation in relative prominence of these cues may prevent a listener from privileging any cues over others. How might one go about making sense of this passage? Because the cuing of a metric schema is dependent in part upon patterns of accent on the musical surface available to the listener, I suggest we follow John Roeder s strategy and employ an accent taxonomy to help us decide which phenomenal cues to privilege. This approach may also help clarify which metric schema one may prefer at the composition s outset. 80

101 Example 2.5b. Adams, Hallelujah Junction, mm. 1-15, composite score with adjusting projection schema. Example 2.6. Adams, Hallelujah Junction, mm , composite score. 81

102 Accent as Metrical Informer and Perpetuator What precisely is meant when someone uses the term accent? Cooper and Meyer (1960) define accent as...a stimulus (in a series of stimuli) which is marked for consciousness in some way. 59 In practice, performers mark an event for consciousness by emphasizing or stressing the particular event and making the event more prominent than those which surround it. Accent is relative: it is a comparative measurement of prominence between two or more relatable stimuli. Thus, an accented event attracts more attention than an unaccented event. We could go so far as to say that accent describes any deviation or change that can be sensed or noticed, for any change in an environment is able to draw more attention than the immediately prior unchanging state. This more general definition of accent might imply that any articulated sound inherently holds accent: such is the transient nature of sound. But if every sound holds accent, how can we intelligibly differentiate between the changing sounds? Theorist John Roeder (1995) eloquently summarizes the problem of relativity: One aspect of this problem is that at any given timepoint, accents may arise in many simultaneously varying musical dimensions, and it is unclear how to sum the contributions from all these dimensions. More fundamentally, there is no consensus over how accent should be evaluated even within a single dimension. It can be argued, for example, that the strongest dynamic accent does not always accrue to the loudest moment of a piece, but rather to loud moments which are preceded by relatively quiet music. To evaluate the weight of an accent it also seems necessary to take into account the properties of the events preceding and succeeding the point of climax. 60 Roeder suggests, then, that we require three relatable events to distinguish accent: a general state of un-accent, a relatively more accented event, and a lesser accented event which allows the previously accented event to be perceived as such. Thus, the measurement of accent is not universal but, rather, dependent upon the context in which it is perceived. 59 Cooper and Meyer 1960, Roeder 1995, 3. 82

103 In an analysis of compositions by Steve Reich, Roeder (2003) classifies a number of contextual accent types attributable to the onset of pitch events that affect the perception of rhythm and meter. Most basic is an accent of attack, whereby the onset of any audible event initiates a phenomenal change and therefore exhibits accent. Accents of climax characterize the onset of events whose pitch is higher than those of preceding and succeeding events. Inversely, accents of nadir characterize the onset of events whose pitch is lower than those of preceding and succeeding events. Both climax and nadir accent types are related to melodic contour, and while any change in contour will exhibit a climax or nadir characteristic, contour changes spanning wider melodic intervals have the potential to exhibit a greater degree of accent. Accents of duration (interonset) are found at the onset of pitches whose duration exceeds that of the prior pitch duration. An accent of group beginning characterizes the onset of a repeated motive. Accents of climax, nadir, duration, and beginning cannot be understood without surrounding context; the becoming of these accent types results from a relative comparison between both a preceding and succeeding event. For example, it is not until a melody recedes from a climax or ascends from a nadir that we can understand that an accent attributable to contour change has occurred. That is to say that such accents are observed in a post hoc fashion. Changes of harmony also exhibit accent, which Roeder terms subcollection shift. The triadic nature of tonal music suggests that movement to an adjacent pitch in a diatonic scale indicates a change in harmony, whereas tertian motion is often merely an expansion of the presently sounding harmony. 61 In spite of this wonderfully descriptive taxonomy, Roeder remains aware of the constraints that the subjectivity of perception places on a universal application of such a taxonomy:...i do not intend their formality to suggest that all these accents are aurally salient in 61 Roeder 2003,

104 all music. Nadir accent, for example, is arguably negligible in the more usual styles of music that present a given melody only once or twice. 62 Joel Lester (1986) is also aware of the seeming impossibility of using these factors to objectively measure accent: First, not all accent-producing factors operate with equal importance. Some factors by themselves can produce a more powerful accent than several other factors combined. Second, accents in a musical passage occur in a metric context. The meter is both the product of certain patterns of accentuation, and also, once established, a strong influence on accentuation. Third, our ability to perceive different kinds of accents in a passage depends in part on how well we know that passage--that is, on how well we are able to appreciate certain types of accentuation as they occur instead of after they have passed. 63 Each of these points is worthy of consideration and discussion, but the second point captures the difficult communion between phenomenal accent and metrical accent. Lester seems to suggest a circular connection between accent and meter, whereby meter itself, though initially a derivative of phenomenal accent, becomes a feature that influences the subsequent perceptual salience of accent. Roeder similarly suggests this situation by including in his taxonomy pulse stream accent: Regularly repeating durations marked by accent induce a pulse stream, which itself accents timepoints metrically. 64 Pulse stream is not itself a type of phenomenal accent but, rather, a cognitive construct that grows out of the phenomenal accents of the musical surface; Roeder s pulse stream accent better compares to metric projection as theorized in Chapter One. In this way, when we say accent, we often mean both the quality of a discrete event (the phenomenal side of accent) and a cognitive timepoint in a series of projected pulses (the metrical side of accent). 62 Ibid., Lester 1986, Roeder 2003,

105 As mentioned at the start of this Chapter, in A Generative Theory of Tonal Music (1983) Lerdahl and Jackendoff distinguish between phenomenal accent and metrical accents. Their definition of phenomenal accent aligns with the one we have so far developed:...any event at the musical surface that gives emphasis or stress to a moment in the musical flow. 65 The authors define metrical accent as any beat that is relatively strong in its metrical context. 66 Lerdahl and Jackendoff also believe that, to some degree, metrical accent arises from phenomenal accent. Recall this quote from Chapter One: Phenomenal accent functions as a perceptual input to metrical accent that is the moments of musical stress in the raw signal serve as "cues" from which the listener attempts to extrapolate a regular pattern of metrical accent. If there is little regularity to these cues, or if they conflict, the sense of metrical accent becomes attenuated or ambiguous. If on the other hand the cues are regular and mutually supporting, the sense of metrical accent becomes definite and multileveled. 67 The authors continue: In sum, the listener s cognitive task is to match the given pattern of phenomenal accentuation as closely as possible to a permissible pattern of metrical accentuation 68 It is here that Lerdahl and Jackendoff clarify that, while metrical accent has some relationship to phenomenal accent, it eventually takes its own course. By coining the term metrical accent, Lerdahl and Jackendoff call attention to the cognitive metric hierarchy developing within the listener. Should events holding significant phenomenal accent coordinate with pulses in the metric hierarchy, the events will be cognitively assigned greater perceptual significance. Lester also discusses the coincidence yet independence of phenomenal and metrical accents: Whereas other types of accent require some factor to mark off the accented point in time, the metric hierarchy has a life of its own. Once that [metric] structure is established, we 65 Lerdahl and Jackendoff 1983, Ibid. 67 Ibid. 68 Ibid.,

106 as performers or listeners will keep it going as long as it is not strongly contradicted by other evidence....we, as listeners, will maintain that [metric] organization as long as minimal evidence is present, even in the face of accentual factors that would give rise to a different metric structure in the absence of such an imposed meter. In addition, accentual features which by themselves would not be able to establish a metric grouping can often serve to maintain an already established meter. 69 Metrical accent sometimes coincides with but is not the same as phenomenal accent. In fact, one might even say that metrical accent cannot be an object of sensory perception. Instead, it is a mental construct that resides within each individual listener. Its cognitive function is biased to privilege an unchanging pattern of predictable pulses further organized in a hierarchical manner. As demonstrated in Chapter One, however, the metric hierarchy is not strictly a static construct: a listener has the ability to adjust metric schemas according to the emerging novelty of a musical surface. This raises the questions: to what degree does phenomenal accent inform the metric schema? Will a listener privilege an event with relatively weak phenomenal accent but a strong metrical accent in order to perpetuate metric projection? Inversely, the analyses in Chapter One questioned: What degree of phenomenal accent is required to overwhelm an entrained metric schema? I am not convinced that there is a firm answer to this question. That is, listeners may vary greatly in their relative conservative or radical attitudes toward metric projection and phenomenal accent; these attitudes and listening practices may also vary greatly between genres and styles. It is, albeit, still possible to use an accent taxonomy to show empirical evidence that supports a listening experience characterized by varying degrees of metric stability. Example 2.7a shows the opening figure of Hallelujah Junction with the addition of an accent chart akin to those employed by Roeder (2003). The chart tracks accents of duration, 69 Lester 1986, 42,

107 climax, nadir, beginning, and harmony change. I must mention four important notes regarding liberties I take in adapting Roeder s methodology: 1) I have made minor adjustments to the score in order to be faithful to the phenomenal experience. Eighth notes are reinterpreted as sixteenth notes when a sixteenth note is articulated before the completion of the eighth-note duration, as the resonance of the eighth note is subsumed by the ensuing sixteenth-note articulation. Similarly, rests are assumed to add duration to a preceding event. 2) As in earlier Examples I have moved the accidentals to an E Major key signature for ease of reading; mm are entirely diatonic to the E Major pitch collection. This is not intended to implicate any harmonic function. 3) I assign accents of climax and nadir slightly different than Roeder. As both accent types are relative to their surrounding events, we must consider the distance from both nadir to nadir and climax to climax, as well as the contour surrounding the event in question. For example, were a series of pitches to move from low to high and were those high pitches to be successively repeated, only the first of those high pitches would exhibit climax. Thus, climax accents are only be assigned when a locally high pitch immediately succeeds a lower pitch, unless that high pitch remains lower than a relatively recent higher pitch. Leaps by intervallic motion are privileged over stepwise motion. These rules apply inversely to nadir accents. 4) A pulse stream can arise from any number of simultaneously occurring accent types. In the passage in question I assert that a strong pulse stream will arise from events with a sum of at least three accent types. I hold that this is due to the large degree of available accents and the frequency with which three or more accents occur simultaneously. Two simultaneously occurring accents may still induce a pulse stream, but the perceptual salience will generally be subsumed by events with a greater number of accents. As hypothesized in the analysis of mm (ex. 2.5a and 2.5b), the recurring interval of six sixteenth notes will probably allow the listener to derive a regular pulse. The accent analysis in Example 2.7a reinforces this perception: eighth-note durations exhibit the greatest number of accent types, making them most likely to be perceived as the most strongly accented events. Example 2.7b shows a simplified projection map with pulses corresponding to relatively strong attack points. 87

108 Example 2.7a. Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart. Example 2.7b. Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart inducing a three-pulse projection schema. Do these measures exhibit further metric qualities? Recall that musical meter is more than a series of single beat pulses; stable meters feature multiple levels of hierarchically related pulse patterns. Example 2.7c shows a map which uses the remaining accentual features to organize the opening measures into groups of three eighth-note pulses. Despite the silence at m. 2, beat 2, the listener can easily project eighth-note pulses to cognitively construct a metric hierarchy. The A 4 sixteenth notes further expand the hierarchy, although sixteenth-note pulses may be beyond the threshold of perception I will discuss thresholds of perception related to rate of articulation more thoroughly in Chapter Four. 88

109 Example 2.7c. Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart inducing a three-pulse projection schema and stable metric hierarchy. This six-sixteenth-note pattern pervades the opening measures of the composition; Example 2.8a shows mm with strong accents giving rise to a pattern of six sixteenth notes. Remember the slight interruptions in this pattern present in mm. 4, 7, and 12 (Example 2.5b). Listeners who firmly entrain to the six-sixteenth-note pattern should notice that these moments require a slight adjustment to their pattern of entrainment. To use the aforementioned accent terminology, listeners must adjust their cognitively projected metrical accent to correspond with the pattern of phenomenal accent on the musical surface. Should the listener strictly maintain the initially adopted pattern of metrical accent? Or should the listener follow the new information of the phenomenal accents? In Example 2.8b downwardly-pointing vertical arrows show where regular dotted quarter-note metrical accent would be projected. Notice that the metrical accents (downward arrows) quickly become seriously displaced from the accrued phenomenal accents (in boxes). From mm. 5-7 only accents of beginning align with the vertical arrows; in mm. 8-12, projected metrical accents align only with climax and harmony. The strongest phenomenal 89

110 accents realign with the maintained metrical accent pattern in m I believe that this conservative position of maintaining dotted-quarter-note metric projection is not viable in the face of conflicting phenomenal cues: the susceptibility of the listener to entrain to the strongest phenomenal information will make entraining to the single accents of beginning in mm. 5-7 nearly impossible. However, if the listener undertakes the occasional truncating adjustments shown in Example 2.5b, metrical accent (metric projection) adjusts to remain congruent with phenomenal accent, thereby resulting in a more comfortable entertainment experience. 71 This passage exhibits qualities similar to the phase shifting found in music by Steve Reich. Indeed, if one were to shift the opening measure upon itself by one quarter note in either direction, the resulting figure would be that which is heard when both pianos are playing. 90

111 Example 2.8a. Adams, Hallelujah Junction, mm. 1-14, composite score with accent chart and no adjustment to dotted-quarter-note entrainment. 91

112 Example 2.8b. Adams, Hallelujah Junction, mm. 1-14, composite score with accent chart; entrainment adjusting to phenomenal surface. 92

113 I expect that a listener will notice the metrical character of these opening measures to be mostly regular; the frequent occurrence of the six-sixteenth-note pattern seen in Example 2.8a clearly conveys this. The same cannot be said of the passage that follows from mm in Example 2.9. Beginning in m. 15, the six-sixteenth-note projected metrical accent becomes greatly attenuated, first through an overabundance of phenomenal accent at the interval of two sixteenth notes, then through a brief passage of phenomenal accents that accrue at four and eight sixteenth-note intervals, whereafter in m. 16 the six-sixteenth-note interval briefly returns. Notice especially that after m. 18 the distances between accumulations of three or more accents is frequently greater than the duration of a notated measure. Not coincidentally, I find it nearly impossible to entrain to any regular metric projection from mm ; the analysis eloquently shows the connection between metric ambiguity and irregular patterns of phenomenal accent. This is not to say that this passage does not exhibit metric qualities: it is possible to perceive an incipit of a regular pulse streams. Remember, though, that meter is more than a series of regular pulses. All scholarship upon which I ve drawn agrees that stable meters feature a multileveled hierarchy which privileges the relative strength of recurring pulses. Example 2.10 shows mm where we can see a pattern in which two accents occur at the interval of two sixteenth notes, perhaps suggesting a nascent metric hierarchy. However, the lack of any relatively strong accents (three or more coinciding phenomenal accents) prevents a clear hierarchy from developing. Furthermore, the pattern deteriorates in m. 23 as accent irregularity once again pervades. Example 2.11 shows a similar pattern at mm

114 Example 2.9. Adams, Hallelujah Junction, mm , composite score with accent chart and cardinal numbers indicating distance between strong accents. Example Adams, Hallelujah Junction, mm , composite score with accent chart showing potential for eighth-note projection. 94

115 Example Adams, Hallelujah Junction, mm , composite score with accent chart showing potential for eighth-note projection. Roeder s accent taxonomy beautifully captures a snapshot of the phenomenal experience of Hallelujah Junction s opening passage: metric regularity can be momentarily observed where accents occur with the greatest degree of regularity (mm. 1-15), while locations featuring little regularity may leave the listener noticing metric ambiguity (mm ). The phenomenal accents of the musical surface influence the listener s construing of a metric schema and its subsequent stability. Beyond cuing mere local meter, these changing patterns of phenomenal accent or lack thereof also inform the listener of formal boundaries spanning the whole of the composition. I will follow this hypothesis to its end in the coming Chapters, but for the moment let s look how we can classify regions with varying accent qualities. Accent and Metrical Dissonance Harald Krebs (1999) suggests that varying patterns of accentuation that characterize formal sections give rise to metrical consonance and metrical dissonance. By analogy to consonance and dissonance in pitch space, metrical consonance and dissonance describes the degree to which simultaneous pulse layers are in accord with one another. Layers, much like 95

116 pulse streams, arise in a composition as a result of recurring patterns of rhythmic motion. Krebs discusses how three specific types of layers contribute to the presence of meter: The layers that contribute to the meter of a work can be divided into three classes: the pulse layer, micropulses, and interpretive layers. The pulse layer is the most quickly moving pervasive series of pulses, generally arising from a more or less constant series of attacks on the musical surface. (The omission of a few pulses here and there does not seriously disrupt the pulse layer once it is clearly established.) More quickly moving layers, or "micropulses," may intermittently be woven into the metrical tapestry of a work as coloristic embellishments. Of greater significance are series of regularly recurring pulses that move more slowly than the pulse layer. These allow the listener to "interpret" the raw data of the pulse layer by organizing its pulses into larger units. The pulses of each "interpretive layer" subsume a constant number of pulse-layer attacks; an interpretive layer can therefore be characterized by an integer denoting this constant quantity. I refer to this integer n as the "cardinality" of the layer, and to an interpretive layer of cardinality n as an "n-layer". 72 Despite small differences in nomenclature, Krebs s terminology points to the familiar hierarchy of pulse and meter suggested by earlier authors. Furthermore it supports Roeder s suggestion that patterns of accent give rise to pulse streams (pulse layers) in a hierarchical manner (micropulses and interpretive layers) and respects the autonomous nature of the pulse stream (cognitive projection). Assigning cardinalities to these various layers allows Krebs to precisely calculate a metrical consonance or dissonance. According to Krebs, metrical consonances occur when the subdivisions of rhythms align with the notated meter signature, more precisely meaning that layers are both aligned and factorial (eg. 2/4 3/6, 2/6, 3/9, etc.). Passages with the greatest amount of metrical consonance feature a pulse layer aligned with the tactus of the notated meter signature, micropulses that are subdivisions of that meter signature, and interpretive layers that occur at rates which conform to the hypermetrical organization suggested by the meter signature. Metrical dissonances occur when layers which share a common cardinality are misaligned or 72 Krebs 1999,

117 when layers with differing cardinalities are co-prime (eg. 2/3, 4/7, 4/5, 3/4, etc.). Krebs classifies these two scenarios as displacement dissonance and grouping dissonance, respectively. 73 Grouping dissonances can occur in either direct or indirect fashions: direct grouping dissonances are created through superposition of co-prime cardinalities (eg. duple and triple concurrently occurring), while indirect grouping dissonances occur through the juxtaposition of co-prime cardinalities (eg. duple succeeded by triple). As explained by the concept of projection and the nature of metrical accent, indirect dissonance results when the listener cognitively projects an established pulse even if it has been discontinued on the musical surface. The actual duration of an indirect grouping dissonance may vary from listener to listener and passage to passage, depending upon whether the listener chooses to adjust their pattern of entrainment to coordinate with the new pulse rate or maintain the original metrical projection through the dissonant rhythm. Krebs s dissonance taxonomy deviates from a purely phenomenological approach to meter in that he privileges the notated meter signature, which he calls the primary metrical layer. 74 By assigning a pulse layer to a composition s notated meter signature Krebs allows for the possibility of an imperceptible, subliminal dissonance. Such subliminal dissonances arise when pulse layers incited by accentual features suggest a pulse layer that disagrees with that indicated by the notated meter signature. 75 For example, if a composer chooses to write a pervasive three-eighth-note pulse layer in 3/4 time, a subliminal dissonance will arise if there are not two- and six-eighth-note pulse layers coordinated with the notated measure. Importantly, this type of dissonance can only be understood by viewing the notated score. And because Krebs 73 Ibid., As with his phenomenal accent types, Krebs adopts this term from Lester. 75 Ibid.,

118 believes composers have purposefully notated such dissonances, it should be the job of the performer to convey them: The perceptibility of layers, of course, depends to a large extent on the performer, who must decide which layers to bring out. Frequently, the composer's notation will make clear which layers are intended to be prominent. Some writers have suggested that there is in fact nothing that the performer can do to actualize subliminal metrical conflicts that they are merely notational curiosities, symbolic rather than real. Composers who so tortuously notated such conflicts, however, surely did not mean them to be a secret between themselves and the performer, but rather wished performers to communicate them. The performer must encourage listeners to join him or her in sensing a subliminal metrical dissonance instead of simply giving themselves over to the new and different state of consonance that the musical surface suggests. 76 Beyond using the information available to the performer in the score, Krebs does not detail strategies to communicate subliminal dissonances, and it seems especially challenging to do so when a subliminal dissonance begins a composition. Note, for example, the subliminal dissonance which begins Hallelujah Junction: the primary pulse layer has a cardinality of 6 (sixteenth notes), while the pulse layers indicated by the meter signature (of 3/4) suggest cardinalities of 4 and 12. As Krebs suggested earlier, interpretive layers drawn from phenomenal accents (rather than subliminal factors) hold the greatest significance for the perception of metrical dissonances: I conceive of interpretive layers not as components of an abstract, "given" metrical grid, but as perceptible phenomena arising from the regular recurrence of musical events of various kinds. Interpretive layers often arise from a regularly spaced succession of what Lerdahl and Jackendoff call phenomenal accents. Dynamic accent, shown in scores by markings like sf, fz, rf, and by various hairpins and carets, is a particularly obvious type 76 Ibid., 29, 47. It is important to consider subliminal dissonance from an historical perspective: it is rare to find compositions from the post-renaissance and pre-romantic eras in which patterns of phenomenal accent do not conform to the notated meter signature (primary metrical layer). Krebs s analyses focus primarily on the piano works of Robert Schumann, in which relatively short-lived subliminal dissonances are perceptually salient due to unchanging meter signatures. In this way, it is likely that the modern listener may perceive the resolution of a subliminal dissonance as a change of meter signature. 98

119 of phenomenal accent. Such accentuation is, however, also produced by the placement of long durations among short ones (agogic or durational accents), thickly textured events among more thinly textured ones (density accents), by dissonant events among consonant ones, by registral high and low points (registral accents), by the affixing of an ornament to a note, by changes in harmony and melody (new-event accents) in short, by any perceptible deviation from an established pattern. A succession of accents occurring at regular intervals that is, the highlighting of every nth member of the pulse layer results in the establishment of an interpretive layer of cardinality n. As listeners we inevitably focus on those interpretive layers that are most perceptible and filter out those that are relatively hard to hear. In many cases, the most perceptible layers are those that are articulated by the greatest number of musical features. A layer expressed, for instance, only by pattern repetition will be less perceptible than one delineated by pattern repetition in conjunction with accentuation. 77 Krebs comments seem to support my using Roeder s accent taxonomy to uncover interpretive layers. Let s return to this opening passage and apply Krebs s methodology using my adaptation of Roeder s accent taxonomy to determine the cardinality of the apparent layers. Example 2.7a shows that the strongest pulse layer projected in mm. 1-3 spans six sixteenth notes and contradicts the notated 3/4 meter signature, which implies hierarchicallyrelated pulse layers of four and twelve sixteenth notes. Thus, the layer of six sixteenth notes create a subliminal grouping dissonance G6/4, in which the cardinal numbers describe that we have groups of six sixteenth notes where we expect groups of four. The accent chart in Example 2.7a also allows us to determine other layers informed by phenomenal events. In Example 2.12 we notice three discrete events characterized by a six-sixteenth-note pulse layer: the sixteenthnote anacrusis, the E major chord, and the Eb4 sixteenth note. These events functioning together create displacement dissonances D6+2+3, sharing common cardinalities but uncommon points of beginning. 77 Ibid., 23,

120 Example Adams, Hallelujah Junction, mm. 1-3, composite score with accent chart showing cardinality of various phenomenal layers. Applying Krebs s dissonance taxonomy further reveals aspects that are particular to this composition and perhaps to Adams s individual compositional language. First, because of the great number of phenomenal cues, it seems that dissonances can arise from any of the pitch events occurring on the musical surface. This is related to the earlier discussion surrounding accent: while accent is a ubiquitous property of any change in the environment, it must be measured relative to its context. The recursive nature of Hallelujah Junction s opening passage allows a six-sixteenth-note pulse layer to be assigned to all five events in the canonic pattern beginning at m. 4, beat 2. Do these combine to create displacement dissonances? These displacement dissonances do not seem to be characterized by a feeling of discomfort. Maury Yeston even argues that when the pulse layers (strata) share congruent cardinalities their interaction is resultantly consonant. 78 Second, it seems that subliminal dissonances, which complicate visual perception of the pulse streams or layers, are irrelevant to a listener without the score. Indeed, the first perceptible disruption of metric stability does not occur until we encounter the first grouping dissonance at 78 Yeston 1976,

121 m. 4, where the six-pulse layer is interrupted by a four-pulse figure; notice in Example 2.8a the similar disruptions at mm. 7 and 12. This dissonance can be more precisely labeled an indirect grouping dissonance, as the four-against-six dissonance occurs in a manner of succession. As this relates to Examples 2.5a and 2.5b, the conservative metrical projector may perceive this dissonance for a longer duration, while the radical projector may immediately succumb to the force of the dissonance and adopt a new entrainment pattern. Third, and perhaps most important, the homophonic texture of Hallelujah Junction poses significant problems to perceiving layers when hearing the composite of both piano parts. Although we may readily perceive certain layers when examining the individual parts, the timbrally similar and/or identical pianos sound as one combined phenomenal stimulus. The inability to clearly perceive individual layers means that not all layers are equally salient for perceiving meter and/or metrical dissonances. To further explain the challenge posed by a phenomenally skewed analysis let s examine mm Example 2.13a shows the two pianos performing parts that are visually distinct: the first piano continues to perform the original theme with relative regularity, being interrupted only two times at mm. 22 and 27, while the second piano performs an ostinato-like passage of sixteenth notes. Let s assign cardinalities to the layers that arise from these motives from each independent part. The most apparent layer in the first piano is associated with the climax and duration accent of the eighth-note chord; the layer lasts six sixteenth notes, with a four-sixteenthnote variation in mm. 22 and 27. The second piano shows two apparent layers associated with the right-hand climax and left-hand nadir accents, each lasting four sixteenth notes. These foursixteenth-note layers are, however, short lived and interrupted by either quarter or eighth rests, resulting in larger patterns which last 16, 14, 16, 18, and 16 sixteenth notes, respectively. While 101

122 each pattern beginning can be assigned a cardinality, and there is clearly some type of grouping dissonance occurring, the lack of pattern consistency makes it challenging to precisely define a metrical dissonance in Kreb s taxonomy. Furthermore, and most critically, these two parts, which are visually distinct but timbrally similar or identical, are essentially indistinguishable when heard performed together. Notice in Example 2.13b that when the four treble clef staves are condensed into a single staff, it becomes quite apparent that the two pianos share a similar pitch space, greatly diminishing the possibility that a listener may be able to perceive the two individual parts as distinct. Example 2.13a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers. 102

123 Example 2.13b. Adams, Hallelujah Junction, mm , composite score. How do these interpretive layers relate to the variety of phenomenal accents analyzed earlier? Example 2.13c shows an accent chart relating the phenomenal experience of the combined pianos from mm while also indicating the interpretive layers of the separate parts in Example 2.13a. Here we see that all of the formerly described strong accents are supported by at least one articulation of the interpretive layer. Perhaps then, it is not surprising to find that the only instance of the interpretive layers aligning m. 24, the & of beat 2 coincides with a relatively strong accent. However, when deriving interpretive layers from the individual piano parts, we notice discrepancies between where a layer seemed to exist in Example 2.13a and where phenomenal accent types appear in Example 2.13c, perhaps suggesting that these layers do not exist at all. This is a more formal demonstration that the timbral and registral similarities of the two pianos hinder the perceptual salience of each individual part. 103

124 Example 2.13c. Adams, Hallelujah Junction, mm , composite score with accent chart showing strong accents and numerical labeling of visual layers from Example 2.13a. A similar problem characterizes mm The original score shown in Example 2.14a shows the second piano performing two three-sixteenth-note interpretive layers displaced by one sixteenth note, each seemingly characterized by a nadir and climax accent. On a larger level we notice these three-sixteenth-note layers are a part of a larger pattern of sixteenth notes, generally spaced at the interval of 14 sixteenth notes. As was noticed in the previous example, the pattern is occasionally adjusted; in mm. 52, 58, and 62 we find the beginning of interpretive layers that span either 16 or 12 sixteenth notes. How do these layers align with those in the first piano? I am hard pressed to find a meaningful interpretive layer in the first piano from mm ; the eighth-note articulations occur in such a random manner that it is difficult to discern any recurring motive. The first piano s texture changes significantly at m. 56 as it returns to performing the opening motive, where again its normal duration is six sixteenth notes. As with 104

125 the opening motive, this pattern also features slight durational adjustments: we find an eightsixteenth-note iteration in m. 58 and a four-sixteenth-note iteration near the end of m. 61. Example 2.14a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers. 105

126 How perceptible are these layers when considering the composite sound produced by both pianos? Example 2.14b shows the score in a single part with both an accent chart corresponding to the phenomenal experience of the combined pianos and a labeling of interpretive layers drawn from Example 2.14a. Between mm. 51 and 55 it seems there is very little connection between interpretive layer onset and accent accumulation. On several occasions, interpretive layers even coincide with no apparent phenomenal accent! There seems to be a stronger argument for the perceptibility of interpretive layers from mm Here, strong accents are found to better coincide with the beginning of interpretive layers. Only those strong accents labeled in mm. 61 and 64 remain unaccompanied by the onset of such layers. However, the three-sixteenth-note layer is shown to be significantly less accented than the six-, eight-, twelve-, and fourteen-sixteenth-note layers, once again suggesting that this layer is not perceptually salient. What does this particular insight mean for the perceptibility of other layers? Even a passive listener to mm who is not actively keeping time is likely to notice the changes in texture that occur at m. 44, where flowing and resonating sixteenth notes give way to percussive and pointillistic attacks, and at m. 56, where the unpredictable staccato eighth notes give way to the return of the opening theme. Here, I expect that a listener will prefer to entrain to the six-sixteenth-note interpretive layer associated with the opening theme, occasionally adjusting entrainment where necessary. Why might the listener be able to perceive and entrain to the six-sixteenth-note layer from mm but not at the similarly composed section from mm ? 106

127 Example 2.14b. Adams, Hallelujah Junction, mm , composite score with accent chart showing strong accents and numerical labeling of visual layers from Example 2.14a. I offer two reasons. First, the perceptibility of the opening theme seems highly dependent on the clarity with which the listener can distinguish the highest pitch in the pattern. As was formerly suggested, the pitch range of the second piano between mm in Example 2.13a precisely envelops that of the first piano, rendering the highest pitch of the first piano s pattern (E 5) indistinct. In Roeder s terminology, the climax accent is no longer heard as a climax. During the later iteration beginning at m. 56 in Example 2.14a, we see that the second piano plays no higher than a Bb4, while the motive in the first piano continues to reach E 5, allowing 107

128 for accents of climax to more often be attributed to the motive. Notice in Example 2.14b that beginning at m. 56 accents of climax always coincide with the onset of the interpretive layer attributed to the main theme. Second, if the listener metrically interprets the opening theme in the manner I suggest, they will be susceptible to entraining to the restatement of the opening theme at m. 56 in a similar manner. Earlier I shared Lester s claim that familiarity with a work can influence listeners to differently perceive patterns of accent, for we can appreciate certain types of accentuation as they occur instead of after they have passed. 79 In this way, it is possible that prior conditioning may allow the listener to require fewer accents to re-entrain to the original metric schema; the learned familiarity with the main theme allows the listener to privilege its associated metrical accent, despite the metrical accent s weaker phenomenal underpinning. There is great potential for further cognition research regarding metric entrainment in situations where the listener has varying degrees of familiarity with the score, ranging from a second hearing to complete memorization. Metrical Dissonances and Changing Textures in Hallelujah Junction Metrical dissonances in Hallelujah Junction are often associated with changes in texture. Take, for instance, Example 2.15a which shows the broad passage from mm that closes the first movement; a blending of four motives progresses gradually from expansive and flowing sixteenth notes, to angular and truncated sixteenth notes, to irregular staccato eighth-note chords, and finally to more rhythmically irregular chords of eighth-note triplets. Adams achieves these transitions using an approach that typifies the entire composition; instead of changing textures in 79 Lester 1986,

129 both piano parts suddenly, each part changes textures independently, thereby creating dovetailed transitions. Example 2.15a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers. 109

130 Example 2.15a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers, contd. 110

131 Example 2.15a. Adams, Hallelujah Junction, mm , original score with visual analysis of individual layers, contd. Example 2.15a shows that at m. 214 the pianos perform similar motives of ascending and descending sixteenth notes; the first piano performs in a lower tessitura ranging from C2 to F4 while the second piano performs in a higher tessitura ranging from C4 to D 6. A short time later 111

132 in m. 220 the second piano joins the texture of the first piano before assuming its own unique rhythmic motive from mm Meanwhile the first piano abandons its arpeggiated sixteenth-note texture in m. 228 and assumes an aggressive and irregular pattern of staccato eighth-note articulations. Finally in m. 237 the second piano joins the texture of the first as the two parts perform a hoquet-like passage of eighth-note chords until the movement s conclusion. The final texture change occurs in m. 257 where we notice the first piano beginning to incorporate triplet figures that compete with the pervasive eighth-note texture. These interleaved changes of texture contribute to mounting intensity as the movement nears its conclusion. Notice, also, in Example 2.15a how each of these transitions is characterized by varying durations of motivic patterns. The patterns indicated in Examples 2.15a and 2.15b are similar to, but not synonymous with, interpretive layers. The numerical designation is used only to show the duration of the patterns which are visually apparent in the separated piano parts of the original score. Beginning in m. 214 the second piano performs a pattern lasting twenty-two sixteenth-note durations, which can be divided into durations of seven and fifteen sixteenth notes. The first piano performs a pattern lasting thirty-two sixteenth-note durations beginning in m. 217, divisible here into durations of fourteen and eighteen sixteenth notes. The second piano s change to a less continuous sixteenth-note texture in m. 223 is accompanied by a transition in pattern length which varies in duration from thirty-two, thirtyfour and thirty-eight sixteenth notes, depending upon the inclusion of eighth rests between or during statements. Likewise, the first piano s change in texture at m. 228 results in patterns that span either twelve, sixteen, or twenty sixteenth notes. 112

133 Example 2.15b. Adams, Hallelujah Junction, mm , composite score with numerical labeling of visual layers from Example 2.15a. 113

134 Example 2.15b. Adams, Hallelujah Junction, mm , composite score with numerical labeling of visual layers from Example 2.15a, contd. While these patterns are not necessarily indicative of phenomenally-derived interpretive layers, it is fair to say that each pattern exhibits an accent of beginning. In Chapter One I argued that pattern beginnings are especially significant for cognizing metric projection. When viewing the original score with the two pianos on separate lines, the beginning of patterns appear to be the only outstanding features available for metric analysis. When viewing the combined score shown in Example 2.15b this visual clarity is lost. As the combined score better represents a phenomenal hearing of the passage, it makes sense that the onsets of the previously labeled layers are phenomenally indistinguishable between mm The pattern of accentuation beginning in m. 228 may be more perceptible, as the texture surrounding the onset of each layer 114

135 is more transparent, either due to a great leap in interval or a prior accent of duration resulting in a present accent of beginning. Beyond m. 237 the second piano joins the texture of the first, the composite result being a relatively homophonic style. Here rhythms occur in such an irregular manner that it is impossible to find any consistent pattern giving rise to an interpretive layer. And as the first piano begins performing triplets, the composite rhythm becomes even more irregular and metrically indecipherable. How might we calculate metrical dissonance in this passage? I suggest this to be an impossibility. In fact the apparent randomness of this passage seems to be proportionately calculated. Adams s use of rhythmic proportionality in temporal space is a critical insight uncovered by Kyle Fyr (2011). Example 2.15c shows mm with whole numbers showing the duration between eighth-note articulations. We find 3 occurrences of a half-noteduration (four eighth notes), 10 occurrences of a dotted-quarter-note duration (three eighth notes), 21 occurrences of a quarter-note duration (two eighth notes), and 35 occurrences of an eighth-note duration. This is equivalent to increasing factors of approximately 1.5, 2, and 3, respectively. Thus, while the irregularity of cardinality prevents calculating a dissonance as Krebs does, we can still understand the passage to be highly dissonant. The great degree of metrical dissonance exhibited by mm signals a climactic conclusion to the opening movement. 115

136 Example 2.15c. Adams, Hallelujah Junction, mm , composite score with numerical labeling of eighth-note onset interval. Formal Structure as Related to Metrical Dissonance Earlier I suggested that these shifts from relative metrical consonance to dissonance communicate larger formal structures throughout Hallelujah Junction. As in the opening movement, the climax of the second movement also features a relatively large amount of metrical dissonance. Example 2.16a shows this climax from mm having three distinct rhythmic ideas: the first piano s recurring dotted eighth note rhythm serves as a stable body against which the second piano s recurring eighth notes (mm ) change to eighth-note triplets (mm ). Metrical dissonance can be more precisely calculated in this passage and has strong phenomenal salience; interpretive layers are highly regular and these layers are clearly perceivable due to their varying tessituras. Calculating dissonance here is not, however, entirely 116

137 straightforward. Because of the presence of triplet eighth notes in opposition to a passage of sixteenth notes a direct grouping dissonance cardinality must be determined by using a common factor. In this case, a quarter note can be assigned a duration of 12, thereby accommodating both three triplet eighth notes and four sixteenth notes. Through this method we can calculate the apparent direct grouping and displacement dissonances, as well as the indirect juxtaposition dissonance. First a grouping dissonance of G12/18 exists from m. 339, beat 3 through m. 342, beat 1. As the second piano s eighth notes change to triplet eighth notes at m. 342, beat 2, this grouping dissonance changes to G8/18. The transition to triplet eighth notes is itself an indirect dissonance, juxtaposing 12 against 8. Furthermore, each of the three rhythm types creates a displacement dissonance with itself due to the alternating pattern of right and left hand articulations. Krebs would classify the dissonances exhibited by the eighth notes, dotted eighth notes, and triplet eighth notes as D12+6, D18+9, and D8+4, respectively. Example 2.16a. Adams, Hallelujah Junction, mm , original score with cardinalities of interpretive layers. To further show the great degree of metrical dissonance exhibited by this passage, let s treat each quarter note duration as if it were 12-event modular space. In Example 2.16b each articulation is assigned a whole number (between the two piano parts) signifying the location of 117

138 its onset within that 12-event space. Below both parts, I indicate the distance between the onset of each unique event. This labeling clearly shows how the passage s composite rhythm becomes more dense as the second piano transitions from eighth notes to triplet eighth notes in m. 342: while the prior eighth-note-against-dotted-eighth-note figure features a regular pattern of events spaced by two 6 and two 3 MOD-12 units, the dotted-eighth-note-against-triplet-eighth-note figure exhibits an irregular pattern of 1, 2, 3 and 4 MOD-12 units. Interestingly, Adams seems to reserve these instances of extreme metrical dissonance for particularly climactic moments, constructing passages of textural and metrical saturation that contribute to a metaphorical Gordian knot. I suggest that the relationship between metrical dissonance and formal structure is not unique to Hallelujah Junction. In fact, it seems apparent that this relationship can be especially relevant in situations where formal functions are more strictly defined. In the following Chapter, I explore how varying states of metrical consonance and dissonance delineate formal boundaries in the exposition of Robert Schumann s Symphony in C Major, Op. 61. These varying states of metrical consonance and dissonance suggest a phenomenal metric interpretation of the exposition's themes which may differ dramatically from assumptions that are used by the performer. 118

139 Example 2.16b. Adams, Hallelujah Junction, mm , onset points in a MOD-12 space (per quarter-note). 119

140 CHAPTER THREE Metrical Dissonance in Schumann s Symphony in C Major, Op. 61 Robert Schumann s Symphony in C Major, Op. 61, exhibits many metric peculiarities. This is a fair characterization of Schumann s oeuvre in general, and in this vein Krebs provides the most comprehensive study. 80 Of particular relevance to the model of metric theory proposed in this document is the exposition of the Symphony s first movement. A quick glance at the score shows the Allegro ma non troppo exposition notated in an unchanging 3/4 meter signature. However, as was the case with Hallelujah Junction, the notated 3/4 meter signature does not accurately reflect a phenomenal metric entrainment experience. Rather, patterns of phenomenal accent frequently give rise to metrical dissonances that contribute to perceiving an unstable metric hierarchy. Frequently, these patterns of accent (interpretive layers) misalign with the notated measures (primary metrical layer) giving rise to subliminal dissonances. Furthermore, interpretive layer durations tend to change without prompting, which results in a juxtaposing of layer durations and an upsetting of the metric hierarchy s stability. A deeper analysis reveals that each of the exposition s classical-style formal functions exhibits different patterns of accent. These insights support two ideas posited in the earlier Chapters: musical meter is not always well represented in notation and a composition s form can be effectively communicated through phenomenal meter. Before diving into some analyses, I d like to briefly return to the phenomenon that Krebs calls subliminal dissonance, as this particular type of dissonance plays a central role in the metrical organization of the Symphony s exposition. As described in the previous Chapter, 80 See Krebs, Fantasy Pieces

141 subliminal dissonances occur when the sum of the phenomenal cues suggests a metric schema that disagrees with the primary metrical layer. While this document has focused primarily on the perspective of the scoreless listener, it may be valuable here to consider the perspective of the informed listener and the performer, both of whom being familiar with the score. 81 The performer, in the interest of maintaining a steady tempo, may choose to metrically interpret a passage containing subliminal dissonance by employing a pattern of counting which corresponds to the notated measure, as opposed to a pattern of counting that corresponds to the audible phenomenal accent. For example, imagine a passage notated with a 3/4 meter signature where the notated beat 2 features the strongest phenomenal accent. While the scoreless listener is likely not to know they have entrained to the notated beat 2, the scored listener or performer is likely to feel a displacement dissonance between the metric accent of beat 1 and the phenomenal accent of beat 2. In this way seeing the score and hearing the score suggest two different musical experiences. Thus, I might ask the following: do the informed listener and performer perceive a meter relative to the visual unchanging metric grid, whose pattern of metrical accent serves as a conceptual anchor for overlying phenomenal rhythms? Or do they perceive meter informed primarily by the phenomenal rhythm and accent of sounds that are heard? Or perhaps informed listeners and performers blend these perspectives. The question of subliminal dissonances at this intersection of experience is rather interesting: is entrainment affected primarily by conditioning or by the phenomenal? I have been arguing that musical meter does not exist in notation and only becomes an apparent phenomenon when a listener perceives recurring patterns of accent as the 81 This distinction is important: unlike the music of Thom Yorke, scores for Schumann s music exist and a readily available. It is common for anyone studying Schumann or his music to listen to a recording while following a score. 121

142 music unfolds. However, performers are likely aware that the production of music generally relies upon practiced meter. There are many valuable reasons that a performer may wish to maintain a cognitive metric schema that corresponds to notated meter signatures rather than phenomenal accent. Consider the following: the number of musicians who would perform Schumann s C Major Symphony is far greater than the number needed to perform the compositions analyzed in Chapters One and Two. The electro-pop tunes and the piano duet utilize only a small number of musicians, making it easier to coordinate a performance of rhythmically differing parts. Conversely, a full orchestra of over seventy musicians performing a score consisting of over twenty independent parts is clearly more complicated to synchronize and nearly always requires the leadership of a conductor. Conductors typically show a pattern that is relatively congruent with notated measures, similar to the manner in which performers typically learn their music. This pattern is generally synonymous with what Krebs and Yeston call the primary metrical layer the pulse layer that best corresponds to the notated meter signature. For performers, cognitively entraining to the primary metrical layer is essential for both maintaining a consistent tempo and ensemble cohesion. This holds significant implications for the perception of subliminal dissonance: cognitively producing metrical accent that perpetuates the primary metrical layer allows the performer to perceive subliminal dissonances as especially strong metrical dissonances. Compare this experience to that of the unconditioned listener, whose unfamiliarity with Schumann s music might range from total unfamiliarity to unfamiliarity with only the specific composition being heard. In the previous Chapter I stated that subliminal dissonances are perceptually irrelevant to a listener who is unfamiliar with or not presently reading the notated 122

143 score. Provided no other audible cues, a subliminal dissonance by itself will not be perceived as metrically uncomfortable or ambiguous. That is, a subliminal metrical dissonance is only dissonant against the notation. Thus, performers and listeners may perceive different and unique phenomenal experiences, each being entirely dependent upon the degree of familiarity with the notated score and stylistic conventions. Even, perhaps, the first-time listener following along with the score may deliberately project metrical accent (cognitive) according to the notation they see. As earlier suggested, there would be great value in an empirical study of metric entrainment as related to aural familiarity with a composition and specific knowledge of the score. There may also be value in studying how a performer s metric schemas may be dominated by metrical accent (cognitive projection) compared to phenomenal accent. From the listener s perspective, phenomenal accent first induces metrical accent. Thereafter, both the cognitive and phenomenal may influence one another, albeit, phenomenal accent maintains greater power to dictate changes in metrical accent than does the inverse relationship. For the performer, metrical accent is frequently a reflection of the meter signature and can be imagined as a grid upon which phenomenal accent is placed. These perspectives are clearly at odds with one another. My analysis of Hallelujah Junction showed that the perception of differing metrical states frequently correlates with a composition s formal features. While I am still focused on assessing the listener s phenomenal metric experience, knowing that the primary metrical layer is three quarter-note pulses in duration (indicated by a 3/4 meter signature) serves as a significant analytical reference for many listeners and for nearly all performers. Because so much of the metrical dissonance in the exposition of the Schumann s C Major Symphony is in fact subliminal dissonance, passages which do not exhibit subliminal dissonances can exhibit maximum metrical consonance. 123

144 Main Theme and Codetta To demonstrate how varying states of metrical dissonances delineate formal areas in Schumann s C Major Symphony, let s first examine the exposition s main theme. Example 3.1 shows a piano reduction of this passage as it spans eight measures from mm , after which it is repeated from mm in expanded orchestration. The main theme s dotted-eighthsixteenth-, double dotted-quarter-sixteenth-note rhythm is a prominent (pervasive) rhythmic figure throughout the passage. How might a listener entrain to its musical surface? It seems certain to me that a listener may notice a clear pulse that corresponds with the quarter note. Notice in Example 3.2a that, despite the dots and the long duration of the double-dotted quarter note, the rhythm of the main theme alone can give rise to a quarter-note pulse. The first two sounds initiate a dotted-eighth-note projection (level A), but the third sound suggests that this projection does not match the musical surface. If we perceive the sixteenth note that precedes beat 2 to be a subdivision of a larger pulse, the events occurring on beats 1 and 2 give rise to quarter-note projection. Although this projection is not reinforced by phenomenal accent on beat 3, a listener who continues to project a quarter-note pulse will be rewarded with congruent event articulations that occur on beats 1 and 2 of the following measure. This, of course, suggests that the sixteenth-note anacrusis preceding m. 51, beat 1 must be interpreted as a pulse subdivision, as was done in the opening measure. 124

145 Example 3.1. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition main theme. Example 3.2a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , melodic rhythm giving rise to quarter-note projection. As the music continues, the main theme s dotted-eighth-sixteenth-, double dottedquarter-sixteenth-note rhythm repeats in a three-pulse pattern. This recurring pattern suggests the presence of a stable metric hierarchy. But despite how a meter signature s implied metric hierarchy begins with the onset of a measure, it is clear in this case that the strongest phenomenal accent in each measure consistently accrues to the longest duration, the double dotted quarter 125

146 note on beat 2. Example 3.2b includes this three-pulse projection (level C), further fleshing out the metric hierarchy. Notice in Example 3.1 how beat 2 is articulated with the most prominent accent of duration and further supported by a change of harmony. This three-pulse metrical interpretation suggests the main theme exhibits a subliminal displacement dissonance of D3+1, meaning the three-pulse interpretive layer begins one pulse removed from the primary metrical layer (meter signature). The listener without the score might have assumed the music began with a pick-up and that the longest duration coincides with the metrical accent, for all of the phenomenal cues point to perceiving beat 2 as the downbeat. It therefore seems likely that the listener will not notice any metric ambiguity and remain unaware of the subliminal dissonance. Example 3.2b. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , melodic rhythm giving rise to quarter-note projection with three-pulse hierarchy. Although other accent types are noticeable most obviously accents of registral climax I suggest their ability to influence the perception of meter to be largely negligible. Once the listener has entrained to the three-pulse hierarchy shown in Example 3.2b, I suspect that the listener s own sense of metrical accent will have been so strongly reinforced by the main theme s duration accent that other phenomenal cues will lose their ability to affect change upon the listener s metric schema. For example, the climax accent that occurs in m. 52 on the final sixteenth note of beat 1 is clearly the most prominent phenomenal cue in a very local context. It is perhaps the case that because it occurs only one sixteenth-note duration removed from the strongest perceived accent in the metric hierarchy, this climax actually serves to further solidify 126

147 the suggested metric schema, being heard as a clear anacrusis to the succeeding strong metrical accent; the same can be said for the similarly-placed climax accents in mm. 56, 60, 64 and 65. Climax accents are also found on beat 1 of mm. 53 and 61. These accents occur on relatively strong pulses of the perceived metric hierarchy and further enhance the perception of hearing beat 2 as the phenomenal downbeat. As the exposition draws to a close, listeners may hear the same motive giving rise to a different metric schema. Example 3.3a shows the passage of the exposition s codetta from mm , with the incipit of the main theme reappearing in m To understand how this theme may be perceived differently, let s examine how the material leading into m. 100 prepares a different metric schema. Notice how in mm. 92 and 93 contour changes on beat 2 in both the highest and lowest pitches suggest a three-pulse metric schema, with beat 2 once again heard as the downbeat. Beat 2 of the following measure (m. 94), however, notably has no supporting phenomenal accent. This lack of metric reinforcement in the midst of contrary chromatic motion appears enough to raise a doubt over the continued viability of this projection schema. Indeed, in the following measure (m. 95) contrary motion on beat 1 provides significant phenomenal accent. Hearing the downbeat of m. 95 as the onset of a three-pulse metric schema is greatly reinforced by the harmonic and melodic motion leading into the downbeat of m. 96. Notice the strong tonic-dominant-tonic harmonic motion in m. 95 on beats 2 and 3 resolving on the downbeat of m. 96. This arrival is accompanied by a melodic landing on the tonic pitch G, the descent to which having been initiated earlier in m

148 Example 3.3a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition codetta. If the listener perceives the downbeats of mm. 95 and 96 to be metrically strong events, it should follow that the listener will project a three-pulse metric schema through the measures that follow. Example 3.3a shows that mm are a restatement of mm , this time with the bass voices articulating the three-quarter-note motive. Despite its earlier metric interpretation, I assume the listener will interpret the harmonic changes on the downbeats of mm. 97 and 98 to support the new metric schema. Further support comes from phenomenal cues heard in the first violins, shown here in Example 3.3b. After two measures of playing successive sixteenth-note G5 s, the first violins move on the downbeat of m. 98, confirming a three-quarter-note pulse projection. The measure-long dominant to tonic motion from mm further supports the metric schema. Example 3.3b. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , first violin melody. 128

149 Having entrained to this schema for several measures, the orchestra s homophonic iteration of the main theme s rhythm in m. 100 can be easily perceived in the new metric context. Other factors also contribute to this perception. First, the theme s rhythm is presented for only one measure. Immediately following in m. 101 the orchestra begins a series of nine consecutive dotted-eighth-sixteenth note articulations. Second, the harmonic rhythm of mm clearly 6 emphasizes the three-pulse pattern, with beats 2 and 3 of m. 102 sounding a ii 5 - V 7 leading to a tonic resolution on the downbeat of m This same harmonic pattern is repeated in mm , with the three-pulse metric schema further supported by a the two-beat duration of the melody s B5 beginning at m Rather fittingly, the exposition s penultimate measure features a staccato articulation of the main theme s rhythm, more clearly perceivable in the metric schema consistent with the notated primary metrical layer. What seemed to function more like a subliminal dissonance in mm is now clearly heard as a displacement dissonance in m. 100 and following. The metric context which was formerly amenable and flexible to phenomenal cues is now more rigid and inflexible, giving rise to a clear displacement dissonance based on the difference between metrical accent and phenomenal accent. It may not be possible to perceive the main theme s subliminal displacement dissonance during the first iteration. If the exposition is repeated, then it may be made apparent at the beginning of the exposition s second iteration as a displacement dissonance. How might this be? Consider the strong three-pulse projection pattern that is cued by the exposition s conclusion. If a listener were to continue this projection schema through the repeat to the exposition s beginning, as shown in Example 3.4, they would notice that although the downbeat of m. 106 supports the projection schema, the phenomenal duration accent on beat 2 is in direct conflict. Here we might want to invoke again the paradigm of the radical and conservative listener. Upon noticing 129

150 this conflict and hearing the repeated iteration of the main theme, the radical listener is likely to immediately adjust their pattern of entrainment to reflect what was sensed upon first listen. This adjustment requires adding one extra pulse to the three-pulse hierarchy. Conversely, the conservative listener may be likely to perpetuate the codetta s three-pulse metric schema through the conflict. In this way, the codetta orients the conservative listener to the primary metric layer, which allows for entrainment to the main theme in accord with notation. In either the radical or conservative case, sustaining the three-pulse schema through the exposition s repeat will make it apparent that metric peculiarity exists around the main theme. Example 3.4. Schumann, Symphony in C Major, Op. 61, first mvmt., mm. 102 to 53, end of exposition and beginning of second statement. The rhythm of the main theme appears in several forms throughout the Symphony. For example, as the first movement nears its conclusion, the theme is manipulated so that the pervasive pulse shifts to the notated downbeat. Example 3.5 shows mm , where, after a 130

151 brief manipulation of the rhythm, beginning in m. 360 duration accents occur consistently on the notated downbeat. A similar situation is found at the opening to the Symphony s final movement. Example 3.6 shows mm. 5-8 of the closing movement, where after three movements of thematic development, the heroic theme is stated in its properly resolved form; duration, climax, nadir, and change of harmony accents are firmly present on each measure s downbeat. In this way, phenomenal accent has been brought into alignment with metrical accent. No longer does one find subliminal or displacement dissonances, but instead we hear a celebration of hardwon metrical consonance! Example 3.5. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , melodic rhythm. Example 3.6. Schumann, Symphony in C Major, Op. 61, fourth mvmt., mm

152 Interestingly, the subordinate theme from the opening movement of Schubert s Unfinished Symphony, D. 759 uses a similar melodic rhythm to that of the Schumann s main theme. Example 3.7 shows how the melody in the bass voice from mm generally receives the same duration accent on beat 2. A phenomenal hearing of this passage, however, does not seem to give rise to the same beat-two-as-downbeat gestalt that we hear in Schumann s theme. Here the two measures which precede the melody allow the listener to establish a clear threepulse metric hierarchy which can be maintained through the displacement dissonance D3+1 exhibited by the melody. Articulations on the downbeats of mm. 42 and 43 and metrically regular syncopations establish the meter. Thereafter, harmonic changes occur on notated downbeats, perpetuating the original three-pulse interpretive layer and preventing subliminal or displacement dissonances from arising. Example 3.7. Schubert, Symphony in B minor, D. 759, first mvmt., mm

153 Transition to Subordinate Theme The sixteen measures of the main theme are strongly unified by the recurring pattern of durational accent that emphasizes beat 2. Unsurprisingly, the transition linking the main theme to the subordinate theme is characterized by a gradual unraveling of this accent pattern. Let s examine the phenomenal accent exhibited by the transition music shown in Example 3.8 (mm ). To show how meter continues from the main theme into the transition, Example 3.8 includes the final measure of the main theme (m. 65) and the three-pulse pattern oriented to beat 2 below the score. Despite m. 66 exhibiting a dramatic change in texture (from full orchestra to woodwind choir), the pattern of durational phenomenal accent continues to emphasize beat 2. The following measure (m. 67), performed by the strings, likewise supports hearing beat 2 as the phenomenal downbeat, reinforced by a nadir accent in the melodic contour. However, the disappearance of the formerly ubiquitous duration accent draws attention to a slight change of metric character. Where we are used to hearing long pitches, we now hear rapid sixteenth-note motion. It is important to note that the sixteenth-note motion doesn't begin on beat 2; the misalignment between the beginning of the sixteenth-note runs and the beginning of beat 2 obscures both the phenomenal (durational) and metrical (projection) accents that have thus far favored beat 2. Another alteration to the former pattern of phenomenal accent immediately follows. Notice for example the strong accent of climax and duration that is heard on the downbeat of m. 68 where the strings arpeggio peaks. These changes in metric character begin to untie the strong metric projection schema that unified the main theme, while at the same time diluting the former metrical dissonance. 133

154 Example 3.8. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , transition to subordinate theme with change from three- to two-pulse hierarchy. How do these changes affect overall entrainment and the perception of meter? For the conservative listener who continues to project a three-pulse metric schema, there should be no significant change. The phenomenal accent exhibited in m. 66 to the downbeat of m. 68 is congruent with the former three-pulse projection schema, despite the strong accent on what would be perceived to be the third pulse. As the figure is repeated in mm. 68 and 69, again alternating between woodwinds and strings, the original three-pulse metric schema can continue to be maintained. Note the slight rhythmic variance of the upper woodwinds on beats 2 and 3 in m. 68. Though not affecting entrainment, these small metric changes continue to influence the morphing of the passage s rhythmic character. Are these changes in rhythm and phenomenal 134

155 accent enough to influence the radical listener s pattern of metric entrainment? I suspect not. The durational accent on beat two returns in mm , despite the uncertainty of mm The three-pulse pattern does not, however, seem to be reinforced by the phenomenal surface in the immediately succeeding passage. From m. 70, beat 2 to m. 73, beat 1 all phenomenal cues point to a four-pulse metric schema. Beginning with the anacrusis to m. 70, beat 2, the composite rhythm articulates seven consecutive sixteenth-dotted-eighth note figures. This alone is not enough to call the former three-pulse projection schema into question, as one could theoretically continue projecting a three-pulse metric hierarchy. What other phenomenal accents do we find? Likewise beginning at m. 70, beat 2 we hear a strongly articulated two-pulse harmonic rhythm leading to the arrival of the subordinate key of E Major on the downbeat of m. 73, whose cadence is easily the strongest we have thus far encountered. This two-pulse harmonic rhythm is further supported by the initial pulse-wise alternation between the highly pitched woodwinds and the deeply pitched strings. Notice how the woodwinds articulate m. 70, beat 2 and m. 71, beat 1, while the strings articulate m. 70, beat 3 and m. 71, beat 2. Also notice how the strings are scored with notated accents. Hearing a quarter-note pulse alternating between highly pitched woodwinds and accented, deeply-pitched strings seems to resemble the backbeat schema, which holds particular relevance for the modern listener familiar with contemporary popular music. After four pulses of alternating articulations, the woodwinds and strings join to perform four pulses in homorhythm, leading to the eventual cadence on E Major. How might this change from a three-pulse metric schema to a four-pulse metric schema be represented with a projection graph? Notice in Example 3.8 how the continuing of the threepulse schema from the main theme aligns with the arrival of the four-pulse schema at m. 70, beat 2. During the first listen, it is impossible to know that a four-pulse schema will arrive at this 135

156 point, although the changing rhythmic character in mm may allude to a potential change. I suspect that accented strings at m. 70, beat 3 may be the first sign of a potential change. That we hear a two-beat repetition of the texture change supported by a two-pulse harmonic rhythm indicates a hemiola-like figure. Krebs would most certainly label this a grouping dissonance G2/3. The radical listener might choose to abandon the three-pulse schema and entrain to some type of two- or four-pulse hierarchy, and they would be duly rewarded by hearing the ensuing phenomenal accent pattern at m. 71, beat 3 that supports a two- or four-pulse metric hierarchy. The conservative listener who attempts to maintain the original three-pulse metric schema will be sore to notice a particularly strong cadence (m. 73, beat 1) occurring on what they perceive to be a phenomenal beat 3. However, I doubt that even the most conservative listener would prefer to maintain a three-pulse metrical accent through the phenomenal two-pulse harmonic rhythm; the impossibility of continuing with a three-pulse schema intensifies as the cadence approaches. In this way, we notice a stark contrast between the metrical character of the main theme and the metrical character that concludes the transition to the subordinate theme. Fittingly, it is a gradual process through which the phenomenal three-pulse schema morphs into a phenomenal four-pulse schema. Schumann unravels the subliminal/displacement dissonance, but in so doing does not give us a metrical realignment with notation, but rather gives us a grouping dissonance. In this way, Schumann gives us one dissonance in place of another instead of unravelling a dissonance into metrical consonance in the subordinate theme area. Especially note the particular difference between phenomenal accent concentration in the main theme and the conclusion of the transition. While the main theme features only one strong phenomenal accent in a three-pulse cycle, the conclusion to the transition features either two or four strong phenomenal accents in a four-pulse cycle. The sense of expansiveness cued by the sparse phenomenal accent of the main 136

157 theme is sharply contrasted by the rhythmic impetus of more rapidly occurring phenomenal accents at the transition s conclusion. In sum, these passages metric qualities move from a state of relative relaxation (main theme) to a driving forward (transition) into the onset of the subordinate theme. In the introduction to this Chapter I mentioned the significance of subliminal dissonance. As of yet, my suggested patterns of metric entrainment strictly avoid congruence with the notated 3/4 meter signature. Despite the main theme s three-pulse metric schema matching the meter signature s duration, phenomenal accent suggests hearing beat 2 as the phenomenal downbeat. Differently, but similarly incongruent, it is obvious that the four-pulse metric schema which concludes the ensuing transition does not match the 3/4 meter signature. As there have yet to be any phenomenal cues that suggest for adopting the notated meter signature as a metric schema, the patterns of phenomenal accent indicate a subliminal three-pulse displacement dissonance D3+1 and a subliminal two-pulse grouping dissonance G2/3. While these dissonances may be unable to be sensed by the listener as uncomfortable or ambiguous metric situations, it seems quite valuable from the analytical perspective to notice that the conclusion of the transition and the beginning of the subordinate theme cadences on the notated downbeat at m. 73. That this notated downbeat coincides with both an especially strong phenomenal event, as well as an especially strong beat of the listener s cognitive metrical accent, is particularly significant for the formal arrival of the subordinate theme. Subordinate Theme The initial metric perception of the subordinate theme will be directly influenced by the metric schema that a listener, using projection, carries forward from the transition s conclusion. 137

158 Example 3.9a shows the passage from mm containing the subordinate theme. Conveniently, the pattern of phenomenal accent exhibited by the subordinate theme s melody strongly supports a continuation of the transition s concluding metric hierarchy. Notice how the melodic pattern of four ascending eighth-note pulses creates accents of climax at two-pulse intervals (m. 73, beat 3, m. 74, beat 2, and m. 75, beat 1). The climax accent is briefly suspended in m. 75, but having experienced four two-pulse-accent intervals suggests a continuation of this two-pulse interval. This would suggest for us to find accent at m. 75, beat 3. Although notably absent of climax accent, m. 75, beat 3 does feature a harmonic resolution to a consonant pitch of the tonic E Major chord, providing a salient accent of harmony. Two quarter-note pulses later, the accents of climax resume with the ascending melodic pattern (m. 76, beat 2 and m. 77, beat 1) perpetuating the entrained hierarchy of four-pulses. Again, it would be impossible for a listener who is unfamiliar with the score to know that this passage features a strong subliminal grouping dissonance G2/3. However, even a score-less listener should notice that the two-pulse schema of the subordinate theme contrasts with the three-pluse schema of the main theme. Example 3.9a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition subordinate theme. 138

159 Example 3.9a. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , exposition subordinate theme, contd. Thus far in the subordinate theme we sense a regular pattern of two-pulse phenomenal climax accent. Notice, however, the interruption of this accent pattern in m. 77. Example 3.9b shows how the descending pattern of eighth notes in m. 77 does not stop until the downbeat of m. 78 where we notice a climax accent from the octave leap; instead of two quarter-note durations between climax accents, we find three durations. How will this conflict be interpreted? I believe that the lack of expected accent on beat 3 of m. 77, resultant from a continuation of the two-pulse projection schema, may contribute to a slight sense of metric insecurity; this is the first two-pulse projection in the Example not supported by an outstanding phenomenal accent type. The lack of accent on beat 3 of m. 77 does not on its own derail the two-pulse projection, but strong accent on the immediately following pulse (m. 78, beat 1) and a second statement of the subordinate theme s ascending four-eighth-note pattern may cue in the listener the recent memory of entraining to a two-pulse schema aligned with climax accents. Despite it lying within a thicker texture of repeated B s, the pattern s familiar climax accents are rather audible at m. 78, beat 3, m. 79, beat 2 and m. 80, beat 1. While the conservative listener may attempt to maintain a half-note pulse through mm , the strongly conflicting pattern of phenomenal accent leads me to believe that a more comfortable entrainment experiences requires adjusting the metric schema by one quarter-note pulse. Though the listener will have to insert a measure of 139

160 one or three pulses to make the adjustment, the consistent subdivisions allow for relatively easy adjustment because only the metric projection need be momentarily suspended. Example 3.9b. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , interruption of two-pulse projection. The closure of the subordinate theme s second statement in the measures that follow (mm ) similarly repeat the material heard in mm , leading us once more to a situation where the two-pulse projection schema loses phenomenal support. Notice at m. 82 in Example 3.9a how the four eighth-note pattern becomes a six eighth-note pattern. Unlike the previous situation, where the four-pulse pattern returns at m. 78, mm. 83 and 84 feature a continuation of a pattern of six eighth notes. This six-eighth-note layer is supported by phenomenal climax and harmonic accents: note the upward leap to B 5 at m. 83, beat 1 and the chromatic ascent to D6 at m. 84, beat 1. Both of these points also outline significant harmonic functions; m. 83, beat 1 sounds the presently tonic E Major chord, while m. 84, beat 1 sounds a D Major chord, functioning as dominant in the modulation to the key of G Major at m. 85. The projection map accompanying mm in Example 3.9c shows how a listener may adjust their pattern to the changing pattern of phenomenal climax and harmonic accent. In this instance it may be that the listener, having already heard a similar two-quarter-note pulse-defying pattern of six eighth notes in m. 77, might more easily adjust to the musical surface. 140

161 Example 3.9c. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , change from two- to three-pulse hierarchy. A third statement of the subordinate theme at m. 85, now in the traditional subordinate key, re-invokes the familiar two-pulse metric schema. Example 3.9d shows how the four-eighthnote pattern receives additional phenomenal support from the quarter note motion in the bass voices at m. 86, beat 2 and, likewise, at m. 88, beat 2. That this two measure passage is performed twice (mm and mm ) seems to suggest for a stable metric hierarchy of two quarter-note pulses organized into three-hyperpulse groupings. Notice this hierarchy as it relates to the musical surface in Example 3.9d; consistent two-quarter-note pulse groupings are clearly defined, with the bass voices performing hypermetrically-supportive duration and nadir accents on the downbeats of mm. 85 and 87. These downbeats are made to be clearer due to the anacrustic quarter-note motion on beat 2 of mm. 86 and 88 and contribute, in part, to a greater subliminal grouping dissonance of G2/3. 141

162 Example 3.9d. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , stable metric hierarchy. The measures that conclude the subordinate theme (mm ) are arguably the most metrically confusing in the entire Symphony. Were a listener to perpetuate the strongly supported pattern of two quarter-note pulses from the earlier passage at m. 85, they would find little phenomenal support. Clearly the most prominent phenomenal accent types in mm are accents of climax and duration, with both coinciding in mm. 90 and 91 to create especially strong moments of emphasis. Notice in Example 3.9e how continuing two-pulse projection results in no congruence between projections and these prominent phenomenal accent types. How might a listener perceive meter in this passage? Perhaps the conservative listener would simply maintain a pattern of quarter-note or half-note pulses as perceived in mm In this particular instance, I am more keenly interested in the potential experience of the radical listener. The first conflict with the previously entrained two-pulse pattern occurs at m. 89, beat 2, where we notice an accent of climax in the melodic voices. The immediately following eighthnote pulse also introduces conflicting accent, here with accents of nadir and pattern beginning coinciding. It seems that it is around this point that the listener may try to grasp for a new 142

163 phenomenal accent pattern. Example 3.9f shows attempts at establishing projection according to these accents. As the music unfolds, the next most prominent accent is clearly the phenomenallystrong duration and climax accent heard on the quarter note articulated on the offbeat of m. 90, beat 1. A projection of irregular eighth-note durations (five) seems required to align the accent of climax in m. 89 with that of m. 90, while a familiar projection of two quarter-note durations comfortably aligns the nadir accent in m. 89 with the climax and duration accent in m. 90. Listening further reveals another presentation of the ascending eighth-note passage in both m. 90 and 91. Thus, it seems reasonable that a listener might defer the two-quarter-note projection and entrain to a three-quarter-note projection. A three-pulse metric hierarchy oriented to the nadir and beginning accents does not, however, seem to best reflect the phenomenal surface. Example 3.9e. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , phenomenal conflict in a continued two-pulse projection schema. Example 3.9f. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , adjusting entrainment to match phenomenal nadir accent. 143

164 Perhaps a metric hierarchy coordinated to the climax and duration accent of the notated quarter note on the offbeat of beat 1 better reflects phenomenal meter. Example 3.9g demonstrates how such a projection schema might come to be. In this example, I suggest a more conservative approach to how this schema may develop. Instead of immediately adjusting to the nadir accent in m. 89, let s imagine the listener steadfastly entrains to quarter-note pulses. The climax accent at m. 89, beat 2 supports continued quarter-note projection despite the accent s misalignment with the two-pulse hierarchical level. A lack of phenomenal support for this twopulse level at m. 89, beat 3 perhaps weakens the schema s overall stability. But a phenomenal event still occurs here and may allow the schema to continue projecting. The arrival of the strong climax and duration accents on the off-beat at m. 90, beat 1 further supports a breakdown of the metric schema, while also serving as a strong perceptual cue to begin a new metric schema. Krebs does not use fractions of numbers to calculate metrical dissonances, but if he did the displacement dissonance from mm would be labelled D This dissonance is clearly quite strong; it may even be possible that the listener simply gives up on meter (projection) and is satisfied with merely tracking the subdivisions of the pulse. Example 3.9g. Schumann, Symphony in C Major, Op. 61, first mvmt., mm , adjusting entrainment to match phenomenal climax accent. 144

165 Isn t it interesting that at the height of the subordinate theme s development we experience the most confusing metric environment yet? Consider from a phenomenal perspective how we have arrived at this point. First, the main theme presents a perceptually-unambiguous three-pulse metric pattern. This pattern loosens itself into the subordinate theme as it gives way to a driving, two-pulse cadential pattern. The subordinate theme maintains the two-pulse pattern, but only with significant caveats. In this way, the subordinate theme s metric instability draws the main theme s metric stability into contrast. Upon the conclusion of the development, the codetta reasserts metric predictability, and even goes as far as to eliminate all metrical dissonances by, at last, cueing metrical projection that conforms to the notated meter signature. In sum, these contrasting metrical characters convey thematic areas in a phenomenal manner that seems consistent with a cognitively-derived model of metric theory. Perceiving Changes in Phenomenal Meter Related to the Tactus While clearly exhibiting metrical dissonance in an analysis of the score, the local, subliminal nature of several of the apparent dissonances is likely to go unnoticed by the listener who can only infer meter from the phenomenal surface. This is not to imply, however, that no subliminal dissonances can be phenomenally observed. Return to mm. 77 and 82 where we noticed a change from the two-quarter-note projection schema to a three-quarter-note projection schema. Krebs calls the juxtaposition of two differing interpretive layers an indirect grouping dissonance. Because metric projection tends to self-perpetuate, the succession of altering interpretive layer durations should be especially salient for perception. Take, for example, the successive changes in phenomenal meter throughout the passage containing the subordinate theme. My analysis suggests that mm (Example 3.9a) can elicit 145

166 at least five changes of phenomenal meter: mm. 77, 78, 82, 85, and 89 or 90. It suspect that listeners will not perceive all of these adjustments with the same degree of discomfort or comfort. Based on our multi-leveled understanding of projection, I posit that the degree of sensed discomfort experienced while adjusting one s metric schema is precisely related to the specific hierarchical pulse level at which the phenomenal meter adjusts. For example, when changing between the two-pulse hierarchy and the three-pulse hierarchy from mm , the quarter-note tactus level continues uninterrupted. Only the metric schema s hypermetric interpretation must adjust, in this case from two pulses to three. The type of change required in mm. 89 or 90, however, results in a shift of the hierarchy below the tactus level. In order to entrain to the accented quarter note in mm. 90 and 91, listeners must shift their metric schema by one eighth note. Because a quarter-note pulse is the most salient tactus, its mid-pulse interruption results in a severe disruption of metric projection, affecting every level upwards in the hierarchy. Within the metric hierarchy, the tactus is the most perceptually salient level of entrainment. Therefore, any adjustment that affects the projection of the tactus level can lead to the collapse of the entire metric hierarchy. Although a stable hierarchy can serve to strengthen projection of tactus and allow a listener to projection hearing a stronger tactus iteration many pulses into the future, any adjustment of that tactus will result in an adjustment of the entire hierarchy. Thus metrical discomfort is experienced most greatly when the metric schema reorients at a level below the tactus level. When adjustment occurs at levels above the tactus level, the perpetuation of the tactus and its subdivisions provides a consistent pulse underneath the changing hypermeter. When shifts occur below the tactus level, the tactus itself must be reoriented, and subsequently, so too do we adjust the upper hierarchical levels that are informed by that tactus. 146

167 Let s examine three situations in Example 3.10 which I would expect to elicit varying degrees of metric discomfort. In Example 3.10a, two recurring measures of a phenomenal surface that cues a three-pulse meter are followed by a measure of a two-pulse meter. Here the projection of both the tactus and its subdivisions continue unaffected. Example 3.10b presents a situation where the tactus is interrupted: while the first hypermetric group suggests for a projection of six eighth-note durations, the phenomenal rhythm that follows features only five eighth-note durations. While projection of the subdividing eighth-notes continues uninterrupted, both the quarter-note tactus and three-pulse hypermetric levels must be reinterpreted. Example 3.10c imagines a situation where the adjustment occurs even more deeply below the tactus. The first measure in this example shows two levels of phenomenal subdivision below the tactus. Notice how the second measure of Example 3.10c lasts only eleven sixteenth-note durations, instead of the expected twelve. While sixteenth-note projection can continue unaffected, we must adjust not only the tactus level, but also a level of subdivision below that tactus level. Example 3.10a. Three-pulse metric schema with a quarter-note tactus requiring adjustment at the quarter-note level. Example 3.10b.Three-pulse metric schema with a quarter-note tactus requiring adjustment at the eighth-note level. 147

168 Example 3.10c. Three-pulse metric schema with a quarter-note tactus requiring adjustment at the sixteenth-note level. These examples imply that it is especially important to determine which pulse rate we perceive as the tactus. The uncomfortability experienced by the sixteenth-note adjustment in Example 3.10c would be dramatically reduced were the eighth note instead the tactus. How, then, does a listener come to perceive a tactus level within many hierarchically related pulse rates? Research presented by Justin London (2012) suggests that adult humans tend to gravitate towards pulses of moderate tempo near approximately 100 beats per minute. I explore the relationship between this unique tactus preference and phenomenal metric perception in the next Chapter. Representing Phenomenal Meter It is worth thinking about the degree to which the score-less listener s metric interpretation differs from that of the performer. The difference is the result of cognition: the performer gives some privilege to regularly recurring metrical accent, even if that accent is primarily indicated visually instead of aurally. For the performer this metrical accent is clearly represented by the meter signature indicated in the notated score. Since the 3/4 meter signature does not accurately reflect phenomenal meter, is there a clear way to notate the exposition s phenomenal meters? Perhaps we can reinterpret Schumann s notation with meter signatures that 148

169 reflect what is heard. In this way, we may eliminate subliminal dissonances and marry what is heard by the score-less listener and what is conveyed in a performer s interpretation. Remember that a meter signature itself does not have the power to prescribe tactus. Tactus perception is entirely tempo dependent. For example, in an andante 6/8 measure we may perceive six tactus pulses, while a vivace 6/8 measure likely elicits two tactus pulses. In this way, any note value can represent the tactus. However, it is frequently the case that the meter signature s denominator indicates the likely tactus duration while the numerator indicates the number of those tactus durations that comprise a hierarchical level. This is commonly understood by performers. Thus, while a meter signature does not exclusively designate a tactus, it makes sense for a phenomenal metric representation to do so. Example 3.11 shows one possible presentation of the phenomenal metric experience from mm In this example, the meter signature s denominator signifies the tactus-level pulse and the numerator indicates the number of pulses per metric cycle. Notice the discrepancy between longer measure durations of 4/2, 3.5/2, and 3/2 and the shorter durations of 5/4, 3.5/4, and 3/4. I suggest that different scorings help to represent the shifting sense of phenomenal tactus, as it frequently moves between the half note and the quarter note. 149

170 Example Schumann, Symphony in C Major, Op. 61, first mvmt., mm , hypermetric reinterpretation with meter signatures reflecting phenomenal meter. The subordinate theme s four-eighth-note pattern suggests for an opening hypermetric grouping of four half-note durations, followed by three and a half half-note durations. Because the subordinate theme s four-eighth-note melodic motion continues, it makes sense perceptually to continue projecting a half-note pulse into the following notated measure. As described earlier, this half-note pulse cannot be comfortably projected through the entire passage. However, in the passages where the conflicts are noticed (Example 3.11, mm. 2 and 4), a half-note pulse still seems to be most perceptually salient. In this way, it seems reasonable to represent the 150

171 interruption of the half-note pulse using a meter signature 3.5/2. As the phenomenal accent that gave rise to the half-note pulse disappears toward the end of m. 4 and through mm. 5 and 6, the quarter-note seems to become the more salient pulse level. Here, then, I suggest the transition to G major can be represented with two 3/4 measures. As the theme s four-eighth-note pattern returns in m. 7. I indicate this by returning to a meter signature with a denominator of 2, while the phenomenal cues indicate a hypermetric grouping of three half-note pulses G3/2. What I have notated as mm requires the most liberal reinterpretation. In these measures I suggest a compromise between privileging the melodic nadir accent which begins the ascending eighth-note pattern (m. 9, beat 2) and the quarter notes accents of climax and duration (mm. 10 and 11, beat 1). Alternating between measures of 3.5/4 and 3/4 places the phenomenal accent of the melodic contour in the strongest metric positions within each measure, while allowing the previous 3/2 measure (m. 8) to complete its cycle. This awkward notation correlates well to this particular passage s phenomenal metric awkwards. Notice the different position of the ascending melodic pitches that begin mm. 8 and 9. At the downbeat of m. 8 the prior ascending eighth notes arrive on the pitch B5 after which the same pitch is immediately repeated as the beginning of the succeeding four-note ascending pattern. In m. 9, B5 remains the melodic arrival pitch, but here it is not repeated to begin a new four-note pattern. Instead, the eighth notes unexpectedly continue their ascent. Because of this it seems impossible to perceive m. 9 in the same metric context used to perceive the two preceding measures. As was earlier described, the phenomenal accent in the opening measure of the codetta (Example 3.11, m. 12) outlines a three-quarter-note pulse. Though this schema can be continued into the follow measure, there is not significant phenomenal support to perceive more than one clear cycle. Thus, the descending chromatic pitches in m. 13 suggest for an extended period of 151

172 deferred quarter-note pulses, leading me to suggest for a measure of 5/4. The strong phenomenal accent on the notated downbeat at m. 14 cues the three-pulse schema that is supported through the remainder of the excerpt. Interestingly, the final measures of the exposition are the only measures in which my phenomenal metric reinterpretation matches Schumann s original notation. The phenomenally-based reinterpretation, then, conveys the morphing metrical dissonances that are associated with specific functions of the subordinate theme area. In this way, we might be able to clearly represent the unique metrical characters of the exposition s formal areas, which, though originally available to only audible perception, are now made readily available to visual perception. Imagine the implications of the performer using this reinterpretation in performance: by internally entraining to what we see in Example 3.11, the performer may be inclined to especially emphasize the changing sense of phenomenal meter, possibly creating stronger accents that contribute to more readily perceiving the phenomenal metric changes. I do not mean to suggest that this become a standard practice. Rather, I intend for it to highlight how the unique relationship between phenomenal accent and metrical accent transcends itself to interact with larger-scale form. 152

173 CHAPTER FOUR Phenomenal Meter in Rodney Roger s Lessons of the Sky The opening passages of Rodney Roger s Lessons of the Sky for soprano saxophone and piano offer the listener a variety of options for interpreting meter. Example 4.1a shows the composition s first nine measures, where we see irregular patterns of rhythm spanning five changes of meter signatures. Though these traits are not typically associated with stable pulse and meter, both listeners and performers may notice that it is possible to maintain a steady quarter-note tactus from mm Example 4.1b shows a reinterpretation of these opening measures that reflects perceiving a phenomenal quarter-note tactus. In the original passage (Example 4.1a) both the saxophone and piano generally begin measures with either an accented articulation or the first sixteenth note of a recurring passage. The reinterpretation in Example 4.1b exhibits similar qualities, with especially strong pulses heard at the newly-notated mm. 2, 4, 7, 10 and 12 (circled in Example 4.1b). Why precisely might someone favor perceiving an unchanging quarter-note tactus, as in Example 4.1b, when Rogers original notation (4.1a) appears to convey a distinctly different metrical character? Much of Lessons of the Sky lends itself to being perceived with an unchanging quarternote tactus despite notated meter signatures that may suggest otherwise. My analyses from earlier Chapters show that this may be the simple result of a pulse s proclivity for selfperpetuation. In this particular composition, however, I suspect there to be more at play. Justin London (2012) suggests that tactus selection strongly correlates to the rate at which phenomenal accent is observed. Research shows that adult humans tend to assign tactus-level projection to pulses occurring at moderate rates. A cognitive bias for selecting a moderately-paced tactus may 153

174 lead a listener to prefer one possible metric interpretation over another. This knowledge, in conjunction with what I have already shared regarding the hierarchical nature of metric projection, allows for metrically reinterpreting Rogers composition to reflect a phenomenal metric experience that reduces or eliminates the changing meter signatures. Example 4.1a. Rogers, Lessons of the Sky, mm Example 4.1b. Rogers, Lessons of the Sky, mm. 1-9 renotated to reflect quarter-note pulse. 154

175 Perception of the Tactus The stability of a metric schema is directly related to the regularity of perceived phenomenal accent. When phenomenal accent patterns are highly regular, the listener can clearly sense multiple hierarchically-related pulse durations. Once established, this hierarchy is then reinforced by a listener s cognitive projection of metrical accent (a metric schema). While a listener may project pulses at multiple levels of the hierarchy, metrical accent is projected at one especially prevalent level, the tactus. How does perception of the tactus relate to perception of the entire metric hierarchy? I suggest that the perception of a metric schema is deeply related to the rate at which phenomenal accents are perceived. Justin London (2012) presents significant findings that relate a human s ability to cognitively organize phenomenal stimuli (e.g. pitch events) to the rate at which those stimuli occur. These findings uncover the temporal limits to what adult human can reasonably perceive. In his first chapter, London surveys studies that examine the relationship between the duration of time a listener has entrained to a pulse and the attentional energy that listener must give to comfortably maintain entrainment, that is, cognitively synchronizing to a perceived stream of pulses. As one might expect, a strong correlation exists between the length of an entrainment period (time allowed for synchronizing to phenomenal stimuli) and the ability to accurately project a pulse and predict future events. 82 London says the following: How we attend to the present is strongly affected by our immediate past; once we have established a pattern of temporal attending we tend to maintain it in the face of surprises, noncongruent events, or even contradictory invariants. 83 Although London is clearly drawing on others (Hasty and Lerdahl 82 London 2012, Ibid. 155

176 and Jackendoff), by relating his comments to cognitive research he shows that projection is not merely a theoretical gestalt but also a cognitive function. So while perpetuating quarter-note tactus in Example 4.1b is already supported by both Hasty and Lerdahl and Jackendoff (a pulse tends to perpetuate itself, even in the face of contradicting stimuli), consistent tactus projection is further supported by cognitive research. London argues in his second chapter that metric selection is largely dependent on the tempo of a performance. 84 Supported by research in the field of cognitive science, London explains that the cognitive limits to temporal perception significantly affect how listeners attune to stimuli. Research shows that for two pitch events to be perceived as separate events, they must be separated by an interonset interval (IOI) of 2 milliseconds; events separated by an IOI shorter than 2 ms are heard to be articulated simultaneously. 85 For listeners to accurately determine the order in which two different pitch events are articulated, the events must be separated by an interval of at least 20 ms. 86 However, events with IOIs of 20 ms (thirty attacks in the space of one second when nominally isochronous) are still far too rapid to be cognitively organized into a metric schema, that is, grouped into patterns of two or three pulses (simple and compound meter). London determines the fastest rate for a perceptible meter by drawing on a number of significant corpus studies. He notes that the shortest IOI at which a human can vocally articulate two successive pitches is approximately 120 ms. He cites research that shows jazz drummers can articulate successive ride cymbal notes at an IOI no faster than approximately 100 ms. London further shows that trained musicians can accurately tap every fourth pulse only when those 84 Ibid., Ibid., Ibid. 156

177 pulses occur with IOIs of 120 ms or longer. Thus London suggests that a significant IOI boundary for perceiving meter lies near approximately 100 ms. 87 An IOI of 100 ms does not, however, accurately reflect a rate at which a listener can perceive a tactus. Due to meter s hierarchical nature, London hypothesizes that hearing a beat requires at least the potential of hearing a subdivision, for the tactus cannot also be the fastest subdividing level in the hierarchy. Recall the discussion of traditional metric theory in the opening Chapter which heavily referenced the interaction of different levels of pulse rates (Yeston s strata ). London further supports this by citing the phase/antiphase nature of time keeping, walking, and general human activity. Using 100 ms as the IOI for fastest perceivable subdivision, London therefore argues that a boundary for tactus perception exists between an IOI of 200 ms (simple subdivision) and 300 ms (compound subdivision). 88 London similarly draws on recent corpus studies to determine the least rapid rate for perceptible meter. He cites how synchronization with a pulse becomes difficult when IOIs are between 1.5 and 2 seconds and that successive pitch events separated by more than 1.8 seconds do not sound as a continuous entity. 89 Synchronizing to pulses with IOIs greater than 2.4 seconds becomes a reaction activity; when the IOI exceeds 3.4 seconds, subjective rhythm disappears entirely. 90 These recent findings reinforce rudimentary research predating the twentieth century which describes a tempo of 30 beats per minute (bpm), with an IOI of 2 seconds, as too slow to be useful. 91 London therefore suggests an envelope for tactus duration to exist near 2 seconds. Because metric schemas convey more than just a series of tactus pulses, London calculates 87 Ibid., Ibid., Ibid., Ibid. 91 Ibid.,

178 perceptual envelopes for two-tactus (simple) and three-tactus (compound) meters to exist between 4-6 seconds. Therefore, London argues that the most stable tacti occur at tempos between 240 bpm and 45 bpm. 92 Lastly, and most crucially for my analyses, London discusses research regarding maximal pulse salience, that is, an average pulse rate that is perceived to be neither too fast nor too slow, but just right the Goldilocks interval if you will. London shares that adult humans have a particular tendency to entrain to pulses with IOIs near 600 ms, or 100 bpm. 93 At tempos faster than 100 bpm, listeners tend to underestimate the perceived tempo, while at tempos slower than 100 bpm listeners tend to overestimate the perceived tempo. 94 How do these findings relate to cognitive thresholds for the most and least rapid perceivable meters? Interestingly, a tempo of 100 bpm (IOI of 600 ms) falls near the middle of London s spectrum: approximately three times as fast the fastest perceivable tactus duration (200 ms IOI) and three times as slow as the slowest perceivable tactus duration (1800 ms IOI). London therefore argues that tempos which feature pulses and subdivisions with IOIs in near 600 ms are especially significant. To support this hypothesis, London creates a schematic of possible simple and compound meters and simple and compound subdivisions, as shown in Figure He first assigns the central note (tactus) an IOI and then calculates IOIs for each respective. He then totals the number of periodicities that fall within the boundaries of his suggested perceptual envelopes. London argues that tempos are perceptually (thus, cognitively) more stable and useful when the number of possible meters and subdivisions that fall within his 100 ms to 4-6 second 92 Ibid., Ibid., Ibid. 95 Ibid.,

179 IOI envelopes (the boundaries of metric perception) is greatest. Table 4.1 lists the number of nodes containing periodicities that fall within those boundaries. It further calculates the number of possible meters for a range of IOIs by multiplying the number of measure nodes that fall below the respective upper boundary by the number subdivision nodes that fall above the respective lower boundary. 96 It is compelling to see that a tempo of 100 bpm (600 ms IOI) offers the greatest range and variety of meters and subdivisions within those envelopes! While this may not be conclusive evidence for defining a universal Goldilocks interval, London s work shows that a 100 bmp tactus is reinforced by the greatest number of perceivable subdivision and metric groupings. Indeed, this implies that a direct link exists between tempo and metric hierarchy! 96 Ibid.,

180 Figure 4.1. Hierarchical periodicities for a 100 bpm quarter note. Using a quarter-note tempo of 100 bpm (600 ms IOI), London calculates the duration of potential meters (above) and subdivisions (below). The drawn lines designate the values within London s perceptual envelopes Ibid.,

181 Table 4.1. Possible hierarchic configurations as tempo changes. This table shows the number of nodes (see Figure 4.1) that fall within London s perceptual envelopes. Notice the greatest balance of node types (column 3) leads to having periodicities near the Goldilocks interval, indicated in column 4 with Y or N. Note that 50, 180 and 200 bmp, which also features periodicities near the Goldilocks interval are hierarchically related to the cluster between bmp. Perceptual Boundaries in Context; Projection and Perception at 100 bpm: I London s tempo-dependent perceptual envelopes play an important role in perceiving meter in Lessons of the Sky. Rogers indicates an opening tempo of eighth note = While it is possible to entrain to an eighth-note tactus with a 300 ms IOI, London s findings suggest that it may be cognitively easier to entrain to a hierarchically-related quarter-note tactus with a 600 ms IOI instead. Indeed, a renotated tempo marking of quarter note = precisely surrounds the 100 bpm Goldilocks interval. Reconsider, then, Examples 4.1a and 4.1b: perceiving the opening measures according to Example 4.1b s reinterpretation is, thus, likely 161

182 because of two interacting rationale. First, because phenomenal accent strongly supports immediate quarter-note projection, the hierarchical nature of metric projection encourages continued quarter-note projection. Second, the quarter note projects itself near the cognitivelypreferred tactus rate of 100 bpm. Entraining to a quarter-note tactus near 100 bpm early in the composition conditions the listener to perceive a similar tactus throughout the composition. For example, look at mm shown in Example 4.2a. Despite being notated with frequent changes of meter signature the entire passage can be perceived with a quarter-note tactus. As it is reinterpreted in Example 4.2b, a quarter-note pulse begins in m. 61 before being interrupted at the passage s conclusion. The absence of four-sixteenth-note groupings in mm is particularly interesting (Example 4.2b, system 2, mm. 2-5): the saxophone presents a three-sixteenth-note pattern, while the piano performs a five-sixteenth-note pattern. Having already entrained to eight preceding quarter-note pulses encourages continued quarter-note tactus projection; this tactus projection is confirmed by the alignment of the saxophone s eighth-note Bs in the originally notated mm One could conceivably entrain to a dotted-eighth-note tactus beginning in m. 63. However, I find this particularly challenging as there has been a relatively short entrainment period (m. 63, twice repeated), and it is nearly impossible to aurally perceive that the piano is even playing a fivesixteenth-note pattern due to the previously explained proclivity to entrain to a four-sixteenthnote grouping. 98 This is not to say that one cannot entrain to pulses with durations other than 98 When listening to mm , it is impossible to concomitantly entrain to both distinct patterns of the saxophone s three sixteenth notes against the piano s five sixteenth notes. Traditional theory would suggest this passage to be polymetric. London explains that polymeter is purely a theoretical construct, for it is not possible for a human to cognitively entrain to two simultaneously occurring meters (2012, p. 67). Whether the listener privileges the saxophone or piano may depend on multiple factors, including balance (in live performance or recording), previous training as a soloist or a pianist, and so on. 162

183 four sixteenth notes. I will explain soon that a variety of metric schemas can be selected provided that a listener perceives substantial corresponding phenomenal accent. Example 4.2a. Rogers, Lessons of the Sky, mm Example 4.2b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note pulse. 163

184 Projection and Perception at 100 bpm: II A similar analytical process can be applied to mm in Example 4.3a, which shares many characteristics with Examples 4.1: the piano alternates between several passages of sixteenth-note melodies and sixteenth-note ostinato chords, while the saxophone alternates between punctuated melodic flourishes and longer lyrical phrases. Similarly, I suggest that the phenomenal surface also encourages entrainment to a quarter-note tactus; Example 4.3b shows a reinterpretation that reflects such a hearing. Though strong accents are not coordinated with every quarter-note tactus, this hearing is supported by phenomenal accents at the downbeats of Rogers originally-notated mm. 269 (climax and duration), 274 (harmony), 279 (climax and duration), 281 (duration), and 282 (climax and duration). Example 4.3a. Rogers, Lessons of the Sky, mm

185 Example 4.3b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note tactus. Measures exhibit a particularly interesting entrainment challenge: phenomenal nadir and climax accents in the piano s changing pattern of sixteenth-note articulations seem to conflict with quarter-note projection. Notice in the second system of Example 4.3b that the piano s right- and left-hand pitch clusters articulate an irregular pattern of sixteenth-note groupings: 3-3-2, 3-3-2, 3-2, 3-2, One might suggest that without the 165

186 quarter-note entrainment period available in the first system, a listener would have a difficult time discerning a stable tactus. However, when performed at the notated tempo of 100 bpm, cognitive tactus preference strongly implies entraining to a quarter-note tactus duration. As before, this quarter-note tactus may not be reinforced in every measure of Example 4.3b s second system, but notice that the pattern s onset articulates a nadir accent at a regular hyper-tactus interval of two quarter-note pulses. Once again, quarter-note projection in the face of contradictory evidence is rewarded; in m. 279 both the saxophone and piano perform phenomenal accent that strongly reinforces quarter-note tactus projection. Observe how in the fourth system of Example 4.3b quarter-note projection breaks down after the saxophone s F#6. As the ties crossing bar lines suggest, phenomenal accent no longer reinforces quarter-note projection, and shortly thereafter a pattern of five sixteenth notes arises in both the saxophone and piano. This pattern is entirely incongruent with the renotated meter and possibly indicates that a listener should instead entrain to a five-sixteenth-note tactus. While meters featuring five subdivisions are generally unconventional, entraining to a five-sixteenthnote tactus in this particular instance requires relatively little cognitive effort. In m. 284 the former four-sixteenth-note subdivision is attenuated, thereby allowing the five-sixteenth-note phenomenal accent pattern to establish a hierarchy with no conflicting cues. It may also be useful to note that at Rogers indicated tempo a five-sixteenth-note tactus recurs at an IOI near at 750 ms, well within the boundaries of pulse perception and very near the Goldilocks interval. Despite the awkwardness of a five-based metric grouping and its uneven 3+2 or 2+3 subdivision grouping cognitive preference shows this to be a perceptually salient meter. 166

187 Projection and Perception at 100 bpm: III Example 4.4a shows a later passage (mm ) that conveys a markedly contrasting rhythmic and metric character of recurring sixteenth-note patterns and an expansive, lyrical saxophone melody. As in earlier examples, Example 4.4a exhibits a phenomenal meter that does not reflect Rogers notated meter signatures. Despite featuring a more regularly-notated metric organization, the changing contour patterns of the successive sixteenth notes and notated interpretive accents give rise to a sonic space that allows the listener a variety of options for interpreting meter. When following the score, it is relatively simple for experienced listeners or performers to adjust their metric schema to match notation by changing counting patterns with the changing meters. I suspect, however, that listeners removed from the score would have a difficult time agreeing collectively on only one interpretation of the tactus and metric hierarchy. In a phenomenal hearing of the passage, the three meter changes at mm. 230, 233, and 234 are essentially imperceptible, and the patterns of climax, nadir and harmonic accent attributed to the piano s sixteenth-note and eighth-note motion make it possible to readily perceive eighth-note, dotted-eighth-note, quarter-note, and dotted-quarter-note pulse streams. London suggests that passages of this variety are metrically malleable, meaning that phenomenal characteristics of the music allow listeners to interpret the music using a variety of metric schemas. 99 In Example 4.4a, do we entrain to the repetitive, three-sixteenth-note groups (with an IOI near 450 ms) in the piano during mm and project the pulse forward, or do we entrain to the downbeats of the notated 3/8 meter signatures (IOI near 900 ms)? 99 London 2012,

188 Example 4.4a. Rogers, Lessons of the Sky, mm London s findings suggest that listeners may cognitively prefer a more radical approach and adopt a quarter-note tactus which disregards the notated meter signatures. At the notated tempo it is cognitively easier to entrain to a quarter-note tactus near the Goldilocks interval than a dotted-eighth-note tactus with an IOI near 450 ms or a dotted-quarter-note tactus with an IOI near 900 ms. Example 4.4b shows this passage renotated to reflect a potential 168

189 hearing of a pervasive quarter-note tactus. 100 Phenomenal accents strongly reinforce perceiving a quarter-note tactus; note these points circled in Example 4.4b. If a listener or performer entrains to a quarter-note tactus and projects it through the passage, they would notice that phenomenal harmonic and/or textural accents at mm. 230, 232, 233, 234, 240 and 243 reinforce quarter-note projection, leading to a very comfortable metric experience. 101 Example 4.4b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note pulse. 100 In my experience it is very tricky in this passage to entrain to anything other than a quarternote tactus. 101 If we related these phenomenal accents to their location in Example 4.4a, we see that these accents could also reinforce a dotted-quarter-note pulse projection beginning in m. 230 (the notated 3/8 meter) at an IOI near 900 ms. This tactus would require an adjustment in m. 233 to maintain congruence with phenomenal accent. Regardless, when multiple pulse streams are concurrently supported by the musical surface it is cognitively more challenging to entrain to a tactus with an IOI near 900 ms than a tactus whose IOI is near 600 ms Goldilocks interval. 169

190 Example 4.4b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note pulse, contd. Perceiving meter in this passage may be further influenced by a strong preference for avoiding both a dotted-eighth- and dotted-quarter-note tactus. In Table 4.1 we see that tactus rates with IOIs between roughly ms and ms distinctly lack subdivisions and hypermetric periodicities within the ms Goldilocks interval range. Note how these ranges relate to specific note durations at Rogers notated tempo. At eighth note = 200 bpm, quarter-note pulses span approximately 600 ms, while eighth-note pulses span approximately 300 ms. Therefore, a dotted-eighth-note pulse spans approximately 450 ms and a dotted-quarternote pulse spans approximately 900 ms. Interestingly, these two durations fall squarely in the ms and ms wasteland that lacks a periodicity near our Goldilocks interval (Table 4.1, column 4). 170

191 Why else might we prefer a quarter-note tactus to other phenomenally supported pulse streams? One could potentially argue that because the eighth-note stream in the piano s left hand continues beyond the saxophone s quarter notes, it should, therefore, exhibit a more entrainable tactus than the quarter note. In relation to London s boundaries for tactus perception, we see that an eighth-note tactus whose IOI is near 300 ms is certainly above the 250 ms fastest-tactus limit. Table 4.1 also shows that a subdivision/meter schematic for a 300 ms IOI exhibits periodicities near the 600 ms Goldilocks interval. Remember though, that the presence of perceivable phenomenal accent is critical for perceiving any pulse. While the stream of eighth notes in the piano s left hand occurs for a substantial number of pulses, it is very challenging to audibly distinguish this stream as a distinct line due to the registration of the piano accompaniment. Though one might infer climax accents from the Es and C#s in mm , the eighth-note line s lower pitches are lost in the pitch space of the right hand s repetitive sixteenth-notes, weakening the potential to perceive a distinct eighth-note pulse stream. Recall this issue as it was related to the registration of the two pianos in Adams s Hallelujah Junction. Projection and Perception at 100 bpm: IV Example 4.5a shows mm which, like Example 4.4, features a repetitive piano ostinato of sixteenth notes and a flowing saxophone melody of longer durations. Again, London s findings on the salience of a 600 ms pulse support the perception of a quarter-note tactus; Example 4.5b shows a reinterpretation that reflects this hearing. In this reinterpretation, the saxophone exhibits new-found metric congruence in the final measure of the first system. Congruent, also, are the four-eighth-note patterns in the piano s left hand, both here and throughout the second system. Note, however, the inconsistencies that remain, especially the 171

192 now syncopated saxophone figure in the final system, groupings of three eighth notes in first system of the piano s left hand, as well as the pervasive three-sixteenth-note figure in the piano s right hand. While it is wholly possible to experience mm as reinterpreted in Example 4.5b, this does not best reflect the phenomenal accent exhibited by the passage. 102 Example 4.5a. Rogers, Lessons of the Sky, mm It is possible that listeners entrained to the pulse exhibited in Example 4.5b may find the metric incongruence of the saxophone melody in the third system heard as syncopation to be a pleasing experience. I suspect, however, that when given a choice listeners would prefer hearing the two melody statements with the same metric interpretation. I ll discuss later how our preference for parallelism in grouping structure can influence one s hearing. 172

193 Example 4.5b. Rogers, Lessons of the Sky, mm , renotated to reflect quarter-note tactus. Let me offer a second reinterpretation, shown now in Example 4.5c. In this Example, the opening bars remain unchanged, reflecting the phenomenal harmonic accent present on the notated downbeats. This decision is informed by events that I circled in Example 4.5c, highlighting the climax, beginning, and nadir accents of the piano s left and right hands. Although harmony does not change on the downbeats of the first 8 measures, the F4 eighth notes of the piano s left hand are audible articulations of tonic in the local harmonic context. Phenomenal accent on the downbeat is further reinforced by nadir and beginning accents in the right hand s C4 sixteenth notes. Alternatively, one could argue that each measure s second eighth note exhibits the pervasive pulse accent, as suggested in Example 4.5d (mm ). 173

194 The E4 quarter notes of the piano s left hand exhibit considerable duration accent, supported by climax accent in the right hand s A4 sixteenth-note. As the Example shows, however, the phenomenal accent attributed to the saxophone s entrance significantly conflicts with this projection model. A similar situation occurs in the final system of the first reinterpretation in Example 4.5b: despite a stable underlying quarter-note projection, the saxophone is incongruent with the projected quarter-note tactus. If we insert one dotted-quarter-note tactus in a location reinforced by accent in the piano s left hand, we now find pulse congruence between both the saxophone s quarter-note motion and the piano s duration and climax accents at the originally notated m. 142; observe the realignment of both quarter and eighth notes in the final system of Example 4.5c. Example 4.5c. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal tactus. 174

195 Example 4.5d. Rogers, Lessons of the Sky, mm , showing tactus shifted. This passage also demonstrates London s argument for eschewing a tactus whose IOI is between 350 and 500 ms. Here, the piano s right hand performs a repeating pattern of three sixteenth notes, theoretically projecting a dotted-eighth-note pulse stream (IOI near 450 ms). Were this pulse stream given perceptual priority, we might hear something akin to what is reflected in Example 4.5e (mm ). Recall, however, Examples 4.4a and 4.4b, where I noted the perceptual difficulties associated with entraining to a dotted-eighth-note tactus (IOI near 450 ms) when concurrently presented with a highly-entrainable quarter-note tactus (IOI near 600 ms). This problem is uniquely exacerbated in Example 4.5e by the figure s notated pianissimo dynamic, which encourages the listener to attune to the better-audible left hand and timbrally-distinct saxophone. 175

196 Example 4.5e. Rogers, Lessons of the Sky, mm , renotated to reflect dotted-eighth-note tactus. Tactus Disrupted by Phenomenal Accent After entraining to a stable tactus, how might the conservative and radical listeners be affected by newly conflicting phenomenal cues? Let s look at a situation where adjusting entrainment during a passage of regular sixteenth-note motion might possibly occur due to changing phenomenal cues. Return to Example 4.4a and its reinterpretation in Example 4.4b. In listening to a performance of the passage, one might notice that it can be tricky to passively maintain the quarter-note tactus through the iterations of m. 237 (corresponding to Example 4.4b, system 3, mm. 3-7). Here, dynamic and climax accents on the originally-notated third and sixth sixteenth notes of the measure strongly conflict with Example 4.4b s renotated quarter-notetactus interpretation. A passive listener who has not steadfastly entrained to a quarter-note tactus or a radical listener who is strongly attuned to the changing phenomenal surface may perceive something akin to the reinterpretation suggested in Example 4.4c (mm ). Notice how the previously described accents become the beginnings of a dotted-eighth-note tactus. The binary nature of the right hand s climactic D5 and semi-climactic B4 may further contribute to hearing a hypermetric dotted-eighth-note pulse, divided by one strong downbeat pulse and one weak subdividing pulse, as shown in Example 4.4d. Lerdahl and Jackendoff support this claim with their tenth metrical preference rule, MPR 10, Binary Regularity : prefer metrical structures in which at each level every other beat is strong. 103 The return to a quarter-note tactus is grounded in conditioning and the latency of patterned ideas. Notice in Example 4.4a how mm. 234, 235 and 237 share the same pattern of notated accent. The conditioned hearing of the two accented 103 Lerdahl and Jackendoff 1983, 101. It may be important to note that this specific strong/weak climax-accent pattern is latent in m. 232, as well as other earlier passages (mm. 80-1, 92-6, 103-5, 205, 207, and 221-2). 176

197 sixteenth notes in Example 4.4d, m. 3, system 2 allow the listener to recall and re-adopt the former pattern of entrainment. Example 4.4c. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal tactus. Example 4.4d. Rogers, Lessons of the Sky, mm , strong/weak tactus representation. 177

198 Hypermetrical Organization above the Tactus Level Thus far, my reinterpretations have strictly represented tactus perception, where each reinterpreted measure represents one tactus pulse. As defined earlier, however, metric schemas are more than just a series of tactus pulses; listeners cognitively organize these pulses in a hierarchical manner according to the perception of reinforcing phenomenal accent patterns. This suggests that more must be done to transform my reinterpretations into true representations of phenomenal meter. Lerdahl and Jackendoff state that tonal music generally features two to four levels of metrical structure above the tactus. 104 Example 4.4d shows, then, a critical first step in better representing phenomenal meter, as system 2, mm.1-3 joins two tactus pulses into single groups. Though music from the Classical era can generally be grouped into regular patterns of two, four and eight measures, longer hypermetric durations in post-classical repertoire are often less strictly organized. Lerdahl and Jackendoff agree, stating that except in the most banal music, these levels are commonly subject to a certain degree of irregularity. According to the authors, this irregularity is often the result of parallelism among groups. 105 I suggest these parallel features are perceptually salient due to the phenomenal accent they exhibit. To demonstrate revisit Example 4.5a and its reinterpretation in Example 4.5c. In Example 4.5c I circle several points that exhibit particularly strong phenomenal accent. Of the seven circled points, three exhibit Bb2 nadir accents and five coincide with the beginning of a short eighth-note pattern. Interestingly, these circled points occur at relatively equidistant points throughout the passage. Example 4.5f shows how this passage could be renotated provided these points are given metric primacy. The parallelism to which Lerdahl and Jackendoff refer is quite 104 Ibid., Ibid. 178

199 apparent in Example 4.5f: mm. 1, 2, 3 and 5 are defined by a self-generating, eighth-note-based pattern with gradually expanding contour, and mm. 4, 6 and 7 feature variations on an ascending arpeggio which begins on Bb2, the passage s lowest notated pitch. And despite the saxophone s relatively miniscule contribution to phenomenal accent, melodic and rhythmic parallels are obvious in mm. 4 and 6. Example 4.5f. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter. Observe that mm. 3 and 5 are distinctly non-parallel, where in m. 5 the four-note pattern is offset by one eighth-note. This is the result of the necessary decision to create one measure of 5.5/4 meter. Similar to my Schumann reinterpretations, I prefer the unconventional meter signature of 5.5/4 to the signature 11/8, as x/4 represents the quarter-note as tactus. As I demonstrated in Examples 4.5b and 4.5c, the uneven number of eighth-note subdivisions that 179

200 exist between the two statements of the saxophone melody requires some metric adjustment to maintain congruency between the tactus and phenomenal accent. While there may be several modes of perception that reconcile this problem, I suggest two straight-forward possibilities. The first is already shown in Example 4.5f, which prioritizes phenomenal accent (points circled in Example 4.5b). Example 4.5g displays another rendering, which better prioritizes parallelisms between motives. Notice in Example 4.5g how m. 5 now reproduces the pattern presented in m. 3. However, in my own experience, it is difficult to perceptually prefer Example 4.5g (preferring parallelism) to Example 4.5f (preferring phenomenal accent). In m. 4 the long duration of the piano s left-hand F6 in m. 4 enhances the ensuing F4 s beginning accent, which implies perceiving a beginning to a new metrical pattern. This argument is made stronger considering its parallel association with mm. 1, 2, and 3, where the beginning of a new eighth-note pattern informs the onset of a hypermetric grouping. Thus, while the grouping of tactus pulses may be dictated by parallels between groups, these parallels themselves are informed by the phenomenal accent which gives them definition. Therefore, it seems that preference is given to phenomenal accent when determining larger hypermetric levels in the same manner that it governs tactus perception. 180

201 Example 4.5g. Rogers, Lessons of the Sky, mm , renotated to reflect parallelism. Metric theory holds that for a strong sense of meter to exist, phenomenal accent must be evenly distributed at each level of the hierarchy. 106 Conveniently, the phenomenal accent that informs the reinterpretations in both Examples 4.5f and 4.5g is distributed rather evenly (from notated measure to notated measure), with groupings containing between 4 and 6 tactus pulses. Consider the music in Example 4.3b, which features one particularly long passage without significant phenomenal/parallel accent, and its reinterpretation in Example 4.3c. While Example 4.3c shows that this passage s strongest phenomenal accents can be generally represented by four-tactus groupings, this relatively accent-less passage creates one disproportionately long grouping of eleven tactus pulses. In the absence of phenomenal cues it is our inclination to perpetuate an already-entrained-to metric schema. Because we aim for relatively even pulse 106 See Lerdahl and Jackendoff s 1983 Metrical Well-Formedness Rules (MWFRs). London 2012 adapts these in his own way (pp ). 181

202 distribution, I suggest for a grouping, indicating the listener will attempt to stay entrained to a 4/4 meter, projecting weakly-confirmed four-tactus groupings until the accent of the saxophone s C6 informs them that this meter must be adjusted. 107 Example 4.3c. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter. How might this method of organization inform further metric reinterpretation in Example 4.1? While Example 4.1b shows several instances of significant phenomenal accent occurring at 107 It is interesting that, while I mentioned we typically give preference to phenomenal accent over parallelism, the parallels between the saxophone melody and the beginnings of Example 4.3c s measures are rather obvious: on four occasions the saxophone begins or is near the beginning of a relatively high pitch, and on four occasions the saxophone begins or is near the beginning of a relatively long pitch duration. Here, parallelism is the result of similar phenomenal accent types being spaced at relatively regular intervals. 182

203 inconsistent intervals of two and three tactus pulses, Example 4.1c shows how the opening measures of the composition may be reorganized to project consistency. If the first pulse is treated as an anacrusis, the passage can be neatly organized into five-tactus groupings. However, accurately representing phenomenal meter must consider the accents circled in Example 4.1b on the piano s G3 and D5 quarter notes (m. 2, beat 3) and the saxophone s notationally-accented, tonic-sounding C5 sixteenth note (m. 3, beat 4). Notice how I account for this in Example 4.1d by notating changing meter signatures that correspond with these accents. Example 4.1c. Rogers, Lessons of the Sky, mm. 1-9, renotated to reflect phenomenal meter. 183

204 Example 4.1d. Rogers, Lessons of the Sky, mm. 1-9, renotated to reflect phenomenal meter locally. Lerdahl and Jackendoff suggest that metric levels often extend above measures and that those levels must also exhibit an alternation between strong and weak beats. Unfortunately, the irregular pattern of two and three tactus pulses in Example 4.1d does not allow for precise alternation between strong and weak beats. However, if each tactus grouping begins with a strong beat, then strong and weak beats are to a certain extent already evenly distributed. London discusses the potential for metric stability in patterns of unevenly-accented pulse cycles in Chapters 7 and 8 of his book. London calls these meters non-isochronus, meaning beat accents occur at uneven intervals. Perhaps we can marry Lerdahl and Jackendoff s and London s ideas. Example 4.1e shows the 2/4 and 3/4 passages joined into 2+3/4 and 3+2/4 groupings. Drawing on strong/weak pulse alternation, this notation suggests that one could perceive a strong two-beat tactus grouping 184

205 followed by a weaker three-beat tactus grouping, repeated inversely thereafter. Whereas Example 4.4d represented the strong beat/weak beat schema on a micro level, its hierarchical relative is seen here on macro scale. Example 4.1e. Rogers, Lessons of the Sky, mm. 1-9, renotated to reflect phenomenal meter with strong/weak pulse relationship. Example 4.4 can be reinterpreted further using a similar approach. Example 4.4e shows the passage s basic phenomenal accent grouping, and Example 4.4f shows how these measures may be joined to communicate even greater unity. Though I have spoken relatively little about harmony, it is interesting to see how each of the measures in Example 4.4f so closely outline the passage s harmonic organization. This structure may relate to the idea posited in the previous Chapter that a passage s formal function may be phenomenally apparent; here phrases are generally grouped by their harmonic homogeneity. 185

206 Example 4.4e. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter. Example 4.4f. Rogers, Lessons of the Sky, mm , renotated to reflect phenomenal meter with strong/weak pulse relationship. 186