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Each phrase follows a path from predictable meter to a state of unpredictability. The phrase-level process is also psychological: While theories of meter often focus on surface-level rhythms and brief cognitive processes like entrainment, there is also significant precedent for a meter-focused analysis of larger formal units.

Authors generally concede that meters cannot create progressions as rich and varied as those generated by tonal harmony, but they nevertheless note similarities between forms suggested by metric changes and those based on tonal structures Lewin , —65; Leong ; Murphy Metric shifts between stability and conflict often outline familiar forms on local and global levels Malin , For instance, the Baroque hemiola is a metric means of designating the end of a formal unit phrase and a cadence.

Composers thus metrically outline large-scale forms Krebs , — Two higher-level forms in Vortex Temporum I , Reh. First, there is a stage-level form, unfolding over the course of five discrete stages consecutive sets of phrases. Example 14 summarizes this form, noting the beginning meter 1 , metric process of growing conflict 2 , and closing meter 3 of each stage.

Each stage begins with the same pitch collection, register, and meter—the eight-pulse tactus meter from Example 1. The stages become progressively more metrically complex as Grisey gradually introduces new types and degrees of conflict in successive phrases. Finally, each stage concludes with a phrase that pushes metric complexity to new heights, frequently including a new meter that conflicts with existing meters.

This new meter then helps to mark the formal boundary between sections by contrasting with the eight-pulse tactus meter when it returns at the start of the next stage.

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Or in cognitive terms, the first section passes from easy and rare entrainments to more difficult and frequent entrainments as metric variety increases and phrases are compressed. This form develops non-linearly because it embeds the stage-level form, which includes cyclical returns to the initial meter. Nevertheless, my analysis will show how the overall motion toward increased complexity and more difficult entrainment obtains despite this cyclicity.


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In fact, the stage-level form enhances the entrainment-based perspective on the overall form that I advance in the following analysis, a claim I will return to in my conclusion. I provide text descriptions of the relevant metric developments in each stage. Readers may refer to Example 6 for details about the specific meters, displacement types, and displacement durations in each phrase.

Note that each phrase generally corresponds to a rehearsal number in the score, excepting stage 5, where multiple short phrases occur between rehearsal numbers. For a broader perspective, refer to Example 14, which summarizes my prose description by noting the key aspects beginning, process, ending of each stage. Each shorthand example depicts an entire stage; each line within these examples corresponds to a single phrase. Tactus durations, corresponding to each labeled and uniquely colored box, represent the meter of each phrase.

A new color in the final box of a phrase designates a final bar meter shift. By following the proportional notation while listening, readers can get a sense of the dynamic psychological changes and meter-driven form arising from changes in our relationship with entrainment. See Example 15 for a proportional shorthand graph of this section.

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Unpredictability increases as phrases end in successively longer jolts. After the first phrase refer to Example 1 fulfills its eight-tactus length—creating the phrase-level expectation—the second phrase ends three pulses too early, the third sheds seven pulses, and the fourth loses thirteen.

At first a minor bump, by the fourth phrase these jolts elide more than an entire tactus duration. The third phrase also incorporates a final bar meter shift, from an eight- to a six-pulse tactus.

Arpeggio Studies Ex. 31-45 Sheet Music by John Griggs

Together, these factors mean that listeners must entrain anew in each phrase even though the first four phrases share a meter. Finally, the fifth phrase begins in a new meter, with a six-pulse tactus. Unpredictability peaks as listeners encounter a phrase with a new meter thirteen pulses before they expected a new phrase to begin. The fifth phrase ends with a one-pulse deferral, another new metric conflict increases variety and sets up the advent of the second stage by slightly delaying the return to the stable eight-pulse tactus meter.

Whereas the phrases of the first stage remain largely within one meter eight-pulse tactus , in the second stage every phrase has a different meter. See Example 16 for a summary of this section. Moreover, periodic and partially aperiodic meters are juxtaposed in successive phrases as tactus durations shrink from eight to five pulses. This juxtaposition throws into relief the differences between the periodic subdivisions of the meters with even tactus durations eight and six pulses and the aperiodic subdivisions of the prime tactus meters seven and five.

Tactus durations and metric displacements no longer follow a logical ordering.

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Rather, meters flip from relatively fast to relatively slow tacti in successive phrases, and displacements vary widely in length from phrase to phrase. Listeners can no longer anticipate the type of meter in an upcoming phrase, nor can they predict what type or length of displacement will end the current phrase. Vortex Temporum I , Reh. First and last bars of each prhase piano part are shown in each line. Example 18 reduces these phrases to their first and last bars. The first bar of each phrase shows the meter of that phrase, while the last bar demonstrates the displacement that ends each phrase.

This reduction highlights the frequent shifts between even, periodic meters Reh. Displacements vary in similar fashion: Moreover, three of these phrases Reh. This NI meter contrasts with the preceding meters, which have all included a periodic tactus layer. Moreover, the NI meter points up how the temporal window for entrainment has decreased: Listeners can now expect phrases half the length of the first phrase and must also entrain in half the time, and twice as often, as they did earlier.


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Phrases consistently end early, with significantly longer jolts in some phrases. In fact, many phrases end nearer their midpoints than their expected endpoints, with the result that listeners can barely finish entraining to one meter before they encounter a new one. Entrainment occupies more of their experienced time and begins to draw their conscious attention. The initial phrase of this stage establishes this new status quo when it ends twenty-four pulses early Reh. The extreme jolt in this prominent position immediately signals to listeners the degree of conflict they can expect in this stage.

Shift between different metric levels, Reh. Moreover, meters with vastly different tactus durations appear in succession, creating a sense of stratified motion between temporal levels. A meter change like this may seem more like a shift between metrical levels—between a subdivision and a tactus layer—because the shorter tacti two and three pulses share the durations of the slower meters subdivision units seven, six, five, four pulse meters.

The non-metrical string parts more frequently and completely occlude the metrical information of the piano and wind parts. In fact, the phrase length expectation is attenuated if not eliminated in this stage because almost every phrase ends shortly after it begins. The ninth and fastest meter arrives two-pulse tactus and stratified motion continues to be suggested by pairings of meters with short and long tacti.

The increased metric complexity of this late stage is reflected in the prominence of meters with short two- and three-pulses , prime seven- and five-pulses , and aperiodic NI tacti. The piano breaks from the sixteenth-note pulses, striking a low sustained E; then the winds, violin, and piano ascend a scale and arrive at a unison chord to mark the end of the first section on the downbeat of Reh. Heard in isolation, the fifth stage would not seem to promote metric hearing, with its extreme conflicts and constantly changing meters.

How can listeners continue to entrain and attend to meter when phrases end abruptly, meters change every few seconds, sustained tones obscure the metric evidence, and most meters are rapid and at least partially aperiodic? Grisey points up metric hearing in the cyclic structure of the stages. By returning to the opening meter at the beginning of each stage, he reminds his listeners of their earlier metric experience and the easy entrainment they had at the start of the piece. When he opens even the fifth stage with this meter, he is prompting his listeners to engage with the following complicated meters via entrainment, to reflect on the challenges and relate their experiences to the way entrainment felt in the first phrase of the piece.

If listeners respond to this aspect of the meter-driven form, they will gradually become more aware of entrainment. Musically, the first section progresses from simple, slow meters to complex, faster meters; rhythmic structure moves from simplicity to complexity; and tension generically increases as meters accelerate and conflict with their immediate context. Psychologically, the section begins with simple and subconscious experiences: As the section unfolds, entrainment must happen more quickly, and requires active reasoning and attention to parse the increased variety of meters and possible changes.

We could therefore think of the opening of Vortex Temporum I both as an intriguing meter-driven formal process and as a demonstration or enactment of a cognitive principle. In truly spectral fashion, Grisey communicates how entrainment and what metric differences to perceive in his formal processes, gradually revealing the psychological nature of a temporal experience to his audience. Louis Department of Music Forsyth Blvd.

Louis, MO joseph. Defining Constraints on Musical Communication. Who Said What about Whom? Contemporary Music Review 2: Louis, MO, November Music and the Psychology of Expectation. Metrical Dissonance in the Music of Robert Schumann. Metric States, Symmetries, and Systems. Lerdahl, Fred, and Ray Jackendoff. A Generative Theory of Tonal Music. A Critique from Cognitive Aesthetics. Studies in Temporality in Twentieth-Century Music , ed.

Psychological Aspects of Musical Meter. Analytic Engagements with Musical Tendencies. Metric Manipulations in Haydn and Mozart: Chamber Music for Strings, — Mirka, Danuta and Kofi Agawu, eds. Communication in Eighteenth-Century Music. Information Theory and Aesthetic Perception. University of Illinois Press. Pressnitzer, Daniel, and Stephen McAdams. The Stratification of Musical Rhythm. I am grateful to the attendees of that conference for feedback. Additionally, I wish to thank Paul Steinbeck, Robert Snarrenberg, Benjamin Duane, and the two anonymous reviewers at Music Theory Online for their helpful comments on earlier drafts of this article.

Thanks are due to the Washington University in St. Louis Music Department for their help with licensing costs. Finally, all examples are copyright: See Moscovitch and Anderson for summaries of the history of Spectralism and the interactions of these composers as the movement developed. Cross discusses the thorny question of what music counts as spectral music.

Spectral compositional techniques are summarized in Toiviainen , —11; Rose ; Moscovitch ; and Fineberg For this reason, Grisey and Murail often disavow the spectral label, preferring instead adjectives such as liminale or transitoire Grisey a. Grisey , —43 discusses grouping and rhythmic notation. Roeder describes how timepoints accrue accents through these grouping factors, framing the perception of pulse streams analogous to metric entrainment in terms of comparisons of accent strength and recurrence.

The fulfillment of expectations leads listeners to form higher-level expectations, corresponding to slower periodic layers, by grouping surface projections together. For a few alternative views of the derivation of meter from a surface, see for instance Yeston ; London , 60—78; and Mirka For example, the 4 4 time signature describes that meter in terms of its tactus quarter note and its grouping into larger bar units four tacti per bar. I will similarly summarize meters by their tactus values in my analysis below. See also Yeston ; and Lerdahl and Jackendoff , 12— More generally, Radvansky and Zacks , 51—52 discuss the differences of event salience and expectations at various timescales.

Local events at faster timescales are less salient and engender weaker expectations than events at slower timescales. For example, expectations for tonal closure at the end of a phrase are stronger than expectations for tonal resolution within that phrase. In fact, Grisey notated this phrase in 4 4 , probably the most common meter in Western art and pop music.

But from the perspective of perceptual salience, it is the notated half note eight pulses and not the quarter note four pulses that corresponds to the tactus of this phrase. It is perhaps possible to hear my subdivision layer as a tactus, especially since its rate of motion fits better within the range of tempi judged ideal in tapping studies e. Parncutt ; see also London , ch. I thank the anonymous reviewer for this observation. The resemblance between this meter and familiar Western meters is not a coincidence: Huron unpacks the significance of prediction and anticipation at pre-conceptual levels.

We do not need to be aware of an expectation for it to have a profound musical and cognitive effect. Importantly, this progressive increase in ambiguity and conflict obtains at multiple temporal levels. As my analysis section 4 will show, Grisey enacts this process at the phrase, stage, and section levels. Cohen , 65—74 describes music-focused research into Gestalt psychology and information theory, while Thaut , — specifically examines rhythmic-temporal communication.

Grisey discusses predictability in terms of the flesh of time in Grisey , — See also Hennessy , 26— London asserts that meter facilitates greater rhythmic understanding because the predictability afforded by meter allows listeners to discern finer gradations of surface rhythm and project expectations over longer spans of time London , 48; London Goldman expounds the antagonistic relationship between Boulez and the spectralists.

For an alternative, language-focused perspective on communication and expectation, see Bourne , [8. The tempo is consistent at quarter note equals BPM throughout the first section. In a more conservative reading Imbrie , the faster meters four, three, two could be interpreted as variations on the eight, six, and four pulse meters, in which case the later phrases would shift back to slower meters and metric variety would be somewhat reduced.

While this hearing is certainly possible, several factors support designating the faster meters as full-fledged meters. First, the extremely salient lowest pitches and pitch cycles clearly differentiate four, three, and two from eight, six, and four. Moreover, these meters are structurally distinct: However, choosing my reading or the conservative reading does not change my analysis of metric conflicts between successive meters: I thank the anonymous reviewer for suggesting the alternative reading. I thank Robert Snarrenberg for suggesting the term metric decay, which dovetails with the acoustic metaphors favored by Grisey in his writings.

At the same time, the string crescendos reinforce the phrase layer when they culminate their crescendo in a loud, short burst at the downbeat of the new phrase. Hasty defines deferrals as unexpected continuations or expansions of existing durations, where we instead expected a new duration to begin. Further, jolts affect the perceived past we only recognize them after they have happened while deferrals affect the perceived present we recognize them when something should have happened but did not.

This would reflect a more flexible hearing of the passage. And in fact, performers may take advantage of this flexibility for practical or aesthetic reasons. If you would like to authenticate using a different subscribed institution that supports Shibboleth authentication or have your own login and password to Project MUSE, click 'Authenticate'.

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