Beginnings

The idea of showing proportional, integer-related durations through the shapes of the notes on the page can be traced back to the thirteenth-century treatise Ars cantus mensurabilis (“The Art of Measured Song”) by Franco of Cologne.¹ This was a watershed moment in the history of Western music, because proportional notation freed composers from their former reliance on a limited set of stock rhythmic formulas, the rhythmic modes of the ars antiqua, and opened new possibilities for rhythmic flexibility, complexity, and innovation. Since the time of Franco, proportional rhythmic symbols have been used in all of the leading systems of music notation in the West, from the mensural notation of the late Middle Ages and Renaissance to the modern notation that emerged in the seventeenth century and remains in common use today.

There is no denying that proportional rhythmic notation presents a simplified image of musical time, because the durations in actual performances rarely form integer relationships with each other. Aware of this limitation, some scholars began seeking new, performance-sensitive representations of rhythm by the early twentieth century.² Among these, the most influential was Carl E. Seashore, a pioneering figure in the field of music psychology who assembled a research team at the University of Iowa during the 1920s and 1930s to analyze commercial recordings and live performances. One of the main goals of Seashore’s work, summarized in his 1938 textbook on the psychology of music, was to demystify the artistic dimensions of performance, which he equated with “deviation from the fixed and the regular: from rigid pitch, uniform intensity, fixed rhythm, pure tone, and perfect harmony.”³ Seashore sought a practical visual representation of these artistic deviations, and his solution is a graphic format that he calls the “performance score.”⁴ Figure 3.1 is his performance score showing measure-level durations in two renditions of the beginning of Chopin’s Polonaise in A Major, Op. 40, No. 1, as performed in the lab by the concert pianist Harold Bauer.⁵

This graph maps the relationship between two aspects of musical time. The horizontal axis shows what we might call “score time”; it divides the performance into discrete increments based on the barlines in the score, a chain of units of time that are conceptually equal to each other.⁶ The vertical axis shows these units’ duration in “clock time,” which can be thought of as a continuous timescale, a free-flowing stream of musical time. In this type of graph, the changes in vertical position from measure to measure represent expressive timing. Duration is the inverse of tempo, so downward slope means acceleration and upward slope deceleration.

Bauer was asked to play through the entire passage twice with the same expression, and this figure is meant to show that he could do that rather well. The main expressive features here are a “phrase arch” – that is, an acceleration followed by a deceleration – within mm. 1–8, another phrase arch in mm. 9–16, and an acceleration all the way through mm. 17–24.

To give a clearer sense of what the performance score represents, Multimedia example 3.1 includes an excerpt from another recording of the A-major Polonaise from the 1930s, along with my own Seashore-style analysis. The recording is by Artur Rubinstein, who repeats mm. 1–8 as prescribed in the score but does not repeat mm. 9–24.⁷ To track Rubinstein’s recording, I located measure onsets in Sonic Visualiser, a widely used, open-source performance analysis program.⁸ Like Bauer’s performance in the lab, Rubinstein’s recording has a clear phrase arch in mm. 1–8 as well as a ritardando in m. 16, but in place of Bauer’s eight-bar phrase arch in mm. 9–16, Rubinstein gives a pair of four-bar phrase arches that reflect this material’s antecedent-consequent design. Another difference is that Rubinstein has a further phrase arch in mm. 17–24 when the opening material returns, in place of Bauer’s race-to-the-finish-line accelerando.

Some might object that this performance score misrepresents the musical experience, because our attention may very well be drawn to expressive features in the recording that are not shown in the graph. For example, we might be struck by certain rhythmic details within the measure, such as the elongated eighth notes and the compressed sixteenths, as well as certain details of dynamic accentuation, not to mention the wrong notes in the repeat of the opening phrase. Why aren’t these features shown in the graph? Seashore would not deny their importance or discourage us from exploring them empirically, but he places a premium on the clarity that can be obtained when variables are isolated. Because the graphs discussed above only show durations at the measure level, they draw our attention to phrase-level patterns that are subtle but nonetheless readily audible and, moreover, musically meaningful given their close correspondence to the phrase structure of the composition and their stability across repeated performances. For these reasons, I find it helpful to avoid thinking of these graphs as representations of the overall effect or “essence” of a performance. Instead, I prefer to think of them as lenses or filters that draw attention to certain expressive features that the analyst considers meaningful.

After Seashore’s retirement in 1946, a second wave of empirical performance analysis began to emerge in the 1960s and continued into the 1980s and 1990s.⁹ New technologies such as MIDI and the personal computer greatly facilitated this type of research. Most empirical studies from this period echo Seashore in equating performance expression with measurable deviations from exact regularity, and they often feature graphs that closely resemble Seashore’s performance scores. Innovation during this period lay mostly in the development of new theoretical models that map aspects of musical structure onto patterns gleaned from performance data. Among these generative models of performance, as they came to be known, the one cited most often in recent literature is Neil Todd’s computational model of phrase arching, the expressive device we observed in the timing graphs discussed above.¹⁰ The model represents phrase arching as sets of nested parabolic curves, and it encompasses both expressive timing and variations in loudness (Figure 3.2).¹¹ The embedding depth of each curve is proportional to the unit’s hierarchical position in the grouping structure, which is itself modeled through the rule system given in Fred Lerdahl and Ray Jackendoff’s Generative Theory of Tonal Music.¹² The model predicts that the changes in tempo and dynamics at section boundaries will be more extreme than those at phrase boundaries within a section, which will in turn be more extreme than those at subphrase boundaries within a phrase. As supporting evidence, Todd provides graphs of timing and dynamics from performances by professional pianists, with patterns similar to those predicted by the model (see the broken lines in Figure 3.2).

In a thoughtful review of Todd’s work, Luke Windsor and Eric Clarke compare human and computer-generated recordings of a Schubert Impromptu and conclude that the model does not provide a complete explanation of how pieces are performed but may nonetheless be useful as a general expressive baseline or norm against which the nuances of a particular performance can be interpreted.¹³ This could be described as a deductive (theory-driven) approach to performance analysis. Bruno Repp offers a complementary inductive (data-driven) approach in a series of corpus studies from the 1990s.¹⁴ Here models are extracted from large sets of performance data through statistical methods (mainly averaging and principal components) and are then used as frames of reference for the interpretation and comparison of individual recordings.

In addition to this work by music psychologists, some historical research involving empirical methods of performance analysis began to emerge in the 1990s. One especially noteworthy contribution in this vein is José Bowen’s 1996 article on tempo fluctuation in recordings of orchestral music from the Classical and Romantic eras.¹⁵ Bowen tracked the tempo by tapping along on a computer keyboard while listening to recordings, a more efficient but less accurate method than those used by music psychologists.¹⁶ Bowen presents the tempo data in a series of graphs, again mainly in the style of Seashore’s performance scores but now at several levels of scale, proceeding from entire movements to individual phrases. He uses this data to support some rather broad interpretive claims about historical trends in performance practice over the course of the twentieth century and about the performing styles of several well-known conductors. For example, he shows that the tempos within the first movement of Tchaikovsky’s Sixth Symphony have become more extreme over time (Figure 3.3),¹⁷ but he also points out that the earlier recordings tend to have wide tempo fluctuations within each section, citing as a prime example the Concertgebouw Orchestra’s 1937 recording conducted by Willem Mengelberg (Multimedia example 3.2).¹⁸ Within the latter recording, Bowen highlights a series of three rather extreme phrase arches, beginning in m. 100, as well as a high level of tempo volatility from the beginning of the exposition to the first climax at m. 38.

Another tapping study from the mid 1990s, an essay by Nicholas Cook on Wilhelm Furtwängler’s recordings of Beethoven’s Ninth Symphony, forged a connection between empirical performance analysis and the discourse of music theory: Cook shows several correspondences between Furtwängler’s interpretation and the analysis in Heinrich Schenker’s monograph on the Ninth Symphony, which Furtwängler knew well.¹⁹ This sort of disciplinary cross-fertilization was a hallmark of research by scholars associated with the Centre for the History and Analysis of Recorded Music (CHARM), of which Bowen and Cook were founding members. First established at the University of Southampton in 1994, within a decade CHARM grew into a multi-institutional enterprise that hosted a series of international interdisciplinary conferences during the first decade of this century.

Recent Innovations

Over the past twenty years, partly as an outcome of CHARM and other related initiatives, the empirical literature on rhythmic aspects of performance has continued to flourish. During this period, traditional timing graphs have continued to appear routinely in performance analysis contributions, for instance in my own work on prolongational boundaries and metric dissonance in recorded music, and in writings on performing style and expression by Daniel Leech-Wilkinson and Dorrotya Fabian.²⁰ However, several new types of analytic figures have also been proposed, often in conjunction with observations about subtle qualities of motion that arise from interactions between performing nuances and aspects of grouping and meter. To illustrate this recent trend, I’ll now turn to some musical examples from five representative publications, all of which deal with recordings of nineteenth-century music for solo piano. (This repertory has probably been emphasized because piano note onsets are percussive and therefore relatively easy to locate in a sound file, and because tempo rubato tends to be more prominent in recordings of music from the nineteenth century as compared to other periods.)

In a 2012 essay on three recordings of the first movement of Beethoven’s “Moonlight” Sonata, Elaine Chew superimposes curved arrows on timing graphs to show what she calls “high-level phrase arcs” (Multimedia example 3.3).²¹ Chew’s conception of the phrase arch phenomenon differs fundamentally from Todd’s, because she does not invoke Lerdahl and Jackendoff’s rule system, whose input consists of compositional features such as melodic parallelism and cadence.²² Instead, her conception of the phrase rests entirely on expressive timing: for Chew, a “performed phrase” is simply a segment of the recording that begins and ends with minima in the timing graph.²³ She compares the rhythmic effect of different recordings by considering the relationship between these performed phrases and the metric notation within the first fifteen measures, showing that Daniel Barenboim and Maurizio Pollini slow down at the barlines, whereas Artur Schnabel does so only at formal boundaries marked by cadences. Chew notes that prior to these cadence points, “the consistent ebb and flow across the bar lines creates a sense of continuity, perhaps a clue as to how [Schnabel] creates these long, long lines.”²⁴ The arrow notation in Multimedia example 3.3a highlights the high-level performed phrases in Schnabel’s recording. In a follow-up study, Chew considers the formal implications of the high-level phrase arcs in other recordings. For example, she describes the differences between Schnabel’s and Pollini’s interpretations (Multimedia example 3.3b) in the following way:

Thus Chew’s annotated tempo graphs point not only to differences in performing style, but also to different ways of hearing the succession of formal functions in this passage.²⁶

Phrasing hierarchy is also the subject of an article on recordings of Chopin’s Mazurkas by Mitchell Ohriner, who builds on the traditional timing graph in two ways.²⁷ As a first approximation, he uses simple annotations to show how different performances resolve a grouping ambiguity in different ways: in Multimedia example 3.4, the third beat of m. 24 in the Mazurka in C Major, Op. 24, No. 2, can be heard either as (a) the ending of a group that begins in m. 21 or (b) the beginning of a group that ends in m. 28.²⁸ Recordings by Frederic Chiu and Vladimir Ashkenazy tip the balance one way or the other. Furthermore, at the measure level (not shown here), Chiu accelerates in m. 25 but Ashkenazy does not, so there is also a difference in the salience of different levels of the grouping hierarchy in these two performances, as shown through the boldfacing in Multimedia example 3.4c.²⁹

The second method involves using an algorithm to describe grouping similarities and differences within a large set of recordings at three levels of scale. As shown in Multimedia example 3.5, the segments of a timing graph are translated into durational contours, that is, ordered sets that represent the durations within the segment as integers from 0 (the shortest unit) to n–1 (the longest unit), where n is the number of units.³⁰ Next, a contour reduction algorithm checks for group-final lengthening (GFL) – that is, a slow ending – at the 2-, 4-, and 8-measure levels, in reference to a hypothetical grouping structure of (2+2)+(2+2) measures.³¹ The algorithm omits the midsized duration within a rolling 3-unit window, and this process is repeated recursively until the contour cannot be reduced further. Only contours that reduce to a pattern of decelerating throughout (<01>), or accelerating and then decelerating (<102> or <201>) are considered GFL-reflective. The results are displayed visually as a set of boxes that are either thick or thin, indicating the presence or absence of GFL. Multimedia example 3.6 shows the results for 6 of the 29 recordings of the C-major Mazurka that Ohriner analyzes with this method.³²

Because the reference structure is the same for all recordings, the second method is less sensitive to fine grouping boundary differences than the first, but it can nonetheless reveal more fundamental differences in the pacing of the recordings. For example, it shows that Brailowsky’s recording differs from both Chiu’s and Ashkenazy’s insofar as it lacks GFL at the four-bar level, giving it a markedly different feel from the other two. (Multimedia example 3.6 includes an excerpt from the Brailowsky recording.) The basic idea here is that the hypothetical (2+2)+(2+2) pattern exists in reality only to the extent that its constituent parts are performed in a GFL-reflective way. In this sense Ohriner’s approach (like Chew’s) offers a less deterministic representation of grouping than earlier generative approaches grounded in Lerdahl and Jackendoff’s rule system.

Meter rather than grouping forms the framework for a new visualization strategy proposed in a paper by Olivier Senn, Lorenz Kilchenmann, and Antoine Camp, who prepared a note-by-note durational analysis of the first four measures in Martha Argerich’s recording of Chopin’s Prelude in E Minor (Multimedia example 3.7).³³ In this case, units at various levels of the metric hierarchy – measures, half-measures, beats, and beat subdivisions – are represented in both the horizontal and the vertical axis, resulting in a series of nested squares.³⁴ The left and right hands are tracked separately, and asynchronies at points aligned in the score are shown as positive or negative integers in the middle of the graph, between the two streams of nested squares. Performance expression is multileveled, and it is very helpful to be able to survey several levels of activity at a single glance. This hierarchical representation is thus a significant advance over the traditional timing graph, which tracks durations on only one level of scale at a time, typically the measure or beat level.

Senn et al. point out some recurring patterns in the Argerich recording: the second half-measure is consistently longer than the first, there are consistent “melody leads” (i.e., the melody notes begin a few milliseconds before the chords), and there are beat- and subdivision-level ritardandos into the first three barlines as well as the midpoint of m. 4.³⁵ The authors describe the overall effect of these patterns as follows: “Argerich seems to make a new effort to gain some speed with each new bar (hence the faster first halves), but this effort is lost [and] as the bar progresses, the tension relaxes and the music seems to stagnate.”³⁶ From the loose fit between Argerich’s performed durations and the rhythmic notation, Senn and his co-authors offer this musical narrative-in-miniature as a speculative “inverse interpretation” that is presented “strictly from the listener’s point of view.”³⁷ This is a preliminary study of modest scope, limited to four measures from one recording.³⁸ However, it has the special distinction of having inspired several creative applications, in the form of a series of compositions by the American composer Richard Beaudoin. In these compositions, the durations in the score are derived from various nested-squares performance analyses, including the one in Multimedia example 3.7.³⁹

Some other recent studies have explored phrase-level patterns of dynamics as well as expressive timing. Jörg Langner and Werner Goebl explore interaction between these two domains of performance expression through innovative “performance worm” animations involving a string of overlapping discs in a space defined by tempo on the horizontal axis and loudness on the vertical.⁴⁰ Multimedia example 3.8 demonstrates this approach through an analysis of a recording of Chopin’s Etude in E Major, Op. 10, No. 3, by the pianist Mauricio Pollini.⁴¹ As noted above, Todd’s generative model suggests that both tempo and dynamics tend to increase and decrease in arch-shaped patterns over the course of a phrase. If the dynamic and tempo arches were congruent, then one would expect the performance worm to move diagonally most of the time, first to the northeast and then to the southwest. However, Langner and Goebl observe that in many recordings, such as the Pollini recording in Multimedia example 3.8, the tempo tends to increase before the dynamic level in the approach to a phrase’s climax, and counterclockwise patterns are prevalent overall.⁴² Though based on a relatively small sample of recordings, this observation suggests a possible refinement of our understanding of phrase arching. Follow-up studies have sought further recurring patterns of motion in the tempo-loudness space, including performer-specific patterns as well as generic patterns, through artificial intelligence models.⁴³

Dynamic and temporal aspects of phrase arching are explored in a different way by Nicholas Cook in the chapter on recordings of Chopin’s Mazurkas that forms the centerpiece of his broad-ranging book on performance analysis, Beyond the Score: Music as Performance.⁴⁴ Cook presents the data in the form of “scape plots” in which beat-level data are represented at the base of a triangle and adjacent beats’ values are averaged recursively at higher levels, as shown schematically in Figure 3.4.⁴⁵ The loudness triangle is inverted and aligned with the timing triangle along the base, creating a diamond shape overall, and the numerical information is color-coded in such a way that flame-like patterns emerge at the loudest and slowest points in an excerpt. Multimedia example 3.9a is Cook’s analysis of a recording that he describes as a textbook example of phrase arching, namely Heinrich Neuhaus’s recording of Chopin’s Mazurka in C-sharp Minor, Op. 63, No. 3.⁴⁶ Through graphs of several other recordings of this piece, Cook demonstrates that the technique of phrase arching, though prevalent since the Second World War (especially among Russian-trained pianists), was less widely used in the first half of the twentieth century, when a more rhetorical and improvisatory style of performance prevailed. In Ignaz Friedman’s recording (Multimedia example 3.9b), for instance, Cook hears a “focus on moment-to-moment expression” arising from features such as the unusually long opening upbeat and abrupt changes of tempo and articulation later in the excerpt.⁴⁷ Thus Cook uses his flaming diamond graphs to support the argument that phrase arching is a historically and culturally specific practice, not a universal or “hard-wired” psychomotor phenomenon, as Todd’s model and other generative models seem to imply.

Despite their sharp differences in design, all of the recent examples I have surveyed help bring into focus musically meaningful patterns that might otherwise escape our notice, as well as the qualitative effects that these patterns help to generate. Like Seashore’s performance scores, then, these new types of graphs can be thought of as analytic lenses or filters that help us reach a clearer understanding of the music we are actually hearing, as distinct from the music we read on the page. Another way in which these new graphs resemble Seashore’s performance scores is that they use the notational meter – the beats and measures shown in the score – as a frame of reference: durations are tracked at the level of the notated beats, and measure numbers are shown either on the horizontal axis (Chew, Ohriner, Cook), at the top of the nested squares (Senn et al.), or on the face of the performance worm (Langner and Goebl). In my view, this is not problematic in the particular examples surveyed above, because these beats and measures can easily be heard and felt in these recordings. However, reliance on the metric notation limits the generalizability of these analytic approaches, because many other recordings do not conform to that notation – especially those by performers trained at the beginning of the twentieth century, a period when many musicians and critics railed against the “tyranny of the barline.”⁴⁸ I consider the variations in meter within such non-literalist recordings to be a distinct type of “artistic deviation from the regular” – in other words, a special type of performance expression.

Visualizing Metric Variation in Performance

Let’s consider two versions the opening of Chopin’s Mazurka in A Minor, Op. 17, No. 4, one recorded by Ignacy Paderewski in 1923 and the other by Guiomar Novaes in 1954.⁴⁹ According to the score (Multimedia example 3.10a), this excerpt consists of eight bars in triple meter, grouped as 4+4.⁵⁰ Timing graphs of the two recordings are very similar (Multimedia example 3.10b), and the corresponding timescape graphs are nearly identical (Multimedia example 3.10c).⁵¹ However, these illustrations conceal some fundamental differences between the recordings in the domain of experiential meter, by which I mean the meter felt by the listener, as opposed to the meter represented in the score.⁵² To get at these differences, I will borrow some concepts and analytic symbols from Christopher Hasty’s Meter as Rhythm, which casts meter not as a structure or schematic framework but instead as a dynamic, emergent aspect of the listening experience.⁵³ Hasty represents experiential meter through static images, but I will instead use animations in an effort to represent more directly the temporal qualities that the theory describes, an approach suggested in John Roeder’s review of Meter as Rhythm.⁵⁴ I created Multimedia example 3.11–3.15 using Adobe Animate, a multimedia computer animation program.

For Hasty, the basis of the experience of meter is projection, the feeling that once a duration has been articulated, it will immediately be reproduced: after hearing two sounds, we expect a third sound at a specific future moment (Multimedia example 3.11a).⁵⁵ A complex projection occurs when a long projection is coordinated with two or more shorter projections (Multimedia example 3.11b),⁵⁶ and within a complex projection, the dominant beginning (|) launches a projection that remains active for more than one beat, while a continuation (\) initiates a shorter projection between two dominant beginnings.⁵⁷

One of the benefits of Hasty’s analytic notation, as compared to traditional metric symbols like time signatures and barlines, is that it allows for more subtlety and flexibility in tracking changes of meter. Hasty defines three types of metric change relevant to the excerpts discussed below: denial occurs when a timespan does not fulfill its projected duration and instead launches a new projection that is considerably shorter or longer than the one before it (Multimedia example 3.12a),⁵⁸ deferral happens when an extra continuation postpones a dominant beginning and therefore alters the metric type, for instance by shifting from duple to triple meter (Multimedia example 3.12b),⁵⁹ and interruption occurs when the ending of a projected duration is pre-empted by a new, early beginning (Multimedia example 3.12c).⁶⁰

Hasty’s own analyses center on compositional applications of these processes of metric change, mainly in music from the twentieth century, and he uses scores rather than recordings as his objects of analysis. I often hear these same metric processes when listening to recordings of earlier, common-practice repertoire, including the Paderewski and Novaes excerpts discussed above, to which I now return.⁶¹

Multimedia example 3.13a shows the variations in meter that I hear in the Paderewski excerpt.⁶² Up to the boundary of the introduction and theme, I hear a projection of triple meter without anacrusis. This triple meter is so well established that I feel a fifth downbeat during the sustained chord that begins in the fourth experiential measure, but the mazurka theme enters before this fifth measure is completed – a clear case of interruption. A further complication occurs during the first three beats of the theme, which Paderewski plays as weak–strong–weak. This can be heard either as a syncopation or an anacrusis in its local context, but once the theme’s second downbeat is articulated the ambiguity resolves in favor of syncopation. From that point on, I hear an unambiguous triple meter until the end of the excerpt.

Multimedia example 3.13b maps the hypermeter in Paderewski’s recording, which is equally flexible. Although a potential for duple hypermeter can be sensed throughout the excerpt, this potential is fully realized only toward the end.⁶³ Duple hypermeter is ubiquitous in Romantic piano music, so we might reasonably expect it at the outset of the recording. Instead, the second hyperdownbeat is deferred until the sustained chord at the end of the introduction, which gives the fourth experiential measure a much stronger sense of “beginning” than the third, and suggests that the recording begins with triple hypermeter. This effect is enhanced by the increased tempo and rhythmic activity leading up to the sustained chord, which bring a sense of anacrusis, of into-the-beginning. However, the projection of this triple hypermeter is interrupted by the entry of the mazurka theme, which itself projects the duple hypermeter expected from the outset.

In Novaes’s recording, I hear a fundamentally different metric process involving a shift from duple to triple meter at the surface level (Multimedia example 3.14a). Duple meter is projected throughout the introduction, and the triplet figure near the end of the introduction sounds like a quarter-note triplet, not an eighth-note triplet as in Paderewski’s recording and in the score. This quarter-note triplet foreshadows a shift from duple to triple meter at the beginning of the theme, and the ritardando that surrounds the triplet helps smooth out this metric transition. In conjunction with this ritardando, a shift from duple to triple hypermeter can also be discerned in the introduction (Multimedia example 3.14b). The accents on the first and fifth chords of the introduction project duple hypermeter, and the triplet figure defers the next hyperdownbeat until the sustained chord. Then, as in Paderewski’s recording, this triple hypermeter is interrupted by the entry of the mazurka theme, which itself projects duple hypermeter throughout.

The Paderewski and Novaes example show some ways in which an investigation of experiential meter may shed light on distinctive and meaningful patterns that would be obscured in an analysis framed by the notational meter. To demonstrate this approach further, I will now turn to a more metrically intricate passage from Debussy’s 1913 piano roll recording of his piece D’un cahier d’esquisses (“From a Sketchbook”) as reissued on CD.⁶⁴ The passage includes a cadenza that features two gong-like bass notes, each of which is followed by a rapid ascending passage and a short melodic fragment. The cadenza is preceded by a short transition and followed by a coda.⁶⁵ Multimedia example 3.15a maps the metric fluctuations that I hear in this excerpt.

A clear and continuous tactus can be felt throughout the excerpt, but the underlying (measure-level) projections are considerably less clear and continuous: a slow duple meter is established at the outset, but two processes of metric change, namely, denial and deferral, can be traced within the cadenza. Because of the slow tempo, these processes do not bring as much tension as the interruptions and deferrals in the Paderewski and Novaes recordings. Instead, they bring a subtle ebb and flow of metric determinacy over the course of the excerpt.

The melodic fragment after the cadenza’s initial gong stroke includes an extra continuation (“weak beat”) that defers the next downbeat until the second gong stroke. This establishes a potential triple meter, but when the melodic fragment returns soon after the second gong stroke, its rhythm is altered in such a way that the gesture now encompasses four beats instead of three. The result is that the tentative triple meter recedes from our field of awareness,⁶⁶ and the referential duple meter is restored.

By the end of the excerpt, the tempo has become slow enough to undermine the duple meter, which seems to dissipate shortly after the coda begins. However, there is a feeling of rhythmic continuity between the tactus pulses of the cadenza and the triple measures of the coda. In other words, there seems to be a transition from a slow duple to a moderate triple meter toward the end of the excerpt, similar to what happens in the Novaes excerpt discussed above.

The score and recording are again quite remote from each other in this case. The cadenza is mostly unmeasured in the score (Multimedia example 3.15b), and one of its main focal points in the recording, the second gong stroke, is altogether absent.⁶⁷ A few beats later, at the beginning of the coda, the score indicates a change from $\begin{array}{c}6\\ 8\end{array}$ to $\begin{array}{c}6\\ 4\end{array}$ meter, as well as a tactus shift (dotted quarter = quarter), indicating that the coda should proceed at only one-third the tempo of the previous $\begin{array}{c}6\\ 8\end{array}$ measures – a dramatic discontinuity in comparison to the seamless connection heard in the recording.

These animations provide a new way to visualize and interpret the nuances of rhythm and timing in recordings that lack a consistent metric framework. As such, they fill a gap in the literature surveyed in Parts I and II of this chapter. Further applications might explore recordings of Gregorian chant and other types of monophonic singing in free rhythm, as well as free-flowing instrumental music like the unmeasured keyboard preludes of the French baroque. These are just a few of seemingly endless possibilities for research in performance analysis, a field that will surely continue to grow and develop as new technologies become available in the years ahead – thereby helping to further expand our appreciation and understanding of the rhythms of performance.

Figures and Multimedia examples for Chapter 3

These figures appear on the website www.rptm.ca/essays/vrp

1. 3.1 Timing in the opening of Chopin’s Polonaise in A Major, Op. 40, No. 1, performed by Harold Bauer (Seashore, Psychology of Music)

2. 3.2 Graphs of human and computer-generated performance expression:

   1. (a) Timing in an excerpt from Haydn’s Sonata in E-flat Major, Hob. XVI/49 (Todd, “A Computational Model of Rubato”)

   2. (b) Loudness in an excerpt from Chopin’s Prelude in F-sharp Minor, Op. 28, No. 8 (Todd, “The Dynamics of Dynamics”)

3. 3.3 Average tempos in various recordings of the first movement of Tchaikovsky’s Symphony No. 6 in B Minor, Op. 74 (Bowen, “Tempo, Duration, and Flexibility”)

4. 3.4 Generalized scape plot design (Sapp, “Computational Methods”)

These multimedia examples appear on the website www.rptm.ca/essays/vrp

1. 3.1 Timing in the opening of Chopin’s Polonaise in A Major, Op. 40, No. 1, performed by Artur Rubinstein

2. 3.2 Timing in the first movement of Tchaikovsky’s Symphony No. 6 in B Minor, Op. 74, performed by the Concertgebouw Orchestra conducted by Willem Mengelberg (Bowen, “Tempo, Duration, and Flexibility”)

3. 3.3 Timing and high-level phrase arcs in the opening of Beethoven’s “Moonlight” Sonata, Op. 27, No. 2

   1. (a) performed by Artur Schnabel (Chew, “About Time”)

   2. (b) performed by Artur Schnabel (top half) and Maurizio Pollini (bottom half) (Chew, “From Sound to Structure”)

4. 3.4 Timing and grouping in an excerpt from Chopin’s Mazurka in C Major, Op. 24, No. 2 (Ohriner, “Grouping Hierarchy and Trajectories of Pacing”)

   1. (a) performed by Frederic Chiu

   2. (b) performed by Vladimir Ashkenazy

   3. (c) effects of performance strategies upon grouping structure

5. 3.5 Durational contour in Chopin’s Mazurka in C-sharp Minor, Op. 63, No. 3, performed by Stanislav Bunin (Ohriner, “Grouping Hierarchy and Trajectories of Pacing”)

6. 3.6 Group-final lengthening in six recordings of Chopin’s Mazurka in C Major, Op. 24, No. 2 (Ohriner, “Grouping Hierarchy and Trajectories of Pacing”)

7. 3.7 Timing and notational meter in the opening of Chopin’s Prelude in B Minor, Op. 28, No. 4, performed by Martha Argerich (Senn et al., “Expressive Timing”)

8. 3.8 Timing and dynamics in Chopin’s Etude in E Major, Op. 10, No. 3, performed by Maurizio Pollini (Langner and Goebl, “Visualizing Expressive Performance”)

9. 3.9 Timing and dynamics in the opening of Chopin’s Mazurka in C-sharp Minor, Op. 63, No. 3 (Cook, Beyond the Score)

   1. (a) performed by Heinrich Neuhaus

   2. (b) performed by Ignaz Friedman

10. 3.10 Timing and notational meter in the opening of Chopin’s Mazurka in A Minor, Op. 17, No. 4

    1. (a) score

    2. (b) timing graphs of two recordings, performed by Ignacy Paderewski and Guiomar Novaes

    3. (c) scape plots of the two recordings

11. 3.11 Animations for projection, as described in Hasty’s Meter as Rhythm

    1. (a) simple

    2. (b) complex

12. 3.12 Three types of metric change discussed in Hasty’s Meter as Rhythm

    1. (a) denial

    2. (b) deferral

    3. (c) interruption

13. 3.13 Experiential meter in the opening of Chopin’s Mazurka in A Minor, Op. 17, No. 4, performed by Ignacy Paderewski

    1. (a) preliminary analysis (two levels)

    2. (b) full analysis (three levels)

14. 3.14 Experiential meter in the opening of Chopin’s Mazurka in A Minor, Op. 17, No. 4, performed by Guiomar Novaes

    1. (a) preliminary analysis (two levels)

    2. (b) full analysis (three levels)

15. 3.15 Experiential and notational meter in the cadenza from Debussy’s D’un cahier d’esquisses

    1. (a) performed by the composer

    2. (b) score

Rephrasing Rhythm

At the 60 percent point of this essay, I wonder what I have offered to our conversation about rhythm. We’ve spoken of rhythm in transactional terms as the tension between an underlying pulse, cycle, or meter and its surface inflections. We touched on the panrhythmic spectrum, whereby rhythm is expressed in both temporal and non-temporal terms. We engaged in some problem solving with cyclical and non-cyclical polyrhythms, and discussed the temporal ebb and flow embedded within nested polyrhythms. But surely you could have figured these things out for yourself!

And, while this is important prefatory material, none of it touches the fundamental concept that rhythm is the musical product of the interface between mind, body, and sound. If that’s a definition so broad as to be useless, I take comfort from centuries of musical thinkers who have felt similarly, from Pythagoras, who expressed rhythm as universal and interdependent harmonies, to Karlheinz Stockhausen, whose seminal comments in Vier Kriterien der Elektronischen Musik (1973) speak to the relationship between rhythm and tone and annotate one of his most memorable musical moments: the dizzying seven-octave descent of a complex tone and its ultimate dissolution into rhythm in the extraordinary Kontakte (1959).

What is clear from Pythagoras, Stockhausen, and others is that rhythm is a physical state of musical material, standing in relation to tone, form, and texture just as ice is a physical state of water related to vapor and liquid water. For many mid-century modernists, searching not just for an answer but for the answer, treating rhythm as an inseparable component of an inclusive musical continuum was the brave new world. Their vision was a unified temporal field consisting of regular or irregular oscillations, perceived at the upper end of frequency as pitch and at the lower end as form. Rhythm falls in between. They proposed that serial technique, the great equalizer in the realm of pitch, could be applied to rhythm (as time-point technique in the language of East Coast Americans) and to form (as Moment Form in Stockhausen’s nomenclature.)

At its best, rhythmic serial technique provided the mechanics of permutation, in which variations were rooted in a single source code that created consistency. If all rhythms were related to the same set, then they were, by definition, related to each other. (If A=B and B=C, then A=C.) For this listener, this creates a powerful sense of organicity and inevitability in Olivier Messiaen’s modal approach to rhythm, but in lesser composers often produces an unappealing uniformity. As tantalizing as a grand unified theory of rhythm might seem, it often fails for the same reason it’s not possible to swim in steam. The fundamental material may be the same, but a change of state necessitates a change in usage.

So, we are left with a koan: rhythm consists of the same stuff as pitch and form, and comprises with them the pitch-form continuum. But boundaries of rhythm are delimited by perception, which firmly distinguishes it from pitch and form. Take a pitch and gradually slow the speed of the oscillations until we hear discrete impulses. We call this rhythm. And, getting slower yet, soon we can no longer tell whether or not the oscillations are equal. We call this form. On the fast end of the spectrum, our rhythmic sense is limited by the speed of cognition and on the slower end by the capacity of memory.

If rhythm is a property of human perception, then Pythagoras’s search for the cosmic origins of music was partly in vain. This also means that our delight at Gérard Grisey’s musical rendering of the regular rhythms of distant pulsars in Le noir de l’étoile (1990) celebrates not the universality of pulse in the cosmos but the miraculous ability of a human listener to track patterns in time. It turns out that the music of the spheres is really music of the ears.

Most rhythmic patterns fall on the slow side of the spectrum between pitch and form. Even the fastest moments in virtuosic piece like Bone Alphabet fall short of the range of audible tone. (A stroke speed of 1035, on the border of playability, is a meager 17.25 cycles per second, not usually audible to humans.) On the slow end of the spectrum, we can track a regular pulse up to a period of around eight seconds (in an utterly unscientific self-study). Here, questions of expectation and foreshadowing, repetition and reformulation emerge. This is the poetry of rhythm: Have we heard this before? When do we expect the next impulse? And, depending on whether this satisfies or derails expectations, what emotional or artistic impact results?

So, rhythm – or at least our perception of it – is ultimately a quality of memory. This works in the aesthetic as well as psycho-acoustic realm. The reason a reassigned downbeat in a Beethoven Scherzo (see practically every symphony!) surprises us lies in memory. We recognize and remember the downbeat and then are surprised when it is reassigned. (Beethoven’s method adds stress to a weak beat and when he is sure we perceive that as the downbeat, he shows us the true downbeat.) If we did not remember – and consequently develop an expectation – we could not be surprised.

The language of memory – corporeality, aurality, and the creation of and deviation from pattern – is also the language of rhythm. So “rephrasing” rhythm as an aspect of memory leads us gratefully away from the calculations of nested polyrhythms and the anxiety that even very fine musicians feel when faced with thorny non-cyclical polyrhythms. Even though I have made thousands of such calculations and still see the value of accurate solutions to complicated rhythmic problems, I argue that rephrasing rhythm as memory aligns us, as performers, with the mindset of listeners. It informs the late pieces of Morton Feldman, where forgetting is poignant, and governs the vocal/instrumental works of Roger Reynolds and Georges Aperghis. Each contains “complex rhythms,” but viewing rhythm as memory, each offers music on a human scale and with human rewards.

Rhythm as Memory

In the last decade of his life, Morton Feldman composed the longest and most evocative pieces of his career. Musicians are advised to know all three of the late trios for flutes, piano and percussion, but to this performer his For Philip Guston (1984) is the most beautiful. On the level of rhythm, it is also his most complex. Measures occupy equal physical space, but because the meters are different in each part, vertical alignment rarely equals simultaneity. Without significant calculations, the performers frequently do not know with what notes in the other parts they are to align.

4.4 Feldman, opening of For Philip Guston.

Morton Feldman For Philip Guston|für Flöte, Schlagzeug und Klavier.

© Copyright 1985 by Universal Edition (London) Ltd., London/UE17967, used with kind permission

In the age-old trade-off between reading and memory – think of the amount of music and literature committed to memory in the Ancient World due to the lack of written texts – if you cannot track points of alignment by reading the score, you must necessarily remember them. Thus, maintaining the correct rhythmic relationship with your partners in Guston means that tempo must be internalized (memorized as an inner click-track) and rhythm converted scrupulously to physicality (memorized as corporeal impulse). For the percussionist, the $\begin{array}{c}3\\ 32\end{array}$ note has a shorter stroke than the $\begin{array}{c}3\\ 8\end{array}$. We know that stroke length varies in response to dynamics and the surface quality of the instrument to be played. But in For Philip Guston, where the dynamics are fixed and low, and the instruments, aside from the chimes, do not require radically different stroke types, the corporeal memory of striking is a nearly exact map of rhythm.

In Michael Gordon’s XY (1998), also from the percussion repertory, corporeal memory works similarly in the service of rhythmic accuracy. If you are having trouble with your 6:5, try holding the stick playing quintuplets at a height that is $\begin{array}{c}6\\ 5\end{array}$ greater than the sextuplet stick. Then play with the same stroke speed. Other instrumentalists and vocalists will devise their own versions of these mechanical strategies.

On the level of form, we engage a different sort of rhythmic memory. Major points of arrival in tonal forms are so far apart that composers tag them with memorable characteristics – a burst of instrumental color or a memorable theme. However, in For Philip Guston, such markers are largely absent, and with a duration of nearly five hours, even the performers can forget where they are in the form. In Feldman, the poetry of remembering is contradictory: on the local level the execution of rhythm is a matter of remembering the speeds of notes, but on a global level the received sense of form is often the product of forgetting.

Note the “irrational meters” in Josh Levine’s Four Places, Many More Times (2011). By irrational, we refer to measures that contain subdivisions that do not add up exactly to the denominator of the time signature. Measure 134 in Example 4.5 is four quintuplets in length. Measure 138 is $\begin{array}{c}2\\ 3\end{array}$ of a half note. Ferneyhough and many others would name these measures as a $\begin{array}{c}4\\ 10\end{array}$ measure and $\begin{array}{c}2\\ 6\end{array}$ measure, respectively. (There are ten eighth-note quintuplets in a measure; four of them make a $\begin{array}{c}4\\ 10\end{array}$ measure. A quarter-note triplet is a “sixth note” – thus a $\begin{array}{c}2\\ 6\end{array}$ measure.) Conducting this difficult passage, I find that subdividing is less useful than simply remembering the speed of the duple, triple, and quintuple subdivisions and accessing them through corporeal memory.

4.5 Levine, from Four Places, Many More Times.

Used with kind permission of the composer, Josh Levine

Here and There (2019), a Roger Reynolds composition for percussion solo with text by Samuel Beckett, also relies on memory to stabilize rhythm. The work consists of three types of material: monologues for voice alone, arias where the performer speaks and plays simultaneously, and purely instrumental episodes. Reynolds doesn’t represent Beckett’s text in notated rhythms or meters – here the cadence of speaking provides the rhythmic template. Conversely, the instrumental passages are written in meter, and in these moments Reynolds ingeniously provides light cursive writing below rhythms. These words (unspoken in performance) are drawn from the Beckett text, and provide the rhythmic architecture of the instrumental passages. In order to remember the rhythms, just remember the text! (See Example 4.6.)

4.6 Reynolds, from Here and There.

Used with kind permission of C. F. Peters Corp., Henmar Press

An ideal performance of Here and There melds words with music in a fertile in-between world (zwischen immer und nie, “between always and never,” to use the words of novelist André Aciman). So rhythmic sense, like everything in the piece, is rooted in the memory of both words and sounds. In practice, I found I could quickly solidify the rhythm of the text, which then served as a platform not just for consistency of delivery but also as a rhythmic template for the rest of the piece. Text became the “memory palace” of Here and There: a place where the linkage between spoken word and instrumental sounds is stabilized by a shared temporal memory.

Here and There was a recent confirmation, but we intuit that many rhythmic problems are really memory problems. You can play 16th notes at MM60 without the need for a calculator because you have done so repeatedly. You remember not only how fast they are, but also the quality of tension between them and the underlying pulse. You might have difficulty with 13:5, because you have comparatively little experience with that rhythm. You have not yet memorized it. In its many guises, memory is an organizing principle: in an ethical life (see St. Thomas) or as the way we know what to buy at the store (see Hermann Ebbinghaus’s studies in the degradation of short term-memory). But memory achieves a kind of apex moment in the execution of rhythm in which corporeal, aural, mechanical, and even visual modes of memory collude.

Temporal tension directs the utility of both memory and rhythmic sense toward the future. Memory is not about the past any more than rhythm is about the present. Both require that we metabolize past and present time to give direction to the future. Whether that future is remembering a wedding toast or the impetus to lift a mallet (or bow) to play a note in a nested quintuplet is merely a matter of application. The goal in both is felicity of expression.

Pulse

An element that has become a distinctive structural component in Western classical music in the past fifty years is pulse – specifically, pulse independent of meter. Terry Riley’s In C (1964), the composition that jump-started the minimalist movement, is performed with a pianist pulsing on the instrument’s top two Cs throughout the piece. In C is made up of fifty-three modules of varying length through which performers move at their own pace. Robert Carl, in his book Terry Riley’s In C, explains that the “major stumbling block” in early rehearsals for the piece “was rhythm; as soon as the divergence of modules began, it became difficult to maintain a common tempo or metric reference point, and the work fell apart.” Steve Reich was a participant in these rehearsals and Carl quotes Reich as saying, “once a drummer always a drummer, I said we kind of need a drummer here, but since drums would be inappropriate, what about use the pianos; so Jeanie [Brechan] played some high Cs just to keep us together, and Terry said, ‘Let’s give it a try’ or something like that, and we tried it and voila everyone was together.” Carl concludes, “And so the Pulse was born.”¹

Reich began including a pulse in his own compositions, and other composers followed suit. To delineate the pulse, Reich used various instruments: maracas in Four Organs (1970) and later in Tehillim (1981); a pulse clave in Music for Pieces of Wood (1973); combinations of pulse played on marimbas, xylophones, pianos, strings, winds, voices, and maracas in Music for 18 Musicians (1976); and even the handles of percussion mallets tapped together in portions of Sextet (1984) and The Desert Music (1984). Reich continues to employ a pulse in his compositions, both explicitly and implicitly, including the appropriately named Pulse written in 2015. Two other prominent composers who began writing pulse-based music in the 1960s and 1970s are Philip Glass, who used repeated arpeggiated figures and an additive rhythmic process in his early compositions to outline a steady pulse, and John Adams, who used pulsing, repetitive cell structures in his solo piano work Phrygian Gates (1977) and placed the pulse in the wood block in his orchestral work Short Ride on a Fast Machine (1986).

Classical musicians are not generally trained to play strict pulse-based music. Orchestral players, for example, develop an association with pulse from the historical development of the repertoire and learn to adjust regular attacks in music according to conductors’ motions, the amorphous attacks of entire sections of strings, the varied articulation of wind and brass players, and the more precise attacks of harp, piano, and percussion instruments. Although In C was written for a large chamber ensemble rather than a symphony orchestra, the performers face the same problem of creating a unified approach to attack placement and ensemble coherence. However, musicians who perform In C have the additional issue of achieving this amid a web of overlapping rhythmic patterns. The person who plays the upper two Cs on the piano keyboard throughout In C must keep the pulse as steady as possible while being swayed by other players as they wind their way through the work’s fifty-three modules. This responsibility is not unlike that of a jazz drummer and bass player working together to outline a rhythmic grid while soloists improvise freely but with a constant sense of the pulse. The pianist in the Riley work has no one else with whom to maintain a steady pulse and must rely on some internal mechanism to keep playing regularly for an hour or more.

Time Feel/Inner Pulse

At a rehearsal of Steve Reich’s Six Pianos (1973) for a 2007 concert in Toronto, pianist Gregory Oh, who was playing the piece for the first time, remarked to percussionist Bob Becker, who had played it many times with the Reich ensemble, “How can you play with such steadiness but appear so motionless?” Becker replied, “Because I have a sense of inner pulse.” Inner pulse is something we can all develop and is an ability that is essential in the performance of pulse-based music. Becker was trained as a classical musician and like most percussionists has played forms of pulse-based music in various non-orchestral formats. His sense of inner pulse was heightened, however, by his experiences with West African drumming and North Indian tabla drumming.

West African drumming ensemble music is structured around a cyclic rhythmic pattern played on an iron bell. The twelve-unit cycle of the most commonly used bell patterns can be felt with a steady pulse of 1, 2, 3, 4, 6, or 12, although the drummers, singers, and dancers generally perform with a pulse sense of four within the twelve-unit cycle in order to stay together and have a unified feel. Ghanaian master drummer Abraham Adzenyah, the West African drum teacher at Wesleyan with whom both Becker and I studied, used the terms hidden beat and invisible conductor to describe the performance subtleties that are required to play this music correctly. The hidden beat is the inner pulse that is felt communally by all the performers, and the invisible conductor (one of my favorite descriptive phrases) is the time feel sense that is generated in the music.

Tabla drummers of North India, as well as their counterparts in South India, use an additive, or sometimes subtractive, system of rhythmic construction in creating complex mathematical rhythmic patterns. These rhythms are played within a tala cycle that establishes a structural framework for the music. The inner pulse for these drummers is an inviolable sense of the tala cycle, and their rhythms are placed within and across the cycles. Philip Glass first experienced Indian music when he worked with Ravi Shankar on a soundtrack for the film Chappaqua in Paris in the mid-1960s; he later visited India and studied with tabla master Alla Rakha. This introduction to a new way to view rhythm had a significant impact on Glass’s early musical output. Although he did not adopt the concept of tala, Glass did incorporate the idea of cyclic structure in his music. His systematic use of additive rhythms is most clearly seen in 1+1 (1968) and Two Pages (1969). In the score of 1+1, Glass wrote two rhythmic units: (a) two sixteenth notes and an eighth note, and (b) an eighth note. In his instructions in the score, he stated that “1+1 is realized by combining the above two units in continuous, regular arithmetic progressions.” In Two Pages, Glass wrote two, three, and four eighth-note groupings with varying numbers of repeats. Wes York, in an analysis of Two Pages in his essay “Form and Process” (1981), writes, “Ultimately, the structure of Two Pages can be understood as first, the exposition and juxtaposition of two sets of opposing processes, and then, the coordination of all shapes which both emerge from, and reflect back on, those processes. … With respect to the various processes at work one finds two types which are responsible for creating motion and change within the composition. One of these is a subtractive process; the other is additive.”²

In Les Moutons de Panurge (1969), composer Frederic Rzewski uses the additive/subtractive process to the extreme. The piece begins with any number of musicians playing the notated score in a sequence of 1, 1-2, 1-2-3, 1-2-3-4, and so on until reaching note 65. They repeat the complete cycle of 65 notes, then begin subtracting notes from the beginning: 2 through 65, 3 through 65, 4 through 65, and so on until reaching 65 alone. Even though Rzewski allows for errors on the part of the musicians in sequencing the notes, he is clear that a pulse must be established and maintained by all players.

In the works of Glass and Rzewski described above, there is no explicit pulse played by any instrument and the success of the performance depends on a sense of pulse felt by the performers. While expertise in non-Western music is not an imperative for Western musicians attempting to play these compositions, a familiarity with the additive process provides a level of comfort in keeping a steady pulse while navigating the requirements of the scores. As with Becker in his performance of Six Pianos, experience with non-Western music and the sense of inner pulse in the midst of a quilt-work of pulse options helps Western-trained classical musicians play pulse-based music with confidence.

Jazz is also primarily a pulse-based music, and great jazz musicians have this same sense and control of inner pulse. Drummer Kenny Clarke was influential in creating the bebop form of jazz. He changed the style of jazz drumming by moving the basic cymbal pattern from the hi-hat cymbal to the ride cymbal, thus freeing his left hand to play fills on the snare drum. He then played syncopated accents on the bass drum instead of delineating each beat of the meter, thereby creating a flow to the music. Prior to Clarke’s innovations, jazz drummers played the bass drum on every beat so the other musicians could hear where the time was in each measure. Clarke’s random bass drum bombs, as they were called, and his ride cymbal rhythm, obscured the sense of barline and created a feeling of longer phrases stretching over several measures at a time. Clarke was able to create this flow by keeping a sense of time in his head when he played. In an interview with Helen Oakley Dance in 1977, Clarke remarked that musicians in his band complained and said, “Kenny keeps breakin’ up the time. Why doesn’t he keep four beats on the bass drum?” Clarke replied, “If you’re playing, the tempo should be in your head. Don’t depend on me. Depend on yourself. Because if you’re playing music, the tempo you’re playing in is in your head.”³

Steve Reich has often said he admired Clarke’s time feel and that he tries to achieve this same feel in his own music. In a conversation with me in 2003, Reich explained, “That feeling of time and time sense is getting it ‘right’ which might show up on an oscilloscope as ‘slightly wrong.’ A lot of very good players lack that magic because they are very concentrated on being right … some people have that quality of magic in their playing and others just don’t – but you know it when you hear it.”

Time feel is an essential ingredient in the performance and appreciation of jazz as well as in many kinds of pulse-based music. A performer’s ability to control attack placement in relation to a steady pulse becomes an expressive element in the music. When an ensemble has a unified time feel, it can create the lilting sense of “magic time” that Reich admires.

Repetition

In her book titled On Repeat: How Music Plays the Mind, Elizabeth Hellmuth Margulis writes, “In a famous essay, [Gilles] Deleuze references [David] Hume to the effect that ‘repetition changes nothing in the object repeated, but does change something in the mind which contemplates it.’ Deleuze identifies repetition as a phenomenon well-suited to exposing the elements that the mechanisms of perception bring to an experience over and above the elements that literally exist in the world.” Margulis continues, “Since two iterations are never precisely repetitions in their deepest essence – they’re composed of different atoms or occur at different time points – it is perception that abstracts both a relationship of shared identity and a relationship of difference. At a minimum, a repeated element will sound different from its initial presentation by virtue of coming later and having been heard before.”⁴

In the music of Adams, Glass, Reich, Riley, and Rzewski, repeating the same pattern for a long period of time can be perceptually interesting for the player, but it is also fraught with challenges. For the performer, nothing should change in the pattern that is to be repeated because it is the musician’s job to present a soundscape that allows something to change in the mind of the listener who contemplates the music. The performer is the facilitator, and as such must find a way to control muscle memory in order to articulate each repetition similarly, yet keep mental control.

In West African drumming, musicians who play the drum and bell parts that form the accompaniment to the master drum, singers, and dancers repeat their patterns for extremely long stretches of time. At a harvest festival celebration in a Ga village near Accra during the summer I spent in Ghana in 1971, I witnessed two bell players playing the same interlocking pattern throughout the night (although the two original players were relieved by others as the hours passed). But even under normal performance circumstances, a Ghanaian drummer might repeat the same pattern for an hour or more.

Abraham Adzenyah instructs his students to move their bodies in order to transmit the right feeling while playing repeated patterns on the drums. He also teaches them to hear checkpoints in the music so they know how to recover if their mental control falters and they stray from their patterns. These same instructions apply to the performance of repetitive, pulse-based Western music. The performers must first repeat the assigned pattern until muscle memory begins to take over. Body movements, in this case, cannot be excessive, but subtle motions can be utilized to facilitate muscle memory. Once the pattern is secure, the players can expand their sonic perception to hear connections between the patterns they are playing and other patterns that they hear; in particular, the players can locate points of coincidence in the composite pattern relationships. These checkpoints add security by indicating where to get back on track if a mistake is made.

Early minimalist music moves from event to event, so performers are relieved of the necessity of counting numbers of repeats; they have to know the overall structure of the piece and then pay attention as the process unfolds. This frees their minds from the restriction of keeping track of the number of repeats and allows them to become immersed in the musical details of the performance style. The performers learn to play their patterns as evenly as possible to enable listeners to hear a seemingly unchanging stasis in different ways. In other words, the listeners’ minds are allowed, and in fact encouraged, to hear something change.

Endurance

This repetition of the same pattern for several minutes creates an endurance problem for the performers. String players are accustomed to playing almost continuously throughout an hour-long symphony, chamber piece, or solo work. However, in this conventional kind of music they vary their musical patterns, dynamics, articulation, and other musical expressions. Shem Guibbory, violinist in the early rehearsals and performances of Music for 18 Musicians, said:

Audrey Wright, a violinist commenting on her preparation to play Music for 18 Musicians for the first time, said she did not realize “how physically and psychologically demanding it would be, nor how rewarding. With my violinist mentality, I originally scanned the score thinking, ‘OK, this is long. But playing some sustained fourths here, a few pages of repeated eighth notes there, and catching a bass clarinet cue once in a while is a piece of cake compared to a Mahler symphony.’ … As it turns out, it is not a piece of cake.” Wright continued:

Other string players, wind players, and vocalists have commented on the need to prepare for the physical demands of Reich’s music. Even the seemingly simple maraca parts in Reich’s compositions create endurance issues. In Four Organs, Music for 18 Musicians, and Tehillim, the maraca player is the pulse-keeper in the ensemble for long periods of time and must play strongly with the maracas placed high in the air in order to produce the desired volume. Gary Schall, who was the regular maraca player in the Reich ensemble, said that prior to performances of these pieces he practiced maracas thirty minutes a day for a month in order to develop stamina.

In playing Reich’s music and West African drumming, I use something I call energy-shifting, a technique I developed from practicing yoga postures and doing breathing exercises. I find that it is necessary to have a certain amount of tension while playing in order to produce a strong, centered sound, especially when I have to generate this sound with consistency over long periods of time. In order to prevent tension buildup, I keep up the necessary energy by shifting the tension around in my body. For example, if I feel tension mounting in my wrists, I concentrate on releasing the tightness by focusing my breathing in that area and allowing my arms or another part of my body to absorb the tension. By practicing this technique, I developed the ability to energy-shift quickly and easily.

Musicians often make small changes in technique to help their endurance. Stephanie Chua, a pianist in a 2018 performance of Reich’s Six Pianos in Toronto, told me that to help with relaxation she uses fingerings that force her to change positions in the middle of a pattern. Sometimes it is a very small physical change like Stephanie made that enables performers to endure long periods of repetition of the same pattern and thus heighten their ability to concentrate on musicality.

Concentration

Repeating a pattern for long periods of time requires physical adaptation, but the hypnotic state it induces in listeners creates a concentration problem for performers. I often find myself succumbing to a trance-like state while playing In C, Music for 18 Musicians, Drumming, and other pulse-based compositions. In order to enjoy this sensation and still maintain my sense of equilibrium, I use techniques I learned while playing in West African drum ensembles. First, I determine the placement of the general “one” of the ensemble or the beginning of the most audible phrase, and I place my part accordingly. I then relate the pattern I am playing to other patterns that are easily discernible. Finally, I find checkpoints in the music that are easy to detect so I can return to my proper rhythmic alignment if I happen to stray off course. When these elements are in place, I feel secure enough to allow my mind to enjoy the mesmerizing aspects of the music. I rely on my muscle memory to perform the many repetitions of my pattern, but I periodically check in mentally to make sure I am playing correctly.

Commenting on his concerns in performing Reich’s Six Pianos, Gregory Oh said, “The challenge for the performer is managing two states of being: one must be conscious of the notes, the precision of the notes and the time and pacing and fitting in, while simultaneously suppressing any thoughts of control or ego. Rather than ‘Just Do It,’ it’s more like ‘Just Let It Happen.’ The biggest danger in this piece is getting in one’s own way.”⁷

To keep from “getting in one’s own way,” some performers may find that practices such as yoga, tai chi, or meditation are helpful in developing the ability to concentrate. These disciplines train their practitioners to block out unnecessary thoughts and focus on breathing and staying grounded. However, it is important for the performers to maintain awareness of the form of the compositions. It is incumbent on the musicians to hear and understand the progression of events in order to move from one section to the next in proper sequence. All these factors are important in maintaining concentration when performing music that frequently shifts the player’s sense of perception.

Metric/Perceptual Ambiguity

In his performing directions that accompany the score to In C, Riley states, “The group should aim to merge into a unison at least once or twice during the performance. At the same time, if the players seem to be consistently too much in the same alignment of a pattern, they should try shifting their alignment by an eighth note or quarter note with what’s going on in the rest of the ensemble.” This shift of eighth-note alignment occurs early in the piece in modules 4 and 5. Each of the modules consists of three ascending eighth notes, E, F, G, but module 4 is an eighth rest followed by the three eighth notes and module 5 begins with the three eighth notes followed by an eighth rest. In his instructions, Riley further encourages this ambiguity when he says, “Each pattern can be played in unison or canonically in any alignment with itself or with its neighboring patterns. One of the joys of IN C is the interaction of the players in polyrhythmic combinations that spontaneously arise between patterns. Some quite fantastic shapes will arise and disintegrate as the group moves through the piece when it is properly played.” Riley’s intentional pulse displacement allows the listener (and performer) to hear the same composite patterns differently.

A rhythm by itself can be intriguing, but when it is heard against a pulse or another rhythm it can become metrically ambiguous. The metric implications of standard notation in Western music, such as strong impulse at the beginning of a measure, are replaced by options for the listener. Rhythmic groupings and pulse placement are created by the listener, allowing the same music to be heard differently as the listener’s mind changes its perceptual vantage point. David Locke, author of the chapter on rhythm in West African music in this volume, coined the term Gestalt flip to describe the sensation of hearing a rhythmic pattern one way in relation to a pulse and then changing perception of the same pattern to hear it a new way in relation to a different pulse.

Reich’s use of canons played on like instruments creating interlocking composite patterns makes it problematic for the musicians to tell who is playing the parts that they hear. This is especially true in Drumming when players facing each other across bongo drums see their sticks moving in conjunction with their playing partner but can have trouble determining whose sticks are making which sounds. Leslie Tilley, in her chapter on Balinese gamelan in this book, describes a similar effect in Balinese gamelan playing when she writes, “Many non-Balinese musicians are first drawn to Balinese music … because of a distinctly Balinese brand of interlocking, where a melody or rhythm is seamlessly shared between two or more performers such that the resultant composite is a single, smooth strand of music. One often cannot discern which musician has performed which note in an interlocking passage; the perceptual effect is of a group of musicians each playing the entire passage in perfect synchrony, much faster than humanly possible.”⁸

In the amadinda xylophone music of Uganda, musicians play separate patterns on either side of a twelve-note xylophone, alternating their attacks. To achieve the virtuosity required, both players feel their pulses as downbeats. In Music for 18 Musicians, an on-beat and off-beat pulse is played for most of the composition by pianos, marimbas, and xylophones. Philip Bush, who played in the Reich ensemble for many performances of Music for 18 Musicians, was the pianist who played the off-beat pulse. He told me that in order to play the pulse consistently, he had to (Gestalt) flip his sense of pulse so that he felt that his off-beat pulse was on the beat. Consequently, Bush heard the entire composition in a parallel universe from most other people.

In Indian music, rhythmic sequences are overlaid across tala cycles and inserted within tala cycles. The juxtaposition of the irregularity of these rhythms against the regularity of a cycle creates metric ambiguity. Glass uses the process of additive rhythms to create an unexpected conflict with the usual regularity of expectation in Western music. The metric intrigue in all these genres creates perceptual ambiguity for the listener that is first unsettling, but then becomes aesthetically satisfying.

Rhythmic Expressivity

In his book Repeating Ourselves: American Minimal Music as Cultural Practice, Robert Fink writes that in the early days of minimalism, critics such as Donal Henahan and Harold Schonberg (in the New York Times), Samuel Lipman, and Christopher Lasch described the music as a kind of social pathology.⁹ Pwyll ap Siôn, in his article “Moving Forward, Looking Back,” notes that “German critics also seized upon other aspects [of Reich’s music], including what they identified as the music’s mimetic representation of industrial production and machine-like processes. Worse still, some went further and accused Reich of creating a form of musical fascism that ‘suppressed social criticism and manipulated the listener’s emotions.’”¹⁰

One of the implications of this criticism was that minimalist composers were dictating every aspect of the music and that the performers had no room for expression or creativity. The reality within the composer-based ensembles of Riley, Glass, Reich, and others was that the performers not only had input into the composition of the music, but they also established the expressiveness and feel of this pulse-based music.

Riley spent much of his early career as a pianist playing ragtime, blues, and jazz. He also visited Morocco where he heard traditional maqamat music. All these types of music use improvisation and Riley’s music allows for the sense of improvisation that characterizes these various musical styles. In six paragraphs of instructions that accompany In C, Riley outlines ways the performers can improvise their way through the composition.

Glass’s early ensemble was made up primarily of electric keyboards, woodwinds, strings, and voices. The nature of the attack of these instruments and voices gives Glass’s music a less precise rhythmic feel, but still imbues the music with a time feel that is different from that of more traditional classical music.

Reich taught Drumming to the percussionists in his ensemble by rote, demonstrating each part until the players learned the patterns and where the patterns fit into the ensemble. Once the percussionists, all of whom were classically trained, learned the music and had an idea of Reich’s stylistic approach, they developed a rhythmic nuance, time feel, and sense of centered sound that became the identifying characteristic of Reich’s music. This rhythmic expression had its basis in classical training but added elements from other pulse-based music including music of Africa, India, Indonesia, and jazz. As Reich’s compositional output increased to include strings, winds, and additional vocalists, the other musicians imitated the percussionists’ sound and feel. A detailed description of the early rehearsals and development of the performance style for Drumming and other early Reich works can be found in my book Performance Practice in the Music of Steve Reich.¹¹

Attack placement is an essential element in giving feeling to any music, but it is of critical importance in playing rhythms expressively in pulse-based music. Musicians often talk about playing behind the beat, in the middle of the beat, or ahead of the beat, and some musicians instinctively place their attacks in one of these ways, giving their performance a certain style or character. Other musicians are able to alter their attack placements according to the musical expression they wish to convey. Regardless of which of these ways the interpretation of attack is deployed by musicians who are performing pulse-based music, the aim should be to have a unified sense of feel from the entire ensemble.

Sense of Ensemble

When the minimalist composers’ ensembles were created, they rehearsed compositions for long periods of time before the premieres. Rehearsals of Drumming, for example, took place weekly for nearly a year prior to the first performances. The lengthy rehearsal periods gave the musicians an opportunity to develop the techniques that resulted in a distinctive sound with a unified time feel in the music. The structure of the music and the inner communication that was required to perform it helped create a bonding experience in the ensembles and create a sense of purpose and community. Steven Schick, in his book The Percussionist’s Art, says:

Robert Carl describes the sense of ensemble in Terry Riley’s composition this way:

The ensembles that were created by Riley, Reich, and Glass in the 1960s and 1970s brought a new approach to the performance of contemporary music and resulted in tightly knit groups that were well rehearsed. The composers and musicians combined to develop a special sound and feel to the compositions that allowed the musicians to have a sense of ownership as important contributors to the final product. The sense of community that Schick and Carl mention helped the performers and composers establish the techniques that formed the basis of the new virtuosity.

Conclusion

The techniques I have described are not, individually, exclusive to the performance of pulse-based music, but collectively they form a performance practice that requires a different kind of virtuosity than has been a part of the traditional training of a Western classical musician. Now, more than fifty years since the premiere of In C, second and third generations of musicians are assimilating these techniques into their musical development, and gradually this different kind of virtuosity will become standard practice for musicians who have an interest in rhythm and a desire to play pulse-based music.