Minimalist compositions thwart most attempts at analysis given their remarkable simplicity; their structure is often deliberately obvious. The experience of a minimalist piece, however, is anything but simple. These compositions encourage the listener to ignore the past and the future, memory and expectation, and explore an extended present. This temporal experience, which Jonathan Kramer describes as vertical time, can create a sense of depth and complexity that is difficult to derive from the score. Previous scholarship has defined this temporality, but the concept of vertical time has only ever been applied in a general way. If an analytical approach to minimalism is to include temporality, which influences greatly the experience of the music, the musical elements that create vertical time must be defined. Too often vertical time is seen merely as the absence of linear elements instead of something that is actively cultivated in a variety of ways. By analysing the creation of vertical time, and by allowing for the inclusion of some linearity, it is possible to explore the temporality of minimalism in a meaningful and specific way. The potential of this analytical technique will be demonstrated through a study of Tom Johnson's An Hour for Piano (1971).
Early minimalist music is simple. It lacks the network of hierarchical interactions and complexity found in even basic common practice music. One glance at a score reveals all there is to know about how a piece is organised. In short, at least from a traditional analytical viewpoint, it is obvious and boring. Yet any admirer of the genre knows that it is precisely this lack of surface-level activity that creates an engaging listening experience - what initially seems static and monotonous becomes interesting and complex. Jonathan Kramer describes such an experience at a performance of Satie’s Vexations:
When I first entered the concert, I listened linearly. But I soon exhausted the information content of the work. It became totally redundant. For a brief period I felt myself getting bored, becoming imprisoned by a hopelessly repetitious piece. Time was getting slower and slower, threatening to stop.
But then I found myself moving into a different listening mode. I was entering the vertical time of the piece. My present expanded as I forgot about the music’s past and future. I was no longer bored. And I was no longer frustrated, because I had given up expecting. I had left behind my habits of teleological listening. I found myself fascinated with what I was hearing. The music was not simply a context for meditation, introspection, or daydreaming. I was listening. (1988, p.379)
Minimalist composers were well aware of this listening mode, and actively cultivated it in their work. As Philip Glass wrote concerning his Music in Twelve Parts (1971-1974):
The music is placed outside the usual time-scale substituting a non-narrative and extended time-sense in its place.…When it becomes apparent that nothing “happens” in the usual sense, but that, instead, the gradual accretion of musical material can and does serve as the basis of the listener's attention, then he can perhaps discover another mode of listening - one in which neither memory nor anticipation…have a place in sustaining the texture, quality or reality of the musical experience (Mertens, 1983, p.79).
In a sense, these early minimalist compositions are boring for all the right reasons. Their simplicity allows the listener to engage with the music in a way that is foreign to most Western music, and it is in this type of listening where the music lies. Unfortunately, while listeners and composers alike understand this, theorists have little tangible material on which to pin an analysis. This listening experience may create a sense of depth, but the notes on the page do not.
Ian Quinn encapsulates the problem. He writes, “Traditionally, analysis aims to reduce the information content of a piece - productively, and however provisionally temporarily, and contingently - by parsing it relative to some well-understood system of formal conceptual categories. Process music comes to the table already digested” (2006, p.292-293). Jonathan Kramer is even more blunt, writing that this music can “scarcely be analysed, in the usual sense of the term.…It is essentially pointless to explicate” (1988, p.388).
Still, many scholars have pushed forward with standard analyses, and despite the apparent difficulties, they have yielded some effective results. Steve Reich's early phasing pieces have notably been the subject of analysis by Richard Cohn (1992), whose beat-class approach to rhythmic structures has also been adopted by Roberto Antonio Saltini (1993) and John Roeder (2003). Paul Epstein also produced one of the first meaningful analyses of the events between the stages of phasing in his article “Pattern Structure and Process in Steve Reich's Piano Phase” (1986).
On the other hand, others see these formalist analyses as empty and incomplete when trying to describe this music. Jonathan Bernard writes that minimalist music demands “that the prospective investigator be willing, not to abandon quantitatively oriented methods, but to de-emphasise them somewhat in favor of taking seriously the connections between minimal music and minimal art and treating them, rather than simply as an avenue for metaphorical comparison, as a way of 'seeing' the music, or as if one could see it” (1995, p.266). Yet Robert Fink questions any such connection to art when he writes that “the ascetic formalism of minimalist music simply bounces back off the formalist asceticism of 1960s visual art, highly-polished mirrors reflecting each other's cultural emptiness in infinite analytical regress” (2005, p.18). Erik Christensen employs a spatial approach without explicit connection to the visual arts. He argues that music, through the creation of time in combination with particular listening dimensions, establishes a virtual timespace (1996). Unfortunately, his analyses have been criticised for vagueness and a lack of depth (Bigand, 1997).
Despite the variety of attempted analytical approaches, few have considered with specificity what Kramer and Glass describe as an essential aspect of this alternative mode of listening - time. Both Vexations and Music in Twelve Parts seem to draw the listener into the present moment; memory and expectation become largely irrelevant given such unchanging or slowly changing information. In the face of such present-orientation, the traditional theoretical approach is weakened given its emphasis on discovering structural underpinnings and on the development of a piece over time. Likewise, while those authors who prefer to draw connections to a spatial dimension may avoid such a pitfall, their analyses become more abstract and less particular to any given piece. If an analysis is to effectively describe the listening experience created by a particular minimalist composition, it must therefore consider temporality in some concrete way.
Nearly all Western music exists in linear time, which Kramer defines as “the temporal continuum characterised by [the] principle of composition and of listening under which events are understood as outgrowths or consequences of earlier events” (1988, p.453). This musical temporality is closely linked to Western culture, where the idea of time being linear is the dominant view (Slife, 1993). Moreover, the concepts of ‘before’ and ‘after’, which are inherent to memory and the mind (Fraser, 1975), are so powerful that some have argued that time itself should be considered anisotropic (Denbigh, 1972). Still, it is not only the direction of time that is important in Kramer's definition, but also teleology - that what is being heard in the present is somewhat defined by events that occurred previously, making memory and anticipation integral to linear time. This is the temporality that became the de facto perspective in Western music for listening, performance, analysis, and composition.
By the early twentieth century, this temporality had become so entrenched that composers had to actively work against it if they were to present any alternative. One of the first to recognise such attempts was Leonard Meyer, who described the music of Stockhausen and Cage as “directionless, unkinetic” and “anti-teleological” (1967, p.72). He wrote that “because it presents a succession rather than a progression of events, this art is essentially static. There are no points of culmination or focus. All events are equally important and time, as we ordinarily conceive it, dissolves. There is only duration” (1967, p.81).
Jan Pasler also recognised this development, though without Meyer's stern critiques. By contrasting it with the narrative of common practice music, she developed several classifications:
Antinarratives are works which rely on the listener's expectation of narrative, but frustrate it through continual interruption of a work's temporal processes.…Nonnarratives are works that may use elements of narrative but without allowing them to function as they would in a narrative.…A third type of innovation constitutes works without narrativity, those that shun any organizing principle, whether an overall structure or preordained syntax, and thereby try to erase the role of memory [nonnarrativity] (1989, pp.244-248).
These classifications demonstrate how composers thwarted linear time in vastly different ways. She argues that Stravinsky and Stockhausen trended more toward antinarratives, which Kramer refers to as “multiply-directed time” or “moment time” (1988, p.50-52), while many Cage works may best be associated with nonnarrativity due to chance elements. Minimalist works fall under her second classification of nonnarratives, yet due to their particularly stark contrast with linearity, Kramer prefers the term vertical time - “a single present stretched out into an enormous duration, a potentially infinite 'now' that nonetheless feels like an instant” (1988, p.55).
This concept of vertical time works well with minimalism because it easily articulates the differences with linear time, and helps explain the dissonance that can occur between the musical time of minimalism and objective clock time. Still, the term remains vague as Kramer applied it equally to minimalist music and to music by Stockhausen, Cage, Wolff and others. While the elongation of the present seems to be the common thread (1988, p.387), by his own accounts the perception of elapsed clock time can vary wildly (1988, pp.379-380). To better clarify the term with respect to minimalism, it is important to understand generally the effect that minimalist music has on temporal perception.
Sense and memory are highly receptive, and every impression they absorb slows down the time we are experiencingIn minimalist music, the extremely gradual rate of change draws attention to minute details, which in turn affects the perception of time. Stefan Klein describes the effect, “Once we recognise the value of a moment we are experiencing, we try to savor every aspect of it. Sense and memory are highly receptive, and every impression they absorb slows down the time we are experiencing. The effect is even more dramatic when we focus on the very smallest changes” (2006, pp.68-69). Michael Flaherty, through studies on altered experience of time has derived five sequential factors that produce this effect, which he refers to as protracted duration (italics are the author’s own):
First, there is a context that is characterized by extreme circumstances. The situations that engender protracted duration fall into one of two categories: those with unusually high levels of overt activity, and those with unusually low levels of overt activity.…Second, when people are confronted with extreme circumstances, they experience an increased emotional concern for understanding the nature of their situation.…In other words, the individual cares deeply about the situation and his or her place in it. Third, the shock of transition to extreme circumstances heightens cognitive involvement with self and situation.…Fourth, heightened perspicacity towards one's own subjectivity and surroundings generates stimulus complexity even if the setting is not characterized by a wealth of overt activity.…Fifth, increased stimulus complexity fills standard units of temporality with a density of experience that far surpasses their normal volume of sensations (1999, pp.91-95).
In the context of these factors, it is clear how minimalism is able to create a sense of protracted duration. The extremely low level of overt activity encourages a heightened awareness, particularly of small details, which in turn creates a sense of time lengthening.
This interpretation of vertical time in minimalism, however, seems to contradict a common experience - that a composition feels as though it was far shorter than the clock indicates, not longer, as protracted duration might imply. Flaherty refers to this effect as temporal compression, and notes that it is purely “a facet of retrospection. In contrast with protracted duration and synchronicity, both of which are primarily phenomena of the present, temporal compression is uniquely associated with the past” (1999, p.104). Anytime one makes an evaluation of how much time has elapsed, it requires recollection. It is not surprising, given the reduced activity within the time span of a minimalist composition that it feels as though little time has actually passed when evaluated afterwards, even when the experience of the present feels elongated. Other music that creates a sense of vertical time does not necessarily produce this same combination of protracted duration and temporal compression, making it a distinct characteristic of minimalism.
The Creation of Vertical Time
The concept of vertical time does much to illuminate the temporality of early minimalist works, but the problem of analysis still remains. All too often vertical time is seen as merely the absence of linear elements. Kramer mentions the lack of phrasing, or at least phrasing that refuses to organise into larger hierarchical levels as a sufficient cause of vertical time (1988, 55), while Iannis Xenakis and Roberta Brown simply discuss the absence of change as sufficient cause “[i]f events were absolutely smooth without beginning or end, and even without modification or 'perceptible' internal roughness, time would likewise find itself abolished” (1989, p.87). It is true that minimalist pieces often exclude musical elements of common practice music and linear time, but the creation of vertical time is not accomplished by omission only - it is created actively by elements such as repetition, pulse, and audible structure.
Repetition is “perhaps the most stereotypical aspect of minimalist music, the tendency that audiences superficially associate with its stuck-in-the-groove quality” (Gann, 2001). One reason so many minimalist compositions employ repetition is that it is one of the most powerful tools for moving beyond initial meaning in favor of creating vertical time. As the following examples show, repetition is capable of drastically altering the context in which the information is received. Initially, the receiver takes a particular meaning from the information, but as the number of repetitions increase, any linear implications become irrelevant. It is no longer germane to anticipate what the next phrase may be, and thus the context is reoriented from the linear to the vertical, forcing the receiver to consider the information differently.
Several pieces demonstrate this point through their use of language to create musical effect, such Steve Reich's Come Out (1966) and David Borden's Music (1972). Come Out is based on a recording of Daniel Hamm, who had been beaten in a Harlem police precinct house. The piece begins with him saying “I had to like open the bruise up and let some of the bruise blood come out to show them” (Reich, 2002a, p.22). These last five words became the ostinato for Reich's phasing and canonic processes, and through repetition, these words lose their meaning as language is stripped down to rhythm and sound. A similar transformation takes place with Borden's Music. Here Borden samples a recording of the song “Music! Music! Music!” (1949) written by Stephen Weiss and Bernie Baum. The song begins “Put another nickel in / In the Nickelodeon / All I want is loving you / And music, music, music.” Borden loops this last word continuously for the duration of the 13-minute piece, and it becomes the rhythmic and harmonic basis of the composition.
The effect that both composers rely on is known as semantic satiation. First noted over a century ago (Severance and Washburn, 1907), semantic satiation is the effect of a word losing its meaning after repeatedly speaking, writing, or hearing it. More recently, Kounios, Kotz, and Holcomb have demonstrated that the effect is semantic in nature, and not rooted in sensory or perceptual loss (2000, p.1377). As the meaning of the words diminishes in Come Out and Music, the listener is able to perceive other characteristics, such as rhythm and pitch, enabling Reich and Borden to use them as musical elements.
Pulse is another important generator of vertical time, and it is particularly relevant to minimalist piano music due to the instrument's inability to sustain pitches for a significant length of time. Pulse is found in all common practice music as it is necessary for meter, and it need not be overtly present in the music as the listener can easily derive it from the piece's rhythmic organisation. What distinguishes pulse in minimalism is that composers articulate it with a steady, often invariant rhythm, and it does not act as a baseline for higher levels of organisation. Moreover, pulses are rarely grouped beyond the metrical level, and even then metrical ambiguities are common. With this musical element brought to the forefront and limited to its most basic uses, it helps create a sense of vertical time. “Pulse and goal-directed movement evoke two kinds of temporal experience which are qualitatively different.…Continuous pulse has no definite beginning and end” (Christensen, 1996, p.14,50).
A third source of vertical time in minimalism is audible structure, which is perhaps the most important factor in vertical time creation. Repetition and pulse are invariant, so to maintain vertical time while still allowing for some change (which can imply linear time) requires an audible structure. Reich's statement that “once the process is set up and loaded it runs by itself” (2002b, p.34) is a natural fit for audible structure, but other, less rigorous pieces still have this characteristic. Tom Johnson describes audible structure as being where “the pitches, rhythms, and colours presented in the first few minutes usually define a specific kind of music, and the remainder of the piece will not depart very far from that” (1989). A piece does not need to be completely predictable so much as the material must lie “right on the surface” (Gann, 2001). This level of transparency allows the listener to disregard memory and anticipation and focus on the present, where, as Glass wrote, “the attention of [the] audience is mainly drawn to the sound itself” (Mertens, 1983, p.71).
Vertical time is created by more than the absence of musical elements that evoke memory and expectation, such as melodic and harmonic relationships within a tonal system or phrasing and periodicity. Musical characteristics that actively thwart linearity - repetition, pulse, and audible structure - are also a factor. The nature of vertical time is also not uniform; it can and does vary with and within each composition. Moreover, compositions that are devoid of all linear elements are relatively rare. La Monte Young’s Composition 1960 #7 may be one such example, but more often minimalist pieces do have elements of linearity. For example, Philip Glass does not set up a completely predictable process in Two Pages (1968), despite the rigor of the additive technique, and Steve Reich moves through three harmonic areas in the otherwise predictable Piano Phase (Potter, 2000, p.29). The experience of vertical time remains dominant in these pieces, but it is also coloured by linear elements, providing the listener with a rich experience that is not readily apparent from the score.
At first, it may seem to be an impossibility that vertical time and linear time can coexist, but these terms are rather fluid. J. T. Fraser, one of the foremost scholars in the area of time, writes that “time refuses to conform to the law of contradiction” (1975, p.45). This in turn led Kramer to conclude that “these categories of time…readily overlap; and, because time is exempt from the law of contradiction, they can coexist” (1988, p.8). This may seem counterintuitive, but by allowing for a more flexible interpretation and overlapping of these seemingly discrete categories of time, a more meaningful and concrete analysis of minimalist music becomes possible. Tom Johnson’s An Hour for Piano (1971) makes a remarkable case study for such an analysis as the experience of this piece is deeply rooted in temporality - from the title and intriguing program notes to the way in which linear elements colour the perception of vertical time.
An Hour for Piano
Tom Johnson has said that he considers An Hour for Piano, which began as improvisational accompaniment for a modern dance class at New York University in 1967, to be the first minimalist work that he composed.
I would sometimes write down a few measures of some particular kind of music I had found that seemed worth remembering. I had a drawer where I would keep these sketches. Many months after this started I went through the drawer and really liked seven…basic textures, so I took them home and started writing them out in detail, expanding them and composing transitions from one texture to another. All of them were comfortable at one tempo, so I could fit them all together.…I was still never thinking about what might come before and what might come after. These were just timeless textures.
When I had quite a mess of isolated sections, I began connecting them, which often required writing a few additional pages. The pages and the sections began piling up, and it was clear that this might go on for a while, so as a way of setting a goal I came up with the title “An Hour for Piano.” Gradually I linked all of these two-page and four-page and twelve-page sections together into one piece without interruption, but there was still no before and after. All the sections were just floating in the same time frame. The only reason the piece starts the way it does is that I never found anything that seemed to lead to that (Johnson, 2010).
Rather than being highly regular, these textures, as Johnson calls them, appear in no particular order, are freely altered, and even combined with one another. Exact repetitions are also few, even though each texture is short, creating an ever-changing landscape for the listener. Moreover, elements of linearity are also present, from transitions that anticipate the arrival of a new section to the creation and resolution of melodic tension.
An Hour for Piano is based on six primary textures (figures 1-6), as well as an additional transitional texture (figure 7) that appears frequently in loose combination with other textures. All musical material in the piece is derived from these textures, which are essentially one-bar ideas. According to the score, the entire piece is played at a strict tempo of 59.225 beats per minute to the quarter note in order to achieve a performance that lasts for exactly one hour. Dynamics are to be “extremely subtle,” ranging “between mezzo forte and forte” (Johnson, 1971) and while the meter is not notated, it does not often deviate from common time.
Rather than employing any specific compositional techniques to help this material span an hour, Tom Johnson composed with “intuition” to determine the placement of, variations within, and transitions between these textures (2010). The piece is roughly divided into seventeen short sections, each employing only one of the primary textures. Toward the end of each of these sections, the texture is often broken down slowly until only a single note or pattern is left, while the next texture is likewise built up. This makes the beginnings and endings of sections impossible to define with complete accuracy. The damper pedal also contributes to a blurring between sections, as it is held down for the entire piece, and Johnson's transitional writing is designed to accommodate this effect.
Sources of Vertical Time
Johnson's organic compositional approach does encourage a somewhat different listening experience than other minimalist compositions. In other pieces, much of the purpose of creating vertical time is to allow for listener freedom. Large numbers of exact repetitions give the listener ample opportunity to explore the sounds as there are relatively long stretches of time where no new information is presented. The difference with repetitions in An Hour for Piano is that minute changes that take place within a thematic section continuously offer the listener something new, while not changing the underlying material. Repetition remains at the forefront of the listening experience and acts to create vertical time even with such variation. This does not mean the listener is so drawn to the immediacy of the pitches that exploration of particular notes, intervals, or overtones is not possible, but rather that it is not explicitly necessary. In pieces where there are many exact repetitions, such exploration is not only necessary but also the intended focal point; in An Hour for Piano it is only one possibility.
Consider for example the third texture as presented in figure 3. This texture could be expressed as only bar 122 - the quarter-note pulse on G in the right hand, the adding of Eb and Db to the texture in quavers, the low D in the left hand on downbeats, and the left hand chord presented on the second and fourth beats of each bar. This basic material is invariant from bars 121 to 136 and lasts for one minute and eight seconds. However, the pattern is never exact. The low D may be present on successive downbeats, or may have as many as two bars between iterations. There are also usually no more than two lone crotchets in the right hand before the addition of either Eb, Db, or later Cb. But there is no pattern to how frequently or in what order those quavers are included; the listener is constantly presented with slightly different information even though the music never significantly changes. Once the collection of pitches is established with the listener, the duration of this passage, combined with the exact repetition of G5 every beat and the left hand chord every two beats, severely limits the usefulness of memory or expectation, drawing attention to the present. The sense of vertical remains, but it is coloured by subtle, unpredictable changes.
A strong sense of pulse, which is another contributor to vertical time, is present in this piece, reinforcing the sense of common time with the content of each texture. There is rarely any organisation above the metrical level into either phrases or hypermetric groupings, and the sense of pulse remains steady throughout. Notice that each texture, with the exception of the sixth, emphasises G5 at either the crotchet or minim level, while the sixth texture has a clear C4 pulse on the first and third beats of each bar. This emphasis of pulse acts like the ticking of a clock and even undermines to an extent the organisation at the metrical level.
Finally, the audible structure of An Hour for Piano is another significant contributor to vertical time. As was previously mentioned, Johnson’s concept of audible structure concerns itself less with complete predictability and more with generally unchanging content. Here, the seven textures, even while distinct from one another, are similar in pitch content, rhythm, and register. Thus, when Johnson varies these textures, dismantles or builds them up in transition, and occasionally combines them with one another, the linearity of these compositional devices is diminished. Moreover, these changes do not usually carry with them linear implications; Johnson wanders from one “timeless texture” to another without any seeming to imply that another should necessarily follow.
To perpetuate the sense of vertical time, Johnson reserves bigger changes for gradual transitions. These transitions tend to be long as Johnson mirrors the slowly fading sounds of the previous texture, sustained with the ever-held damper pedal, slowly reducing the musical material. Not surprisingly, the length of transitions is often over a minute, with one of the longest transitions, centred on bars 632-662 (figure 8), lasting well over two minutes.
This transition has no sense of regularity or exact repetition. The accompaniment of the fourth texture is reduced to a descending third, occurring sporadically, while the melodic material merges with the transitional texture. The right hand does not seem to make a complete transition as the harmonic perfect fourth of the fourth texture, E5-A5, appears as late as bar 653. This combination is maintained until the third texture's left hand chord is brought in at bar 660; the right hand only later takes on characteristics of the third texture. Most transitions in the piece tend to function in this manner, and there are even a few deceptive transitions within thematic sections where a texture is broken down only to be built back up and continue. Still, the effect of these transitions is not propulsion forward into a new idea, but rather an emphasis on the present due to the slow rate of change and the frequent inclusion of the transitional texture.
Melodic Sources of Linear Time
As part of his intuitive compositional process, Tom Johnson also opened up the possibility for instances of linear time. While Johnson retained an overall sense of verticality, he did allow events to take place that at times surprise, create tension and resolution, and alter the pacing of the piece. Not unexpectedly, the effect of this combination of linear and vertical elements varies considerably, from becoming dominantly linear on a local level to altering subtly the sense of vertical time on the macro level. In some ways, this interaction is not wholly unlike other early minimalist pieces, inasmuch as linear time is considerably less prominent at larger hierarchical levels. However, An Hour for Piano differs in that these linear elements are more prominent and not byproducts of a rigorous compositional process.
Melodically, the first and second textures naturally carry with them the potential for more linearity than the others since the end of each one-bar and two-beat unit, respectively, has an unresolved Ab. That is, the basic unit of each of these textures spirals into its next iteration. In most cases, this sense of propulsion is negated by repetition, but there are some instances where this characteristic is exploited to create a sense of tension and resolution. For instance, the first texture, which opens the piece, is not heard again for over 33 minutes.11 Its arrival is therefore aurally distinct, and Johnson adds to the effect by allowing for linear time on the local level.
Figure 9 marks the return of the first texture, and there are several ways Johnson creates tension in these six bars. First, his use of the octave Gs in bar 529, which do not proceed to the expected Ab and Bb, creates a sense of anticipation within the listener. The same thing happens again in the following bar, but the tension is even further heightened by an additional beat, delaying the downbeat and complete return even further. There is also an omission of a semiquaver rest at the start of beat five in this bar, further propelling the listener to the downbeat of bar 531. Just a few bars later, Johnson creates another source of tension by a different alteration to the texture. In bar 535, the octave Gs that are expected to fall on the downbeat are replaced with a Bb as the lower note. This unexpected substitution creates melodic tension as the line from the preceding bar presents G-Ab-Bb-Ab-Bb-Ab-Bb-Ab before finally resolving to G on the downbeat of bar 536. Granted, there are many alterations to this texture as it is presented each time, most of which carry little to no linear implications, but the long absence since it was last heard, combined with the pronounced tension of these changes here creates localised linear time.
More remarkable instances of linear time are created by the sixth texture. By itself, this texture does not have the same forward momentum as the first or second. Instead, the semiquavers in the right hand tend to wander among the white notes between F and C while the left hand maintains its steady, quaver accompaniment. The potential for linearity comes as notes of longer durations in the right hand tend to be on A5, creating dissonance over the C-G accompaniment. In figure 10, the tension of A5, which had been weakened earlier by elements of verticality, is brought to the forefront through longer note values before it resolves to G in bar 348. To further emphasise the dissonance of A and resolution to G, Johnson repeats this motion as if it were a cadential extension. From the pickup to bar 351 until bar 353, there are four repetitions of A of increasing duration, until the section is brought to a close in bar 353. From here, the stability of G is used to transition into the third texture.
While there is a strong sense of linear motion through this tension and resolution, the sense of closure provided by the G seems significant because of its structural position in the work. The sixth texture is the last to be introduced, and from this point on, Johnson returns to earlier textures with more alterations. As such, this sense of stability acts as a structural marker for the entire work. It also has harmonic implications for the piece. The pitch G is certainly the anchor for every texture, to the extent that Kyle Gann wrote that this piece “never deviates from the key of G, though some dissonant motives wash through from time to time” (2004). However, Johnson chose to have this passage, the only cadential moment in the entire work, end with an emphasis of C in the bass.
In the context of the verticality of the piece, two observations arise from this instance of seemingly large-scale linearity. First, despite Johnson’s lack of “before and after” in the compositional process (2010), he still conveys that this is the last texture he will introduce. However, since there is no sense of forward motion from one texture to another, the effect of this cadence is minimised; it becomes a localised phenomenon despite its potential large-scale implications. Second, by allowing the sixth texture to briefly anchor itself to C, Johnson helps to diminish the linear potential of the most prominent pitch, G. If G is not the anchor12 of the piece, then its potential as a destination for linear motion is diminished. Moreover, since Johnson makes the appearance of a stable C extremely brief in the context of the entire composition, any dominant role of G is greatly diminished. Thus, G cannot function as either tonic or dominant for almost the entire piece, greatly minimising linearity on anything but a localised level, and contributing to the overall sense of vertical time.
Harmonic Basis of the Primary Textures
Despite the prominence of the pitch G in An Hour for Piano, its harmonic function is consistently minimised. Johnson avoids any overt thirds within chords to limit harmonic linearity, and, outside of the aforementioned cadence, does not allow G to function in a consistent role. Also, as can be seen through analysis of each individual texture, Johnson subtly implies the importance of the pitch C, though rarely confirms its role as the potential anchor of the piece.
The melodic content of the first texture is centered on the G-Ab-Bb-Ab motion, which without repetitions creates a strong drive to G. Yet while the Ab does strengthen the role of G, that does not inherently imply that G is a large-scale anchor. To be interpreted as such, it would have to be in Phrygian mode, but this modality is never established. The same situation arises in the second texture with the inclusion of Ab in the arpeggio-like figuration. However, unlike the first texture, the left hand increases the importance of C. Notice that the left hand accompaniment is an inversion of the accompaniment for the first texture. Therefore, instead of outlining the fifth from G up to D, it outlines the fifth from G down to C. This is an important difference as the harmonic role of G remains minimized.
The third texture presents a great deal of ambiguity concerning its harmonic basis. G continues to be important in the right hand, but now it is combined with Eb, Db, and Cb, which have not yet been seen. Moreover, the accompanying chords are dissonant against these pitches, consisting of a D2 followed by the E-A-B chords, and there are no C-naturals in either hand. The complete collection of pitches, then, is G, A, B/Cb, Db, D, Eb, E, which by itself seems to defy categorisation.13 If the hands are separated from one another, two observations can be made. First, the left hand seems to be working with one quintal harmony - D, A, E, B - and the right hand functions in a whole-tone scale. Thus both hands deny any pitch as being central, which can be seen as a further minimising of a harmonic role for G.
The fourth texture does seem to emphasise the importance of G both with the frequency of the note in the right hand and the G to B third in the left. However, the melodic content of this texture, derived from the top notes, emphasises C, with its frequent placement on the downbeats, and G a perfect fourth beneath. Granted, the C is always heard as part of a minor seventh with the D below, as Johnson continues to keep potential harmonic linearity in check. The fifth texture also hints at a quasi-dominant function of G. Here, the accompaniment has a prominent minor seventh from G to F, but omits the third lest it create linearity by functioning as a V7. To further imply the importance of C, there are a few deviations from the basic form that resolve melodically to that pitch, which can be seen in figure 11.
In a sense, the sixth texture, with its accompaniment figure centered on C, can be seen as a resolution of the aforementioned minor seventh, though the movement between these two textures is so gradual that the repetition all but eliminates any sense of linear harmonic motion. What is significant about the sixth texture is that it is the only one to clearly emphasise C. It is important to note, however, that there is no sense of linear time created by this harmonic construct. There is no great building of tension where the listener anxiously awaits the resolution of the first five textures. There is no hint of dominant function until the fifth texture, and even then it is weakened to the point of irrelevance. The effect of de-emphasising the importance of G, while at the same time subtly hinting at the importance of C is therefore twofold: the textures seem to float aimlessly, without an anchor, contributing to vertical time, and it provides a loose structure for the piece, which slightly alters the overall sense of vertical time.
Protracted duration, temporal compression, and synchronicity
An Hour for Piano creates a sense of temporality that is different from many early minimalist pieces. Repetition, pulse, and audible structure all act to create vertical time, yet the constantly changing surface does not inherently create the same level of listener freedom. Likewise, the sense of linearity is present on multiple hierarchical levels as Johnson freely includes linear musical elements with his organic compositional approach. Yet the most significant distinction between this piece and other minimalist compositions is the manner in which An Hour for Piano interacts with clock time.
Clock time does not concern the experience of time but rather the objective sense of time as Western society has most commonly come to accept it. In music, clock time is not often a factor. An audience may wish to know the approximate duration of a composition beforehand, but ordinarily the composer does not usually intend for that knowledge to directly affect the perception of the music. This is not the case with An Hour for Piano. Even while the title does not necessarily inform the listener that the performance should be exactly one hour, he or she understands that it should at least be approximately so. This understanding in turn can lead to a greater sense of conflict between protracted duration and temporal compression.
These two experiences of time are an integral part of minimalist music: while the sense of time slowing is felt in the present, time usually seems to have passed quickly when the listener compares the perceived duration of the past with the elapsed time according to the clock. By including the length of the piece in the title, Johnson seems to encourage the listener to check the clock at least occasionally. This feeling in turn makes the listener much more aware of the disparity between these temporal experiences than in most minimalist music, or taken to an extreme, it can potentially eliminate the sense of compressed temporality.
To help the listener avoid this pitfall, Johnson has included lengthy program notes14 that are “to be read while hearing” (1974). He wrote the notes three years after he completed the piece and described them as being “strictly an afterthought, almost independent from the music” (2010). Yet they can be an important part of the listening experience. The notes are self-explanatory and mirror the minimalist aesthetic of the piece with repetition and slight variation. The notes, as they state often, are “intended to increase your ability to concentrate on the music,” and as such also encourage the reader to “not read further” if this goal is not achieved (1974). They also contain prompts for the listener, asking him or her to consider the pacing of their reading in relation to the music, the effect of reading the same passage at different times with different music, and even how the text impacts the work's perceived form. Given the freedom the reader has concerning pacing, pauses, and even whether or not to read the notes, they can help the listener focus more intently on the music and avoid the danger of clock watching.
At the same time, Johnson seems to be interested in creating the experience of synchronicity during the piece as well. Synchronicity is where “the perceived passage of time is nearly synchronised with the standard temporal units of clocks and calendars” (Falherty, 1999, p.98). This is not the normal experience with minimalist music, as has already been shown, but the program notes encourage the reader to consider the remaining time of the piece in relation to the remainder of the text.
The effect of the program notes is threefold: they help to avoid the perils of clock watching by providing another source of stimuli, they increase concentration on the music through its free structure and prompts, and they encourage perception of all three types of temporal experiences - protracted duration, temporal compression, and synchronicity. Such wide-ranging interactions with time clearly distinguish this piece from other minimalist compositions as it not only seeks to create a rich sense of musical time but also incorporates clock time as well.
An Hour for Piano represents a different approach to minimalist piano writing than other compositions of the time. The intuitive construction of the work creates a strong sense of vertical time even if the level of listener freedom or exploration does not seem to be the same. With this composition, Tom Johnson in some ways explicitly writes out what other composers would have required listeners to explore individually, but repetitions, pulse, and audible structure remain fundamentally intact, creating vertical time.
While large-scale linear implications do exist, they only slightly alter the overall sense of verticality. The elements of vertical time creation consistently undermine such implications, though they cannot wholly eliminate them. Therefore, while elements of linear time are clearly a part of the listening experience at various hierarchical levels, they do not negate the dominant verticality but instead help create a richer temporal experience.
The addition of clock time to the listening experience further enhances the overall temporality, a characteristic that makes An Hour for Piano distinctive from its counterparts. Through the apt title, constant pulse approximating the ticking of a clock, and inventive program notes, Tom Johnson gives clock time a vital role in the piece. It is not included at the expense of protracted duration and temporal compression, but rather added alongside these temporal constructs. Understanding the rich and unique temporal experience of An Hour for Piano can only be fully understood with respect to temporality, and any analytical approach that fails to address the issue of time therefore remains incomplete.
The potential for this type of analysis extends beyond An Hour for Piano.15 By considering vertical time as something actively cultivated, it is possible to compare the temporality of different minimalist compositions. Moreover, allowing for the coexistence of vertical time and linear time further illuminates the temporal experience. Minimalist music may seem simple on the page, but the experience of it is not. The depth of minimalism lies primarily in the complexity of its temporality, and an understanding of vertical time creation and the role of linear elements creates a meaningful avenue for analysis of this music.
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 These three were chosen for their particular suitability to minimalist piano music. Other
elements, such as drones, actively create vertical time, but are not necessarily applicable to a discussion of An Hour for Piano
 Kyle Gann places such importance on this factor that he even defines minimalism as “The Era of Audible Structure, 1960-1980,” in “Minimal Music, Maximal Impact” (2001).
 Aristotle, the first to try to prove the law of contradiction, defines it as such: “The same attribute cannot at the same time belong and not belong to the same subject in the same respect - presupposing, of course, such qualifications as may be necessary to guard against logical objections” (2007, p.123).
 Johnson also cites the Four Note Opera (1972) as one of his first minimalist pieces (Warburton, 2007).
 It is also rather different from the music Johnson would go on to compose. Since ca. 1979, his compositions have tended to be far more rigorously process-oriented and/or mathematical (Decroupet, 2001). For example, his Chord Catalog (1986) for piano consists of all 8178 chords possible within one octave.
 Two- to four-bar examples were chosen to show both the essential character of the different textures as well as some of the different ways they are presented.
 Of the 870 total bars, only 61 (7.0%) are not in common time, and those bars vary from 2.5 to 7 beats in length. The total number of beats for the piece is 3553.5. This number is then divided by 60 to calculate the precise tempo indication.
 16 bars multiplied by 4 beats/bars multiplied by the duration of each beat, which is just over one second (60/59.225 seconds/beat).
 The only texture that does not always emphasise the metre is the transitional texture on the repeated crotchet G. Even there, the grace notes and occasional blending with other textures usually provide a downbeat emphasis when present.
 The case for an even longer transition could also be made as the fourth texture starts breaking down as early as bar 620 and the right hand quavers of the third texture do not appear until bar 671.
 The first texture ends around 3’06” and does not return until 36’13”. (All timings have been calculated based on a metronomically consistent performance, as per Johnson's performance indications.) This is the longest amount of time between complete instances of a particular texture in the entire piece. The next longest is in the third texture, from the end of its second instance, 18’24”, until its return at 45’34”, a length of 27’10”. Yet even this texture returns at least partially in transitional material during that span. The long absence of the first texture is substantial, especially considering the aural importance it receives by starting the piece.
 The word tonic is deliberately avoided as there is no true sense of a tonic-dominant
relationship in the piece.
 Some pitches would have to be considered dissonant and thus omitted to have this texture fit within a recognisable scale, but the material does not seem to give any
 The program notes can be found in their entirety from: http://www.lovely.com/albumnotes/notes1081.html
 This analytical approach is applied to other minimalist and postminimalist pieces in the dissertation The Interaction of Linear and Vertical Time in Minimalist and Postminimalist Piano Music, upon which this article is based. The dissertation is available in full from:
ABOUT THE AUTHOR
R. Andrew Lee, DMA is an avid performer and researcher of minimalist music and has been described as an "an indispensable new-music pianist" (Kyle Gann, 3rd International Conference on Minimalist Music). Lee has recorded multiple albums for the Irritable Hedgehog label, including the first perfectly-timed recording of Tom Johnson’s An Hour for Piano and Dennis Johnson's 5+ hour epic, November. Dr. Lee currently teaches at Regis University in Denver, Colorado, and was most recently Artist-in-Residence at Avila University. He received his doctorate in Piano Performance from the University of Missouri-Kansas City Conservatory of Music, where he studied under the direction of Prof. John McIntyre. He has also earned a masters in piano performance from UMKC and a bachelors in piano performance from Truman State University.