Time-Space Synthesis: The Relationship of the Grid to Twentieth- Century Music
"The grid is an emblem of modernity by being just that: the form that is ubiquitous in the art of our century....."
Thus art critic and historian Rosalind Krauss describes the importance of the grid as a structure within the visual arts of the twentieth century. Krauss goes on to state that: "In the flatness that results from its coordinates, the grid is the means of crowding out the dimensions of the real and replacing them with the lateral spread of a single surface."
The foregoing observations about the grid and its relationship to twentieth-century visual arts could be applied to the temporal arts as well, particularly music. It is in the synthesis of musical time and space that the grid and its geometricized order operates. In modern music, the grid allows for the concept of equivalence of musical space and contributes to the abolishment of the opposition between the horizontal and the vertical aspects of this space.
In the visual arts the grid acts as a "mapping of the space inside the frame onto itself." The mapping of the musical space first begins to make its appearance in twentieth-century music in the late serial works of Anton Webern. In these works the unification of all the elements of a composition is regarded as ideal.
As a visual structure, the grid is anti-narrative and rejects sequential reading. This absence of discourse, and rejection of both thematicism and hierarchical order, is explored in music in the works of the so-called Minimalist composers (Riley, Reich, Glass, Feldman, Young). A number of these composers have either employed grids as a concrete element of their music (for example, Feldman's early graphic music scores) or have worked with visual artists whose motif is the grid (for example, Philip Glass and Sol Le Witt). In the work of these composers time has ceased to be ordered according to what composer Philip Glass describes as the "traditional concepts of recollection and anticipation."
It is the ambiguity of metric accent (strong-weak) and the absence of the traditional dependency on periodicity in this music that contributes to this loss of directionality. (Again, this characteristic can be traced back to the music of Webern, where the barline begins to serve more as a frame for the structuring of the musical space than a metrical division.)
Finally, it is the repetitiousness and unity of the grid that exerts strong influence on the musical space. This paper will explore the grid as a spatial model in twentieth-century music. It will trace the development of this relationship beginning with the music of Anton Webern and extending through present-day minimalist composers.
Grids have become important modes of organization for recent art. Rosalind Krauss writes that: "By discovering the grid, Cubism, de Stijl, Mondrian, Malevich ... landed in a place that was out of reach of everything that went before. Which is to say, they landed in the present and everything else was declared the past."
For these painters the grid emphasized a drawing style that stressed the materialness of the physical qualities of the painting's surface, thereby affirming the space of art as self-contained. Igor Stravinsky takes an interestingly similar stance in regard to the autonomous nature of music when he states: "Music expresses nothing but itself." Yet, while the grid does focus on the material nature of art, it has become so ingrained in modernist sensibility that it has also come to stand as an aesthetic symbol, what Rosalind Krauss describes as an "emblem of modernity."
Considered this way, Rosalind Krauss notes how the grid can be viewed as a mapping of the physical qualities of the surface onto the aesthetic dimensions of the same surface. It is the dual nature of the grid (material and aesthetic) that partly accounts for its power. As a symbolic representation of contradiction (material/aesthetic) embodying well-contained forces (horizontal/vertical), the grid has `burrowed down' into our culture.
In its capacity to emblematize the Modern, the grid's presence emerges not only in the pictorial arts but in music as well - "the result not of imitation but of aesthetic decree." Art critic John Elderfield writes that: "While it is satisfying to recognize what look like formal continuities, what we really need to distinguish is what belongs to the established traditions of a genre and what special ways its topics perform in special situations." The examination and justification of function and character of the grid within both the fields of modern music and art is an attempt to decipher meaningful structural functions and directions that lay behind the music and art of the twentieth century.
The basic principle underlying the present analysis is recognition of what English sociologist Michael Lane describes as "the logical priority of the whole over the parts." This happens to be a structuralist approach. Lane writes "Probably the most distinctive feature of the structuralist method is the emphasis it gives to wholes, to totalities ...The essential quality of the structuralist method, and its fundamental tenet, lies in its attempt to study not the elements of a whole, but the complex network of relationships that link and unite those elements ... structuralism seeks its structures not on the surface, at the level of the observed, but below or behind empirical reality."
A distinguishing feature of structural analysis is the rearrangement of the sequential features of a story (particularly myths) to form a spatial organization that represents the temporal dimension of the story. In the application of this method, features of the story are arranged into vertical columns. In this way the diachronic and synchronic dimensions of time are represented by the horizontal and vertical axes of the grid. The horizontal axis indicates temporal succession, the vertical axis indicates the non-temporal relations of comparison and contradiction. In this sense the structuralist approach bears a relationship to the formal structure and character of the grid. Distinguishing features of this relationship include:
1) the priority given to the whole over the parts;
2) the priority given to relationships among the parts;
3) the rejection of narrative or sequential reading;
4) the attempt to deal with contradiction, and
5) the basic organization of the horizontal/vertical grid as the matrix of order.
Suzi Gablik, in Progress in Art states that "parallel developments in two completely different fields of culture allow one to suspect the existence of a general principle which has operated on the entire evolution of culture; this concerns the unfolding of `laws of development' within the cognitive processes.
Certain conditions (`laws of development') have combined to bring about the grid as the modern paradigm. One of these is the concept of equivalence. For Mondrian and the de Stijl, the straight line, the plane and the angle were the primary elements of art. Ultimately for Mondrian, the right angle, consisting of an equivalent but unequal opposition of vertical and horizontal lines, was the purest of relationships. In 1926 Mondrian drew up a list of general principles in which he states: "There must be an equivalence of plastic means. The duality of opposing elements in the plastic medium is also required in the composition. Constant equilibrium is achieved through opposition and is expressed by the straight line (line of the plastic means) in its principal opposition in the right angle."
The de Stijl development in Holland parallels the musical developments in the Second Viennese School in Germany. Between 1920-23 Schoenberg composed his first twelve-tone work governed by the `total system of relations of the sounds with one another."
In an essay which appears in Style and Idea entitled "Composition with Twelve Tones," Schoenberg discusses the essence of the musical idea and its disposition:
A musical idea though consisting of melody, rhythm and harmony, is neither the one nor the other alone, but all three together. The elements of a musical idea are partly incorporated in the horizontal plane as successive sounds and partly in the vertical plane as simultaneous sounds ... The mutual relation of tones regulates the succession of intervals as well as their association with harmonies; the rhythm regulates the succession of tones as well as the succession of harmonies and organizing phrases. The unity of musical space demands an absolute and unitary perception. In this space ... there is no absolute down, no right or left, movement of tones has to be comprehended primarily as a mutual relation of sounds, of oscillatory vibrations, appearing at different places and times.
For Schoenberg musical space has no limits or boundaries. The musical work is a relational structure that the composer applies to limitless space. Interestingly, the concept of fixed space as only one in an unending series of spaces appeared in the art of the Cubists at approximately the same time. The cubists recognized the equivalence of a multitude of perspectives, spatial frames, and points of view that can be applied to an object. The equivalence and relativity of a number of points of view in space is best represented by what art critic Clement Greenberg describes as the "all-over, decentralized" or "polyphonic" picture. It is a picture that dispenses apparently with beginning, middle, and end." The use of the musical term "polyphonic" is no accident for Greenberg, who believes there is a specific parallel between Mondrian's term "equivalence" and Schoenberg's twelve-tone composition.
Greenberg observes: "Just as Schoenberg makes every element, every sound in the composition of equal importance - different but equivalent - so the `all-over' painter renders every element and area of the picture equivalent in accent and emphasis. Like the twelve-tone composer, the `all-over' painter weaves his work of art into a tight mesh whose scheme of unity is recapitulated at every point."
The grid itself is a synthesis of this `all-over' extension in space. Its appearance in the modern period as a model for spatial relationships has been concisely summarized by Suzi Gablik:
The geometric grid which the Renaissance, with its Euclidean turn of mind, imposed on the process of representation was systematically dismantled and `de-constructed'by the cubists:the tendency since then, in the twentieth century has been to use the elements of geometry abstractly, in the manner of a `universal grammar'; exploring and permuting its internal relationships.
Charles Rosen speaks of a similar dismantling that took place in music, of the harmonic and tonal conception of large form:
The renunciation of the symmetrical use of block elements (i.e. cadential formulas) in working out musical proportions places the weight on the smallest units, single intervals, short motifs.
The unequal but equivalent opposition of the horizontal and vertical aspects of pictorial space that Mondrian speaks of is similarly addressed by Pierre Boulez in regard to the twelve-tone series and its effect on musical space; "The series dilutes the opposition between the horizontal and vertical, just as it creates a universe where consonance and dissonance are abolished." 
Each piece of music possesses its own special form of coherence and is a unique presentation of the relationship (part to part and to whole) of musical time and space. The examination and juxtaposition of the function and character of the grid within both the fields of modern art and music is an attempt to decipher meaningful structural functions and directions. In order to show the connection and continuity that exist between both pictorial and musical conceptions of space in the modern period, some detailed analysis is necessary.
In Composition With Lines Mondrian creates an `all-over', `polyphonic' picture, (similar to late impressionism) entirely from short horizontal and vertical lines (and a few squares) of varying size that fill the canvas with almost uniform density. Some lines occur singly or are paired as equal parallels. Some touch or cross one or two others; in these places, asymmetric clusters of up to six bars are formed. In all cases, these modular units stand in contrast to and are surrounded by a white background. In this way, variations on a basic unit result in a decentralized unified texture. The particular disposition of the individual units appears at first to be highly irregular. However, this apparently random disposition of elements on closer examination yields a number of surprising consistencies:
1. The greater the degree of density of the units the less frequent their occurrence.
2. A certain degree of formal symmetry is achieved by the series of single vertical lines that run parallel to both the left and right edges of the painting.
3. The weight of the composition seems to lean towards the lower, denser half of the surface as opposed to a slight decrease in density towards the upper half.
Thus in Composition With Lines Mondrian seems to have combined both a systematic and intuitive approach in order to produce a unified composition. Meyer Schapiro notes of this painting how "the simple and regular compose a surprisingly irregular design." (The reader is referred to Meyer Shapiro's discussion of Composition With Lines in the previously cited volume.)
This can also be said to be true of the Checkerboard paintings of 1919. In these paintings, the grid is explicit; in fact, it seems to have been produced almost mechanically. However, the placement of colors seems to defy analysis except on the basis of intuition. The use of intersecting bars as a means of creating varying densities in Composition With Lines is replaced in the Checkerboard paintings by areas of the same color that touch horizontally or vertically (sometimes even diagonally). In a manner similar to Composition With Lines, the greater the density of colored boxes, the more limited their occurrence tends to be. An important difference lies in how they are comprised. Composition With Lines is produced through an additive or cumulative process. The grid of the Checkerboard paintings is subtractive; that is, the lines are a method of dividing the pre-existent surface. However, the use of color fields in this painting adds another dimension and it is this element that must be applied additively. (It is interesting to note how the horizontal and vertical bars of Composition With Lines represent plus and minus signs and, as a result, these paintings are often referred to as such.) The additive and subtractive nature of the grid has been particularly influential in the methods and concepts of both painter/sculptor Sol Le Witt and composer Philip Glass.
The methods of repetition and modification employed by Mondrian in these works find their parallel in music in the techniques of imitation and variation. In both Webern and Mondrian's cases, these principles are applied as a means of achieving limitless results from limited means.
For Webern, the grid had symbolic meaning and illustrates his ideas of balance, symmetry and the equality of horizontal and vertical musical space. To illustrate these ideas Webern used the Latin palindrome ("The sower Arepo keeps the work circling.")
In the analysis of the op.21 which follows, it becomes clear that both the horizontal (duration) and the vertical (registral) dimension receive the most careful attention, and it is clear that Webern is seeking to equalize their structural importance. In fact, the strongest aural impression of Webern's music is that everything is of equal importance. The Symphony in this sense must be viewed whole. Through octave displacement, careful measurement of durations and rests and the compression of lines, the Symphony takes on an `all-over' texture. Contributing to this `all-over' effect particularly in the first movement of the Symphony is the absence of any solo/accompaniment function - each instrument in the ensemble is equal. In the same way, each pitch receives its own dynamic and articulation.
Twelve-tone compositional technique gave Webern the means to deepen and clarify the unity he was striving for. Through the use of contrapuntal techniques, primarily canon, Webern maintains his equality of voices and the `all-over' quality of his music. Webern thought of canon as representing the vertical presentation of themes (themes heard simultaneously) and sonata and variation forms as representing the horizontal presentation (themes heard successively). It was through the use of such traditional techniques and practices that Webern achieved thematic unity in his twelve-tone compositions. The Symphony is Webern's crowning achievement in his personal synthesis of the horizontal and vertical dimensions of music. (See example 1.)
In Composition With Lines the exposed untouched surface of the white
canvas is as actively important an element of the composition as the
pointillistic bars. In a similar way,
silence, both contextual and absolute, plays an active role in Webern's Symphony. In the
The totality of the registral space receives equal emphasis, and by the unique positioning of sounds, both registrally and temporally the individual areas remain unique. Areas of varying density are created by the piling up of the individual parts or by what Boulez terms "a constellation of events." An important aspect of Composition With Lines is the frequent intersection of the bars. In the Symphony the intersection of lines (voice crossings) is a frequent occurrence that reinforces the autonomy of the individual lines.
Similar to Mondrian's use of restricted and divided color areas in the Checkerboard paintings, the orchestration of the Symphony can best be described as pure. That is, the simultaneous orchestral doubling or mix of a sound is rare. Webern is concerned with exploring the way a pitch sounds in a specific instrument, in a specific register. When the same pitch is transferred among instruments, the aim is the distinctiveness of each sound. As in Mondrian, beneath what initially may appear in the Symphony as a random disposition of sounds, is a highly systematized approach.
In the opening of the Symphony the serial flow is mainly carried out linearly, in four-part texture, as a double canon in contrary motion. What is interesting is how Webern chooses to subvert both of the rigid systems he has adopted for the opening. The row for the Symphony consists of two equal asymmetric six-note subsets, of which the second half is a transposition in retrograde of the first half (A-F#-G-A flat-E-F/B-Bflat-D-C#-C-E flat). This limits the number of transpositions to 24. In the canon of the first movement, four forms of the row always appear simultaneously. The result of this vertical alignment is the frequent repetition of pitches. Actually, the twelve-note chromatic does not complete itself until the 24th note is reached. In this sense, Webern uses the twelve-note system as a means of defeating itself. In addition, the canon is completely masked by the frequent voice crossings, uses of Klangfarben and pitch repetition. Thus, Webern's method proves to be rich enough in implication to encompass this contradiction between the technique and its means of application.
Example 1: Anton Webern, Symphony op.21, I, mm. 1-14.
Copyright 1929 by Universal Edition. Copyright renewed. All rights reserved. Used by permission of European American Music Distributors
Corporation, sole U.S. and Canadian agent for Universal Editions.
In the unfolding of the different row forms the registral placement of the individual notes is fixed and is based on a six-note rotational chord of five perfect fourths and its mirror form, revolving symmetrically around the axis pitch of A natural. The opening of the Symphony therefore can be viewed as one of the most extreme examples of minimalist technique. That is, the first 26 measures are nothing but the arpeggiation of a single registrally fixed twelve-note chord! Traditionally, in tonal music, the arpeggio served primarily as an accompaniment figure and as such was treated not in note-by-note fashion but as a unit (i.e. the triad). In the Symphony, the arpeggio's role is no longer that of accompaniment; rather, it is all that is displayed. Here the arpeggio receives an `all-over' distribution of emphasis and it is the means by which Webern can make coherent the note-by-note unfolding of the series.
It is the arpeggiation of this chord that unifies the various techniques that are applied to the opening. It can therefore be considered as the composite harmonic field of the piece.
Example 2: 12-note rotational chord from Webern's op.21.
The rhythmic flow of the opening can be viewed in the same way. From measures 1-14, each of the individual parts, when taken as a composite, yield but one single rhythmic pattern: quarter, half, quarter. The second half of the opening (mm.13-26) yields another pattern of four quarter notes. Thus, though the individual parts seem highly disjunct rhythmically, the composite rhythm takes a highly singular and conjunct approach.
The context or mode of organization in Webern's Symphony is both systematic and repetitive. For Webern, the concept of the grid acts as the invisible means of organization and coherence. In Mondrian, the concept of the grid is the intentionally visible means of organization. In the works examined thus far, both symmetry and repetition play an important role, and, in Webern in particular, symmetry is the predominant form of repetition. In fact, the second section of the first movement of the Symphony consists solely of two mirror-like halves (mm. 26-44).
A similar use of symmetry as a form of repetition occurs in the opening measures of the Variations, op.27 (1936). Here the musical gesture or motive is treated as a repeatable modular structure. As in the Symphony, beat and meter act as a scaffolding or frame on which to construct symmetries of pitch and rhythm.
Copyright 1937 by Universal Edition. Copyright revewed. All rights reserved. Used by permission of European American Music Distributors
Corporation, sole U.S. and Canadian agent for Universal Editions.
George Rochberg writes: "Although metric uniformity was still characteristic of Webern's style, the loss of an evident melodic shape took its toll on the directional force inherent in rhythmic periodicity. For the first time since the beginning of non-tonal pitch orders (atonality and twelve-tone) periodicity and the phrase were brought to a state of near equilibrium. One of the techniques that reinforced this condition of virtually suspending rhythmic motion was structural symmetry, for which Webern had a strong penchant."
The beats that make up each measure are of equal value and are referential points in the mapping of time. This approach is similar to Mondrian's almost mechanical mapping of space in the Checkerboard paintings.
The `all-over' distribution of emphasis that both musical time and space and painterly space now receive is often characterized as being non-relational in effect. That is, individual entities, forms or motives are not presented against any background and are not related to each other in a hierarchical sense. In clarifying the formal difference between non-relational (non-hierarchical) and relational (hierarchical) art, Lawrence Alloway, former curator of the Guggenheim museum, notes that, although relationships, in fact, still do persist in non-relational art: "the relations are those of continuity and repetition rather than of contrast and interplay."
Although both Mondrian and Webern eventually moved away from the non-relational surface structure in their works (for example, Mondrian's use of asymmetry and Webern's use of stratified harmonic areas, already present in the second movement of the Symphony, it is this aspect of their work that has had the greatest influence on a younger generation of individual artists in diverse media who are often referred to as "minimalists."
Indeed, repetitive, self-contained, modular structures now permeate modern art and music alike. And the grid itself has surfaced as the prototypical mode of organization. John Elderfield writes: "For the minimalist sensibility, the grid system supports the flatness and delimitation of the flatness in the painting in question, its `minimal conditions' for being a painting." Although much has been written concerning the presence of the grid as both structure and symbol in the work of minimalist sculptors and painters (i.e. Sol Le Witt, Agnes Martin, Carl Andre, Jasper Johns, Philip Guston) little has been said concerning the presence of the grid concept in the music of the composers most closely associated with these artists (i.e. Morton Feldman/Abstract Expressionism and Philip Glass, Steve Reich/Minimalism). Indeed, initially their music met with the most interest and understanding among visual artists.
My examination of a sampling of these composers' works will be more concerned with the music's overall implications and Gestalt than with note-by-note analysis. Through this I hope to emphasize, in relation to the grid, the important relationship that exists between what Pierre Boulez calls "the expression of a form and the content of the composer's thought."
In the late 1940's Jackson Pollock and the Abstract Expressionist New York School of painters (Mark Rothko, Philip Guston,Ad Reinhardt) began to develop their own form of painting that was based on the `all-over' and decentralized drawing style of Mondrian's non-relational paintings. The Abstract Expressionists were very much concerned with preserving the flatness of the canvas - its surface - and, for these painters, the grid served as a means of emphasizing both the materialness of the surface and the "all-over" drawing style. The ideas and works of these artists had an enormous influence on the musical attitudes and methods of composer Morton Feldman. In a similar way, the early non-relational paintings of Mondrian also served as stimuli for Feldman's music, particularly his graphic music (Projections 1-4, Out of Last Pieces, Intersections). Clearly one of the most striking qualities about these works is their resemblance, in terms of sheer visual composition, to the paintings of Mondrian. Indeed, the title "Intersections" is meant to suggest the grid-like character of New York City's streets, where Feldman was living.
Feldman's concept of composition as a way of generating ideas and asking questions embraces both graphic and precise notation and, although he stopped composing graphic scores, his musical intentions and concerns remained highly singular and consistent. Because it is in his graphic pieces that Feldman began to formulate his central aesthetic tenet, this discussion is limited to these works.
In contrast to Earle Brown's graphs of the same period, which are clearly based on the grid structure, for example, December, 1952, Feldman's graphic scores are not meant as visual stimuli for the performer, but are more exacting in their demands. John Cage observed that: "On paper, of course, the graph pieces are as heroic as ever, but in rehearsal Feldman does not permit the freedom he invites, to become the occasion for license." In the Projections series, the parameters of register, number of sounds to be played, timbre and duration are specified. In the same way an action painter such as Pollock literally `projects' his paint onto a canvas, Morton Feldman's aim in the Projection pieces was "not to compose but to project sounds into time."
Feldman's use of the word "sound" instead of pitch is important in that, traditionally, "pitch" implies relationships with other pitches, whereas sound is not associated with a traditional context. This stance is similar to Varèse's stated compositional goal of the "liberation of sound." In the Projections, register and timbre, being determined by the notation, are the real differentiating structural factors - pitch itself is secondary. According to Feldman: "What I did was break down the whole notion of close passage work." These works are clearly non-relational, that is, there are no identifiable motives and the music is a continuous process, beginning and ending without a particular goal being reached. Michael Nyman observes that: "each note, each chord, is a separate weight, composed and heard separately, having no priority over the one coming before or after."
Feldman is also concerned with the surface of his music. He has remarked: "My obsession with surface is the subject of my music." As the Abstract Expressionists sought not to disturb the flatness of the canvas (the space) Feldman wishes to present "time undisturbed in its unstructured existence." Feldman's approach is clearly a more radicalized version of Webern's non-hierarchical treatment of musical time and space. The surface of Feldman's music could certainly be described as flat. The graph is without dynamics or expressive marks. There are no "solo" or "accompaniment" passages and nothing stands out in relief. Indeed, the graph itself lays flat on the page, unlike the graphs of other composers (for example, the graphs of Sylvano Busotti). An unusual aspect of listening to Feldman's music has always been that individual events are not remembered; each event cancels out the one that came before. And although compositional systems are contrary to Feldman's beliefs, he is so singular in his approach to his materials that this consistency can be viewed as constituting a system in itself.
Feldman writes: "It is not a matter of a controlled or a de-controlled methodology. In both cases, it is a methodology. Something is being made. And to make something is to constrain it." Feldman has observed that an important word for his generation was "ambivalence" In describing his music, Feldman has said "My compositions are really not compositions at all: one might call them time canvases in which I more or less prime the canvas with an overall hue of the music,. I prefer to think of my work as: between categories. Between time and space. Between painting and music. Between the music's construction and its surfaces."
For Feldman, the grid, and its many implications, are not only present figuratively in the score and conceptually in the music, but symbolically as well, in terms of his relationship to his work. Feldman writes: "The moment a composer notates musical thought to an ongoing ictus, a grid of sorts is already in operation, as with a ruler." The influence of Abstract Expressionism on Feldman is similar in many respects to the reciprocal relationship that developed between Minimalist painters, sculptors and composers in the 1960's, particularly in New York City. It is by way of what Rosalind Krauss describes as the "mutually accessible space of the grid" that this relationship can best be explored.
John Elderfield notes how the emergence of grids and grid structures in the Minimalist art and sculpture of the '60s (Sol Le Witt, Carl Andre, Anges Martin, Don Lewallen, Kenneth Nolland) objectified Pollock's previous approach. In this sense they no longer serviced other pictorial components, for none existed.
Lawrence Alloway writes: "The field and the module (with its serial potential as an extendable grid) have in common a level of organization that precludes breaking the system. This organization does not function as the invisible servicing of the work of art, but it is the visible skin. It is not, as that is to say, an underlying composition, but a factual display."
A distinguishing characteristic of the Minimalist Music which emerged in the '60s (Steve Reich, Philip Glass, Terry Riley, La Monte Young) lies in its emphasis on musical process. This concern with musical process has led many to refer to this music as "systematic." However, unlike serial music, wherein the process acts more as a hidden structural device, the Minimalist emphasis on process is an audible aspect of the music. Replacing the more traditional models of composition, the concern of these artists is with discovering the right process or operation for generating the structural and organizational principles used in the building of the work. The process or system acts not as the simple arrangement of independent elements but as the structuring of concepts.
The aim of much Minimalist music and art is to project modular sound objects within a given piece by means of a continuous and repetitive process. Lawrance Alloway writes "Here form becomes meaningful, not because of ingenuity and surprise, but because of repetition and extension.". The process itself is often serial in nature, based on additive or subtractive methods (Philip Glass) or some other type of serial methodology (permutation, rotation). However, between the Serialists in music (Babbit, Boulez) and the Minimalists, the emphasis placed on process (the system) has extended itself from a purely underlying structural function to a perceptual one as well. This emphasis on the perceptual necessitates that the material being processed be minimal in nature, that is, clearly defined, explicit, and ponderable.
The intention that the process of the work be perceivable does not necessarily contribute to a loss in complexity or richness of content. The repeated motive is subject to continuous change. This repeated transformation of a configuration (the process) results in a profusion of surface changes that elicit a continuous multiple number of possible readings. Therefore presupposing, as stated by Lawrence Alloway, "a situation of multiplicity (John Cage's continually apt phrase) in which the extension of the module elicits variable readings but without sacrifice of a consistent structure."
Precedents for this type of formulation or mode of reduction in music can be found in the music of Anton Webern, particularly in the Concerto for Nine Instruments op. 24. This work is based entirely on a three-note motive. Through intervallic rotation and transposition this modular unit appears in the twelve-note pitch series in all four possible modes of transformation: prime, retrograde, inversion, retrograde-inversion, and is clearly presented in the opening six-measure introduction.
In addition, in the total of 48 row forms, there exist four transformations (P,R,I,RI) of the 4 x 3, twelve-note grouping that each contain, at the identical pitch level, four different rotations of the three-note modular unit. In that it appears four times at the same pitch level, the available material for the composition is reduced to 24 distinct three tone groups (and their retrogrades).
Example 4: Anton Webern Concerto for Nine instruments op.24 No.1, mm.1-5.
Copyright 1948 by Universal Edition. Copyright revewed. All rights reserved. Used by permission of European American Music Distributors
Corporation, sole U.S. and Canadian agent for Universal Editions.
A. FOUR BASIC KINDS OF STRAIGHT LINES: 1. Vertical 2. Horizontal
3. Diagonal left to right
4. Diagonal right to left
B. SUPERPOSITION: 1. Vertical is the same as in A. 2. is 1 + 2
3. is 1 + 2 + 3 4. is 1 + 2 + 3 + 4
While A is flat, B in tonal.
Example 5: Anton Webern, Op.24 I rotational row forms. Sol Le Witt, plan for Wall Drawing IA & B, IIA & B, 1969.
The rotational process at work here is strikingly similar to that presented by Sol Le Witt in his Wall Markings (1968)  in which the motif is a square divided into four quarters, each one inflected by straight lines running across, down and on left and right angles diagonally.
Rotation is also an important basis for Steve Reich's Piano Phase (1968). Example 8 shows the first of three motivic modules which constitute the piece. One pianist repeats the motive while the other slowly accelerates until he is one 16th-note ahead. This process is continued until both players are back in unison. The different alignments of the module against itself result in a continuously shifting accentual scheme. Clearly the principle that underlies this phase shifting is that of rotation. This is made even more apparent in the very nature of the opening motive itself with its numerous pitch repetitions.
Motive one is composed of twelve notes using five different pitches. The left hand notes repeat every six 16th notes, the right hand notes every four. In this motive, pitch repetition is generated from the rotation of dyads within the motive, every six 16th notes. This rotation generates not only the repetition of the five pitches within the motive but the repetition of the motive itself.
Copyright 1980 by Universal Edition. Copyright renewed. All rights reserved. Used by permission of European American Music Distributors
Corporation, sole U.S. and Canadian agent for Universal Editions.
Both the Webern and Reich works make use of this rotational process. However, in Piano Phase, the process is gradual, literally slowed down and exposed over an extended time span. This process is all that is displayed. It is the essence of the work and as such does not service other musical components. The process is literally the only audible structure of the work. It is the empirical and material means by which we approach the sound.
In Opus 24, aspects other than the rotational process are necessary to the work's substance. The process is thus used as a framework to be departed from. It is the underlying organization, the invisible framework on which to place the more crucially dramatic forms of music (phrase, form, dynamic, melodic direction, articulation, orchestration).
In notes for Music in Twelve Parts, Philip Glass discusses what he sees as the essence of the Minimalist work; he writes :"It is that one would be able to perceive the music as a presence freed from dramatic structure, a medium of pure sound."
In its ability to signify the antidramatic, the antinarrative, the antidevelopmental, the antihierarchical, the antirepresentational, the grid has conferred precisely this sense of presence on twentieth-century art. And it is the grid's capacity to foreclose accepted conventions, to function as an emblem of the minimal, the essential, the irreducible, that has compelled so many twentieth-century artists.
List of Available Reproductions
Kermit Swiller Champa, Mondrian Studies (Chicago: The University of Chicago Press, 1985).
Hans L.C. Jaffe, Piet Mondrian (New York: Harrny N. Abrams, Inc., 1985).
Sol Le Witt, Sol Le Witt, ed., Alicia Legg, essays by Lucy Lippard, Bernice Rose, and Robert Rosenblum (New York: The Museum of Modern Art, 1978).