In modern mathematics, the two-dimensional plane is usually defined as the set of pairs of real numbers; lines and regions are subsets of this set. The plane is thus treated as an infinitely dense grid of locations. This mathematical point of view does not translate into a practical convention for finite computational representations, however. The mathematical correlate of any non-empty shape or line-segment is a set of uncountably many points, and the coordinates of most of these points are irrational numbers which cannot be represented in a finite way.
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Paul Paccione. 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. 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. 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. 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. 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.
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. 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 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:. Thus, in order to compare the parallel structures and functions of the grid in the diverse disciplines of art and music it will be useful to take a structuralist approach.
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 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. 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 unity of musical space demands an absolute and unitary perception.
In this space 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. Its appearance in the modern period as a model for spatial relationships has been concisely summarized by Suzi Gablik:.
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. 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 the paintings that culminate in the entirely abstract Composition With Lines , Mondrian's style moves from an impressionistic and analytical representation of nature to pure abstraction. And in the Symphony op. Both works have had an enormous influence on later generations of artists, who have used the inherent ideas as points of departure for their own works. 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:. The greater the degree of density of the units the less frequent their occurrence. 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. 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.
This can also be said to be true of the Checkerboard paintings of 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 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. 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. 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.
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 Symphony , motives are reduced to an extremely limited number of pitches surrounded by rests. Webern's consistent use of disjunct intervals further serves to isolate the individual sounds.