*What ordertype is universally present wherever there is any order in the world? The answer is, serial order. What is a series? Any row, array, rank, order of precedence, numerical or quantitative set of values, any straight line, any geometrical figure employing straight lines, and yes, all space and all time.*

—Josiah Royce, *Principles Of Logic*

**SERIAL ORDER IS A METHOD,** not a style. The results of this method are surprising and diverse. Edward Muybridge’s photographs, Thomas Eakins’s perspective studies, Jasper Johns’s numerals, Alfred Jensen’s polyptychs, Larry Poons’s circles, dots and ellipsoids, Donald Judd’s painted wall pieces, Sol LeWitt’s orthogonal multipart floor structures all are works employing serial logics. This is not a stylistic phenomenon. Variousness of the above kind is sufficient grounds for suggesting that rather than a style we are dealing with an attitude. The serial attitude is a concern with how order of a specific type is manifest.

Many artists work “in series.” That is, they make different versions of a basic theme; Morandi’s bottles or de Kooning’s women, for example. This falls outside the area of concern here. Three basic operating assumptions separate serially ordered works from multiple variants:

1. The derivation of the terms or interior divisions of the work is by means of a numerical or otherwise systematically predetermined process (permutation, progression, rotation, reversal).

2. The order takes precedence over the execution.

3. The completed work is fundamentally parsimonious and systematically self exhausting.

Serial ideas have occurred in numerous places and in various forms. Muybridge’s photographs are an instance of the serialization of time through the systematic subtraction of duration from event. Muybridge simultaneously photographed the same activity from 180°, 90°, and 45° and printed the three sets of photographs parallel horizontally. By setting up alternative reading logics within a visually discontinuous sequence he completely fragmented perception into what Stockhausen called, in another context, a “directionless time-field.”

Robert Rauschenberg’s *Seven White Panels* and Ellsworth Kelly’s orthogonal eight-foot-square *Sixty-Four* are anomalous works of the early 1950s. Both paintings fall within a generalized concept of arrays, which is serial, although their concerns were primarily modular. Modular ideas differ considerably from serial ideas although both are types of order. Modular works are based on the repetition of a standard unit. The unit, which may be anything (Andre’s bricks, Morris’s truncated volumes, Warhol’s soup cans) does not alter its basic form, although it may appear to vary by the way in which units are adjoined. While the addition of identical units may. modify simple gestalt viewing, this is a relatively uncomplex order form. Modularity has a history in the “cultural methods of forming” and architectural practice. Frank Stella has often worked within a modular set although in his concentric square paintings he appears to have serialized color arrangement with the addition of random blank spaces. Some of the early black paintings, like *Die Fahne Hoch*, employed rotational procedures in the organization of quadrants.

Logics which precede the work may be absurdly simple and available. In Jasper Johns’s number and alphabet paintings the prime set is either the letters A–Z or the numbers 0–9. Johns chose to utilize convention. The convention happened to be serial. Without deviating from the accustomed order of precedence he painted all the numbers or letters, in turn, beginning again at the end of each sequence until all the available spaces on the canvas were filled. The procedure was self-exhausting and solipsistic. Other works of Johns are noteworthy in this context, especially his *Three Flags* which is based on size diminution and, of course, the map paintings. His drawings in which all the integers 0-9 are superimposed are examples of a straightforward use of simultaneity.

An earlier example of simultaneity appears in Marcel Duchamp’s *Nude Descending a Staircase*. Using the technique of superimposition and transparency he divided the assigned canvas into a succession of time intervals. Due to the slight variation in density it is impossible to visualize specific changes as such. Alternations are leveled to a single information which subverts experiential time. Duchamp has said the idea was suggested to him by the experiments of Dr. Etienne Jules Marey (1830–1904). Marey, a French physiologist, began with ideas derived from the work of Muybridge, but made a number of significant conceptual and mechanical changes. He invented an ingenious optical device based on principles of revolution similar to Gatling’s machine gun. This device enabled him to photograph multiple points of view on one plate. In 1890 he invented his “chronophotograph” which was capable of recording, in succession, 120 separate photos per second. He attempted to visualize the passage of time by placing a clock within camera range, obtaining by this method a remarkable “dissociation of time and image.”

Types of order are forms of thoughts. They can be studied apart from whatever physical form they may assume. Before observing some further usages of seriality in the visual arts, it will be helpful to survey several other areas where parallel ideas and approaches also exist. In doing this I wish to imply neither metaphor nor analogy.

*My desire was for a* conscious control *over the new means and forms that arise in every artist’s mind.*

—Arnold Schoenberg

Music has been consistently engaged with serial ideas. Although the term “serial music” is relatively contemporary, it could be easily applied to Bach or even Beethoven. In a serial or Dodecaphonic (twelve tone) composition, the order of the notes throughout the piece is a consequence of an initially chosen and ordered set (the semi-tonal scale arranged in a definite linear order). Note distribution is then arrived at by permuting this prime set. Any series of notes (or numbers) can be subjected to permutation as follows: 2 numbers have only 2 permutations (1, 2; 2, 1); 3 numbers have 6 (1, 2, 3; 1, 3, 2; 2, 1, 3; 2, 3, 1; 3, 1, 2; 3, 2, 1); 4 numbers have 24; 12 numbers have 479,001,600. Other similarly produced numerical sequences and a group of pre-established procedures give the exact place in time for each sound, the coincidence of sounds, their duration, timbre and pitch.

The American serial composer Milton Babbit’s *Three Compositions for Piano* can be used as a simplified example of this method (see George Perle’s *Serial Composition and Atonality* for a more detailed analysis). The prime set is represented by these integers: P = 5, 1, 2, 4. By subtracting each number in turn from a constant of such value that the resulting series introduces no numbers not already given, an inversion results (in this case the constant is 6): I = 1, 5, 4, 2. A rotational procedure applied to P and I yields the third and fourth set forms: Rp = 2, 4, 5, 1; RI = 4, 2, 1, 5.

*Mathematics—or more correctly arithmetic—is used as a compositional device, resulting in the most literal sort of “programme music,” but one whose course is determined by a numerical rather than a narrative or descriptive “Programme.”*

—Milton Babbit

The composer is freed from individual note-to-note decisions which are self-generating within the system he devises. The music thus attains a high degree of conceptual coherence, even if it sometimes sounds “aimless and fragmentary.”

*The adaptation of the serial concept of composition by incorporating the more general notion of permutation into structural organization—a permutation the limits of which are rigorously defined in terms of the restrictions placed on its self-determination constitutes a logical and fully justified development, since both morphology and rhetoric are governed by one and the same principle.*

—Pierre Boulez

*The form itself is of very limited importance, it becomes the grammar for the total work.*

—Sol Lewitt

*Language can be approached in either of two ways, as a set of culturally transmitted behavior patterns shared by a group or as a system conforming to the rules which constitute its grammar.*

—Joseph Greenberg, *Essays in Linguistics*

In linguistic analysis, language is often considered as a system of elements without assigned meanings (“uninterpreted systems”). Such systems are completely permutational, having grammatical but not semantic rules. Since there can be no system without rules of arrangement this amounts to the handling of language as a set of probabilities. Many interesting observations have been made about uninterpreted systems which. are directly applicable to the investigation of any array of elements obeying fixed rules of combination. Studies of isomorphic (correspondence) relationships are especially interesting.

Practically all systems can be rendered isomorphic with a system containing only one serial relation. For instance, elements can be reordered into a single line, i.e., single serial relation by arranging them according to their coordinates. In the following two-dimensional array, the coordinates of C are (1, 3), of *T* (3, 2):

R P D

L B T

C U O

Isomorphs could be written as: R, L, C, P, B, U, D, T, O or R, P, D, L, B, T, C, U, O. An example of this in language is the ordering in time of speech to correspond to the ordering of direction in writing. All the forms of cryptography from crossword puzzles to highly sophisticated codes depend on systematic relationships of this kind.

*The limits of my language are the limits of my world.*

—Ludwig Wittgenstein

Certain terms, not common in an art context, are necessary for a discussion of serial art. As yet these terms, often abused, have remained undefined. Some of the following definitions are standard, some are derived from the above investigations, the rest are tailored to specific problems of the work itself:

*Abstract System*—A system in which the physical units that are to function as objects have not been specified.

*Binary*—Consisting of two elements.

*Definite Transition*—A rule that requires at some definite interval before or after a given unit, some other unit is required or excluded.

*Grammar*—That aspect of the system that governs the permitted combinations of elements belonging to that system.

*Isomorphism*—relation between systems so that by rules of transformation each unit of one system can be made to correspond to one unit of the other.

*Orthogonal*—Right angled.

*Permutation*—Any of the total number of changes in order which are possible within a set of elements.

*Probability*—The ratio of the number of ways in which an event can occur in a specified form to the total number of ways in which the event can occur.

*Progression*—A discrete series that has a first but not necessarily a last element in which every intermediate element is related by a uniform law to the others. (a) *Arithmetic Progression*—A series of numbers in which succeeding terms are derived by the addition of a constant number (2, 4, 6, 8, 10 ) (b) *Geometric Progression*—A series of numbers in which succeeding terms are derived by the multiplication, by a constant factor (2, 4, 8, 16, 32)

*Rotation*—An operation consisting of an axial turn within a series.

*Reversal*—An operation consisting of an inversion or upside down turn within a series.

*Set*—The totality of points, numbers, or other elements which satisfy a given condition.

*Sequence*—State of being in successive order.

*Series*—A set of sequentially ordered elements, each related to the preceding in a specifiable way by the logical conditions of a finite progression, i.e. there is a first and last member, every member except the first has a single immediate predecessor from which it is derived and every member except the last a single immediate successor.

*Simultaneity*—A correspondence of time or place in the occurrence of multiple events.

An odd “free” utilization of series was Allan Kaprow’s *18 Happenings in 6 Parts*. His initial set was capriciously chosen—seven smiles, three crumpled papers and nineteen lunch box sounds. The nineteen lunch box sounds were snapping noises made by purchasing many kinds of lunch boxes or recording a few and then altering them until nineteen variations were obtained. The arrangement of sounds will be 3, 12, 7, 8, 10, 2, 15, 6, 1, 13, 5, 18, 4, 19, 17, 9, 14. While being a completely arbitrary use of row technique it does present an interesting possibility for the routinization of spatio-temporal events. Sports, such as football, are based on similar concepts of sequentially fixed probabilities of random movement.

Alfred Jensen’s involvement would appear to be with an unorthodox appreciation of “Number,” judging from such titles as *Square Root 5 Figurations*, *Twice Six and Nine*, and the recent *Timaeus (Plato’s dialogue on esthetics)*. For Plato, as well as Pythagoras, “Number” had an ideal existence and was viewed as paradigmatic (a concept which has been reintroduced into mathematical logic by Russell and Whitehead’s concept of number as a “class of classes”). Whatever the derivation, order in Jensen’s paintings is defined in terms of progressional enlargement and diminishment of adjacent rectangular spaces. Although his color choices seem arbitrary their placement is not, being arranged in bilateral symmetries or systematic rotations. The checkerboard pattern which he adheres to is one of the oldest binary orders.

*The structure of an artificial optic array may, but need not, specify a source. A wholly invented structure need not specify anything. This would be a case of structure as such. It contains information, but not information* about*, and it affords perception but not perception* of.

James J. Gibson, *The Senses Considered As Perceptual Systems*

Perspective, almost universally dismissed as a concern in recent art, is a fascinating example of the application of prefabricated systems. In the work of artists like Ucello, Durer, Piero, Saendredam, Eakins (especially their drawings), it can be seen to exist entirely as methodology. It demonstrates not how things appear but rather the workings of its own strict postulates. As it is, these postulates are serial.

Perspective has had an oddly circular history. Girard Desargues (1593–1662) based his non-Euclidean geometry on an intuition derived directly from perspective. Instead of beginning with the unverifiable Euclidean axiom that parallel lines never meet, he accepted instead the visual evidence that they do meet at the point where they intersect on the horizon line (the “vanishing point” or “infinity” of perspective). Out of his investigations of “visual” (as opposed to “tactile”) geometry came the field of projective geometry. Projective geometry investigates such problems as the means of projecting figures from the surface of three-dimensional objects to two-dimensional planes. It has led to the solution of some of the problems in mapmaking. Maps are highly abstract systems, but since distortion of some sort must occur in the transformation from three to two dimensions, maps are never completely accurate. To compensate for distortion, various systems have been devised. On a topographical map, for example, the lines indicating levels (contour lines) run through points which represent physical points on the surface mapped so that an isomorphic relation can be established. Parallels of latitude, isobars, isothermal lines and other grid coordinate denotations, all serialized, are further cases of the application of external structure systems to order the unordered.

Another serial aspect of mapmaking is a hypothesis in topology about color. It states that with only four colors all the countries on any map can be differentiated without any color having to appear adjacent to itself. (One wonders what the results might look like if all the paintings in the history of art were repainted to conform to the conditions of this hypothesis.)

In the early paintings of Larry Poons it was not difficult to discern the use of serialization. The rules of order varied, sometimes they appeared to be based on probabilities of position and/or direction and/or shape (dot or ellipsoid). Definite transitions or replacements occurred in some paintings. *Enforcer* appeared to be based on a system of quadrant reversal and rotation.

Although Donald Judd’s chief concerns seem to be “specific objects” he has utilized various modular and serial order types. One of Judd’s untitled galvanized iron pieces, consisting of four hemicylindrical sections projecting from the front face of a long rectangular volume, is based on the progression: 3, 4, 3 1/2, 3 1/2, 4, 3, 4 1/2. The first, third, fifth, and seventh numbers are the ascending proportions of the widths of the metal protrusions. The second, fourth, and sixth numbers are the widths of the spaces between. The numbers are not measurements but proportional divisions of whatever length the work is decided to be. Fascinating progressions of the above kind can be found listed in Jolly’s *Compilation of Series*.

Sol LeWitt orders his floor pieces by permuting linear dimension and binary volumetric possibilities. Parts of his eighteen-piece Dwan Gallery (Los Angeles) exhibition could have appeared at one of three height variables, in one of two volumetric variables (in this case all were open) and in one of two positional variables, on a constant 3 x 3 square grid. The complexity and visual intricacy of this work seems almost a direct refutation of Whitehead’s dictum that the higher the degree of abstraction the lower the degree of complexity. LeWitt’s two-dimensional orthogonal grid placement system is related to the principles of order in mapmaking.

Other artists are currently exploiting aspects of seriality. Dan Flavin’s *Nominal Three to Wm. of Ockham* (author of the law of parsimony) is based on an arithmetic progression. Jo Baer’s paintings, Hanne Darboven’s complicated drawings, Dan Graham’s concrete poetry, Eva Hesse’s constructions, William Kolakoski’s programmed asterisks, Bernard Kirschenbaum’s crystallographic sculpture, Robert Smithson’s pyramidal glass stacks all suggest future possibilities of serial methodology.