Meter and Perspective
Perspective is a particular construction for representing pictorial space. The space represented by a perspective construction is continuous, homogenous, measurable, and represents one moment in time. The pictorial space is behind the picture surface, the side opposite the physical space in which the viewer stands, so that the picture is like a window. The space is created by a network of lines, angles, points, and planes, and is enhanced by shading and gradations in tone. The most important principle of linear perspective is that orthagonals - the lines that represent perpendiculars to the picture surface - should converge on one vanishing point. The converging orthagonals have two important consequences. One, the shape of a figure must be drawn, not with the measurable angles of the physical object, but rather with angles distorted to coincide with the orthagonals. Therefore the surfaces of a cube that recede into the picture space will not be drawn with right angles. Two, the size of a figure will be determined by its virtual distance from the picture plane. The deeper into the space a figure is, the smaller it will be.
Michael Kubovy identifies five functions of perspective. The first function is the obvious one - the rationalization of space, the creation of the illusion of depth, a space in which naturalistic figures can be represented in a variety of complex spatial relationships. A second function is to provide "narrative focus," that is, to draw the spectator's eye to a principal figure or action. A third function is similar to the second, to provide "structural focus" - a way to organize shapes or motifs on the surface in order to focus attention on structural, rather than representational, features. The fourth function is expressed in a phrase of Warman Welliver, a "code for concealing allusion and meaning." Since homogenous space is measurable, proportions in a picture can refer, for example, to the Trinity. The fifth function is Kubovy's contribution, and will be taken up a bit later in this paper. A discrepancy between the actual position of a viewer and a picture's center of projection creates a "dialectical tension" that is essential to the emotional impact of the picture.
Meter is a feature of the rhythmic organization of Western common-practice music. My favorite definiton of meter is Wallace Berry's "accent-delineated grouping." Joel Lester identifies two components of meter - a stream of beats, which "mark off functionally equivalent spans of time," and an organization, by accents, into regular groups. Meter is essentially an interaction between two levels. Carl Schachter hears meter as "the articulation of regularly recurring equal segments of time." The "boundary points" of the equal spans attract attention, resulting in "metrical accents." The equal divisions require some kind of emphasis (or accent) initially, but then "persist in the listener's consciousness." Regardless of the particular definition one uses, three features of meter are common property. First of all, meter involves the regular recurrence of articulated units of time. Theorists of rhythm are divided into two camps over whether meter is, by definition, necessarily regular. (I would like to argue the irregularists' position, but this is not the place.) We can sidestep that discussion because the primary metrical levels of common-practice music are overwhelmingly regular. Secondly, some events and moments in time are relatively strong, and others are relatively weak, or accented and unaccented. Thirdly, the most provocative aspect of meter is its operation at several levels; it is a hierarchy. Several levels have names: the beat, the division of the beat, the measure, and the hypermeasure.
Each topic, meter and perspective, is fascinating in its own right. But correspondences between them are striking, correspondences in their nature, function, and place in the history of their respective arts.
Perspective and meter both provide a framework, or a scaffold, or a grid, on which the painter or composer hangs his figures. Alberti, sometimes identified as the inventor of linear perspective who gave us the notion of the picture surface as a window, and Viator, whose textbook on artificial perspective was enormously influential in the Renaissance, both provided techniques for making a perspective construction (Figure 1). Even a brief glance at their construction diagrams shows one the framework. A study by Leonardo da Vinci for A Flight Of Steps and Horses (Figure 2 below) illustrates the perspective scaffold, with the continuous orthagonals converging at the vanishing point. In the finished picture some of the orthagonals would be articulated as floor tiles, many would be obscured and interrupted by figures, or would only be implied by the short segments of the steps. John White says of Masolino's Dispute of St. Catherine (Figure 3) that it looks like the perspective grid was left in place after the figures were added, "and then congealed into the form of architecture."
Figure 1a: Alberti's construction of a square cast in perspective.
Figure 1b: Viator's construction of a square cast in perspective.
Figure 1c: Viator's construction of a square inscribed in a circle cast in perspective.
The Pierpont Morgan LIbrary, New York, PML 25572.
Figure 2: Leonardo da Vinci"s Study for Flight of Steps and Horses, Simplified line drawing by Don Scharfenberg.
Figure 3: Massolino: The Dispute of St. Catherine
Meter, also, is a scaffold. Once the frame is in place, the rhythmic motives and patterns are heard in their relationship to it. Not all levels of the hierarchy will be articulated at any point in the music. They need not be; we will continue to comprehend the whole metric structure. The opening four bars of the Brahms Clarinet Quintet (Example 1), for example, establish the basic metrical structure by articulating units at the levels that will be normative, that will underlie the first movement. The dotted quarter-note beat, which is implied by the time signature and tempo indication, is articulated by the unified groups of sixteenth notes, by the recurring quarter-eighth rhythm pattern with its agogic accent, and, of course, by the harmonic rhythm. The eighth-note division of the beat is articulated by the quarter-eighth rhythm pattern, even though the second eighth note of the group is not attacked. The sixteenth-note subdivision is the surface value of the second half of mm. 1 and 2, and is the composite rhythm of mm. 3 and 4. The measure is implied by the bar lines, but is articulated by the agogic accents and the pattern repetition of mm. 1-2 and the sequence of mm. 3-4. The two-bar hypermeasure is established by the change in motive and rhythmic patterns in m. 3 and the entrance of the low strings. The next level, the four-measure unit, is articulated subtly by the entrance of the clarinet in m. 5, and is more clearly articulated later in the movement.
Example 1: a) Brahms Clarinet Quintet op. 115, I, mm. 1 - 4. b) Diagram of metrical hierarchy.
c) Diagram of metrical hierarchy after Lerdahl and Jackendoff.
I would like to focus on two functions of perspective from Kubovy's list of five. First, the illusory function of perspective in creating space. The depiction of motion and development of pictorial space are intertwined in the history of Western art. In discussing the "Greek Revolution" of the sixth to fourth centuries B.C., Gombrich says, "Narrative art is bound to lead to space and the exploration of visual effects." When the artist's concern for the "what" - "the timeless function of the potent image" - is extended to the "how" of a single "imaginary fleeting moment in time", the function of the art requires space. White observes that Greek vase painters (from the time of Exekias, see Figure 4) used foreshortening and the three-quarter view to depict only three things - shields, sails, and chariots; the depiction of motion led to the depiction of space. Function swung the other way in the early Christian era. "The Byzantine icon is not conceived as free 'fiction'; it somehow partakes of the nature of a Platonic truth." It is a "timeless reenactment," a potent image, and the surface again is flat. A wonderful example of the equivalence of space and motion in fourteenth-century French manuscript illumination is a pair of scenes described by White. In the first scene a group of personages is climbing a set of stairs to a pulpit (Figure 5). While the stairs support the weight of the climbers they are depicted in a foreshortened frontal setting. In the next scene, the worthies are on the pulpit, the stairs are no longer needed to support the action, so they have reverted to the surface and are flat on the picture plane.
Figure 4: Warring Hoplites: Attic Black Figure Amphora
Staatliche Antikensammlungen und Glyptothek München.
Figure 5: Scenes from the Coronation Book of Charles V. MS. Cotton. Tib. B. ViII. Fols. 63.
and 64. British Museum, London.
By permission of the British Library.
Another kind of motion is the motion of an observer's attention directed through a perspective space. Masolino's Dispute of St. Catherine (Figure 3) is a hole punched into the picture surface. The illusory space is rational, and can be measured by the ceiling panels. Kubovy's second function, narrative focus, comes into play. The vanishing point of the picture is right in front of Catherine's face, at the left hand of the judge. Three of the four heads on either side of Catherine are arranged along an orthagonal. No matter where one looks - at a wall, ceiling, councilman - the eye's attention is dragged back to the spot on the surface between Catherine's head and the judge's hand. That point also serves to create a dramatic interplay between these two characters, even though they occupy different planes in the picture space, and Catherine is not looking at the judge.
John White analyzes an example in which the motion is more complex and multi-directional. In the S. Lorenzo Annunciation of Filippo Lippi (Figure 6 below), the left-most angel is glancing out at the observer, drawing attention into the picture space. Then the angels and Gabriel direct attention over to the Virgin at the right of the picture. All four figures are in the foreground, near the surface, but once attention has reached the Virgin, the strong, unbroken orthagonals of the brightly lit architecture draw the observer deep into the quickly receding space. The vanishing point is just to the right of the central painted pillar. The brightness of that pillar pulls attention right back to the surface, and the cycle of movement begins again.
Figure 6: S. Lorenzo Annunciation of Filippo Lippi. Simplified line drawing and analytical sketch by Don Scharfenberg.
The most striking feature of the opening eight measures (Example 2) is, of course, the increase in surface activity from the half note, to the quarter, to the eighth, and finally to the sixteenth. Along with this two-dimensional motion is motion into the third dimension of metrical depth due to the accretion of levels. From measure 2, where three levels are articulated with some degree of clarity, we hear a gradual accumulation of levels; the density of levels in measure 8 we hear as the greatest depth.
Example 2: Mozart Clarinet Quintet, K. 581, I mm. 1 -8
A second process exposed in these measures is expansion and contraction between metrical levels. The half note level is primary in measure 2, almost disappears in measure 3 in favor of the clear articulation of the whole-note and quarter-note levels, and regains its primacy in measure 4, where the quarter-note level disappears and the whole-note level becomes secondary. The expansion followed by contraction propels the listener's attention through the metrical space.
The motion generated by contraction and expansion is important enough to warrant another, more complex, example (Example 3). The striking change in motive in measure 23 is accompanied by striking motion through the metrical space. The half-note and eighth-note levels are primary in measure 22, while the quarter-note and whole-note levels are tertiary. In the following measure the eighth-note level disappears, the half-note level declines in intensity, and the whole-note, quarter-note, and sixteenth-note levels emerge as primary. The motion expands from each of the primary levels of measure 22 into measure 23, and at the same time contracts.
In fact, the entire first movement is characterized by the unsettled relationships between the three central normative levels - the measure, beat, and division - and the motion of the listener's attention directed between them. The only two points at which the three levels are reconciled - all primary - are the last measure of the exposition and the last measure of the movement; the metrical motion is all directed toward those two points of stability.
Example 3: Mozart Clarinet Quintet, K. 581, I mm. 19 -25
Another function shared by perspective and meter is the creation and control of tension. Kubovy describes what he calls a "dialectical tension" between a viewer's physical position and a picture's center of projection. The center of projection is a geometrical abstraction of a point in the physical space in front of the picture plane, and is inferred by the mind from the location in the picture of the vanishing point or points. Early theorists of perspective believed that the viewer's eye had to be at the center of projection for the spatial illusion to be convincing. Brunelleschi's famous peepshow was a device that insured that a viewer would look at his picture from the center of projection by forcing him to look through a peephole a measured distance from the surface of the picture. But a perspective picture viewed from a spot other than its center of projection does not look wrong; it does not lose its spatial illusions due to what Kubovy calls "the robustness of perspective." Perspective is so robust that it has the psychological effect of moving the viewer. The center of projection of Leonardo da Vinci's Last Supper is more than 15 feet above the floor of the chapel. "Leonardo used perspective to elevate the viewer to an extraordinarily high center of projection, thus achieving a feeling of spiritual elevation."
Several points of view within the same picture create tension. Masaccio's fresco, The Trinity, (figure 7 below) has two points of view, one for the architecture and secondary figures, and another for the figures of the Father and Son. The perspective of the architecture has a very low viewpoint, placing the viewer at the bottom of the picture. The two principal figures, though, are seen straight on. When the viewer's attention moves from the architecture to the figures of God, the viewer is elevated, an emotional reaction that
heightens the spirituality of the picture. In fact, the very existence of perspective creates tension between what Kubovy calls "two kinds of incompatible information" - the scene in three dimensions and the picture surface in two dimensions. A major theme of John White's book is the Renaissance artist's continual struggle to balance the deep space of representation with the decorative requirements of the surface. The painted central pillar in Lippi's The Annunciation is placed right next to the vanishing point, the point of greatest depth in the picture space, in order to balance that very tension.
Figure 7: Masaccio, The Trinity.
Masaccio, La Trinité, Firenza, Santa Maria Novella.
A body of interesting literature is growing on kinds of tension created in music with reference to meter. Especially important among the recent contributions are Joel Lester's discussion on "metric ambiguity," Carl Schachter's rich compendium of such rhythmic "incommensurable levels," and "conflicting metrical patterns;" and Harold Krebs' extension of the notion "metrical dissonance." In my own work I am fond of the term "metrical discord" as a general term incorporating various kinds of lack of accord between different metrical articulations.
A particularly rich example is from the development of the first movement of Brahms' Clarinet Quintet, mm.114 - 120 (example 4). The metrical units articulated by changes in harmony and the recurrence of several motives are out of accord with one another, and nowhere in the passage do they accord with the notated measure. The level of the beat is clearly articulated by the rhythmic pattern of the second violin and viola, but above that level the metrical tension is palpable.
Example 4: Mozart Clarinet Quintet, K. 581, I mm. 1 -8
A last correlation between meter and perspective is historical. Both perspective and regular, hierarchical meter are phenomena of Western art and music from the Renaissance through the nineteenth century. Suzi Gablik attempts to show connections between the development of the history of art and Piaget's stages of cognitive development in an individual. While I am not prepared to embrace her provocative theory wholeheartedly, it does shed some light on the relationship between perspective and meter. The art and music I have been discussing are from the "Concrete-operational Stage", in which space is organized according to "Projective and Euclidian relations."
The previous stage - the "Pre-operational Stage" - is the period of ancient and medieval art in which spatial organization is marked by an absence of depth and lack of unified global space. Miriam Schild Burnim identifies "discoordination" between figure and background as a prime contributor to the flatness. Her definition of discoordination, in which each element "gives the impression of having been composed independently and brought together without a definite systematic or proportional connection between their individual parts," sounds strikingly like a simplified definition of the relationship between lines in medieval polyphony - complete lines composed successively with minimum concern for how they fit together. In Renaissance polyphony the concern for treating dissonance is an important step in distinguishing strong and weak beats - a basic element of meter. In a Gothic manuscript illumination, a small all-over repeated pattern covers the entire background emphasizing a substantial two-dimensional surface, intentionally obliterating even the vestige of stage space characteristic of Byzantine and some Romanesque illuminations. (See figure 8 below.) The flatness of Notre Dame polyphony is similarly based on repeated patterns; modal rhythm was not a groping toward metrical rhythm. As William Waite points out, "A pattern is, after all, something already more subtle, more stylized than a rhythm based on sounds of equal duration."
Figure 8: Initial Letter, Book of Hours, London, British Museum, Egerton MS, fol.47.
By permission of the British LIbrary.
The Modern Period of art, according to Gablik's scheme, corresponds to the "Formal-operational Stage" of cognitive development. Space is governed by "logical systems and by propositional thinking." In a Mondrian painting the right angle is the basic element, and it is replicated and arranged on the picture surface according to its own logic, without reference to any concrete object. Any number of examples of formal-logical and propositional systems for organizing rhythm in twentieth-century music can be identified: Boris Blacher's arithmetical progressions; Messiaen's rhythmic modes; Boulez's serialization of rhythm; or Babbitt's time-point system, in which durations are tied to pitch intervals.
Meter and perspective allow two art forms - painting and music - to encroach upon the other's primary illusion. Painting has real physical space - a canvas or panel or paper of particualer dimensions. But it also has as its primary illusion virtual space (to use Langer's terminology). Music has real physical time - chronometric, or clock time. Its primary illusion is virtual time, or integral time. As we have seen, meter, with its hierarchy and motion and tension, crates virtual space, which Langer calls its secondary illusion. Perspective space permits a picture to capture a moment, and the motion generated by perspective entails virtual time, which could be considered a secondary illusion. Meter and perspective can be seen as two manifestations of a single "spirit-of-the-age" for four hundred years of Western intellectual history.
 This description of perspective is distilled from the following sources: Michael Kubovy, The Psychology of Perspective and Renaissance Art (Cambridge: Cambridge University Press, 1986); John White, The Birth and Rebirth of Pictorial Space, 2nd ed. (Boston: Boston Book and Art Shop, 1967); William M. Ivins, Jr., On the Rationalization of Sight (New York: Da Capo Press, 1973); Miriam Schild Bunim, Space in Medieval Painting and the Forerunners of Perspective (New York: AMS Press, 1970); Radu Vero, Understanding Perspective (New York: Van Nostrand Reinhold Company, 1980); Samuel Y. Edgerton, Jr., The Renaissance Rediscovery of Linear Perspective (New York: Basic Books, 1975).
 Alberti's and Viator's constructions of a square cast in perspective are reproduced in Ivins, Rationalization, as Figures 1 and 2 on p. 15. Viator's De Artificiali Perspectiva, in its 1505 edition, is reproduced in facsimile with Ivins. Viator's second diagram, a square inscribed in a circle cast in perspective, and his twelfth diagram, a church with its perspective grid, among others, illustrate the nature of the framework.
 The hierarchy diagram of 1b is mine. The diagram of 1c uses a notation developed by Fred Lerdahl and Ray Jackendoff in "On the Theory of Grouping and Meter," The Musical Quarterly, (LXVII, No. 4; October 1981) pp. 479-506; and A Generative Theory of Tonal Music (Cambridge: MIT Press, 1983).
 Gothic manuscript illuminations that illustrate the characteristic background pattern can be found in Bunim, Space in Medieval Painting. Figures 38, 39, p. 233; 44, 45, p. 235; and 61, 62, p. 242 (though these last two date from a later period and include the reintroduction of a stage space).