PRÉCIS OF A RENAISSANCE APPROACH TO CLASSICAL
GEOMETRY, PROPORTION & COMPOSITION

PART II

David Mayernik, FAAR, RSA
Assoc. Prof., University of Notre Dame

26 December 2007

CONTENTS

Go to PART I

INTROIT II
MUSIC & THE BODY
Number in Music and Architecture: Dufay and Brunelleschi
Number in Music and Architecture: Francesco Colonna and Bramante
Mathematics and Geometry: Number and Proportion
Corpus ad Quadratum and Ad Triangulum: symmetry and correspondence
Seeing Proportions: Brunelleschi, Alberti, and Palladio
THE JUDGMENT OF THE EYE
Michelangelo, Bernini, and the artist/architect
Disposizione
BRIEF BIBLIOGRAPHY II

INTROIT

In dealing with the seductive possibilities of the relationship between music and architecture in the Renaissance, it is useful to recognize the three distinct aspects, or phases, of this relationship: invention, intention, and perception. Rarely, if ever, do historians of the period distinguish these three distinct operations, implying that they are all effectively the same. I would like to quite clearly make these distinctions, in order to proceed more methodically to the ways in which the relationship between music and architecture might have been understood from the fifteenth through the seventeenth centuries, by artists and their audiences—in other words how the relationship was used, and how it was appreciated.

       Invention for the Renaissance effectively takes the place of our word creation; with its corollary meaning of “discovery,” it suggests a process that draws upon ideas and forms that already exist, and so an invented idea is one with roots, even if it grows into something novel. How artists of the period invented was an exercise with its own rules, method, and mythology. While of some interest to connoisseurs, the artist’s approach to the inventive process was not always meant to be evident to an audience of viewers or listeners.

       Intention, on the other hand, is that aspect of a work of art that is meant to be perceived. It is the embedded message, the formal strategy, or the broader reference intended to be available to someone other than the inventor. While intentional, these sometimes-coded messages were not always successful in being plainly evident.

       Perception, then, is what is actually available and perceived by an audience; of course, no audience is homogeneous, and the relative experience and knowledge of each member conditioned what he or she was able to perceive. Nowhere is this more of an issue than in the question of proportion in Renaissance buildings, or the relationship between musical and architectural harmonies.

MUSIC

Number in Music and Architecture: Dufay and Brunelleschi

First, we have 6 x 4 = 24, yielding the basic "small" module of the cathedral group (the Baptistery sides, the Campanile breadth, also the facets of the Cathedral tribunes). Next appears the operation 6 x 4 x 3 = 72,producing the "intermediate" module of the nave width and height, and the-cupola diameter. Finally we provide multiplication of the full set, 6 x 4 x 3 x 2 = 144, or the length of the entire nave and the height of the Cupola. The numerical set 6.4.3.2 (present in the motet), in other words, contains in the most condensed form possible the entire cathedral (indeed, the entire cathedral group of buildings, including the 144 braccia Campanile), whose virtual completion was marked by the 1436 consecration.^[1]

Perhaps the most direct case of music and architecture sharing similar numeric principles is Dufay’s motet Nuper rosaurm flores written for the dedication of the Duomo in Florence in 1436, when Brunelleschi’s dome had for all practical purposes been completed (it would still need its lantern). [Hear it here.] Scholarly debate has gone back and forth about just how precisely Dufay’s musical structure, expressed in numerical ratios (6:4:3:2), matches the Duomo’s proportions, but I’ve quoted from the most recent treatment by Marvin Trachtenberg, whose common sense approach to the embedded numbers of the great building mostly accords with my own sense about how available proportions are to an audience of reasonably informed spectators. What I would like to add to Trachtenberg’s thorough treatment are the following observations, in a similarly common sense vein, about invention, intention, and perception.

      First, it needs to be said that how we see these ratios is not how they were necessarily conceived. It may very well be that, as Trachtenberg suggests, the dimensions between Baptistry and Duomo generate these precise ratios, but it is not wholly credible that any audience—no matter how visually acute—could actually perceive or know these dimensions. Dimensions presume a unit of measure, and while each community is generally capable of judging the size of its unit (a foot, a palm, a braccia or arm), it is unlikely anyone could visually measure—and thus perceive—seventy-two of them. What we can do is count a unit—an arched nave bay, for example—and comprehend the number of bays long or wide a particular space actually is.

      Following on the first point, then, it may be that Dufay knew these dimensions, and was intrigued by their concordance with musical structure—perhaps deliberately so—and treated the set as a generative idea for his music that would not necessarily be rationally available, but could be somehow sensed, by his audience: his expectations then were for sympathy in his audience, rather than understanding. If, however, he expected his audience to actually perceive his ratios in Brunelleschi’s building (or, better, in the complex of buildings that includes Brunelleschi’s dome) the ratios had to be perceivable somehow. And, to my mind, they would not be if they depend on measure (again, which presumes an invisible unit) rather than proportion (which depends on simple ratios of things that can be seen).

      Architectural proportions may be analogous to notes, chords, or musical structure, but they are no more music than the proportions are “architecture.” What they provide is a rational framework for relating buildings and motets to the structure of creation, and in that sense they anagogical—not ends in themselves, but means of affecting the soul. Music, as the fifteenth century cardinal and philosopher Nicholas of Cusa believed, made its ratios known to the attuned mind and by this rational means affected the soul:

While listening, we perceive with our senses the concordant parts; we measure the intervals and concordances with our reason and with the help of our musical training.... On account of this we are entitled to call our soul reasonable: viz, because it is a measuring and numbering power which grasps whatever requires precise distinction.... Thus reason, seeing that concord is based on number and proportion, invented the rational theory of musical chords, based on the theory of numbers.^[2]

      Nicholas’ concern is the ultimate effect of music on the soul, and Dufay most likely was interested in the same thing. He believed, in other words, that a rational concordance between his music and Florence’s building (a building he well knew had been realized by many hands over many years) was more than an intellectual game, it was an anagogical tuning of his audience to the greater cosmic structure.

Number in Music and Architecture: Francesco Colonna and Bramante

If an order constitutes the mode of a colonnade, then the spacing of the columns must constitute its time.^[3]

      The ways in which music is perceived in architecture is too often limited to discussions of number ratios and proportion, and yet there is some question as to whether—beyond Palladio’s famous discussion of the correct proportions of a room—these were really as significant at the scale of individual bays for Renaissance architects as they are for some architectural historians. On the other hand, there seems to be a direct, evident understanding of rhythm and pace in the Renaissance approach to metrical units like arcades, facade bays, etc. Moreover, as Onians suggests, the particular classical order employed had a role in establishing the mode, or mood, of a composition—somber Doric, for example, or lively Corinthian.

If they say that music satisfies the hearing, which is a natural sense, I say that perspective satisfies the sight, which is more noble inasmuch it is the “first gate of the intellect.”^[4]

      The primacy of eye or ear in Renaissance theory hinges on their nobility, which is conditioned by the nobility of what they perceive. Luca Pacioli, in sympathy with Alberti and Leonardo, privileged the eye as gate of the intellect, presuming then that music was more affective than intellectual in its effect on its audience. Bramante, a lutenist in addition to being an architect and painter, seems to have brought a musical sensibility to his composition of wall surfaces, as it is he who most systematically introduces in the cinquecento the complex bay rhythms that would characterize the next centuries. In the sequence of bays one measures around the interior wall surfaces of his centralized scheme for St. Peter’s, or in the bays of the Belvedere court, accent, rest and a lyrical sense for contrast characterizes his alternating pilaster clusters, plane surfaces and niches. He was, too, immensely sensitive to the orders as modes, most notably his preference for the sonorous doric.

Number and the Body: Leonardo and Vitruvius; Corpus ad Quadraturm and Corpus ad Triangulum

Leonardo’s famous drawing of the Vitruvian man is only one of several interpretations of the Roman architect’s description of the proportions of the ideal human body, and it has become so iconic in part because it so succinctly summarizes the Vitruvian text (most others show only the circle or the square, not both at once as Leonardo did). A side-by-side comparison of Leonardo’s figure with a nineteenth century anatomist’s points up the critical issue of the eight-head module: it is this that generates, more than the circumscribing figures, the intermediate elements like chest, groin, and kneecap. The reason Vitruvius describes the figure at all, though, needs to be borne in mind: since the body was the most perfect creation (and here Vitrivius’s text elides with Genesis), its forms and principles make it the perfect model for architecture, or at least the architecture of the most perfect building type, the temple. What, in the end, are these principles? They include:

1. Symmetry (axial or bilateral, not necessarily biaxial or radial); indeed virtually all multi-cell organisms exhibit some form of symmetry:
   
2. Broad simplicity, local complexity; or, diversity in unity, the idea that the contours of the body are sinuous and continuous, and yet each member (head, hands, legs, etc.) has its own unique, discrete qualities

In so far as a tradition in architecture can be called classical, it must rest on two analogies: of the building as a body, and of the design as a re-enactment of some primitive—or if you would rather—of some archetypal action to which our procedure might refer. From Vitruvius to Boullée, the texts suggest something of the kind, always in different contexts, since such ideas do not contain, or even imply, the repertory of norms and procedures which the constant alteration of circumstances forces you to renew, to rethink and to alter….^[5]

Seeing Proportions: Alberti and Palladio

From the foregoing it should appear evident that problems of proportion in architecture are inherently visual, i.e. they must be seen to be believed. Self-evident as this may seem, the issue has been confused by the association of architectural and musical ratios: while it is certainly true that architects and musicians participated in a common culture that found in mathematical proportions a key to the perfection of the heavens, of a world both beyond this one and at the same time visible from it in the night sky (and occasionally manifest here in meraviglie naturali like shells, stones, and leaves), they were practical and sensible enough to understand music and architecture to be related, not the same.

      The point of this distinction is to recalibrate our proportional minds away from the overlaid geometrical figures on facades and plans that historians once loved and architects still continue to develop almost as an article of faith, and toward the more obvious manifestations of proportional relationships that can be seen in buildings: things arranged in series like arches, perceivable geometric figures like the square (and simple permutations of it like one and a half and two squares) in the end elevations of rooms or the plans of modular spaces like the naves of churches, and compelling platonic solids like cubes, spheres, pyramids and their combinations (one thinks here of Leonardo’s projects for churches).

      Alberti and Palladio, at least since Wittkower’s Architectural Principles in the Age of Humanism, have been the gonfalons of the importance of harmonic proportions in architecture, and yet an attentive reading of what each of them said shows that they spent almost no time discussing a systematic theory of proportions, and much time on the architectural manifestations of them. They took the nature and efficacy of harmonic ratios essentially for granted, and took for granted as well that architects would be well-served by employing them to order their plans and elevations whenever they could. For Palladio, most notably in his famous discussion of the ratios of room plans to elevations, harmonic proportions are a technique for guaranteeing an attractive space (albeit with an understanding that the reason they would be found attractive was our capacity to perceive the underlying geometric structure of creation). For both architects the essential thing is that a well-proportioned plan or facade would be “pleasing to the eye,” and Alberti’s personal emblem of the winged eye suggests the primacy he assigned to that sense, and indeed a primacy accorded it by a wide variety of humanist thinkers.

 
THE JUDGMENT OF THE EYE

Michelangelo, Bernini, and the Artist-Architect

This dominion of the eye finds its haughtiest reign in the minds of the two greatest artist-architects of the sixteenth and seventeenth centuries, Michelangelo and Bernini. It was the former who famously said an architect must have “compasses in his eyes”, and the latter who responded to the question of what proportional system he had used for the Baldacchino at St. Peter’s with “my eye.” These are, of course, not anyone’s eyes, and they are not the sloppy “eye of the beholder” that champions aesthetic relativism. These two masters of the human figure had trained their eyes to perceive and document the beauty of the body in both its real-world manifestations and more particularly in the ideals represented by classical theory and antique precedent. It is precisely their command of the body in drawing, painting and sculpture that gives them eyes to judge and measure, eyes that know whether proportions are “good” in the most complex way possible: relatively. For, as they both knew, the “correct” proportions of a figure are a relative thing: head to torso, hand to arm, foot to leg. It was the juxtaposition of parts that they mastered when they mastered the figure, and it was this that allowed them as masters of disegno to know when the parts of architecture were “right”; context, contrapposto, frame all determine how we perceive proportions, and make any sort of proportional diagram a rather blunt tool, available only to the orthographic viewer. Where ideal proportions are available are in autonomous, prismatic objects like S. Maria della Consolazione in Todi, where geometry self-evidently governs the form.

Disposizione

So we come finally to the great contribution of the Renaissance approach to composition, disposizione. The genius of Renaissance artists lies in their rational capacity to dispose, that is, to refine and distribute the parts—of a painting or a building—toward the realization of a bel composto (a beautiful whole). Numerical proportions are one part of that process, but never a generator of it. Disposition was also more than formal, it was poetic or iconographic, and indeed the distribution of various paintings in a humanist camera or chapel was more often driven by questions of meaning than aesthetics. We read fresco cycles and spaces sequentially, so while we as architects can establish mathematical relationships between parts and the whole, we can only rarely condition precisely how those relationships are to be seen. Some examples of unique conditions where proportions established on planar geometries are actually available to the viewer are Palladio’s two great church facades in Venice, the Redentore and S. Giorgio Maggiore, both seen as two-dimensional compositions from substantial distances that dissolve perspective and become instead virtually orthographic projections.

      In most other cases the experience of architecture is dynamic, and relative. The “relativity” of disposizione is allied to narrative, in the sense that sequences of paintings, walls, and rooms beg the possibility of unfolding, the idea that they lead somewhere. Just as the parts of the body only make sense in relationship to the whole, so proportions (as subdivisions) live in their relationships. Where this becomes meaningful, and therefore valuable, is when they add up to something of significance--when they are not the signified but signifiers, in other words.

As a model of this relationship, we might consider Sebastiano del Piombo’s frescoes in S. Pietro in Montorio. Below, on the semicircular walls, a colonnade dissolves the boundaries of the space, and lashed to the central column is a muscular Christ. In the semidome above a central Christ appears transfigured among the chosen apostles, in the company of Moses and Elijah. The linear connection between the tortured Christ, the column, and the radiant Christ above is overt, and meaningful. On a grander scale, the same juxtaposition of architectural space and an architecture of figures links Raphael’s School of Athens and the Disputà in the Vatican’s Stanza della Segnatura. Here, in this modest chapel on the Janiculum, the relationship is more compact and synthetic. The colonnade is Corinthian, whose diameter of shaft to column height is 1:10; Christ’s proportions of head to overall height of 1:9 is rather elongated by Vitruvian standards (1:8), making Him more elegantly Corinthian himself. The two pairs of columns that flank His are repeated in the number of his torturers, and roughly matched in number and disposition by the apostles and prophets above. As the only upright figure in the Transfiguration He is especially columnar, ennobling thereby the columns of the Temple courtyard below. The reflexive relationship between the two scenes can be understood in terms of number (figures and columns, proportions), anthropomorphism, and exegetically. An analogous juxtaposition existed in another painting in the same church until the late eighteenth century, when Raphael’s famous Transfiguration was relocated by Napoleon. The echoes within Sebastiano’s and Raphael’s compositions therefore were reflected from one work to another, and in that dynamic exchange the Renaissance understanding of proportion and composition—in intention and perception—resonates most clearly.

BRIEF BIBLIOGRAPHY

Nicholas of Cusa: Selected Spiritual Writings, H. Lawrence Bond, tr., (New York: Paulist Press)
Rudolf Wittkower, Architectural Principles in the Age of Humanism, Norton
Kim Woods, ed., Making Renaissance Art, Yale

^[1]Marvin Trachtenberg, “Architecture and Music Reunited: A New Reading of Dufay's ‘Nuper Rosarum Flores’ and the Cathedral of Florence”, Renaissance Quarterly, vol. 54 no. 3 (Autumn 2001), p. 755
^[2]Nicholas of Cusa, De ludo globi, lib. II, translated in Kathi Meyer-Baer, “Nicholas of Cusa on the Meaning of Music,” p. 304
^[3]John Onians, Bearers of Meaning, Princeton, 1988, p. 214
^[4]Luca Pacioli, De divina proportione (pp. 40-41), quoted in Onians, p. 217
^[5]Joseph Rykwert, “The Ecole des Beaux-Arts and the Classical Tradition,” The Beaux-Arts and Nineteenth Century French Architecture, ed. Robin Middleton, MIT Press, 1982, p. 17
 

Go to PART I

******************