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Published onApr 22, 2021

Palladio is sometimes called the most influential architect in history. The Palladian and the neo-Palladian villa, in particular, have dominated important stretches of architectural development from the Renaissance to the present. Certainly no other house designer has enjoyed such prestige and attracted so many direct imitators over so long a period. Many an architect who has built more imposing masterpieces than Palladio’s has been less notable in these latter respects. Today, more books and articles than ever are devoted to Palladio.

It is often observed that Palladio’s villas embody geometrical rules.1 But there is less certainty as to precisely what the rules are. He wrote some of them down and hinted at others, but most have to be extrapolated from his work; and that is where the disagreements lie. Even assuming that agreement may one day be reached as to the nature of these rules, it could still be objected that in searching them out we have devalued the originality and genius of this architecture, that we have reduced Palladio to a game. And we do confess, certainly, that we have not attempted here to evaluate the man’s genius—though of course we completely acknowledge it.

Anyway, much art is gamelike. Numerical analyses, counting, statistics, and the like often accompany artistic greatness. Homer, for example, is a poet with a strong sense of numerical and geometric design. This has been proved by any number of statistical and arithmetical “counts,” counts that involve symmetries within speeches, incidents, or plots in which the poet uses a device called “ring-composition,”2 and in meters and even word order. Sometimes Homer exercises these numerical constraints in an extraordinarily detailed and consistent way, though it is usually something the ordinary reader never notices. Thus Eugene O’Neill, Jr., shows that a word ending in a single short syllable is almost always avoided at position 7½ in Homer’s hexameters and that, in the ninth position in the line, with many possibilities to choose from, his word endings are almost completely limited to either short-long-long-long, or short-long-short-short-long.3 Consciously or unconsciously, Homer obeyed certain numerical rules. He counted, calculated, measured, and mirrored just as so many great artists have done in so many fields. Palladio did the same.

Despite his partial silence on the subject, Palladio does seem to invite us to dig his rules out. By the very fact of publishing the Quattro Libri dell’Architettura (1570), with its plans, elevations, and details of ornament—parts of buildings, whole buildings, and procedures for assembling given parts into new wholes—he presupposes a reader who might want to create his own personal selection from the elements provided. In other words the book itself is a set of rules, a set of possible plans, possible facades, possible details, rules, or possibilities, that are applied rather than articulated. So in the following pages we are simply taking up Palladio’s implicit challenge. Stationing ourselves historically in the late 1560s when, toward the end of his long and splendid career, the architect was readying his book for publication, we watch him play (or replay) forty-odd matches of his architectural game.4 Then we guess the game’s rules.


One popular hypothesis that seems relevant to our project is that the geometrical rules governing Palladio’s plans and facades are assemblages of “ideal” shapes, or that “ideal” or “harmonic” dimensions are used.5 This of course reflects the common Renaissance idea that certain numbers and shapes are nobler than others. A 4:3 rectangle is superior, say, to a 46:39 rectangle because it is a simpler, more basic proportion to calculate. (Four divided by three is 1⅓, 46 divided by 39 is 1.179487195; the relatively commensurable 1⅓ was greatly preferred to 1.179487195, which for practical purposes is incommensurable.) The 4:3 rectangle was also thought to be more harmonious, more beautiful, than rectangles of the latter type. These ideal shapes have been a field of architectural study ever since Rudolf Wittkower published his classic Architectural Principles in the Age of Humanism, which first appeared during World War n.6 Robert Streitz, Colin Rowe, and D. H. Feinstein are among the many authors who have followed in Wittkower’s footsteps.

On a quite different tack, G. Stiny, W. J. Mitchell, and others have shown how Palladian plans can be generated by a “parametric generative grammar,” or shape grammar, named by analogy to the linguistic work of Noam Chomsky.7 Their work is exceedingly novel but the basic idea is an old one. The notion that a recipe or algorithm can generate plans, facades, and designs for entire buildings goes back at least to the Roman architectural writer Vitruvius, who was the undisputed god of Renaissance architecture.8 As Stiny and L. March further point out, the idea was taken up in the Renaissance by Leone Battista Alberti and Leonardo and developed later on by such diverse figures as Goethe, Monge, Froebel, Frege, and Wittgenstein.9 But now the shape grammarians carry the idea further by developing a set of algebraic “productions,” or transformations, that can analyze Palladio’s plans into discrete rules. It is curious that the two groups, those who study proportions and those who study shape grammars, seem so little aware of each other’s work.

Among the Wittkowerians, Deborah Howard and Malcolm Longair have sought to show that Palladio preferred room dimensions reflecting the ratios of the pitch intervals used in musical scales, especially the major fifth (C–G; ratio 2:3, i.e., the lower note, C, vibrates twice to every three vibrations of the upper note, G) and the perfect fourth (C–F; 3:4).10 (The idea makes etymological sense, after all, since the Latin word intervallum means “between the walls.”) Howard and Longair print a list of 34 preferred dimensions using Palladio’s Vicentine foot of 34.7 centimeters. The dimensions range between 1 and 100 and are produced by dividing one musical ratio into another. The authors show that Palladio used these preferred dimensions 65–70 percent of the time; had he chosen at random they would have appeared only 45 percent of the time.11 Hence he really did seem to think that certain lengths and widths were preferable to others, though not exclusively so.

But as the authors themselves point out, many of these same ratios could equally well have come from the purely visual proportions advocated by Vitruvius (De architectura libri decern 6.3).12 There is no real necessity to assume that they were chosen because they were musical. Vitruvius himself says not a word about deriving architectural proportions from musical intervals, despite the fact that some of his proportions do coincide with the musical ones, and though he does talk about musical intervals elsewhere in his book (e.g., De architectura 5.4). More significantly, perhaps, Palladio, in his own discussion of villa designs (Book II of his Quattro Libri), also makes no mention of musical intervals as a source for proportions. No more does his colleague, fellow theorist, collaborator, and patron, Daniele Barbaro, in his annotated edition of Vitruvius—and this despite the fact that he had a strong interest both in proportions and in music.13

Finally, Howard and Longair omit a crucial fact, namely that only four of their eight musical intervals were considered consonant in Renaissance harmony. The others were classified as dissonant, to be avoided except as necessary ways of getting from one consonance to another. The consonances were normally C–C (1:1, the unison), C–F (3:4, the fourth, also known as the diatessaron), C–G (2:3, the fifth or diapente), and C–C’ (1:2, the octave or diapason). More problematic were the thirds and sixths. Today these intervals are considered consonant, but in the Renaissance they were tuned in myriad ways and could sound extremely harsh. The ratios Howard and Longair cite for the major third (C–E, 64:81) and the major sixth (C–A, 16:27), though used, had no particularly privileged status. The other ratios for thirds and sixths that were commonly accepted in the Renaissance, if all incorporated in the Howard and Longair scheme, would have greatly lengthened their list of acceptable ratios. On the other hand, these authors also give ratios that were, in music, definitely to be avoided as harmonies. These are the major second (C–D, 8:9) and the major seventh (C–B, 128:243). In a recent article that supplements the Howard and Longair findings, Branko Mitrovic has added the hopelessly dissonant augmented fourth (C–F#), which is based on the ratio of 1 to the square root of 2.14

It is true that all these proportions, even the last, might well appear in architecture. But an architect could hardly have chosen a good many of them on the grounds that they embodied the ideal qualities of music. Why would he want the visual equivalent of dissonance?15 These intervals are present in architecture despite, not because of, their role in Renaissance music. Yet Howard and Longair, and Mitrovic, never question their assumption that all musical ratios were uniformly desirable.16

The argument that architectural proportions were derived from musical ones, then, is hard to sustain. Yet we have noted that many musical proportions are indeed simultaneously architectural. And it is also true that Palladio and Barbaro, for example, thought in terms of a narrow canon of visual ratios just as the musicians had a canon of musical ones. More than once Palladio, for instance, says that most rooms in a house should have plans that are square, or else should be rectangles equal to 1⅓, 1½, or 1⅔ squares, or at most two squares.17 These fractions work out to the following set of rectangles (with the addition of a 1:21:\sqrt {2} or root-2 rectangle, which he also recommends):

Let us note immediately that four of the six ratios are musical—and consonant. They correspond to the unison (1:1), fourth (3:4), fifth (2:3), and octave (1:2).18 Even the root-2 rectangle (the fearful augmented fourth), which is technically incommensurable, could be rather brusquely rounded off to 2:3 or 3:4.19

Yet Palladio also uses shapes, sometimes even for principal rooms, that are not on this list. Indeed, according to Howard and Longair, he uses his own recommended canon for only 39 percent of his principal rooms.20 For example, the 7:8 rectangle (an acceptable musical interval, by the way, only in the highly theoretical Ptolemaic scale) frequently appears. The main salone of the Palazzo Antonini in Udine, 32 × 28, has this shape (fig. 4.32).21 One very practical reason for breaking with the canon (as we have discovered by doing the research for this book) is that it is often geometrically impossible, using lines or walls of the conventional thicknesses found in Palladio’s printed plans, to split a given rectangle into smaller ones while at the same time sticking to the canonical proportions and never having any space left over.22

The students of proportion rarely tackle the problem of architectural distribution: a given villa’s actual layout and sequence of rooms. Those who work on shape grammars, on the other hand, have concentrated on distribution to the exclusion of proportions and even dimensions.23 And this indeed may be the more promising track to follow. The shape grammarians, furthermore, have been concerned not simply with analyzing but with generating Palladian plans—a thing “harmonic” analysis could never do. And they have produced at least two that to us are convincingly Palladian.24 They do not tell us whether their grammar created these published plans unaided or whether it was coaxed along for the occasion; whether they chose two convincing plans out of a welter of unconvincing ones or whether the shape grammar produced Palladian plans only, or mainly. (That is a question we ourselves will be dealing with.) However, by removing the question of proportion for a later inquiry (not yet undertaken), the grammarians acknowledge the importance of proportion and tacitly endorse the prevailing view that canonical and musical proportions are a critical component of Palladio’s villa plan style. We will probe this view in chapter 4.

The drawback of the grammar-generated plans up to the present is that all are formed of squares or integral multiples thereof. The results range from the possibly Palladian (a) to a pure un-Palladian and unarchitectural grid (b):

What the authors mean is not that Palladio or anyone else would design such a house as the latter, but that this 5 × 3 grid has parameters that can be internally altered but not exceeded. It is a limiting case. Yet plans formed from squares and their multiples can never be truly Palladian. They omit all the intermediate room shapes listed above, formed from squares plus fractions of squares, which comprise Palladio’s only specifically expressed rule on the subject.25 Nor do we learn, via the shape grammar as currently developed, anything about the frequency with which Palladio employed a given shape; nor does the grammar, so far, generate facades—though this could probably be done fairly easily.


It is the purpose of this book to push further the possibilities of all this promising, important, but incomplete earlier work. We will utilize a simple method of plan and facade design adapting a technique Palladio himself uses and describes. We will then propose a set of rules that not only produce Palladian designs but prevent, or at least downplay, non-Palladian ones.

We have said that Palladio is called the most influential architect who ever lived. One reason for this influence, we believe, is that his villa designs can be replicated with set variations such as we will be describing in this book. In a sense what we have done is simply to make explicit what earlier neo-Palladians have done by instinct. History is full of building types that can be similarly treated. One has only to think of Greek Doric temples, or Hindu temples, or Île-de-France Gothic cathedrals—or, for that matter, Cape Cod cottages. The Greek temples, for example, are all, or almost all, rectangular with perimeters marked by one or two rows of columns and an inner chambered compartment. There are formula-based limitations on length and width, on the number and shape of columns, and on everything to do with ornamental detail.

But the strict paradigms and limited variants of these temples and cathedrals have not resulted from a single architect’s having reasoned out a basic idea and then run it through a series of permutations. Instead, in these earlier sets of variations one architect simply borrowed a scheme from another, or from several others, and then created a new variation on it. This can result in a pool of “possible” temples or cathedrals—or possible Cape Cod cottages—comparable to the pool of possible villas we will be looking at. But Palladio’s pool is different from most of these others because it is the work of a single person.

This, in turn, means several things. Palladio’s name belongs with the concept of the Palladian villa in a way quite different from that in which, say, the name of Konrad Roriczer, architect of the cathedral at Regensburg, is associated with the cathedral type to which that building belongs. It would be hard to prove (though not inconceivable) that Roriczer wanted to spell out a range of variations on his chosen theme. But, for whatever reason, he did not do so. The early Renaissance architects Filarete and Francesco di Giorgio come closer to Palladio in this respect, but they provide only a few instances of each type and let it go at that. One could consider their meager collections “pools” and extrapolate rules from them. But any such rules would probably be incomplete and awry. Leonardo, we shall see, was more interested than his predecessors in rule-based architectural variations. But Palladio is the first great architect in European history to work out many variations on a basic theme, and to build a considerable number of them as separatestructures. (His contemporary Sebastiano Serlio published a book roughly comparable to the Quattro Libri in its intention, but he could not be called a “great” architect.)26

Palladio and Serlio, furthermore, actually print their specimen designs in quantities sufficient for us to start speaking of rules and, almost more important, statistical frequencies. For these reasons Palladio is the predecessor of a host of later paradigm-minded architects running from Claude-Nicolas Ledoux (the Paris customhouses) to Le Corbusier (the villas) to Frank Lloyd Wright (the prairie houses). He also anticipates, at a much higher level of achievement, the innumerable essays in prefabrication, modular kits, model housing, etc., that began to appear in the eighteenth century and have since then only grown more numerous and varied.

Let us call this kind of architecture paradigmatic. Paradigmatic architecture generates buildings according to rules expressed by a model in the same way that large numbers of Latin verbs obey the paradigm of amo, amas, amat as it moves through all the endings, tenses, moods, and inner transformations that a Latin verb may have. And not only does the paradigm of amo apply to hundreds, maybe thousands of existing Latin verbs, one could in fact generate new verbs with it. In other words, following the lead of George Stiny, W. J. Mitchell, and others, we are applying the essential notion of a generative grammar to the history and analysis of architecture.27

What led Palladio to take up paradigmatic design? Obviously one factor was that he was faced with many similar programs for similar clients (a large clientele of gentlemen farmers who wanted beautiful villas on their estates). But there were other factors. We will be noting the sheer novelty of symmetry (in its modern sense) in Palladio’s day and will show what great use he made of it. May not one half of a symmetrical building be said to have “generated” the other? Could this in fact not reinforce the idea of an architecture of set variations? And of course Palladio’s book itself was one more in a recently begun paradigmatic series of architectural treatises for which Vitruvius provided the amo, amas, amat. Francesco di Giorgio and Serlio, in particular, had just written books with specimen plans, specimen doorways, specimen windows, specimen columns, and the like. Their approach, like Palladio’s, seems to say: “Conjugate these forms by yourself in accordance with the rules stated here.” It is no coincidence, by the way, that two of the great ages of architectural treatise-writing were two of the great ages of paradigmatic design—ancient Greece with its temples (unfortunately the Greek treatises are lost, but Vitruvius mentions sixty or so), and Renaissance Italy.

One good way of divining the existence of a rule is to watch what happens when it is broken. Our method will be to create villa plans and facades based on Palladio’s ideas; but our designs will lead us gradually from his more obvious rules to his less obvious ones. As we continue the process, each time we make a mistake we will identify and correct it. Eventually we will produce plans and facades that, in our opinion at least, get really close to what Palladio himself would do. We do not do this in order to build new Palladian villas, though that is a perfectly reasonable possibility. Rather, knowing what Palladio would and would not do deepens our understanding of what he actually did do.


We have decided to teach a computer to design Palladian villas rather than doing it ourselves with pencil and paper. At this point we should repeat that, notwithstanding our reliance on a computer, the method we use is itself one that Palladio advocated. There is nothing ahistorical about it. What the computer contributes is simply the ability to calculate a huge number of possible permutations and combinations based on Palladio’s rules. Because of the rapidity and completeness with which it can apply these rules, it can test them on a far wider and firmer basis than could an unaided human being. With the disk containing Planmaker and Facademaker, our software for creating plans and facades, anyone can sit down in front of a Macintosh and generate thousands of Palladian villas. The ultimate number is probably circumscribed only by the operator’s patience. We had thought of calling this book All Possible Palladian Villas, but it has been pointed out to us that the random number generator that stands at the core of our system has limitations. It is not true that even our thousands actually represent all the possibilities.

But without the computer, furthermore, it would be harder to learn from our mistakes. Since we define what Palladio did by discovering what he would not do, we must discover everything he would not do. That is another reason for the phrase “possible villas.” True, our data consist of the 44 villa and house designs in Book II of the Quattro Libri—not a large set of models. Indeed even some of these 44 have been ruled out for reasons that will be discussed. Yet if one multiplies 44 by every possible application of each of our rules, the result is astronomical. Once again, only the computer can ring all the changes. Only the computer can prove that a rule, even if followed in every possible way, will still yield the proper results.

The computer has yet another advantage: it excludes unconscious human prejudice. It eliminates the personality of Palladio’s modern student and concentrates exclusively on the 44 designs. We will see in chapter 4 that certain architects who thought of themselves as imitators of Palladio, for example Lord Burlington, could stray very far from his architectural code. Burlington, who lived two centuries after his idol, translated Palladio’s house plans very much into his own terms. We don’t know whether this was conscious or unconscious. Either way, the computer lacks Burlington’s unacknowledged prejudices. It has no idea what we want it to do; it merely does what we tell it to do. That is the crucial point. It makes every rule explicit and unambiguous.

There are still other advantages in our approach. For one thing, people examine and compare much more carefully when they have before them both an authentic work and a good imitation. The imitation teaches us things about the original that no amount of study of the original alone could do. And with Palladio, there are many such things to be learned. Thus it is not simply a question of rules, even subtle ones, that are always observed. There are things Palladio always does, things he does only in certain specific circumstances, and things he does sporadically—just for the hell of it. The observations we are about to make, then, write a code for the stylistic analysis of plans and facades, a code that consists partly of unbreakable rules and partly of mere tendencies. Statistics establish just how strong or weak a given tendency is.

Our new technique, we hope, will also help create new ways, useful both for the historian and the practitioner, of doing architectural history. Not only can our system be used to judge the precise degree of Palladianism in the work of Burlington and his school, or, say, in that of Palladians in the United States; it can be extended to paradigmatic architecture that has nothing to do with Palladio, and there, presumably, allow equally new insights.

But what will these new insights be? Will we really have made a contribution to architectural history, to understanding a great architect’s mind, or will we simply have produced a video game for architecturally minded grown-ups? Let us answer that with a second linguistic analogy: many highly intelligent people, some of them, indeed, great writers, learn languages, and speak and write them flawlessly, without being able to articulate the grammatical rules that structure those languages. Good writers know right from wrong simply on the basis of the way the words sound. Such people may be able to state certain basic rules, e.g., that a verb has to agree in number with its subject, but they may well never have heard of the frequentative imperfect, that bane of nonnative English speakers.28 Yet good writers and speakers know instinctively when this tense should and should not be employed. It is the same with Palladio. He may have articulated only some of his rules—he might even have been unable to articulate others—but they are nonetheless there, and he obeyed them.

The advantage of articulating these immanent rules is that they etch out, with hitherto unexperienced clarity, the procedures and habits that distinguish this great architect from all others. Knowing them makes it immediately possible to distinguish, in a quantifiable and unquestionable way, the work of imitators like Scamozzi, or Lord Burlington, or Thomas Jefferson, from Palladio’s authentic work. It removes architectural connoisseurship from the realm of instinct and sets it within that of the verifiable. By articulating the rules we newly define and clarify a great man’s individuality. In the end we shall find that Palladio’s rules, expressed and unexpressed, are as elegant as any geometric proof or algorithm. By showing this, by showing to what extent he was a natural geometer, we do not make him less the great architect; on the contrary, we show, in a way that gives more than mere lip service to the proposition, how great architecture may flow from geometry.


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