Alberti took over from Vitruvius the principle that proportion is essential to the beauty of architecture. Indeed, substituting the word beauty for symmetry in Vitruvius's definition of the latter term, he intensified the principle by making proportion identical with beauty. His reverence for number and proportion as the underlying order of the universe was so great that he might have subscribed to the orders expressly because they were ordained with proportions. Consequently, he willingly accepted the Vitruvian notion of a fixed internal proportional system of the orders, although his own experience measuring Roman architecture led him to doubt Vitruvius's formula. Nevertheless, like Vitruvius, he did not justify his system to the reader, so it, too, ends up seeming arbitrary. As regards Alberti's orders, the matter of proportion is left as a complexity that simply has to be dealt with following prescribed formulas.
As regards the proportions of spaces, however, Alberti subscribed to a more systematic understanding of the derivation of proportion than his ancient Roman predecessor. Whereas Vitruvius simply assigned whole-number formulas to certain types of spaces, such as an urban forum or a domestic "Egyptian hall," Alberti acknowledged three different modes of derivation, based on ratios inherent in musical intervals, in geometry, and in arithmetic means. Regardless of which one an architect might adopt, proportion was to be a concern not just for the orders but for all dimensions of a building.
Ultimately all three modes were derived from Vitru-vius, but under Alberti they were developed far beyond his explanations and applied in ways Vitruvius almost certainly never contemplated. For instance, Vitruvius wrote knowl-edgeably about the proportions of musical intervals in the context of acoustics, but he never attached them to architectural dimensions. In his discussion of the amphitheater he described a system of bronze sounding vessels arranged in niches between the seats, which, he held, were traditionally supplied to enhance the acoustical properties of the seating area. How they were to be installed he did not report—to the everlasting consternation of his readers—but in the course of this exposition he did set out, without any attribution of significance, the system of whole-number ratios assigned to the harmonious musical intervals. Whether or not Alberti felt compelled to make positive use of this material only because it appeared in Vitruvius's text, he recognized its parallel source in the doctrines of the Pythagoreans and decided to make something of it. The reason this association mattered is that the observance of proportion in ar chitecture could then be related to the laws of nature, with the implication that harmonious proportions in architecture partake of the harmony of the universe.
This interpretation rests upon the ancient discovery that the strings of a musical instrument produce different tones according to their varying lengths. Correspondingly, the strings producing tones separated by a harmonious interval will have lengths measuring in mathematical proportion to each other. For instance, the interval known as a fifth will be produced by strings whose respective lengths have the proportion of 3:2, because the string sounding the deeper tone will be half again as long as that producing the higher. This phenomenon gains significance from the coincidence that all the other harmonious intervals are also produced by strings whose lengths make whole-number proportions.
Alberti did not elaborate on the broader meaning of this correspondence, but, in the context of a Renaissance culture eager to Christianize the wisdom and learning of antiquity, the existence of a universal relationship between musical tones and numbers was profoundly moving. It was taken to imply that the whole universe is suffused with a rational order based on number, the consequence of which is that numbers are the unseen reality behind all appearances. The extent to which one discerns those numbers is the extent to which one perceives the hand of the Creator in the natural world. Hence to design a building with a system of numerical proportions is to endow it with the aura of divinity. Because the classical orders accompany this understanding of proportion, it is not difficult to see that classicizing architecture was taken to be more expressive of the sacred than any other. The Renaissance predilection for architecture based on the orders, then, was probably prompted less by a taste for columns than a reverent respect for the proportions they embodied.
The proportions of the musical intervals can be applied, according to Alberti (Book IX, chapters 5 and 6), to the dimensions of widths and lengths, for instance those of a temple platform or a city square. Besides the interval of the fifth, with its 3:2 proportion, there is the fourth, 4:3, the octave, 2:1, the twelfth, 3:1, and the fifteenth, 4:1. These ratios can be manipulated beyond the most straightforward adoption of musical intervals. A striking instance is the double fifth, in which the lesser number represents, as usual, the width, but the greater number, which is normally half more than the lesser, is increased by half its own quantity to establish the length. That is, beginning with a ratio of 4:6, the 6 is increased by half its quantity to produce 9, resulting in a proportion of 4:9. That this is, famously, the governing proportion adopted for the Parthenon Alberti probably could not have known.
The arithmetic ratios of these musical intervals are just one of the ways of determining proportions from quantities inherent in the natural world. Another resides in the fixed relationships of geometric figures, employing the proportional dimensions of their sides and often of their diagonals as well. In this system the dimensions are frequently not discoverable in terms of simple numbers, but instead must be derived from the proportional relationships of the parts of the figures expressed in roots and powers. These relationships involve finding the area of a face of a geometric solid, multiplying the length by the width; and the cubic power, multiplying the area by the height. The proportions to be derived from such a relationship are not, however, straightforward adoptions of dimensions based on the side, the area, and the cube. Rather, they consist of the original dimension of the side and the respective diagonals of the area and the cube, both expressed as square root quantities. This is one way of arriving at three dimensions that are geometrically proportional, adaptable to the three dimensions of a room. In such a case the width of the room is the smallest quantity, the side; the length is the greatest, the cube root; and the height is the intermediate quantity, the square root of the area.
A more convenient method for arriving at three proportional dimensions is to employ one of the three ways to determine means. The simplest is the arithmetic mean: add the quantities of two proportional dimensions and divide the sum in half to arrive at the third dimension. Most subtle is the geometric: multiply the quantities of two proportional dimensions and take the square root of the product. More laborious to determine but also more elegant is the musical mean, which is twice as far from the greater of two proportional dimensions as it is from the lesser. This mean is found by subtracting the lesser quantity from the greater, then dividing the remainder by the sum of the smallest proportional numbers contained in each of the two quantities and adding the quotient to the lesser quantity. All three types of means, Alberti averred, have produced excellent proportions in architecture.
The matter of proportion in architecture cannot be treated without considering its relevance to human proportions. Alberti took this relationship from Vitruvius as the determinant for the traditional proportions of columns in the three orders: 1:7 for Doric, 1:8 for Ionic, and 1:9 for Corinthian. These were arrived at through a series of arithmetic means, using two human dimensions in proportion to the height of the full male figure as the extremes. The lesser ratio, 1:6, was discovered by comparing the thickness of a man to his height; the greater, 1:10, by comparing the distance between the navel and kidneys to height. The Ionic is the mean between the two extremes, the Doric between the Ionic and the lower ratio, and the Corinthian between the Ionic and the greater ratio. This relationship between columnar proportions and proportions of human beings underlines the association of the use of proportions in architecture with the unassailable standard of nature. Nature provides both the authority and the justification for this concern of architectural design.
Alberti thus bequeathed to theorists of the mature Renaissance two different sets of proportional standards, the ratios representing the musical intervals and the width-to-height proportions prescribed for the columns of the orders. Ratios were to be applied to the various rooms set out in a plan and also the principal facade. Palladio did not discuss the theory of these ratios but directly demonstrated how he employed them by inscribing dimensions on his plans of villas. Although he varied the ratios from one plan to another, he had certain favorites. As Rudolf Wittkower observed (1962, 131-135), the dimensions of 12, 16, 18, 20, 24, and 30 show up frequently on his plans and are used in various combinations—for instance 18 x 30, or 12 x 20, for a ratio of 3:5, representing the musical interval of a major sixth. A simpler example, 12 x 24, for a ratio of 1:2, represents an octave; a more complex one, 16 x 27, read as 16:24:27, represents a fifth (2:3) plus a whole tone (8:9). The ideal use of proportion would also include a third dimension for the height, itself related in harmonic ratios to the length and width. However, it is not always possible to determine whether or not it was achieved, because Palladio often does not specify that dimension in the Quattro libri. Whether or not the proportions in Palladio's work represent a conscious association with musical intervals cannot be proved because only certain of his designs reveal such a correspondence. As Branko Mitrovic has shown (1990), whereas some of Palladio's proportions can be related to harmonic ratios, he evidently employed some other methods of determining proportions as well, but none consistently enough to ascribe cultural associations to them.
Following Alberti's concern for the different proportions of the three main orders, Palladio applied the Vitru-vian standard of graded intercolumniations respectively to the five orders. Hence the pycnostyle was assigned to the composite (fig. 7-1), the systyle to the Corinthian (fig. 7-2), the eustyle to the Ionic (fig. 7-3), the diastyle to the Doric (fig. 7-4), and the aereostyle to the Tuscan (fig. 7-5). Correspondingly, the piers of arches to which the applied orders can be attached must also vary in their proportions from one order to the next, relative to those assigned to the column. Even the pedestals for each order were given correspondingly different proportions. As an academic exercise it increased the systematic character of each individual order, but the assignment of these respective intercolumnia-tions to the five orders can scarcely have been adopted as a firm rule of practice.
The discrepancies between the proportions prescribed by Vitruvius and those empirically observed in Roman architecture by all the Renaissance theorists was a continuing cause for consternation. They wondered if there really had been a fixed Roman proportional system. Concerned to reconcile the discrepancy, they did not deny that proportions were of the essence, but they sought a consistent basis for determining them. Alberti, Serlio, Vignola, and Palladio all set out independently to establish a clear and definite system, based on the Vitruvian module of half the thickness of the column shaft. Each of them faced the necessity of assigning fractions of a module to the various components of the base and the capital, as well as to those of the entablature. Vignola sought to demonstrate "a definite correspondence and continuous proportion of figures," like that in musical theory (Wittkower 1962, 123). But for as long as theorists adhered to the notion that the proportions of the orders were based on proportions of the human body, there was no apparent way to escape a system employing fractions of modules.
The Gordian knot was cut by Claude Perrault, in Ordonnance des cinq espèces de colonnes (Paris, 1683), when he disavowed any transcendent or symbolic association with proportions in the orders and dealt with them as purely numerical values. Shifting to a module based on one-third of the width of the column shaft, he was able to assign whole numbers to the parts of the orders. Proportion retained its value as an indispensable desideratum, but the use of abstract number—arbitrarily adopted—made proportions simpler to apply.
For as long as the classical orders were the basis for architectural design, the use of rational proportion was one of the architect's key obligations. However, after 1800, when the Greek peripteral temple had been replaced as the ideal by the Gothic cathedral, the cultic adherence to the convention of proportions came to an end.
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