Musical Proportions at the Basis of Systems of Architectural
Proportions both Ancient and Modern

Throughout the history of architecture there has been a quest for a system of proportions that would facilitate the technical and aesthetic requirements of a design. Such a system would have to ensure a repetition of a few key ratios throughout the design, have additive properties that enable the whole to equal the sum of its parts, and be computationally tractable--in other words, to be adaptable to the architect's technical means. The repetition of ratios enables a design to exhibit a sense of unity and harmony of its parts. Additive properties enable the whole to equal the sum of its parts in a variety of different ways, giving the designer flexibility to choose a design that offers the greatest aesthetic appeal while satisfying the practical considerations of the design. Architects and designers are most comfortable within the realm of integers, so any system based on irrational dimensions or incommensurable proportions should also be expressible in terms of integers to make it computationally acceptable

In his book The Theory of Proportion in Architecture, P.H. Scholfield discusses three systems of architectural proportion that meet these requirements: the system of musical proportions used during the Renaissance developed by Leon Battista Alberti, a system used during Roman times, and the Modulor of the twentieth-century architect, Le Corbusier. All of these systems draw upon identical mathematical notions already present in the system of musical proportions. While the Roman system is based on the irrational numbers root-2 and theta=1+ root-2, the Modulor is based on the Golden Mean, phi=(1+ root-5)/2. Both of these systems can also be approximated arbitrarily closely (asymptotically) by integer series, and these integer series can be used to implement the system with negligible error. At the basis of the Roman system is the "law of repetition of ratios" and the geometrical construction known as the "Sacred Cut." An analysis of the Medici Chapel illustrates that both the law of repetition and the Sacred Cut are geometric expressions of the additive properties of the Roman systems and ensure the presence of musical proportions in a design. The discussion concludes with Ezra Ehrenkrantz's system of "modular coordination" based on both musical proportions of Alberti and Fibonacci numbers.