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

In architecture, systems of proportions facilitate technical and aesthetic requirements, ensure a repetition of ratios throughout the design, have additive properties that enable the whole to equal the sum of its parts, and are computationally tractable. Three systems of architectural proportion meet these requirements: a system used during Roman times, a system of musical proportions used during the Renaissance, and Le Corbusier’s Modulor. All three draw upon identical mathematical notions already present in the system of musical proportions. While the Roman system is based on the irrational numbers √2 and θ = 1 + √2, the Modulor is based on the Golden Mean, ϕ = (1 + √5)/2. Both can also be approximated arbitrarily closely (asymptotically) by integer series. Underlying the Roman system is the “law of repetition of ratios” and the geometrical construction known as the “Sacred Cut,” both of which geometric expressions of the additive properties of the Roman systems and ensure the presence of musical proportions. The discussion concludes with a system of “modular coordination” based on both musical proportions of Alberti and Fibonacci numbers.