Radoslav Zuk
McGill University, Montreal, CANADA

The predilection of Renaissance architects and theorists for proportional systems which are based on consonant musical intervals is well known to scholars of that period. Also well known is the skepticism which surrounds the validity of such systems today. Controversial philosophical and scientific evidence, historical misinterpretations, and subjective polemical positions may be the reason for this skepticism. However, the frequently narrow interpretation of the musical intervals analogy may be another cause. To be meaningful, the comparison between consonant musical intervals and architectural proportions must be extended beyond the parallel between a single frequency ratio of two musical pitches and a length-to-width room ratio.

Since architecture is a volumetric entity, its geometric definition involves three dimensions. Bringing the length, width, and height of individual spaces, of total buildings, and even of smaller architectural components into optimal harmonic relationships may be thus considered as an essential design challenge. Alberti and Palladio seem to have been aware of that, and made recommendations for determining proportional dimensions for heights of spaces. Among them were the arithmetic, geometric, and harmonic means between dimensions which they recommended for the lengths and widths of plans for such spaces.

An examination of the resulting length : height : width proportions reveals some striking parallels to the interval relationships in chords which form the constituent parts of musical harmonic structures. It shows also the evolution of Palladio's recommended proportions beyond those of Alberti, in relation to developments in tonal harmony, the Major-Minor harmonic system, in which, according to Donald Jay Grout, "...all the harmonies of a composition [are] organized in relation to a triad on the tonic supported primarily by triads on its dominant and subdominant..." and which "...had long been foreshadowed in music of the Renaissance, especially that written in the latter half of the sixteenth century." Following Palladio's recommendations, proportions based on plan ratios other than those related to the Unison, the Perfect Fourth, and the Octave are equivalent to triads from which either a Major or a Natural Minor tonality can be fully constructed.

While direct analogies can be drawn between the frequency ratios of a musical chord and the dimensional ratios of a single architectural space, the differences in the intrinsic nature of the realms of sound and of space may preclude such exact analogies when larger harmonic structures of musical works and the proportional structures of entire buildings are compared. However, when it is considered that a meaningful architectural experience involves movement from space to space, a broad comparison can be made between such an experience and the experience of tonal musical compositions, where a structured sequence of varied harmonic events (chords) is revealed to the listener. A work of architecture where the spaces and/or volumes intersect at distinct angles and are therefore experienced simultaneously may be compared to a polytonal composition where the superimposition of two or more tonalities is determined by a distinct interval or intervals. In such buildings two or more proportional systems are also superimposed at a distinct angle.

A number of significant twentieth-century projects demonstrate that the dynamism of the current aesthetic preference for geometric shifts and collisions in the built environment, which have their justification in the reality of the prevailing complexities in most spheres of human existence, can be revealed in such rich and apparently discordant, yet ultimately coherent geometric superimpositions. Thus instead of the arbitrary, idiosyncratic, and shallow twists displayed in the configurations of many recent, popularly promoted buildings, equally imaginative, but deeply rooted in the underlying proportional order and therefore inherently profound, spatial compositions can be achieved.


Andrea Palladio: Villa Foscari, Malcontenta, Italy


Richard Meier: Museum of Arts and Crafts, Frankfurt a. M.

This email address is being protected from spambots. You need JavaScript enabled to view it.attended high school and studied music in Graz, and earned his Bachelor of Architecture degree with honors from McGill University in Montreal. He won several prizes, including the Pilkington Traveling Scholarship, the highest award for a graduation design project in Canada. Later he earned a Master's in Architecture from MIT in Boston. Most recently he was awarded an honorary doctorate degree by the Ukrainian Academy of Art in Kyiv. He has taught architecture at the University of Manitoba, the University of Toronto, and McGill University, where he is an Emeritus Professor and a recipient of the Ida and Samuel Fromson Award for Outstanding Teaching in the Faculty of Engineering. A professor and an honorary professor, respectively, at two universities in Europe, he has also been a guest lecturer and guest review critic at various universities in Canada, the United States and a number of European countries. Winner and co-winner of several competition prizes, Radoslav Zuk has designed, among other projects, nine Ukrainian churches in North America and one in Ukraine, in association with or as consultant to a number of architectural firms. Most of these buildings have been recognized in the international architectural press and the projects exhibited in North America and Europe. He has published articles on design theory, cultural aspects of architecture, and on the relationships between architecture and other arts. He is a Fellow of the Royal Architectural Institute of Canada and of several scientific societies, and a co-recipient of the Royal Architectural Institute of Canada Governor General's Medal for Architecture.

The correct citation for this paper is:
Radoslav Zuk, "From Renaissance Musical Proportions to Polytonality in Twentieth Century Architecture", pp. 173-188 in Nexus V: Architecture and Mathematics, ed. Kim Williams and Francisco Delgado Cepeda, Fucecchio (Florence): Kim Williams Books, 2004.