Keywords: Mappings; Partitions; Architecture and Music; Proportion; Isovist; Isovox; Andrea Palladio; Villa Capra
The relevance of music theory as an interpretive framework for the understanding of Palladio’s work has been one of the most debated subjects in the realm of architectural theory and criticism. Typically, the debate is quite abstract and it focuses on possible mappings between the ratios found in Palladio’s plans and corresponding ratios used in contemporary musical temperaments. The project here rather focuses on the actual performance of the ratios found in Palladio’s work and the implications of this performance, melodic and harmonic, for the perception of the space for a situated observer/performer. To that extent the study suggests a model of mapping between space, sound and color and correlates that with polygon partition theory to simulate movement within a space. All examples to test these ideas are based on Palladio’s Villa Capra.
“[The] charm of impossibilities… at once voluptuous and contemplative, resides particularly in certain mathematical impossibilities of the modal and rhythmic domains. Modes which cannot be transposed beyond a certain number of transpositions, because one falls again into the same notes; rhythms which cannot be used in retrograde, because in such a case one finds the same order of values again” (Messiaen, The Technique of My Musical language, 1956). The symmetry properties of the equal temperament scale are computed here using Polya’s theorem of enumeration of non-equivalent configurations with respect to the permutation group of the vertices of the dodecagon. All scales are diagrammatically represented as colored polygons within the regular dodecagon and accompanied by their respective representation in typical western notation in their first transposition. The color in the first set of diagrams classifies the scales in terms of their number of notes and in the second set of diagrams in terms of their symmetry properties.
Acknowledgments. The illustration and sonification of the patterns have been done by Haldoun Kececigil and Heath Washburn.
Sieves
Athanassios Economou and Thomas Grasl
2012
Keywords: Sieves; Xenakis; Group theory; Symmetry; Enumeration
The sieve is a powerful formal tool developed by Xenakis to create integer-sequence generators that can be used for the generation of various numerical patterns to represent pitch scales, rhythm sequences, as well as patterns of loudness, density, timber and so forth. Xenakis used the sieves in a series of musical and architectural works. The key motivation for the work outlined here is to capture and visualize the properties of the algebra of the sieves looking at the decomposition of the moduli that emerge out of the combination of the sieves. Xenakis asserted that his algebra of sieves revealed insights into fundamental aspects of composition. Here these hidden symmetries are identified and foregrounded for any modulus however complex.
The formal tools for the inquiry on sieves are taken from Lagrange’s theorem on subgroups of given order and Polya’s theorem of counting non-equivalent configurations with respect to a given permutation group. Here the theorems are applied within an automated computational framework that automatically generates all the symmetry subgroups of a given symmetry group, calculates the cyclic index of all symmetry groups and subgroups of any finite shape, computes the figure inventory of any number of variables upon the cycle index, produces patches showing the data flow of the moduli and visualizes the result of all non-equivalent configurations with corresponding isomorphic two-dimensional subshapes of regular n-gons.
The sieve is a powerful formal tool developed by the composer / architect Ioannis Xenakis to create integer-sequence generators that can be used for the generation of various numerical patterns to represent pitch scales, rhythm sequences, loudness progressions, density patterns, timber systems and so forth, The decomposition of a n-modulus of the sieve can be modeled as the arrangement of 2 colors x and y upon the faces of a regular n-polygon whereas the color x represents the sieve content and the color y represents the numerical distance between the sieve content.
Here, the work extends the model to capture spatial relations between sieves. The key idea to extend the model is to look at the possible spatial relations between sieves in Euclidean space and primarily in three-dimensional space. The candidate shapes to capture the notion of decomposition of a module are the extensions of the regular n-polygons in the Euclidean plane to the regular prisms in Euclidean space, and even more so, the regular n-dipyramids, the duals of the regular n-prisms. The computation here gives the complete number of all colorings of the faces of 12-dipyramids for up to 5 colors and the emphasis is given in the calculation of 2 colors as the three-dimensional extension of the model for the enumeration of the sieves wrapped around 2 n-polygons in space. More specifically it can be shown that there are 2 ways to color the 2 faces of an 1-dipyramid with 1 color, 6 ways to paint the 4 faces of a 2-dipyramid with 2 colors, 13 ways to paint the 6 faces of a 3-dipyramid with 2 colors, 34 ways to paint the 8 faces of the 4-dipyramid with 2 colors and so forth. The computation here foregrounds the calculation of the coloring of the 16 faces of the 8-dipyramid because this shape presents all possible symmetries found in the 7 infinite families of three-dimensional regular prisms. More specifically, the table shows that 16 faces of an 8-dipyramid can be painted with 2 colors in 2299 distinct ways. In a similar manner there are 2299 polyrhythms that consist of two rhythms played one against the other.
