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A New Geometry of Musical Chords in Interval Representation: Dissonance, Enrichment, Degeneracy and Complementation by Makoto Taniguchi, Miguel Gutierrez (Paperback, 2010)

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The idea emanates from the fact that an intervallic representation of the chord leads naturally to a discrete barycentric condition.

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This monograph covers a fresh and original look at musical chords. The idea emanates from the fact that an intervallic representation of the chord leads naturally to a discrete barycentric condition. This condition itself leads to a convenient geometric representation of the chordal space as a simplicial grid. Chords appear as points in this grid and musical inversions of the chord would generate beautiful polyhedra inscribed in concentric spheres centered at the barycenter. The radii of these spheres would effectively quantify the evenness and thus the consonance of the chord. Internal symmetries would collapse these chordal structures into polar or equatorial displays, creating a platform for a thorough degeneracy study. Appropiate morphisms would allow us to navigate through different chordal cardinalities and ultimately to characterise complementary chords.