A New Geometry of Musical Chords in Interval Representation: Dissonance, Enrichment, Degeneracy and Complementationby Miguel Gutierrez, Makoto Taniguchi
Chords appear as points in this grid and musical/p>
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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.
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As far as I know this is the very first time that an effective mathematical configuration is used to model dissonance. The barycentric model is quite remarkable and simple at the same time. Highly recommended.
This is the first time in the current music theory literature that I have encountered an scholarly written monograph on the subject of dissonance in musical chords with a mathematical approach. Without any doubt the authors of this masterpiece have gone beyond the norms of what is presently available in this subject. I highly recommend this great and well thought scholarly work from its very first chapter Interval Space to Chord Complementation. This book is a must for music theorists as well as mathematicians with a strong appeal to serious music.