1/19/2023 0 Comments Music math image"In some sense, Bach and the Beatles are really exploiting the same geometrical features. "One of the really exciting things about this research is that it allows us to see commonalities among a much wider range of musicians," Tymoczko said. McCartney's pieces make use of a smaller number of motions in the geometrical spaces, corresponding to his more traditional approach to harmony, while Lennon makes use of a much wider set of options, reflecting his roots in rock, Tymoczko said. When they first realized that the shape of two-note chords is a Möbius strip, a fundamental mathematical form discovered in the 19 th century, the researchers were "amazed," Quinn said. "These spaces give us a much better and comprehensive picture of the space of all possible chords." "You can use these geometrical spaces to provide ways of visualizing musical pieces," Tymoczko told LiveScience. Any piece of music can be mapped in these spaces. The space of four-note chords is what mathematicians would call a "cone over the real projective plane," which resembles a pyramid in our 3-D universe. The team found that the shape of possibilities using three-note chords is a three-dimensional ice cream cone, where types of chords, such as major chords and minor chords, are unique points on the cone. For music made of chords containing two notes, all musical possibilities take the shape of a Möbius strip, which basically looks like a twisted rubber band (this was first described by Tymoczko in a 2006 Science paper). The team designed a geometrical technique for mapping out music in coordinate space. Clifton Callender of Florida State University, Ian Quinn of Yale University and Dmitri Tymoczko of Princeton University outlined their "geometrical music theory" in the April 18 issue of the journal Science.
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