In this way, the musicians are able to reduce musical works to their mathematical essence. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information - for instance, what octave the notes are in, their order, or how many times each note is repeated. The trio describes five different ways of categorizing collections of notes that are similar, but not identical. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Understanding music, the authors write, is a process of discarding information. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries." ![]() "But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. There are lots of practical consequences that could follow from these ideas." We could change the way we educate musicians. "You could create new kinds of visualization tools - imagine going to a classical music concert where the music was being translated visually. "You could create new kinds of musical instruments or new kinds of toys," he said. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before." "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. "The music of the spheres isn't really a metaphor - some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form. ![]() The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content." This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.ĭifferent types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. Writing in the April 18 issue of Science, the trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry.
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