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Note: These pages are an updated mirror of the Fractal Music Lab pages at Michael Frame's extensive Yale University Web site on fractal geometry. The material here was originally presented at the August 2004, Fractal Geometry Workshop at Yale.
Harlan Brothers offers Fractal Music Workshops to high schools, colleges, and training institutes around the country. Using the tools of fractal geometry, these workshops offer teachers and students alike the opportunity to explore many of the intimate connections between mathematics and music. Focused presentations and hands-on exercises promote understanding in an intuitive and entertaining manner.
To learn about how your school, mathematics department, or music department can benefit from a Fractal Music Workshop, see:
Purpose: To explore several ways in which music may exhibit fractal characteristics.
Material: The Fractal Music Composer software
Background Some basic music theory, examples of pitch and duration scaling.
Exercises by hand and with the Fractal Composer.
As a final example, pointing the way to future work, we give some examples of structural scaling in music, and a generalization, pulse trains.
Conclusion: Music can exhibit fractality in several ways, from rigid self-similarity of the composition to a psychological superposition of patterns on different scales. Beware of books or webpages about fractal music: many of these are the worst kind of nonsense.