The Shape of Music – How do harmony and melody combine to make music?

Roughly 2,500 years ago, Pythagoras observed that objects, such as the anvils he purportedly studied, produced harmonious sounds while vibrating at frequencies in simple whole-number ratios.

More complex ratios gave rise to more dissonant sounds, which indicated that human beings were unconsciously sensitive to mathematical relationships inherent in nature. By showing that the world could be described mathematically, Pythagoras not only provided an important inspiration for physics, but he also discovered a particular affinity between mathematics and music--one that Gottfried Leibniz was to invoke centuries later when he described music as the "unknowing exercise of our mathematical faculties."

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