Hearing the Irrational

Abstract

Though Greek mathematics would have considered the notion of “irrational numbers” to be inherently contradictory, in the sixteenth century this concept found advocates on musical grounds well before it appeared in the theory of equations. Michael Stifel, the leading German mathematician of the century, first used the term “irrational numbers” in the context of his discussion of music, but then drew back from the infinity of digits implicit in this concept. Girolamo Cardano, the famous physician and mathematician, brought this concept forward in his musical writings and later used it in his treatment of algebra. Nicola Vicentino’s interest in reviving ancient Greek quarter-tones in enharmonic music led him to advocate “irrational proportions.” Each of their involvements with practical music and composition related closely to their mathematical views. Distrust of the irrational, both musical and mathematical, also color controversies about the expressive dissonances used in early opera, such as Giovanni Maria Artusi’s critique of Claudio Monteverdi. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).