The Geometry of Musical Logarithms

The aim of this essay is to create a geometrical link between the music theory and the mathematics of the early seventeenth century by studying and comparing diagrams which directly or indirectly refer to mathematical logarithms. The focus is on the relationships between ratios of numbers referring to sounds and related concepts of perception. The relationship between frequency and pitch is a paradigmatic case of theWeber-Fechner law of psychophysics, stating that equal frequency ratios are perceived as equally sized musical intervals. The Weber-Fechner law maintains that many perceptual phenomena are logarithmic by their very nature. The circular diagrams studied here are by Descartes (1618), Robert Fludd (1618), and Jost Bürgi (1620). Descartes's diagrams have recently attracted the attention of authors from different fields. A second type of geometric diagrams related to musical arithmetic is looked at in the final section of this article.