Structural Transformations in the Relationships between Mathematics and Music up to the Renaissance and the Emergence of the Idea of Number as a Continuous Quantity

摘要

This chapter aims at evincing a conceptual change in the relationships between mathematics and theoretical western music throughout its history from Antiquity to Renaissance, in which structural changes in the conception of ratio associated with the relationships between theoretical music and practical problems played a special role. It identifies situations in the historical development of such a relationship, which show signs of changes on the understanding of music, in which music comes from a arithmetical speculative subject to a sonorous one, associated respectively to music without sounds and with it. In the context of resistance of the Platonic-Pythagorean tradition to changes in theories of ratio and proportion; this chapter will look preliminarily at the evidence of singular transformations in treatises of mathematics and theoretical music of the early modern period. As a result, it is possible to analyse, on the one hand, the emergence of the idea of number in the context of changes on theory of ratios, which borrows semantically distinct but structurally similar attributes from arithmetic analogous structures, as well as, on the other hand, the role of structural problems of theoretical music on the intensification of such a process.