Mathematical structural changes in musical theoretical thought until the Renaissance: the emergence of the idea of modern number in musical theoretical contexts

Abstract

This project investigates mathematical structural changes in the musical thinking until the Renaissance, from a historical/heuristic perspective of the problem of arithmetization of ratios and the consequent emergence of the concept of modern number, as well as the reciprocal effect of changes in theoretical music on the arithmetization in mathematics, a gap in this historiography. In the present proposal, the development of theories of ratios in the context of such changes is investigated from the perspective of the relationship between practical and theoretical knowledge. From this perspective, the role of practical knowledge in transformations of structures of ratio theories until the Renaissance and their repercussions on the concept of number can be further explored by investigating the ways in which mathematical structural problems in music theory were addressed. The concept of modern number gained great impetus from structural problems in theoretical music, such as the need to divide the monochord, divide the tone, and establish a tonal system, problems that prompted a rethinking of the Greek geometric concept of ratio as an identifiable continuum with the line. Such procedures already occur in the treatise "Sectio Canonis" from the 3rd century BC, ascribed to Euclid, as well as in Erasmus Horicius in the 15th century, who, when dividing the tone, used an arithmetic process analogous to the geometric definition of the proportionality of ratios by Eudoxus, closely linked to incommensurable phenomena. In these examples, a mathematical theoretical framework emerges for a virtual space of musical pitch relationships, a rudimentary concept of a continuum of rational numbers, which is a step towards the foundations of the modern concept of number. Considering a synthesis of disciplines such as the history of mathematics and theoretical music, as well as the epistemology of mathematics and heuristics, this is a broad work of abstraction and synthesis. It is also an attempt to find explanations for the course of theories of ratios and proportions, as well as mathematical thinking, through an investigation of the epistemological dimension of the historical development of the concepts involved here. Such an approach directs the more abstract research into historical processes of structural changes in systems of knowledge, which includes the investigation of fundamental cognitive structures of scientific thought and the dependencies of such structures on their respective empirical bases and cultural conditions. The results concerning this more abstract hierarchy will serve to provide in a future moment support for the study of the educational and scientific dissemination potential present in the results of this research, in order to evaluate possible visual and interactive representations of the topics addressed within the scope of digital humanities. (AU)