Structural changes in the theories of ratio in theoretical music in the Renaissance

• Grant number: 15/14711-1
• Support Opportunities: Scholarships abroad - Research
• Start date: February 01, 2016
• End date: January 31, 2017
• Field of knowledge: Humanities - History - History of Science
• Principal Investigator: Oscar João Abdounur
• Grantee: Oscar João Abdounur
• Host Investigator: Jurgen Renn
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Max Planck Society, Berlin, Germany

Abstract

This research deals with structural transformations in the theory of musical proportions in the Early modern times and the consequent emergence of the idea of irrational number in such contexts, as an example of the effect of epistemological principles in the historical development of mathematical ideas. We use the term "arithmetization of theories of ratio" to describe the process in which, particularly in the late Middle Ages and Early modern period, the concept of mathematical ratio lost its geometric aspect, and assumed a structurally similar but semantically distinct one. For example, ratios lose the meaning of comparison between two homogeneous quantities, approaching semantically the concept of number, compounding ratios become multiplication of ratios and proportions between ratios turn gradually into equations involving numbers. By analyzing different structures in the theories of ratios in the argumentation of the treatises on theoretical music from the Early modern period, this research project intends to identify new ideas that on the one hand, have to do with arithmetization and, on the other hand, deal with structural problems in music theory. In the context of the resistance of the Platonic-Pythagorean tradition to conceptual changes in theoretical music; this project will also look at the evidence of transformations of theories of ratios 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 irrational number through borrowing, in the context of theories of mathematical ratios, semantically distinct 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 transformations. (AU)