The transmission of pythagorean arithmetic in the context of the ancient musical tradition from the Greek to the Latin orbits during the renaissance: A computational approach of identifying and analyzing the formation of scales in the De Harmonia Musicorum Instrumentorum Opus (Milan, 1518) of Franchino Gaffurio (1451-1522)

Abstract

On the occasion of the three hundredth anniversary of Leonhard Euler (1707-1783) and the focus of his contribution to the mathematical explanation of scales, the quest arises to examine similar contributions of music theorists that lived prior to Euler. In this paper we selected from the trilogy (Theorica musicae, 1492; Practica musicae, 1496; De harmonia musicorum instrumentorum opus, 1518) of the eminent Renaissance scholar, musician, theorist and composer Franchino Gaffurio (1451-1522) his third work of 1518. In particular, using a modern elementary number theoretic approach, we discovered a number generating function in closed form, which we called a Gaffurio number generating function, that generates the tones of the Pythagorean scale. The number generating function is of the form f (n, m) = p n q m , where p and q are primes, and n and m are elements of /N 0. Here, Gaffurio chooses the smallest primes p = 2 and q = 3, thereby with this ratio p / q denoting the interval of the diapente and he varies the integral exponents n and m from 0, 1, 2, 3, ... and in the process constructing a series of numbers signifying the precise placement of the tones within the gamut. His approach is in line with medieval mathematics, which focuses on elementary number theory and can be seen as a precursor to Euler's Potenzennetz (grid of proportions). This paper embeds our results in Gaffurio's oeuvre and his times and includes a concise of overview of his biography with an emphasis on his pedagogical and literary contributions. Our approach fills a gap in research, because although Latin music theorists from the era of musical humanism frequently include mathematical approaches to the Greek systema teleion and Guidonian system, as projected onto the monochord, they do not disclose the nature and specific details of the mathematical calculations which inform the natural numbers included in both the manuscript and early printed editions. © 2009 Springer-Verlag.