An evaluation of Pietro Aron's understanding of "mode in polyphony" (Trattato della Natura et Cognitione di tutti gli tuoni [1525]) coupled with an analytical examination of Josquin's motet oeuvre leads to a new theory of tonal coherence in sacred vocal polyphony from about 1500. This theory provides a designation for, and identifies the markers of, the tonalities represented in the motets of Josquin; it is based on a nomenclature derived from the technical vocabulary of modal theory and the exigencies of a musical tradition rooted in plainchant, hexachordal manipulation of the gamut, and counterpoint. Three main tonal categories are defined and identified by hexachordal function of final as Ut, Re, and Mi tonalities. Comments by Glarean (Dodecachordon [1547]) support the essentially practical basis of these tonal designations. Each of these tonalities consists of, and is defined by, a distinct collection of "modal types," melodic-contrapuntal paradigms of tonal coherence that are identifiable in a variety of structural guises, from simple contrapuntal contexts to more abstract frameworks. Analytical examples from Josquin's motets illustrate procedural similarities in works belonging to different modal categories and structural distinctions within the same category, as well as demonstrating why diverse analytical approaches have seemed successful with certain works but inappropriate to others.
On the two most extended modal types of operators defined over interval-valued intuitionistic fuzzy sets
