Pairwise Well-Formed Modes and Transformations

TL;DR

This paper explores the historical shift in musical scalar concepts during the Renaissance, using algebraic combinatorics to analyze well-formed modes and their transformations, revealing new structural insights.

Contribution

It introduces a novel algebraic framework for analyzing well-formed and pairwise well-formed scales and modes using automorphisms of the free group on three letters.

Findings

Characterization of scales using Christoffel and Sturmian words

Introduction of positive automorphisms of F3 for scale transformations

Unified algebraic model for ancient and modern modal theories

Abstract

One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of the ancient and medieval world and that of the early modern world, associated especially with Zarlino. We do this at an abstract level, in the language of algebraic combinatorics on words. In scale theory the juxtaposition is between well-formed and pairwise well-formed scales and modes, expressed in terms of Christoffel words or standard words and their conjugates, and the special Sturmian morphisms that generate them. Pairwise well-formed scales are encoded by words over a three-letter alphabet, and in our generalization we introduce special positive automorphisms of $F3$, the free group over three letters.