The art of counterpoint: a Mazzola-type three-voice first-species counterpoint model

We propose a three-voice extension of Mazzola’s algebraic model of first-species counterpoint. A three-voice sonority is represented as $a+{\epsilon }_1\mathrm{.}b+{\epsilon }_2\mathrm{.}c$, where $a$ is the lower voice and $b,c$ are the lower–middle and lower–upper interval classes. Starting from a strong dichotomy $(X\mathrm{/}Y)$ of ${\mathbb{Z}}_{2k}$, we introduce a harmonic mask $H\subseteq X\times X$ and define the admissible three-voice consonances $X_H$. We then construct a Hichert-type successor relation by applying the ordinary two-voice admitted-successor mechanism to the active pairwise projections. In the twelve-tone Fuxian case, the proposed mask has 26 admissible bass-rooted interval pairs, including complete and incomplete sonorities, and yields a finite directed successor structure.