This paper shows how the symmetric group S4 can be used to analyze the manifold ways fingers connect with fretted instruments. S4, visualized as the symmetrical manipulations of a cube, consists of all possible permutations of four elements. Therefore, the operations can be used for the fingers of both the fretting and picking hands. I highlight two types of subgroups in S 4 , dihedral (𝔻8 and 𝔻6) and cyclic (ℤ4 and ℤ3), in order to analytically model the experiential differentiation between grouped and isolated conceptions of finger action, respectively. In addition to finger transformations, I define contextual operations that function componentwise on ordered n-tuples containing representatives of both fretboard and finger spaces ‒ thus, attending to the nexus of instrument and performer. I illustrate the applicability of these transformations with musical examples by Villa-Lobos and Jonathan Kreisberg, culminating in an analysis of jazz guitarist Ben Monderʼs Windowpanes. These explorations show how motions through finger space can both support and diverge from motions in fretboard space.
Smith, N. (2023). Formalizing the fretboardʼs phantasmic fingers. In R. Torres, A. Brandon, & J. Noble (Eds.), Proceedings of The 21st Century Guitar Conference 2019 & 2021. https://digitalcommons.du.edu/twentyfirst-century-guitar/vol1/iss1/20