Blues for Alice: The Interplay of Neo-Riemannian and Cadential Viewpoints

We extend a property of Mazzola's theory of cadential sets in relation to the modulation between minor and major tonalities from triadic to tetradic harmony, using the PLRQ group of Cannas et al. (2017) as the analogue of the classical PLR group. While the PLR group connects triadic cadential sets via the relative morphism $R$, the tetradic case reveals a richer structure: two pairs of cadential sets connected by distinct morphisms forming a ``prism'' in the coslice category over the common chords of the paired cadential sets, and a single pair for those that allow quantized modulations. We demonstrate this structure through an analysis of Charlie Parker's Blues for Alice (1951) and Ray Noble's Cherokee (1938), showing how the prism morphism, PLRQ transformations and quantized modulations organize harmonic navigation in bebop. The categorical framework models a formal correlate of the transformational v\'ecu that musicians experience when navigating between cadential regions.