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

TL;DR

This paper extends Mazzola's cadential set theory to tetradic harmony using the PLRQ group, revealing a richer structure that explains harmonic navigation in bebop jazz through a categorical framework.

Contribution

It introduces a novel categorical model for cadential sets in tetradic harmony, connecting neo-Riemannian and cadential viewpoints with new morphisms and structures.

Findings

Identifies a prism structure linking cadential sets in tetradic harmony.

Demonstrates the model with analysis of jazz standards 'Blues for Alice' and 'Cherokee'.

Shows how the framework captures musicians' harmonic navigation.

Abstract

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 slice category over the tonic seventh chord, and a single pair for those that allow quantized modulations. We demonstrate this structure through 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 captures what syntactic approaches miss: the transformational "v\'{e}cu" that musicians actually experience when navigating between cadential regions.