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

Octavio A. Agustín-Aquino

TL;DR

The paper generalizes Mazzola's cadential-set theory to tetradic harmony using the PLRQ group, revealing a three-paired ABC conglomerate of cadential sets linked by morphisms $R_{42}$, $L_{13}$ and $P_{42}$ within a slice-category framework over the tonic seventh chord. It demonstrates a prism-like organization of six cadential sets and distinguishes quantized modulation from non-quantized modulation, illustrating these ideas with Charlie Parker's Blues for Alice and Cherokee and with related bebop examples. The approach unifies neo-Riemannian voice-leading with Mazzola's global tonality, offering a phenomenological account of how musicians navigate cadential regions that syntactic models miss. The findings illuminate Parker's bebop innovations as navigations through the full conglomerate, providing a mathematically structured account of harmonic motion in jazz.

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 vécu that musicians actually experience when navigating between cadential regions.

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

TL;DR

The paper generalizes Mazzola's cadential-set theory to tetradic harmony using the PLRQ group, revealing a three-paired ABC conglomerate of cadential sets linked by morphisms , and within a slice-category framework over the tonic seventh chord. It demonstrates a prism-like organization of six cadential sets and distinguishes quantized modulation from non-quantized modulation, illustrating these ideas with Charlie Parker's Blues for Alice and Cherokee and with related bebop examples. The approach unifies neo-Riemannian voice-leading with Mazzola's global tonality, offering a phenomenological account of how musicians navigate cadential regions that syntactic models miss. The findings illuminate Parker's bebop innovations as navigations through the full conglomerate, providing a mathematically structured account of harmonic motion in jazz.

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 , 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 vécu that musicians actually experience when navigating between cadential regions.
Paper Structure (12 sections, 3 theorems, 22 equations, 1 figure, 1 table)

This paper contains 12 sections, 3 theorems, 22 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Tetradic Cadential Sets
  3. The PLRQ Group
  4. The Slice Category Construction
  5. The ABC Conglomerate Structure
  6. Analysis: Blues for Alice
  7. Analysis: Cherokee and Quantized Modulation
  8. Section A: Quantized modulations
  9. Bridge: Non-quantized modulation via $P_{42}$
  10. Comparison with Syntactic Approaches
  11. Stylistic Signatures in the Conglomerate
  12. Conclusion

Key Result

Proposition 4.2

The following morphisms connect chords in the cadential sets:

Figures (1)

  • Figure 1: The ABC conglomerate structure of tetradic cadential sets. The dashed lines summarize the morphisms from \ref{['E:Prism']}.

Theorems & Definitions (16)

  • Remark 2.1
  • Definition 3.1: $R_{42}$
  • Example 3.2
  • Definition 3.3: $L_{13}$
  • Example 3.4
  • Definition 3.5: $P_{42}$
  • Example 3.6
  • Definition 4.1
  • Proposition 4.2
  • Theorem 5.1
  • ...and 6 more