Title page for ETD etd-11152012-040803

Type of Document Dissertation
Author Bishop, John
Author's Email Address jbisho8@tigers.lsu.edu
URN etd-11152012-040803
Title A Permutational Triadic Approach to Jazz Harmony and the Chord/Scale Relationship
Degree Doctor of Philosophy (Ph.D.)
Department Music
Advisory Committee
Advisor Name Title
Peck, Robert Committee Chair
Dean, Paul Committee Member
Grimes, William Committee Member
Perry, Jeffrey Committee Member
Shaw, Brian Committee Member
  • direct product
  • geometric duality
  • octahedral symmetry
  • dihedral group
  • cyclic group
  • toroidal polygon
  • permutation groups
  • neo-Riemannian transformations
  • group theory
  • triad
  • reharmonization
  • improvisation
  • chord/scale relationship
  • functional harmony (jazz)
  • Cayley digraph
  • conjugacy class
  • semidirect product
Date of Defense 2012-10-29
Availability unrestricted
This study provides an original triadic theory that combines existing jazz theory, in particular the chord/scale relationship, and mathematical permutation group theory to analyze repertoire, act as a pedagogical tool, and provide a system to create new music. Permutations are defined as group actions on sets, and the sets used here are the constituent consonant triads derived from certain scales. Group structures provide a model by which to understand the relationships held between the triadic set elements as defined by the generating functions. The findings are both descriptive and prescriptive, as triadic permutations offer new insights into existing repertoire. Further, the results serve as an organizational tool for the improviser and composer/arranger. In addition to the ability to describe individual triadic musical events as group actions, we also consider relationships held among the musical events by considering subgroups, conjugacy classes, direct products and semidirect products. As an interdisciplinary study, it is hoped that this work helps to increase the discourse between those in the music subdisciplines of mathematical music theory and jazz studies.
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