A PERMUTATIONAL TRIADIC APPROACH TO JAZZ HARMONY AND
THE CHORD/SCALE RELATIONSHIP
A Dissertation
Submitted to the Graduate Faculty of the
Louisiana State University and
Agricultural Mechanical College
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in
The School of Music
by
John Bishop
B.M., Berklee College, 1990
M.M., University of Louisville, 2004
December 2012
ii
To Quentin Sharpenstein
iii
The harmonic, simple, and direct triad is the true and unitrisonic root of all
the most perfect and most complete harmonies that can exist in the world. It is the
root of even thousands and millions of sounds.The triad is the image of that great mystery, the divine and solely adorable Unitrinity (I cannot think of a
semblance more lucid). All the more, therefore, should theologians and
philosophers direct their attention to it, since at present they know fundamentally
little, and in the past they knew practically nothing about it.It is much employed in practice and, as will soon be seen, stands as the greatest, sweetest,
and clearest compendium of musical composition.This triad I have observed since boyhood (with only God and nature as my guides), I now study it by way of
a pastime, and I hope to see it perfected with Gods help, to Whom be praise forever.
Johannes Lippius, Synopsis of New Music (Synopsis Musicae Novae).
God has wrought many things out of oppression. He has endowed his
creatures with the capacity to createand from this capacity has flowed the sweet songs of sorrow and joy that have allowed man to cope with his environment and
many different situations.
Jazz speaks for life. The Blues tell a story of lifes difficulties, and if you think for a moment, you will realize that they take the hardest realities of life and put them
into music, only to come out with some new hope or sense of triumph.
This is triumphant music.
Modern jazz has continued in this tradition, singing songs of a more complicated
urban experience. When life itself offers no order of meaning, the musician
creates an order and meaning from the sounds of the earth, which flow through
his instrument.
Much of the power of our Freedom Movement in the United States has come from
this music. It has strengthened us with its sweet rhythms when courage began to
fail. It has calmed us with its rich harmonies when spirits were down.
Dr. Martin Luther King, Jr., Opening Address to the 1964 Berlin Jazz Festival.
For musicwe could envisage the question of how to perform abstract algebraic structures. This is a deep question, since making music is intimately
related to the expression of thoughts. So we would like to be able to express
algebraic insights, revealed by the use of K-nets or symmetry groups, for
example, in terms of musical gestures. To put it more strikingly: Is it possible to play the music of thoughts?
Guerino Mazzola and Moreno Andreatta, Diagrams, Gestures and Formulae in Music.
iv
ACKNOWLEDGEMENTS
I would like to express my appreciation for my family and their support and inspiration
throughout this process. My wife, Dim Mai Bishop and my children Sarah Ngc Vn Bishop
and Jonah Ngc Qy Bishop; my parents, John Bishop Jr., Nancy Bishop, V Ng Dng, and
V H Lan Nha; and my siblings, Karen Bishop-Holst and V Ng H Ngc. Without their help,
this work would not be possible. I hope they see it as an expression of my love for them.
I would also like to thank those at Louisiana State University. My advisor, Dr. Robert
Peck for inspiring me to pursue studies in mathematics and for his patience while I was learning.
Also, Dr. Jeffrey Perry and Dr. David Smyth were instrumental in my understanding of
Schenkerian techniques and who were fundamental in my development. Dr. Willis Deloney, Dr.
William Grimes and Dr. Brian Shaw, members of the jazz studies department provided the
utmost support for my studies. I took great pride in working with them.
The concept of using triads as an improvisational tool was first introduced to me my Jon
Damian, Chan Johnson, and Larry Sinibaldi nearly three decades ago; this is the genesis of my
musical problem addressed here. Luthier Abe Rivera changed the course of my life by reinstating
music as my primary focus. I also thank Ann Marie de Zeeuw, my for supported my interest in
music theory, and Dale Garner whose influence instilled in me a great respect for mathematics
and the desire to continue to learn.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS .......................................................................................................iv
LIST OF TABLES ................................................................................................................... vii
LIST OF FIGURES ................................................................................................................ viii
LIST OF EXAMPLES ...............................................................................................................ix
LIST OF DEFINITIONS............................................................................................................xi
LIST OF ANALYSES ............................................................................................................. xii
SYMBOLS ............................................................................................................................. xiii
ABSTRACT ............................................................................................................................. xv
CHAPTER 1. INTRODUCTION, PRELIMINARIES, AND HISTORICAL CONTEXT ........... 1
1.1. Introduction ......................................................................................................................1
1.2. Literature Review .............................................................................................................6
1.2.1. Jazz Literature ............................................................................................................7 1.2.2. Chord/Scale Relationship Literature ...........................................................................8
1.2.2. Triadic Specific Methods for Jazz Improvisation ........................................................8 1.2.3. Triadic Theory............................................................................................................9
1.2.4. Group Theory Literature ........................................................................................... 13 1.3. Mathematical Preliminaries ............................................................................................. 14
1.4. Non-Traditional Triad Usage in a Historical Context....................................................... 22
CHAPTER 2. SET DEFINITION............................................................................................. 30
2.1. Introduction .................................................................................................................... 30 2.2. Diatonic Harmony........................................................................................................... 31
2.3. Modal Harmony .............................................................................................................. 32 2.4. Dominant Action ...............

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