<ul><li><p>A PERMUTATIONAL TRIADIC APPROACH TO JAZZ HARMONY AND </p><p>THE CHORD/SCALE RELATIONSHIP </p><p>A Dissertation </p><p>Submitted to the Graduate Faculty of the </p><p>Louisiana State University and </p><p>Agricultural Mechanical College </p><p>in partial fulfillment of the </p><p>requirements for the degree of </p><p>Doctor of Philosophy </p><p>in </p><p>The School of Music </p><p>by </p><p>John Bishop </p><p>B.M., Berklee College, 1990 </p><p>M.M., University of Louisville, 2004 </p><p>December 2012 </p></li><li><p>ii </p><p>To Quentin Sharpenstein </p></li><li><p>iii </p><p>The harmonic, simple, and direct triad is the true and unitrisonic root of all </p><p>the most perfect and most complete harmonies that can exist in the world. It is the </p><p>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 </p><p>semblance more lucid). All the more, therefore, should theologians and </p><p>philosophers direct their attention to it, since at present they know fundamentally </p><p>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, </p><p>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 </p><p>a pastime, and I hope to see it perfected with Gods help, to Whom be praise forever. </p><p> Johannes Lippius, Synopsis of New Music (Synopsis Musicae Novae). </p><p> God has wrought many things out of oppression. He has endowed his </p><p>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 </p><p>many different situations. </p><p>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 </p><p>into music, only to come out with some new hope or sense of triumph. </p><p>This is triumphant music. </p><p>Modern jazz has continued in this tradition, singing songs of a more complicated </p><p>urban experience. When life itself offers no order of meaning, the musician </p><p>creates an order and meaning from the sounds of the earth, which flow through </p><p>his instrument. </p><p>Much of the power of our Freedom Movement in the United States has come from </p><p>this music. It has strengthened us with its sweet rhythms when courage began to </p><p>fail. It has calmed us with its rich harmonies when spirits were down. </p><p> Dr. Martin Luther King, Jr., Opening Address to the 1964 Berlin Jazz Festival. </p><p> For musicwe could envisage the question of how to perform abstract algebraic structures. This is a deep question, since making music is intimately </p><p>related to the expression of thoughts. So we would like to be able to express </p><p>algebraic insights, revealed by the use of K-nets or symmetry groups, for </p><p>example, in terms of musical gestures. To put it more strikingly: Is it possible to play the music of thoughts? </p><p> Guerino Mazzola and Moreno Andreatta, Diagrams, Gestures and Formulae in Music. </p></li><li><p>iv </p><p>ACKNOWLEDGEMENTS </p><p>I would like to express my appreciation for my family and their support and inspiration </p><p>throughout this process. My wife, Dim Mai Bishop and my children Sarah Ngc Vn Bishop </p><p>and Jonah Ngc Qy Bishop; my parents, John Bishop Jr., Nancy Bishop, V Ng Dng, and </p><p>V H Lan Nha; and my siblings, Karen Bishop-Holst and V Ng H Ngc. Without their help, </p><p>this work would not be possible. I hope they see it as an expression of my love for them. </p><p> I would also like to thank those at Louisiana State University. My advisor, Dr. Robert </p><p>Peck for inspiring me to pursue studies in mathematics and for his patience while I was learning. </p><p>Also, Dr. Jeffrey Perry and Dr. David Smyth were instrumental in my understanding of </p><p>Schenkerian techniques and who were fundamental in my development. Dr. Willis Deloney, Dr. </p><p>William Grimes and Dr. Brian Shaw, members of the jazz studies department provided the </p><p>utmost support for my studies. I took great pride in working with them. </p><p> The concept of using triads as an improvisational tool was first introduced to me my Jon </p><p>Damian, Chan Johnson, and Larry Sinibaldi nearly three decades ago; this is the genesis of my </p><p>musical problem addressed here. Luthier Abe Rivera changed the course of my life by reinstating </p><p>music as my primary focus. I also thank Ann Marie de Zeeuw, my for supported my interest in </p><p>music theory, and Dale Garner whose influence instilled in me a great respect for mathematics </p><p>and the desire to continue to learn. </p></li><li><p>v </p><p>TABLE OF CONTENTS </p><p>ACKNOWLEDGEMENTS .......................................................................................................iv </p><p>LIST OF TABLES ................................................................................................................... vii </p><p>LIST OF FIGURES ................................................................................................................ viii </p><p>LIST OF EXAMPLES ...............................................................................................................ix </p><p>LIST OF DEFINITIONS............................................................................................................xi </p><p>LIST OF ANALYSES ............................................................................................................. xii </p><p>SYMBOLS ............................................................................................................................. xiii </p><p>ABSTRACT ............................................................................................................................. xv </p><p>CHAPTER 1. INTRODUCTION, PRELIMINARIES, AND HISTORICAL CONTEXT ........... 1 </p><p>1.1. Introduction ......................................................................................................................1 </p><p>1.2. Literature Review .............................................................................................................6 </p><p>1.2.1. Jazz Literature ............................................................................................................7 1.2.2. Chord/Scale Relationship Literature ...........................................................................8 </p><p>1.2.2. Triadic Specific Methods for Jazz Improvisation ........................................................8 1.2.3. Triadic Theory............................................................................................................9 </p><p>1.2.4. Group Theory Literature ........................................................................................... 13 1.3. Mathematical Preliminaries ............................................................................................. 14 </p><p>1.4. Non-Traditional Triad Usage in a Historical Context....................................................... 22 </p><p>CHAPTER 2. SET DEFINITION............................................................................................. 30 </p><p>2.1. Introduction .................................................................................................................... 30 2.2. Diatonic Harmony........................................................................................................... 31 </p><p>2.3. Modal Harmony .............................................................................................................. 32 2.4. Dominant Action ............................................................................................................ 47 </p><p>2.5. Tonic Systems................................................................................................................. 61 2.6. Chord/Scale Relationships .............................................................................................. 77 </p><p>2.6.1. The Aebersold/Baker Chord/Scale Method ............................................................... 77 </p><p>2.6.2. Nettles and Grafs Chord/Scale Theory ................................................................... 78 2.6.3. George Russells Lydian Chromatic Concept of Tonal Organization ........................ 79 </p><p>2.7. Triad Specific Methods ................................................................................................... 81 </p><p>2.7.1. Gary Campbells Triad Pairs.................................................................................... 82 2.7.2. George Garzones Triadic Chromatic Approach ....................................................... 85 2.7.3. Larry Carltons Chord-Over-Chord Approach .......................................................... 86 </p><p>CHAPTER 3. GROUP ACTIONS ........................................................................................... 96 </p><p>3.1. Introduction .................................................................................................................... 96 3.2. Scale Roster .................................................................................................................... 96 </p><p>3.3. Symmetries on 4 Elements .............................................................................................. 98 3.4. Symmetries on 6 and 8 Elements................................................................................... 100 </p><p>3.5. Symmetries on 5 and 7 Elements: p-groups ................................................................... 115 </p></li><li><p>vi </p><p>CHAPTER 4. APPLICATION ............................................................................................... 118 </p><p>4.1. p-group Application ...................................................................................................... 118 </p><p>4.2. 3T Systems Revisited..................................................................................................... 119 </p><p>CHAPTER 5. CONCLUSIONS AND ADDITIONAL RESEARCH ...................................... 134 </p><p>5.1. Conclusions .................................................................................................................. 134 </p><p>5.2. Additional Musical Applications ................................................................................... 135 5.3. Additional Mathematical Questions .............................................................................. 137 </p><p>APPENDIX A. MODAL HARMONY .................................................................................... 141 </p><p>APPENDIX B. LEAD SHEETS .............................................................................................. 149 </p><p>APPENDIX C. AEBERSOLD/BAKER SCALE SYLLABUS ................................................ 155 </p><p>APPENDIX D. PERMUTATION LISTS ................................................................................ 157 </p><p>APPENDIX E. DISCOGRAPHY ............................................................................................ 164 </p><p>APPENDIX F. COPYRIGHT PERMISSION.......................................................................... 165 </p><p>BIBLIOGRAPHY ................................................................................................................... 170 </p><p>VITA ...................................................................................................................................... 178 </p></li><li><p>vii </p><p>LIST OF TABLES </p><p>Table 1. T/I conjugation ............................................................................................................ 60 </p><p>Table 2. A4 Cayley table ............................................................................................................ 93 </p><p>Table 3. Scale roster .................................................................................................................. 97 </p><p>Table 4. Alternating group A4 (Oct(1,2), O). .......................................................................... 112 </p></li><li><p>viii </p><p>LIST OF FIGURES </p><p>Figure 1. Cohns Hyper-Hexatonic System................................................................................ 12 </p><p>Figure 2. Symmetries of the triangle .......................................................................................... 21 </p><p>Figure 3. Ionian as (D(), C7): i .................................................................................................. 35 </p><p>Figure 4. Dorian as (D(), C7): r ................................................................................................. 36 </p><p>Figure 5. E := (A, D12) ............................................................................................................... 55 </p><p>Figure 6. Geometric duality and Bemsha Swings final sonority ............................................ 61 </p><p>Figure 7. Symmetries of the square D8................................................................................... 69 </p><p>Figure 8. Group Y ...................................................................................................................... 72 </p><p>Figure 9. Y E4 D ................................................................................................................ 76 </p><p>Figure 10. Bk group J: r ......................................................................................................... 91 </p><p>Figure 11. Tetrahedral symmetry, A4 ......................................................................................... 92 </p><p>Figure 12. Rhomboidal full symmetry group V4 ..................................................................... 99 </p><p>Figure 13. Hexagonal symmetry D12.................................................................................... 101 </p><p>Figure 14. Octagonal group D16 ........................................................................................... 102 </p><p>Figure 15. O := ............................................................................................................. 104 </p><p>Figure 16. Octahedral dualism ................................................................................................. 109 </p><p>Figure 17. (Oct(x,y), O): .................................................................................................. 110 </p><p>Figure 18. Geometric modeling of A4 (Oct(1,2), O) ............................................................... 112 </p><p>Figure 19.1. Octahedral full symmetry group, tetrahedron ....................................................... 114 </p><p>Figure 19.2. Octahedral full symmetry group, cube ................................................................. 114 </p><p>Figure 20. Pentagonal full symmetry group D10 ................................................................... 116 </p><p>Figure 21. Septagonal full symmetry group ............................................................................. 117 </p><p>Figure 22. Torus ...................................................................................................................... 122 </p><p>Figure 23. Toroidal polygon .................................................................................................... 124 </p><p>Figure 24. Fano plane .............................................................................................................. 137 </p></li><li><p>ix </p><p>LIST OF EXAMPLES </p><p>Exampl...</p></li></ul>