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Equal Divisions Doug Wilde a Thesis Submitted EQUAL DIVISIONS DOUG WILDE A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS GRADUATE PROGRAM IN MUSIC YORK UNIVERSITY TORONTO, ONTARIO MARCH 2016 © DOUG WILDE, 2016 ABSTRACT The equal tempered scale may be divided equally into twelve, six, four, three and two equal parts. From each division, symmetrical harmonic environments may be created that sound and function in different ways than diatonic systems. Some composers use these systems exclusively, and some combine them with more traditional harmonic systems. This thesis presents ten original compositions. Each division of the octave is represented by a jazz and a chamber piece. A brief history of equal temperament is included, followed by a chronology of the introduction of symmetrical scales and harmonic material in Western art music and jazz. The methods of several composers, all proponents of symmetry in different ways, are also examined. ii ACKNOWLEDGEMENTS I would like to thank Professors Al Henderson and Matt Vander Woude for their encouragement, guidance and wisdom. The assistance they offered in the preparation of this thesis is greatly appreciated. I would also like to acknowledge the guidance and input provided by Professors Michael Coghlan, Jay Rahn and Sundar Viswanathan. Thank you also to the Graduate Program Assistant, Tere Tilban-Rios. iii TABLE OF CONTENTS Abstract ........................................................................................................................... ii Acknowledgements ....................................................................................................... iii List of Tables ................................................................................................................. vi List of Figures ................................................................................................................ vii Introduction .................................................................................................................... 1 Chapter 1: History and Overview ................................................................................ 2 A Brief History of Equal Temperament ................................................................. 3 History of Equal Divisions ..................................................................................... 7 Equal Divisions and Jazz ...................................................................................... 10 Theoretical Literature ........................................................................................... 12 Problems of Analysis ............................................................................................ 13 Chapter 2: Creation and Application ........................................................................ 15 Symmetrical Scales ............................................................................................. 15 Applications in Classical Music ........................................................................... 19 The Twisted History Of The Octatonic Scale ...................................................... 25 Applications in Jazz: Improvisation ..................................................................... 26 Applications in Jazz: Composition ....................................................................... 28 Chapter 3: Dirty Dozen and Twelve of Us Dirty Dozen .......................................................................................................... 32 Twelve of Us ......................................................................................................... 36 Chapter 4: Six Degrees of Separation and La Tuta Vero Six Degrees of Separation .................................................................................... 42 La Tuta Vero ......................................................................................................... 44 Chapter 5: Octogram and The Eighth Day Octogram .............................................................................................................. 48 The Eighth Day ..................................................................................................... 52 Chapter 6: Chamber Three and Smoke and Mirrors Chamber Three ..................................................................................................... 55 Smoke and Mirrors ............................................................................................... 59 iv Chapter 7: Division by Two and Then Add Red Division by Two .................................................................................................... 62 Then Add Red ....................................................................................................... 67 Conclusion .................................................................................................................... 71 Bibliography ................................................................................................................. 72 Discography .................................................................................................................. 78 Appendix A: Scores ...................................................................................................... 81 Dirty Dozen ..................................................................................................................... 81 Twelve of Us .................................................................................................................... 84 Six Degrees of Separation ............................................................................................... 97 La Tuta Vero ................................................................................................................. 102 The Eighth Day ............................................................................................................. 115 Chamber Three ............................................................................................................. 124 Smoke and Mirrors ....................................................................................................... 130 Division by Two ............................................................................................................ 136 Then Add Red ................................................................................................................ 141 Appendix B: Dirty Dozen with Tone Rows ................................................................ 147 v LIST OF TABLES Table 1: How Composers And Improvisers “Found” Symmetrical Scales ................... 26 Table 2: Compositions That Use Equal Divisions of The Octave ................................. 30 vi LIST OF FIGURES Figure 1.1: Asymmetry Of Major Scale .......................................................................... 2 Figure 1.2: Symmetrical Whole-Tone Scale .................................................................... 2 Figure 1.3: Division Into Twelve Equal Parts ................................................................. 8 Figure 1.4: Division Into Six Equal Parts ........................................................................ 9 Figure 1.5: Division Into Three Equal Parts .................................................................... 9 Figure 1.6: Division Into Three Equal Parts .................................................................... 9 Figure 1.7: Division Into Four Equal Parts ...................................................................... 9 Figure 1.8: The Often-Used Alternate Version Of The Diminished Scale ...................... 9 Figure 1.9: Division of Into Two Equal Parts: Messiaen’s Mode 4 .............................. 10 Figure 1.10: Division of Into Two Equal Parts: Messiaen’s Mode 5 ............................ 10 Figure 1.11: Division of the octave into Two Equal Parts: Messiaen’s Mode 6 ........... 10 Figure 1.12: Division of the octave into Two Equal Parts: Messiaen’s Mode 7 ........... 10 Figure 2.1: Equal division of One Octave into Three Parts ........................................... 21 Figure 2.2: Interpolation of One Note ............................................................................ 21 Figure 2.3: Interpolation of Two Notes ......................................................................... 21 Figure 2.4: Ultrapolation ............................................................................................... 21 Figure 2.5: First Melodic Pattern ................................................................................... 22 Figure 2.6: Transpositions ............................................................................................. 22 Figure 2.7: Major/Minor Relationships ......................................................................... 24 Figure 2.8: Upper Extensions of C7th Chord .................................................................. 27 Figure 2.9: Flexible Architecture ................................................................................... 27 Figure 2.10: Symmetrical Augmented Scale Containing B, Eb and G triads ................ 31 Figure 3.1: Prime, Inversion, Retrograde, and Retrograde Inversion ............................ 33 Figure 3.2: Twelve Tone Matrix created for Dirty Dozen ............................................. 34 Figure 3.3: Three and Four Note Chords from Prime .................................................... 34 Figure
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