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Pitch-Class Set Theory in Music and Mathematics Volume I

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Pitch-Class Set Theory in Music and Mathematics Volume I Portfolio of Compositions: Pitch-Class Set Theory in Music and Mathematics Volume I: Commentary Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy Michael James McNeilis March 2017 Declaration This thesis is my own work, has not been previously submitted for an award at this university of any other institution, and the content is legally allowable under copyright legislation. Signed ……………………………….. Date ……………………. i Abstract This portfolio contains six scores of contemporary works, one electroacoustic piece submitted in concrete audio format and a critical commentary. Three additional scores are included as appendices, which present alternative arrangements of the acoustic works. An accompanying data DVD includes stereo recordings of a concert performance and four workshop performances, as well as stereo and 5.1 versions of the electroacoustic piece. With existing literature primarily focussing on pitch-class set theory as an analytical tool, this study aims to answer the question: Can set theory be used as the basis of a compositional framework to advance the creative process within the interdisciplinary field of music and mathematics? The pieces in the portfolio consider how set theory can be applied alongside mathematical principles like mutation, symmetry and proportion to inform musical networks and topologies based on material such as the octatonic scale, whole-tone scale and the all-interval tetrachords. A particular emphasis is placed on relationships between pitch-class sets in order to produce coherent compositional designs which link micro-level material to macro-level form and structure. Post- compositional evaluation of each work helps to create subsequent designs in the portfolio, and culminates in the piece Aggregation, which assesses the mathematical principles and compositional techniques developed earlier in the portfolio to encapsulate the research within a unified design and large-scale work. The results of the research establish set theory as a viable mathematical language for both the pre- compositional and compositional stages of the creative process, and demonstrate refinement of my own compositional methodology and musical style. ii Acknowledgements and Dedication Firstly, I would like to thank my supervisors Robin Hartwell and Dr Ian Percy for their expertise and guidance throughout my studies. I would also like to acknowledge Liverpool Hope’s Music department for hosting research seminars and providing performance opportunities with RLPO musicians, as well as for its academic and pastoral support. Thanks to my family and friends, who have encouraged and motivated me throughout my time at Hope. Finally, I would like to dedicate this PhD to my Grandfather, John Robinson (1931- 2016). Without his constant support, I could have not undertaken this research journey. iii List of Contents Page Declaration ................................................................................................................... i Abstract ........................................................................................................................ ii Acknowledgements and Dedication ........................................................................... iii List of Examples .......................................................................................................... vi List of Figures .............................................................................................................. viii List of Tables ............................................................................................................... ix Score Details and Media ............................................................................................. x 1 Introduction ............................................................................................................ 1 1.1 Background and Context .................................................................................. 1 1.2 Research Aims and Methodology .................................................................... 7 2 Journey of Research .............................................................................................. 11 3 From Zero ............................................................................................................... 15 3.1 Mutation ........................................................................................................... 15 3.2 Pre-Compositional Design ............................................................................... 17 3.3 From Zero ........................................................................................................ 21 3.4 Summary .......................................................................................................... 24 4 In Equal Measure ................................................................................................... 26 4.1 Symmetry ......................................................................................................... 26 4.2 Pre-Compositional Design ............................................................................... 27 4.3 In Equal Measure ............................................................................................. 31 4.4 Summary .......................................................................................................... 43 5 Unequal Measures .................................................................................................. 45 5.1 Proportion, The Fibonacci Series and the Golden Section .............................. 45 5.2 Unequal Measures ............................................................................................ 47 5.3 Summary .......................................................................................................... 56 6 Space Ripple ............................................................................................................ 58 6.1 Pi ...................................................................................................................... 58 6.2 Pre-Compositional Design ............................................................................... 60 6.3 Space Ripple .................................................................................................... 63 6.4 Summary .......................................................................................................... 66 iv Page 7 Three Minds Fractured ......................................................................................... 67 7.1 The Octatonic Scale ......................................................................................... 67 7.2 Pre-Compositional Design ............................................................................... 69 7.3 Three Minds Fractured ..................................................................................... 72 7.4 Summary .......................................................................................................... 78 8 One to Another ....................................................................................................... 80 8.1 The All-Interval Tetrachords ........................................................................... 80 8.2 Pre-Compositional Design ............................................................................... 82 8.3 One to Another ................................................................................................. 85 8.4 Summary ........................................................................................................ 101 9 Aggregation .......................................................................................................... 102 9.1 Introduction .................................................................................................... 102 9.2 Pre-Compositional Design ............................................................................. 102 9.3 Aggregation .................................................................................................... 104 9.4 Workshops ..................................................................................................... 115 9.5 Summary ........................................................................................................ 115 10 Conclusion ........................................................................................................... 116 10.1 Overall Findings ........................................................................................... 116 10.2 Future Research ........................................................................................... 118 10.3 Closing Thoughts ......................................................................................... 119 Appendix A From Zero – Alternative Design ....................................................... 120 Appendix B From Zero – Matrices for First Iteration Sets (Primary Design) ....... 122 Appendix C From Zero – Possible Set Mutation (Primary Design) ..................... 123 Appendix D In Equal Measure – Initial Sketch of Movement One, Section A .... 124 Appendix E Additional Score – One to Another (Octet Arrangement) Appendix F Additional Score – Aggregation (Finale Excerpt Arrangement) Appendix G Additional Score – In Three Elements Appendix H Data DVD Bibliography .............................................................................................................
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