Mathematical and Computational Modeling of Tonality: Theory and Applications (International Series in Operations Research & Management Science Book 204)

From the Preface: Blending ideas from operations research, music psychology, music theory, and cognitive science, this book aims to tell a coherent story of how tonality pervades our experience, and hence our models, of music. The story is told through the developmental stages of the Spiral Array model for tonality, a geometric model designed to incorporate and represent principles of tonal cognition, thereby lending itself to practical applications of tonal recognition, segmentation, and visualization. Mathematically speaking, the coils that make up the Spiral Array model are in effect helices, a spiral referring to a curve emanating from a central point. The use of “spiral” here is inspired by spiral staircases, intertwined spiral staircases: nested double helices within an outer spiral. The book serves as a compilation of knowledge about the Spiral Array model and its applications, and is written for a broad audience, ranging from the layperson interested in music, mathematics, and computing to the music scientist-engineer interested in computational approaches to music representation and analysis, from the music-mathematical and computational sciences student interested in learning about tonality from a formal modeling standpoint to the computer musician interested in applying these technologies in interactive composition and performance. Some chapters assume no musical or technical knowledge, and some are more musically or computationally involved. Blending Ideas From Operations Research, Music Psychology, Music Theory, And Cognitive Science, This Book Aims To Tell A Coherent Story Of How Tonality Pervades Our Experience, And Hence Our Models, Of Music. The Story Is Told Through The Developmental Stages Of The Spiral Array Model For Tonality, A Geometric Model Designed To Incorporate And Represent Principles Of Tonal Cognition, Thereby Lending Itself To Practical Applications Of Tonal Recognition, Segmentation, And Visualization. Mathematically Speaking, The Coils That Make Up The Spiral Array Model Are In Effect Helices, A Spiral Referring To A Curve Emanating From A Central Point. The Use Of Spiral Here Is Inspired By Spiral Staircases, Intertwined Spiral Staircases: Nested Double Helices Within An Outer Spiral. The Book Serves As A Compilation Of Knowledge About The Spiral Array Model And Its Applications, And Is Written For A Broad Audience, Ranging From The Layperson Interested In Music, Mathematics, And Computing To The Music Scientist-engineer Interested In Computational Approaches To Music Representation And Analysis, From The Music-mathematical And Computational Sciences Student Interested In Learning About Tonality From A Formal Modeling Standpoint To The Computer Musician Interested In Applying These Technologies In Interactive Composition And Performance. Some Chapters Assume No Musical Or Technical Knowledge, And Some Are More Musically Or Computationally Involved.--preface. Introduction. 1. Tonality -- 2. An Abbreviated Survey -- 3. The Spiral Array -- 4. The Ceg Algorithm (part I) -- 5. The Ceg Algorithm (part Ii) : Validation -- 6. Determining Key Boundaries -- 7. Argus Segmentation Methods -- 8. Real-time Pitch Spelling -- 9. Musa.rt -- 10. Visible Humor -- 11. Sensitivity Analysis -- Appendix A. Model Calibration -- Appendix B. Ceg Key Finding -- Appendix C. Glossary. Elaine Chew. Includes Bibliographical References And Indexes. This proposed addition to the ISOR series is perhaps the most novel application of OR techniques ever conceived: the mathematical and computational modeling of tonality. This is the first book ever to apply problem-solving computational analysis to the realm of music, and author Elaine Chew is the one person in the world bringing together the mathematical and computational in a sharply focused coherent text/reference book. An accomplished scholar, pianist, and instructor (OR) at the Viterbi School of Engineering, USC, Chew has also just completed a year-long Fellowship at the Harvard University Radcliffe Institute of Advanced Studies. She is also the inventor of the Spiral Array algorithm for the study of tonality, and that is the cornerstone of her proposed book. Taking interior point optimization methods far beyond the more traditional engineering, financial and logistics arenas, Chew applies OR techniques to the computational analysis of music, technological requirements for digital music storage and retrieval, and the study of human traits inherent in composition, performance, hearing, and analysis. Chew shows how the Spiral Array algorithm can be used in tonal analysis and solutions to music cognition phenomena. Individual sections of the book first introduce the notion of computational modeling of cognition of music structures, and then present the geometric models of tonality, including the Spiral Array, and then look at applications to specific areas like key finding; interactive visual analysis of tonal patterns; audio key finding; and segmentation and similarity assessment. Anyone involved in research in any of these areas will likely know of Chew’s work, and will very likely want this book handy. The research and advances are applicable to a broad range of study, from digital storage and retrieval through human cognition and understanding of musical structure and form. Front Matter....Pages i-xiv Front Matter....Pages 1-1 Tonality....Pages 3-18 An Abbreviated Survey....Pages 19-37 Front Matter....Pages 39-39 The Spiral Array....Pages 41-60 Front Matter....Pages 61-61 The CEG Algorithm (Part I)....Pages 63-71 The CEG Algorithm (Part II): Validation....Pages 73-91 Front Matter....Pages 93-93 Determining Key Boundaries....Pages 95-107 Argus Segmentation Method....Pages 109-130 Front Matter....Pages 131-131 Real-Time Pitch Spelling....Pages 133-156 Front Matter....Pages 157-157 MuSA.RT....Pages 159-178 Visible Humor....Pages 179-190 Front Matter....Pages 191-191 Sensitivity Analysis....Pages 193-222 Back Matter....Pages 223-305