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Teaching Musical Meter to School-Age Students Through the Ski-Hill Graph

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TEACHING MUSICAL METER TO SCHOOL-AGE STUDENTS THROUGH THE SKI-HILL GRAPH Andrea Mary Calilhanna A thesis submitted in fulfillment of the requirements for the degree of Master of Music (Musicology) Sydney Conservatorium of Music, The University of Sydney. 2018 This is to certify that to the best of my knowledge, the content of this thesis is my own work. This thesis has not been submitted for any degree or other purposes. 2 ABSTRACT This thesis, Teaching Musical Meter to School-Age Students Through the Ski-Hill Graph, aims to demonstrate the “pedagogability” of modern meter theory, that is, that new scholarship on meter can translate into a coherent and practically implementable instructional curriculum, with various advantages for school-age students. The curriculum model developed in the thesis is derived from Richard Cohn’s work on and approaches to meter theory, which focuses on “sound rather than notation” and graphic representations of meter through mathematical music theory. The materials set out in the thesis demonstrate ways students might be taught to articulate their experience of meter. A unified approach, it incorporates Cohn’s ski-hill graph and other instruments of mathematical music theory such as the SkiHill app, numbering for counting meter, cyclic graphs, and beat-class theory. Among the outcomes it anticipates is a deeper engagement with music in classroom settings. Through this new foundation, where meter studies fits more compatibly with studies of tonality, students should logically be able to perform, compose, and analyse music both aurally and visually with increased confidence and understanding. 3 ACKNOWLEDGEMENTS Dedicated to Gerard, Catherine and Teresa To my parents Ron and Mary Close for their example, love of music, and the music lessons they provided for me which set a foundation for my own love of music. My sister Ronda for her prayers, support, inspiration, and encouragement over the years; and to all of my family and friends. To my students from whom I learn so much. With thanks to my lecturers for their classes at the Sydney Conservatorium of Music. To all those involved in presenting papers and assisting to organise the three Meter Symposiums at the Sydney Conservatorium of Music between 2016-2018. The Meter Symposiums became one of only two public music theory events in Australia meeting regularly during the years 2016-2018. The inter-disciplinary and multi-university collaborations formed as a result of the symposiums with fellow colleagues and for others have become a great source of inspiration to me over the years. To Dr Alan Maddox, who while I was writing my final thesis chapter, co-organised the third Meter Symposium to be held at the Sydney Conservatorium in November 2018. Special thanks to Dr Michael Webb on becoming my supervisor and for his advice on Music Education. Also, to Michael for the invaluable opportunities to lecture on meter with his students studying to become classroom music teachers, and to collaborate as a research assistant on the Understanding and Teaching Meter Survey, the Understanding and Teaching Meter Survey Report (Calilhanna, 2017) and the article Teaching time: A survey of music educators’ approaches to meter (Calilhanna and Webb, 2018 unpublished). Thank you to Mr Lewis Cornwell for being my Auxiliary Supervisor. With gratitude to Mr David Kulma and Professor Richard Cohn for agreeing to collaborate on the YouTube video “March V Waltz – A Short Intro to Meter and Ski-Hill Graphs” on Music Corner Breve produced by Kulma (2016). 4 To all those in our Teaching Mathematics with Music and Music with Mathematics research project team. It’s been a pleasure and an inspiration to work with such a talented group of scholars while researching and writing my thesis. A special thank you to Dr Andrew Milne for agreeing to develop the prototype of the SkiHill app and then to develop the XronoBeat app. These apps have now become integral resources in my studio and I daily witness the benefits learning meter through visualisations and sonifications bring to students. To Andy in gratitude for asking me to contribute to my first book chapter to discuss my work teaching musical meter through Cohn’s ski-hill graph (Hilton, Calilhanna, Milne 2018). Also, to Andy for inviting me to co-author a paper with him to present at the seventh international conference of the Society for Mathematics and Computation in Music, 2019. I look forward to the SkiHill and XronoBeat apps becoming widely used in classrooms and studios, in fact wherever meter is taught with Cohn’s mathematical music theory to the benefit of students around the world. I particularly wish to thank my Chief Supervisor, Associate Professor Kathleen Nelson for her supervision and lectures on early music. I am also grateful to Kathleen for the opportunity she gave me to assist her research on a medieval manuscripts project. Finally, I wish to thank Professor Richard Cohn, for his inspiring classes in 2015 which prompted my own research and practice and led to the writing of this thesis. Only through teaching meter using Rick’s approach did I notice how much my students and I benefitted from new understandings of meter particularly from graphic representations of meter through the ski-hill graph, cyclic graph, and mathematical music theory. Cohn’s approach to understanding and teaching meter theory is the most innovative and meaningful music curriculum development I have encountered in my more than three decades of music teaching. I will always be grateful to Rick for his generosity as a scholar for sharing his work and ideas. Also, for his support of my project to bring about the “pedagogability” of intellectual ideas and practices for the benefit of young learners in Australia and beyond. 5 Table of Contents ABSTRACT ............................................................................................................................... 3 ACKNOWLEDGEMENTS ....................................................................................................... 4 LIST OF FIGURES ................................................................................................................... 8 CHAPTER 1 ............................................................................................................................ 11 INTRODUCTION AND BACKGROUND ............................................................................ 11 Introduction .......................................................................................................................... 11 Methodology ........................................................................................................................ 13 Organisation of Chapters ...................................................................................................... 14 Review of Texts and Approaches for Teaching Meter ........................................................ 15 Text Books for Tertiary Students ......................................................................................... 17 Recent Research and Thought on Teaching Meter .............................................................. 24 CHAPTER 2 ............................................................................................................................ 36 THE DEVELOPMENT OF THE SKI-HILL GRAPH AND A STUDY OF A MEDIEVAL PREDECESSOR ...................................................................................................................... 36 Part 1: The Ski-Hill Graph ................................................................................................... 39 Origins of the Ski-Hill Graph ............................................................................................... 39 Recent Developments in the Use of the Ski-Hill Graph ...................................................... 48 Part 2: A Medieval Triangular Graph Developed by Johannes Torkesey (d.1340) ............. 53 Brief History ......................................................................................................................... 53 Johannes Torkesey (-d.1340) and Trianguli et Scuti Declaratio De Proportionibus Musicae Mensurabilis (1320-1330) .................................................................................................... 55 Torkesey’s Augustinian Education ...................................................................................... 61 The Triangle, Shield, Square of Opposition, and Circle as Symbols ................................... 62 Popularity of Torkesey’s Work and the Chicago Manuscript 54.1 ...................................... 66 CHAPTER 3 ............................................................................................................................ 69 A CONCEPTUAL FRAMEWORK FOR TEACHING MUSICAL METER TO SCHOOL- AGE STUDENTS THROUGH THE SKI-HILL GRAPH ...................................................... 69 Part 1: Introduction............................................................................................................... 70 Organisation of Materials ..................................................................................................... 70 Context for Teaching Meter With New Understandings in Schools .................................... 71 From Ski-Hill Graph to Cyclic Graphs and Beat-Class Theory .......................................... 71 Cyclicity and Hierarchy ......................................................................................................
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