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Music and mathematics: an overview 1 Susan Wollenberg Part I: Music and mathematics through history 11 1. Tuning and temperament: closing the spiral 13 Neil Bibby 2. Musical cosmology: Kepler and his readers 29 J. V Field Part II: The mathematics of musical sound 45 3. The science of musical sound 47 Charles Taylor 4. Faggot's fretful fiasco 61 Ian Stewart 5. Helmholtz: combinational tones and consonance 77 David Fowler Part III: Mathematical structure in music 89 6. The geometry of music 91 Wilfrid Hodges 7. Ringing the changes: bells and mathematics 113 Dermot Roaf and Arthur White 8. Composing with numbers: sets, rows and magic squares 131 Jonathan Cross Part IV: The composer speaks 147 9. Microtones and projective planes 149 Carlton Gamer and Robin Wilson 10. Composing with fractals 163 Robert Sherlaw Johnson Notes on contributors 173 Notes, references, and further reading 177 Acknowledgements 183 Index 187Library of Congress Subject Headings for this publication: Music theory Mathematics, Musical temperament