## Music and Mathematics: From Pythagoras to FractalsFrom Ancient Greek times, music has been seen as a mathematical art, and the relationship between mathematics and music has fascinated generations. This collection of wide ranging, comprehensive and fully-illustrated papers, authorized by leading scholars, presents the link between these two subjects in a lucid manner that is suitable for students of both subjects, as well as the general reader with an interest in music. Physical, theoretical, physiological, acoustic, compositional and analytical relationships between mathematics and music are unfolded and explored with focus on tuning and temperament, the mathematics of sound, bell-ringing and modern compositional techniques. |

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### Contents

an overview | 1 |

closing the spiral | 13 |

Kepler and his readers | 29 |

The science of musical sound | 47 |

Faggots fretful fiasco | 61 |

combinational tones and consonance | 77 |

The geometry of music | 91 |

bells and mathematics | 113 |

sets rows | 131 |

Microtones and projective planes | 149 |

Composing with fractals | 163 |

Notes on contributors | 173 |

Acknowledgements | 183 |

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### Common terms and phrases

acoustics angle approximation basic called century change ringing channel Chapter clarinet combinational tones composer composition consonance construction corresponding cosmology cyclic design diagram difference set modulo double exchange dual plane equally tempered equally tempered system example Fano plane Figure finite projective plane five bells fractal frequency ratios frets frieze pattern geometrical graph Greek guitar harmonics Helmholtz Iannis Xenakis ideas instruments integer interval Kepler keyboard Kircher length London magic square major third mathematical mathematicians Mersenne Mersenne's method modulo motif music theory musical space musicians notation notes octave oscillograph trace Oxford pairs perfect fifth permutations Peter Maxwell Davies piano piece pitch pitch classes Plain Bob Minimus plain hunting played points possible problem produce Pythagorean comma Pythagorean scale Pythagorean tuning ringers rung Schoenberg semitone sequence sounds Strahle's string structure swing symmetry tion transformations transposition tuning twelve-note University vibrations violin Xenakis