## Foundations of Diatonic Theory: A Mathematically Based Approach to Music FundamentalsFoundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals is an introductory, undergraduate-level textbook that provides an easy entry point into the challenging field of diatonic set theory, a division of music theory that applies the techniques of discrete mathematics to the properties of diatonic scales. After introducing mathematical concepts that relate directly to music theory, the text concentrates on these mathematical relationships, firmly establishing a link between introductory pedagogy and recent scholarship in music theory. It then relates concepts in diatonic set theory directly to the study of music fundamentals through pedagogical exercises and instructions. Ideal for introductory music majors, the book requires only a general knowledge of mathematics, and the exercises are provided with solutions and detailed explanations. With its basic description of musical elements, this textbook is suitable for courses in music fundamentals, music theory for non-music majors, music and mathematics, and other similar courses that allow students to improve their mathematics skills while pursuing the study of music. |

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### Contents

Introduction | 1 |

Mathematics and Music | 2 |

How to Use this Book | 4 |

Spatial Relations and Musical Structures | 5 |

Musical Structures form Geometric Figures | 15 |

An Intervallic Definition | 26 |

Summary and Extensions | 39 |

Interval Patterns and Musical Structures | 45 |

Summary and Extensions | 103 |

Triads and Seventh Chords and Their Structures | 107 |

Maximally Even Triads and Seventh Chords | 108 |

Variety and Multiplicity of Diatonic Chords | 129 |

Summary and Extensions | 141 |

Conclusion | 143 |

For Further Study | 147 |

Notes | 153 |

Interval Patterns in Other Scales | 66 |

Interval Patterns and the Circle of Fifths | 79 |

Structure in Other Collections | 95 |

Sources Cited | 159 |

Index | 163 |

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### Common terms and phrases

adjacent answer approach arrangement ascending melodic minor aspects associated beginning cardinality equals variety Chapter chromatic circle diagram circle of fifths CLOCKWISE DISTANCE Clough common complete concepts corresponding course crossing lines definition determine diatonic collection diminished seventh chord discussed DISTANCE BETWEEN DOTS distinct interval patterns dots around circle DOTS d distance earlier exactly examine example explore Figure five formed four Group of series half steps ideas identify important indicated Interval pattern Number interval patterns appear inversion John keys labeled major mathematics maximally minor scale Music Theory musical structures notes Drawn Number of occurrences observed octave P4 half steps pair pentatonic placed plot possible produce properties questions respect resulting Roman rotating series from exercise series of notes seven seventh chords shown shows Solution structure implies multiplicity Systems tion tones transposed triads and seventh twelve universe