## 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. |

### 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

a c distance ascending melodic minor augmented triad bisector cardinality equals variety Chapter chromatic universe circle diagram circle of fifths collection Distances corresponding deep scale diatonic collection diatonic set theory diatonic theory diatonic triads different interval patterns diminished seventh chord DISTANCE BETWEEN DOTS distance c distance distance Maximally distinct interval patterns dots are spread dots around circle DOTS d distance exercise Initial series four dots Group of series half-diminished seventh chord Interval pattern Number interval patterns appear interval patterns formed interval type inversion inversionally related Jack Douthett Journal of Music labeled M2 half melodic minor scale minor seventh minor triads Music Theory musical structures notes Drawn number of notes Number of occurrences octatonic scale P4 half steps pentatonic collection pentatonic scale properties series from exercise series of notes seven-note shown in Solution structure implies multiplicity transposed series triads and seventh twelve crossing lines variety and structure whole-tone scale