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1 | Harmonic tunings by Bryan C. Mills, 2018. |

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3 | This spreadsheet computes “Harmonic” tunings, a mathematically-idealized just intonation oriented toward instruments playing in just one or two “home” keys. |

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5 | The tuning is parameterized by only two inputs: the note corresponding to the fundamental frequency, and the frequency of A₄ |

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7 | All notes in the tuning are integer multiples of the fundamental frequency. |

8 | Factors of three produce “perfect fifths”. |

9 | Factors of five produce “major thirds”. |

10 | Factors of seven produce “harmonic sevenths” (a.k.a. “subminor sevenths”). |

11 | Factors of eleven produce “neutral thirds”. |

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13 | The first complete diatonic scale begins at harmonic 24, although a close approximation can be found earlier (at harmonic 16). |

14 | The last chromatic note formed exclusively from perfect fifths and major thirds occurs at harmonic 675, but a close approximation to a chromatic scale (using a harmonic seventh instead) begins at harmonic 72, and another occurs at harmonic 96. |

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16 | The harmonics chosen for the diatonic notes agree with those found in J. Murray Barbour's paper “Just Intonation Confuted”. |

17 | Unlike Barbour's chromatic scale and the microtonal chromatic scales of Partch and Johnston, this tuning avoids undertones entirely, staying entirely within the integer harmonics rather than fractions. |

18 | Thus, for example, the E♭ and B♭ in the C chromatic scale are harmonic sevenths above E and C, not major-thirds below G and D. |

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20 | Each note in the tuning has a unique, procedurally-generated name. |

21 | The source code to generate note names and numbers from the given parameters can be accessed via Tools > Script editor in the menu, or at |

22 | https://docs.google.com/document/d/1UryDF2I1Pd1Q_vC28vHefPeaTFVBHYvsg1AmswFjwPY/edit?usp=sharing |

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24 | Four factors of three are almost two octaves and a major third (off by one “syntonic comma”, denoted here by an upward-pointing accidental). |

25 | Three factors of five are almost seven octaves (off by one “diesis”, denoted here by a quarter-tone-flat symbol). |

26 | One factor of seven is almost an octave plus a minor seventh (off by one “septimal comma”, denoted here by a downward-pointing accidental). |

27 | Sevenths are common in folk and jazz music, but the rest of these enharmonic variations are relatively rare in chord-based music. |

28 | Rare enharmonics are indicated by red-highlighted factors, and the “rare enharmonics hidden” filter view hides them. |

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