Harmonic Tuning
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Harmonic tunings by Bryan C. Mills, 2018.
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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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The tuning is parameterized by only two inputs: the note corresponding to the fundamental frequency, and the frequency of A₄
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All notes in the tuning are integer multiples of the fundamental frequency.
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Factors of three produce “perfect fifths”.
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Factors of five produce “major thirds”.
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Factors of seven produce “harmonic sevenths” (a.k.a. “subminor sevenths”).
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Factors of eleven produce “neutral thirds”.
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The first complete diatonic scale begins at harmonic 24, although a close approximation can be found earlier (at harmonic 16).
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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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The harmonics chosen for the diatonic notes agree with those found in J. Murray Barbour's paper “Just Intonation Confuted”.
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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.
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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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Each note in the tuning has a unique, procedurally-generated name.
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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
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https://docs.google.com/document/d/1UryDF2I1Pd1Q_vC28vHefPeaTFVBHYvsg1AmswFjwPY/edit?usp=sharing
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Four factors of three are almost two octaves and a major third (off by one “syntonic comma”, denoted here by an upward-pointing accidental).
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Three factors of five are almost seven octaves (off by one “diesis”, denoted here by a quarter-tone-flat symbol).
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One factor of seven is almost an octave plus a minor seventh (off by one “septimal comma”, denoted here by a downward-pointing accidental).
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Sevenths are common in folk and jazz music, but the rest of these enharmonic variations are relatively rare in chord-based music.
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Rare enharmonics are indicated by red-highlighted factors, and the “rare enharmonics hidden” filter view hides them.
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