Introduction to “Contributions toward a Quantitative Theory of Harmony”

James Tenney presents the introduction to his 1979 essay “Contributions toward a Quantitative Theory of Harmony.” In this introduction, Tenney discusses the history of consonance/dissonance, paying attention to the semantic problem, relations between pitches, qualities of simultaneous aggregates, and contextual as well as operational and functional senses of consonance/dissonance. He also explores the structure of harmonic series aggregates, focusing on harmonic intersection and disjunction, harmonic density, and harmonic distance and pitch mapping. Finally, he considers problems of tonality by analyzing harmonic-melodic roots and the “tonic effect,” along with harmonic (chordal) roots, the “fundamental bass,” and a model of pitch perception in the auditory system. In an epilogue, Tenney describes new harmonic resources as well as prospects and limitations of his contributions.

The Structure of Harmonic Series Aggregates

James Tenney discusses the structure of harmonic series aggregates and provides a detailed explanation of the genesis of his HD function. He describes, through first principles (perception, simple mathematics), what happens when “two or more compound tones are sounded simultaneously.” Using simple properties of relatively prime (reduced) ratios, the harmonic series, and least common multiples and greatest common divisors, Tenney approaches harmony in the way he had suggested some thirty years earlier: “to start if possible at the very beginning, to clear the mind of loose ends whose origins are forgotten; loose ends and means become habits.” After exploring harmonic intersection and disjunction, Tenney concludes with an analysis of harmonic distance and pitch distance.