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From ScratchWritings in Music Theory$
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James Tenney, Larry Polansky, and Lauren Pratt

Print publication date: 2015

Print ISBN-13: 9780252038723

Published to Illinois Scholarship Online: April 2017

DOI: 10.5406/illinois/9780252038723.001.0001

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On “Crystal Growth” in Harmonic Space

On “Crystal Growth” in Harmonic Space


(p.383) Chapter 18 On “Crystal Growth” in Harmonic Space
From Scratch

James Tenney

, Larry Polansky, Lauren Pratt, Robert Wannamaker, Michael Winter
University of Illinois Press

James Tenney talks about his “crystal growth” algorithm, an idea that suggests a new “harmonic syntax” for harmonic space. As a quantitative model, it is both suggestively rich for future composition and plausible as a description of the history of tonal expansion. In this algorithm, sets of points are chosen, one by one, in some n-dimensional harmonic space, under the condition that each new point must have the smallest possible sum of harmonic distances to all points already in the set. That is, at each successive stage in the growth of the harmonic lattice, the next ratio added to the set is one whose sum of harmonic distances to each ratio already in the set is minimal. In relation to this algorithm, Tenney considers the Pythagorean pentatonic scale, which may be conceived as a pitch set that arises when extension into the 3,5-plane is just slightly delayed beyond the point where the algorithm would have begun that extension.

Keywords:   harmonic space, James Tenney, crystal growth, harmonic syntax, composition, tonal expansion, harmonic distance, harmonic lattice, Pythagorean pentatonic scale, pitch

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