On ‘Crystal Growth’ in harmonic space (1993 – 1998) 1

[1] The following article was originally published in a German language version in MusikTexte, 112, February 2007, pp. 75 – 79.

Abstract

The author proposes a method of gradually constructing pitch ‘crystals’ in harmonic space through a simple algorithm where each new pitch is as close as possible to all the pitches already present. Interestingly, the algorithmic growth of these crystals seems to mirror the historical development of pitch resources, including the shift from Pythagorean to just intonation in the sixteenth century. As the crystals continue to grow, they imply the emergence of pitch structures in ‘extended just intonation’. [Editor]

Keywords:

• Harmonic Space
• Crystal Growth
• Just Intonation
• Harmonic Lattices

Notes

[1] The following article was originally published in a German language version in MusikTexte, 112, February 2007, pp. 75 – 79.

[2] Tenney's concept of harmonic space is developed in his 1983 article ‘John Cage and the Theory of Harmony’ (Tenney, 1993 [1983). Harmonic space is comprised of discrete points, each representing a pitch. The frequencies of these pitches are related to one another by the simple integer ratios of just intonation. Each axis in harmonic space represents a different prime number factor: in Figures 1 –  10, for example the vertical dimension represents powers of two (octaves) while the horizontal axis represents powers of three (perfect twelfths). The number of axes (dimensions in harmonic space) is variable depending on the musical context—Figures 11 –  16 replace the vertical octave axis with powers of 5 (introducing just thirds and sixths), and Figures 17 –  20 add a third dimension for powers of 7.

[3] One of the most powerful aspects of Tenney's theory of harmonic space is a metric for measuring harmonic distance. The harmonic distance between two pitches f _a and f _b is calculated by the formula HD(f _a , f _b ) = log₂(a) + log₂(b) = log₂(ab), ‘where f _a and f _b are the fundamental frequencies of the two tones, a = f _a /gcd(f _a , f _b ) and b = f _b /gcd(f _a , f _b )’ (Tenney, 1993 [1983, p. 153). By dividing each frequency by the greatest common denominator (gcd) of the two frequencies, their ratio is reduced to simplest terms.

[4] Pitch-class projection spaces simplify harmonic space by ‘collapsing’ the 2-dimension, thus assuming octave equivalence between identical pitch classes in different registers.

[5] The terms ‘3-limit’ and ‘5-limit’ are drawn from Harry Partch's harmonic theories, and refer to the highest prime factors in a just intonation system. A 3-limit (Pythagorean) system is based solely on factors of 2 and 3, while the expansion to a 5-limit system allows the just thirds and sixths introduced by Renaissance theorists.