A Pitch‐Class Set Space Odyssey, Told by Way of a Hexachord‐Induced System of Genera. (7th January 2013) - Gates, Bernard
Abstract The notion that pitch‐class sets might occupy some form of multi‐dimensional space has engaged a number of theorists over the years. The 'space odyssey' in this article delves into several new issues, such as a methodology for the creation of a comprehensive array of bonded inclusional growth chains, the formation of a diatonic‐to‐chromatic, M5‐induced spectrum of symmetrically arranged set classes and a simple method for establishing levels of affinity between set classes. These investigations lead to the advancement of a sixfold system of genera and to hierarchical distributions of all set classes, furnishing a distinct shape or profile to the whole set‐class universe and to each of the genera. A productive way of producing a three‐dimensional model incorporating all set classes presents itself if four foci, the 'bichromatic' 6–7 (0, 1, 2, 6, 7, 8), hexatonic 6–20 (0, 1, 4, 5, 8, 9), octatonic 6–30 (0, 1, 3, 6, 7, 9) and whole‐tone 6–35 (0, 2, 4, 6, 8, 10), are granted their own spatial 'homes' at distant points on an M‐invariant axis placed perpendicular to a diatonic 6–32 (0, 2, 4, 5, 7, 9) to chromatic 6–1 (0, 1, 2, 3, 4, 5) axis. A transparent globe would seem to be the most suitable vehicle for the representation of this arrangement, where the chromatic and diatonic foci take the 'north' and 'south' poles and the other four take their positions around the 'equator'.