Normal form, successive interval arrays, transformations and set classes: A re-evaluation and reintegration

Abstract

Normal form is an ordering standard for pitch-class sets that facilitates finding structural relationships and properties through comparative analysis. However, the analytical process often fails to correctly identify important relationships or properties, because the normal form algorithm generates misaligned orderings for many pitchclass sets. For example, a comparative analysis of normal form INT1 (Morris 1987, 40-41, 107-109) relationships often indicates pitch-class sets are not inversionally related even though they are members of the same Tn/TnI type.1 The normal form ordering also obscures important structural properties, such as symmetry, for many pitch-class sets. In these problematic cases, a comparative analysis of normal form orderings cannot produce the relevant structural information without the aid of supplemental ad hoc operations. Creating jury-rigged add-on procedures seems to be the accepted solution to working around these informational inconsistencies, whereas modifying the algorithm to eliminate them is an approach that has not been pursued in the literature. In this paper, I will introduce a new normal form conceptualization and a new algorithm that corrects the problems inherent in John Rahn's (Rahn 1980, 31-39) normal form algorithm.2 © 2009 Springer-Verlag.