A linear algebraic approach to pitch-class set genera

Abstract

The concept of interval-class vector (ICV) plays an important role in musical set theory. ICV can be seen as a six-dimensional representative of the intervallic content of a set class (SC), which often forms the basic tool in harmonic analysis of twentieth century music. Interrelations between SCs have been evaluated by means of similarity functions and clustering techniques in many contexts. SCs have also been classified into 'families' called genera. Among others, six-, sevenand twelve-part systems have been outlined. Using the linear algebraic concept of spanning, the six-part genera is unambiguously justified. A group of interval-class vectors, which actually represent points, restrict a finite area in a six-dimensional space. Using the determinant of a matrix, the volume of the area formed by a six-member set class combination1 can be calculated. All possible set groups among TnI-type trichords or hexachords, which produce maximum volumes, are detected. Both the extreme points on the edges of SC space as well as the most neutral set classes in the middle of SC space are recognized using three different methods derived from linear algebra, namely the determinant of a matrix, cosine distance and principal component analysis. A short final demonstration concerns volumes of hexachord combinations used by Finnish composer Magnus Lindberg in his chaconne chains. © 2009 Springer-Verlag.