Mapping the Geometries of Pitch-Class Set Similarity Measures via Multidimensional Scaling

Abstract

A numerical visualization technique called multidimensional scaling (MDS) was applied to the matrices of (dis)similarity ratings from six pitch-class set (pcset) similarity functions to see what types of constructs they actually measure. Three functions that use the interval-class vector (icv) for their computations (Scott and Isaacson’s ANGLE, Isaacson’s IcVSIM and ISIM2) and three that use subset imbedding (Castrén’s RECREL, Rahn’s ATMEMB and AMEMB2) were studied. The ratings of these functions for all trichords, tetrachords, and pentachords (under Tn/I-equivalence) were examined; ratings for each cardinality of set-class separately as well as contiguously grouped together (3+4, 4+5, 3+4+5) were studied. Four-dimensional geometric solutions were found for the icv-based measures while five-dimensional solutions were found for the subset-based measures, based on goodness-of-fit analysis. The configurations show one dimension clearly interpretable as diatonicism vs. chromaticism; other dimensions indicate oppositions such as ”whole-tone” vs. ”anti-wholetone” or ic3- vs. 016-saturation. Still other dimensions are less clear as to good interpretations. The overall results are generally consistent with the idea that these functions all measure constructs relating to familiar scales (diatonic, hexatonic, octatonic, etc.). The results are also compared with several systems of pcset genera. ISIM2 was found to be inconsistent with itself in terms of the geometries it produced. Several set-classes had coordinates near zero along various dimensions in the derived configurations, indicating that in a formal quantitative sense they do not possess the corresponding musical properties being measured; this may raise questions concerning the relative aesthetic worth of some such set-classes.

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