Manuel M. Ponce’s Piano Sonata No. 2 (1916): An Analysis Using Signature Transformations and Spelled Heptachords

Abstract

In the present work an analysis is made of several passages from Manuel M. Ponce’s Sonata No. 2 for piano (Ponce, Sonata No. 2 for Piano, 1916/1968, [1]), employing Julian Hook’s theoretical development of signature transformations and proper spelled heptachords. A signature transformation reinterprets a diatonic object in the context of a different key signature. The signature transformations form a cyclic group of order 84; indeed, the chromatic transpositions (Tn) and the diatonic transpositions ($t_n$) form subgroups of this cyclic group, hence contributing with yet another way of unifying diatonic and chromatic structures. After giving an introduction to the theory behind the signature transformations, we proceed to an analysis of illustrative passages of the Sonata, using units of varying size called diatonic fragments. During this analysis we realized that the classes of proper spelled heptachords, a generalization of the signature transformations, could explain the constant transition between 7-note nearly diatonic scales. These classes also have a clear mathematical structure, with a transposition operator $\tau$ (they are also called $\tau$-classes), and possess some of the symmetries as well as the seven modes of the diatonic class. This analysis made us look for both intra-class transformations, similar to the ones we find in the diatonic class, and inter-class transformations that can explain the fluid movement between classes found not only in this sonata, but in other pieces that are classified as “chromatic” without more detail.