Voice Leading Among Pitch-Class Sets: Revisiting Allen Forte’s Genera

The theory of PC-set class genera by Allen Forte was an important contribution to the understanding of similarity relations among PC sets within the tempered system. The growing interaction between the universes of PC-sets and transformational theories has been explored the space between sets of the same or distinct cardinality, by means of voice-leading procedures. This paper intends to demonstrate Forte’s method along with proposals by other authors like Morris, Parks, Straus, Cohn, and Coelho de Souza. Some analysis demonstrates such operations in passages picked from Heitor Villa-Lobos’s works, like the Seventh String Quartet and the First Symphony.

Generic (Mod-7) Voice-Leading Spaces

This article constructs generic voice-leading spaces by combining geometric approaches to voice leading with diatonic set theory. Unlike the continuous mod-12 spaces developed by Callender, Quinn, and Tymoczko, these mod-7 spaces are fundamentally discrete. The mathematical properties of these spaces derive from the properties of diatonic pitch-class sets and generic pitch spaces developed by Clough and Hook. After presenting the construction of these voice-leading spaces and defining the OPTIC relations in mod-7 space, this article presents the mod-7 OPTIC-, OPTI-, OPT-, and OP-spaces of two- and three-note chords. The final section of the study shows that, although the discrete mod-7 versions of these lattices appear quite different from their continuous mod-12 counterparts, the topological space underlying each of these graphs depends solely on the number of notes in the chords and the particular OPTIC relations applied.

Interval Permutations

This paper presents a framework for analyzing the interval structure of pitch-class segments (ordered pitch-class sets). An “interval permutation” is a reordering of the intervals that arise between adjacent members of these pitch-class segments. Because pitch-class segments related by interval permutation are not necessarily members of the same set-class, this theory has the capability to demonstrate aurally significant relationships between sets that are not related by transposition or inversion. I begin with a theoretical investigation of interval permutations followed by a discussion of the relationship of interval permutations to traditional pitch-class set theory, specifically focusing on how various set-classes may be related by interval permutation. A final section applies these theories to analyses of several songs from Schoenberg’s op. 15 song cycle The Book of the Hanging Gardens.

Thin Partitions: Remembrance and Reflection in Alfred Fisher's Zakhor: Remember

The injunction to remember – in Hebrew, Zakhor – is perhaps the most powerful command in the Old Testament, and memory of the past has therefore always been a central component of Jewish experience. Alfred Fisher's character and music suggest both erudition and practicality as crucial components of the Jewish heritage that has formed an increasingly central part of his life and music during the past decade. In the song cycle Zakhor: Remember, poetic and musical cross-referencing of memories is the framework for a fascinating dialectic that informs the structure and the language of the cycle. In this article, the author studies the different levels of unity in the cycle and discusses its tonal structure, which can be characterized as post-Schoenbergian chromaticism, in which much of the harmonic language is controlled by a limited number of pitch-class sets. Remembrance and reflection are here interwoven with Fisher's emotional responses into the intellectual framework of a profound work of art.