Harmonic Systems

Harmonic communication is fundamentally different from melodic communication in the sense that the “harmonic message” (a sequence of chords), unlike the “melodic message” (a sequence of notes), is generally not stated explicitly in the score (Sect. 7.1). Nonetheless, on the basis of harmonic grammar the analyst is able to infer a harmonic message from a given score, and the receiver a corresponding received harmonic message from the received score. Sections 7.2, 7.3, and 7.4 respectively focus on an aspect of harmonic grammar usually associated with one of the three great luminaries of harmonic theory, Rameau, Riemann, and Schenker. Associated with Rameau is the idea that the vast variety of note sequences describable as “chords” reduce to constructs forming a “stack of thirds from a root,” in particular, “triads” and “seventh chords.” Associated with Riemann is the idea that, relative to a referential triad, triads group into categories or “functions,” based on common elements (similarly for seventh chords). Associated with Schenker is the idea that harmony has an irreducible melodic component in the form of “voice leading,” that is, the manner in which the elements of a given chord are paired with the elements of its successor. Stepwise motion plays an important role in such pairings. The harmonic ideas associated with all three theorists are combined in Sect. 7.5, leading to a theorem concerning the necessary and sufficient conditions for the existence of triad-like and seventh-chord-like harmonic objects.