On Allen Forte's Theory of Chords

Abstract

A theory of harmony may reasonably be considered as one whose analytical units are unordered sets of notes, or chords, under octave-equivalence and transpositional equivalence. Classical harmonic theory enriches this definition by assigning special meanings to certain notes of these sets on contextual or theoretical grounds: the bass and the root, for instance. It also provides valuable extensions of this basic material in its use of functional descriptions, which relate realizations of given chords to a reference note, the tonic or local tonic, and of harmonic progressions, which describe commonly-used sequences of chords. The classical theory is limited, however, in the kinds of chords it can describe. Configurations which cannot be construed in terms of chords built up in thirds must have some elements explained in contrapuntal terms, as non-harmonic of certain specified types, or in some other heuristic way. This century has seen the growth of a repertoire which has yielded to this theory only with difficulty, if at all: that of so-called atonal music. For this music a new theory certainly seems to be called for, one in which any set of notes whatsoever can be given an objective harmonic description. In his recent book, The Structure of Atonal Music [1973], Allen Forte gives us the outcome of a number of years' work on this problem. What he presents is a more-or-less systematic description of all possible sets of notes under octave-equivalence and transpositional equivalence, adding the relations of enharmonic and, surprisingly, inversional equivalence. He does not use any of the apparatus of the functional description, as is quite natural. As means of relating sets of notes, he uses similarity of interval content, common tones (which he calls invariants) and abstract inclusion relationships. The choice of musical examples and the word atonal in the title make it clear that Forte is trying to construct a harmonic theory specifically for music which cannot reasonably be treated with the classical theory on one hand and is free, on the other