Variety and Multiplicity in Diatonic Systems

ample, B, D, F is an instance of the class diminished triad and also an instance of the larger class triad. We shall call the former designation specific and the latter generic. Thus the class including (C, D, E}, F, G, A and (G, A, B} (but not, for instance, D, E, F) is specific (two whole steps) and the larger class including all three-note major scale fragments is generic. The specific point of view is the generally adopted one, based on Forte's classification and developed systematically by Browne and in a historical study by Gauldin. 1 The generic perspective was developed by one of us (Clough) and in more recent work by Thomas Zimmerman. 2 Until now the two approaches have remained unconnected, an uncomfortable and peculiar situation given the similarity of the underlying mathematics. We shall demonstrate here that under the appropriate equivalence relation, there are fundamental connections between the two interpretations. These connections apply to lines as well as chords. We account for these relationships by showing that the embedding of the seven-note diatonic scale in the twelve chromatic notes is one of a special class of