Some Intervallic Aspects of Pitch-Class Set Relations

Abstract

Pitch-class sets are a necessary tool for the analysis of non-tonal music. They provide an organized means of classifying harmonic structures (or any other pitch combinations), but they also pose a problem, one which can most easily be exposed through a tonal analogy. In Example 1 a, the tonal analyst would recognize the structural connection between the first and third chords and describe the second chord as a secondary pitch structure which is linear in origin. This judgment is both facilitated and supported by the lack of a tonal name (for example, triad or seventh chord) for the second chord. In Example ib, the third chord, taken out of context, has a name (minor triad), but in context this is not its meaning. It too is linear in origin and to place the Roman numeral iii beneath it would be erroneous. In the analysis of non-tonal music, pitch-class sets allow us to assign a name to every conceivable pitch combination. Cases like Example l a, therefore, do not arise. Every non-tonal chord succession is comparable to Example ib, but because we lack the structural criteria of functional harmony, we run the risk of considering only the name of a pitch-class set at the expense of its contextual role. How, then, can we evaluate the structural significance of different pitch-class sets in a composition? Such evaluations are invariably based

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