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By Paul Kiparsky, Gilbert Youmans

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Increasing the serial vocabulary to include a third element—say L, for extra strong—only aggravates the distributional problems of the theory; clearly, the insight is that a second level of alternation has been imposed on the first, exactly the kind of situation that motivates hierarchical representation for rhythmic interactions in both language and music (notice that any tightening of the serial theory must be implicitly hierarchical; for example, £ must be characterized as a variety of S). Any further differentiation a m o n g positions, such as has been posited by Kiparsky (1977), Chen (1979), Piera (1980), and others following them, can only add to the distress of serialism.

A representation of the relative structural importance of the events in a piece has come to be known as a REDUCTION of the piece, for reasons that will become obvious in a moment. By contrast with traditional Schenkerian theory, the GTTM theory contains two distinct forms of reduction, differing both in what relationships obtain between more important and less important events and in what temporal domains structural importance is defined. In time-span reduction, the domains are the time-spans determined by grouping and metrical structure.

But, as emphasized in the previous section, the metrical structure of cadences is NOT prescribed. Moreover, cadences are often not identifiable in the musical surface, but only through levels of the time-span reduction, where structurally irrelevant ornamentations have been eliminated. (For a case in which a cadence is spread over the immense span of twelve measures, the Prelude in C Major from Bach's Well-Tempered Clavier, Book I, s e e G T T M : 2 6 1 - 2 6 3 . ) These differences in the units out of which music and poetry are constructed lead to a divergence in the formalisms for expressing metrical rules in the two domains.

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