The multiplicative norm and its implications for set-class theory.

ON PAGES 85-7 of his book Generalized Musical Intervals and Transformations, David Lewin makes a "methodological point," namely, that formal and intuitive truths may not always coincide. As an example he cites from the first movement of Chopin's B[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]-minor Piano Sonata, op. 35, two ordered dyads (see Example 1). "The first melodic dyad of (b), marked y on the figure, belongs to the same interval class as x, the first melodic dyad of (a)," states Lewin, and immediately continues: "This relation . . . is formally 'true' but intuitively problematic" (86).

Two interpretations of Lewin's statement must be immediately discounted. First, the quotation marks enclosing "true" in "formally true" are clearly not meant to suggest that the assertion "x and y belong to the same interval class" is only apparently true, but in fact false. Second, the assertion "x and y belong to the same interval class" is not to be confused with the observation "x and y, as pitch-class intervals, sum up to the perfect prime, modulo the octave." There is nothing, I would think, intuitively problematic about the latter observation, although its analytic import is indeed unclear. What Lewin finds "intuitively problematic," in other words, is not that x and y--a descending major sixth and a descending minor third--are inversionally related as pitch-class intervals; rather, for Lewin it is formally true but intuitively problematic to assert that x and y are equivalent. (1)

The equivalence of inversionally related pitch-class intervals plays a central role in the theory of post-tonal music known as "set-class theory." Moreover, as Lewin's Chopin example demonstrates, a similar equivalence relation is often assumed to play a role in tonal music. After explaining that "intervals larger than 6 ... are considered equivalent to their inversions with respect to the octave," Straus (1990, 8) adds in parentheses: "these equivalences are also observed in many aspects of tonal music and tonal theory." (2) It is not difficult to see that Lewin's "methodological" remark casts doubt not only on the notion that inversionally related pitch-class intervals are equivalent, but also on the more general notion that pitch-class sets of any given cardinality are equivalent, provided they are related transpositionally and/or inversionally. This latter generalization, to which I shall refer as "transpo/inversional equivalence" (henceforth, "T/I equivalence"), is currently the standard equivalence rela tion on pitch-class sets.

In the present study I paraphrase Lewin's remark and argue that it is valid but ad hoc to consider inversionally related pitch-class intervals as equivalent; similarly valid but ad hoc, I shall argue, is the generally assumed equivalence of T/I-related pitch-class sets. By contrast, it is perfectly natural to view inversionally related pitch intervals as equivalent, and the same is true of T/I-related pitch sets. This fundamental difference between pitches and pitch classes, which thus far seems to have been largely overlooked in the theoretical and analytical literature, concerns a mathematical property generally known as "the multiplicative norm." To study this property in a proper musical context requires that we first consider two systems of pitches (or pitch classes) and their intervals. The "interval system," to begin with, is simply a Lewinian pitch or pitch-class GIS that allows for multiplying intervals by integer (or integer-class) operators. The "interval/distance system," in turn, is an interval s ystem with an added distance on the set of pitches or pitch classes.

1. THE INTERVAL SYSTEM

1.1 DEFINITION: Let [Z.sub.1] be the infinite set of all integers, and let [Z.sub.k], k [greater than or equal to] 2 is any integer, be the finite set of integer classes, mod k. Let n be any natural number, and let [G.sub.n] be the abelian group ([Z.sub.n]; +) under usual addition (if n = 1) or addition mod k (if n = k [not equal to] 1). Let [R.sub.n] be the ring ([Z.sub.n]; +, .) under usual addition and multiplication (if n = 1), or addition and multiplication mod k (if n = k [not equal to] 1), and let [M.sub.n] = ([R.sub.n], [G.sub.n], .) be the module [G.sub.n] over the ring [R.sub.n], again, under usual multiplication (if n = 1), or multiplication mod k [(if n = k [not equal to] 1) (3) Finally, let int be a function from [Z.sub.n] x [Z.sub.n] into [G.sub.n] satisfying int (x, y) = (y - x) for any x, y in [Z.sub.n]. We shall refer to any triple ([Z.sub.n], [M.sub.n], int) as an interval system.

We shall refer to the elements of the set [Z.sub.1] as pitches, and to the elements of any set [Z.sub.k], k [greater than or equal to] 2, as (mod k) pitch classes. We shall refer to the elements of the group [G.sub.1] as pitch intervals, and to the elements of any group [G.sub.k], k [greater than or equal to] 2, as (mod k) pitch-class intervals. Finally, we shall refer to the elements of the ring [R.sub.1] as operators, and to the elements of any ring [R.sub.k], k [greater than or equal to] 2, as (mod k) operator classes.