How rare is symmetry in musical 12-tone rows?

A symmetric 12-tone row is equal to a transformation of itself by transposition, retrograde and possibly inversion. (In Berg and early Schoenberg, rows may also be transformed by cyclic shift.) In this paper, we develop efficient recipes for generating and counting symmetric rows, and show (not surprisingly) that Schoenberg, Berg, and Webern used such rows more often than can be attributed to chance.

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