Having your cake and eating it too: the property of reflection in twelve-tone rows (or, further extensions on the Mallalieu complex).(Critical Essay)

I. INITIAL CONSIDERATIONS

PERHAPS NO TWELVE-TONE ROW has garnered more attention than the "Mallalieu" row. (1) Named after its discoverer, Pohlman Mailalieu, this all-interval, [RT.sub.6]-invariant row possesses a unique property: the cyclic-interval patterns of this row generate its own transpositions. (2) Example 1(a) illustrates this property with the [P.sub.0] form of the row. (3) Example 1(b) shows that we can generate P1 by starting with the second pitch class of the row and taking every other note (wrapping around as needed). Similarly, Example 1(c) shows that we obtain [P.sub.4] by starting with the third note of the row and taking every third pitch-class. And so on and so forth.

This property is also evident when the Mallalieu row is partitioned into two-dimensional pitch-class configurations, or cross-partitions. (4) In the broadest sense, a cross-partition is a two-dimensional configuration of pitch-classes that contains an aggregate. A cross-partition might be derived from a row, or it might exist independently of a row; there are a bewildering number of potential realizations for each configuration. Below are a few examples of cross-partitions, the first two "even" and the second two "uneven," owing to null spots:

 
3 4 5 6 1 2  0 2 3 6  0 1 2          0 3 9 6 
0 9 8 e t 7  1 5 7 9  3 4 5 6 7      1 7 5 8 
             4 8 t e  8       9 t e  2 4   e t 

In this study, we stipulate that every cross-partition must (a) be derived from a row, (b) preserve that row's discrete segments in its (vertical) columns, and (c) be a rectangular formation with either two-, three-, four-, or six-note collections in its horizontal lines and columns. (That is, there are no null spots.) Thus, given the row of Berg's Violin Concerto, [P.sub.7] = <7 t 2 6 9 0 48 e 1 3 5>, the following are legitimate 3 x4 cross-partitions (many other designs are possible):

 
7 6 4 1       7 6 e 5       7 6 e 5       2 0 8 1 
t 9 8 3  and  2 9 4 3  and  2 0 8 1  and  7 9 e 5  (etc.) 
2 0 e 5       t 0 8 1       t 9 4 3       t 6 4 3 

We add one additional constraint to the formation of cross-partitions. We call this restriction "equidistant partitioning," or EP. (5) EP requires that each cross-partition strictly maintain the order of its row's pitch classes in its columns. Below are the order-number configurations for 3 x 4 and 4 x 3 cross-partitions derived from EP. (Thus the EP version of Berg's row is the first pitch-class design shown above.)