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COPYRIGHT 2002 University of Washington
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 2 3 6 1 2 3 9 6 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 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 8 1 t 9 8 3 and 2 9 4 3 and 2 8 1 and 7 9 e 5 (etc.) 2 e 5 t 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.)
a 3 x 4 cross-partition a 4 x 3 cross-partition 3 6 9 4 8 1 4 7 t 1 5 9 2 5 8 e 2 6 t 3 7 e
Now let us return to the Mallalieu row. Examples 2(a) and (b) show the arrangement of the row's discrete set classes. (The symmetrical layout results from the row's inherent [RT.sub.6]-invariance). The discrete tetrachords include two members of 4--2[0124] and a central 4--25[0268]; its tri-chords include two instances each of 3--3[014] and 3--11[037]. Example 2(c) shows the 3 x 4 cross-partition that arises from the application of EP. The example displays the set-classes of the row's discrete trichords (the columns of the configuration), and the set classes of its non-segmental tetrachords (the horizontal lines of the configuration). It is crucial to note that the horizontal lines of the cross-partition project the same number and type of set classes as the row's discrete tetrachords. We invoke the term reflection to refer to this property. Example 2(d) shows the 4 x 3 cross-partition that arises from EP. It displays the set classes of the row's discrete tetrachords (the columns of the cross-partition) and its non-segmental trichords (the horizontal lines of the cross-partition). Here, too, the horizontal lines of the cross-partition reflect the same number and type of set classes as the row's discrete trichordal segments. The same holds for the row's 2 x 6 EP-derived cross-partition, whose horizontal lines duplicate the set classes of the row's discrete hexachords, and its 6 x 2 BP-derived cross-partition, whose lines replicate the set classes of the row's discrete dyads. Thus, the Mallalieu row is able to reflect all of its discrete even-sized segments (hexachords, tetrachords, trichords, and dyads) in the horizontal lines of its EP-derived cross-partitions.
The property of reflection has been exploited compositionally by Robert Morris and Andrew Mead. Example 3(a) is taken from Morris's Not Lilacs! (1973), which is based on the Mallalieu row and its M and MI transforms. (6) It extracts the Bb trumpet and alto saxophone lines in measures 60-4 (the piano and drum parts are not shown). Aggregate boundaries are delineated by dotted lines, and row labels are in boldface. The passage opens with [I.sub.5] in the trumpet, which marks the conclusion of an extended solo. [I.sub.5] is followed by an MI transposition of P2, an M transposition of [P.sub.3]'s first hexachord, and an MI transposition of [P.sub.3]. (7) These rows begin an extended sax solo. Example 3(b) highlights several associations among the rows in the passage. [I.sub.5] and the MI transposition of [P.sub.2] share the same (unordered) discrete trichordal collections. We can also observe the property of reflection between the MI transpositions of [P.sub.2] and [P.sub.3]. If we start with the second note of the former row, and take every other pitch class, we generate the latter row. (8)
Example 4 shows a passage from Andrew Mead's Saxophone Concerto (1993). The Concerto is based on a "semi-Mallalieu" row--a row derived from the Mallalieu string that possesses a certain degree of reflection. (9) The passage in question includes the lead-in and the opening measures of the third movement. Variants of this passage appear several times during the movement (measures 22-4; 53-5; 91-5; 119; 162-4), and function as a ritornello of a sort. The distinguishing characteristics of the ritornello are a steady acceleration of harmonic rhythm, an intensification in dynamics and texture, and, especially, the property of set-class reflection. The saxophone leads off with a linear presentation of row [P.sub.3]. (10) The Giocoso proper modulates to [P.sub.8], which is partitioned via EP into a 3 x 4 and then a 2 x 6 cross-partition. (Here, too, dotted vertical lines are used to delineate aggregate boundaries.) The passage continues with 4 x 3 arrangements of [R.sub.4] and [P.sub.6] and concludes with a 6 x 2 re presentation of [P.sub.3] the initial saxophone row. Within the span of four measures, every rectangular configuration of the row is projected: 12 x 1, 6 x 2, 4 x 3, 3 x 4, and 2 x 6.
Example 5(a) shows the set classes of [P.sub.3]'s discrete terrachordal and trichordal segments. Example 5(b) shows the set classes in the horizontal lines and columns of [P.sub.8]'s 3 x 4 cross-partition. (EP in this example is read from the bottom to the top line.) The high, middle, and lower piano lines project 4-Z15[0146], 4-21[0246], and 4-16[0157]. These lines, owing to the properties of the row, reflect the discrete tetrachords of row [P.sub.0]. The vertical trichords of the cross-partition comprise the order-number collections {0,4,8}, {1,5,9}, {2,6,t,} and {3,7,e}. These columns project the same number and type of set classes as the discrete trichords of [P.sub.8]: three instances of [014], and one [015]. Moreover, the third column of [P.sub.8]'s cross-partition, {3,4,7}, recalls the pitch classes of [P.sub.3]'s first discrete trichord, and the first column of the cross-partition, {0,8,9}, echoes the pitch classes of [P.sub.3]'s second discrete trichord. By extension, the alternating trichords in [P .sub.8]'s cross-partition replicate the discrete hexachords of [P.sub.3], namely {034789} and its complement, {1256te}. Further evidence of reflection is shown in Example 5(c), which displays the 4 x 3 cross-partition induced by the application of EP on [R.sub.4]. The horizontal lines of the cross-partition replicate the discrete trichords of [P.sub.0]. The same holds for the cross-partition of [P.sub.6]. These and other types of reflections permeate the Concerto.
The examples above showcase the reflective properties of Mallalieu and semi-Mallalieu rows and help ground this study in musical reality. They also suggest that the earlier investigations into the properties of these rows have focused on the preservation of ordered pitch-class collections at different sets of order numbers in a given row class. The present study extends and generalizes this approach by examining the ways in which unordered pitch-class collections are associated with sets of order number collections in a given row class, without regard to specific orderings. In an effort to constrain our inquiry, we invoke highly constrained order-number partitions that are based on cycles (analogous to diminished-seventh chords, augmented triads, whole-tone scales, and tritones), then select those rows that possess the property of reflection (again, without regard to order within the extracted collections). This approach uncovers a sizeable (and heretofore undisclosed) universe of reflecting rows that might be thought of as "distant cousins" of the Mallalieu family.
Our investigation is motivated by two fundamental questions: how many reflecting rows are there, and what characteristics do they possess? It proceeds in two stages: part one defines the properties of reflection and explores the salient characteristics of the rows; part two enumerates and classifies the universe of reflecting rows. An appendix summarizes the computer program used to enumerate and evaluate the rows.
Before we move on, it is necessary to say a few words about the nature of reflection. We focus exclusively on rows that are able to reflect the same number and type of set classes of their discrete segments in the horizontal lines of their cross-partitions. Thus, every reflecting row is able to reflect at least one of the following: its discrete hexachords in the horizontals of its 2 x 6 cross-partition; its discrete tetrachords in the horizontals of its 3 x 4 cross-partition; its discrete trichords in the horizontals of its 4...
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