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Cross-Partitions
In two recent conference presentations I have advanced a theory of cross-partitions and used this theory to analyze the underlying organization in movements from two of Dallapiccola's works. An example of a cross-partition is displayed below. To the right, the notes of the chromatic scale are represented by integers 0 to 11, where C = 0, C# = 1, D = 2, . . . .
C
Eb
F#
A
C#
E
G
Bb
D
F
Ab
B
0
3
6
9
1
4
7
10
2
5
8
11
This rectangular design projects three-note collections vertically and four-note collections horizontally. No note is duplicated, and all 12 notes of the chromatic scale are accounted for. My analyses reveal that Dallapiccola treats this cross-partition as if it were a slot machine: he reorders the vertical notes while keeping them in their columns. Slot-machine transformations fix the vertical dimension (the harmony), but change the horizontal lines (the melodies). The analyses also show how cross-partitions afford a high degree of harmonic consistency as well as a fertile ground for melodic invention.
Dallapiccola often links cross-partitions to structure entire sections or movements. In several works, the cross-partitions are gradually "horizontalized": the two-dimensional configurations are seemingly flattened into linear rows. In others works, however, no row ever appears as a string: all of the material derives exclusively from the progression of cross-partitions. My theory of cross-partitions builds upon and extends in two ways the current research on Dallapiccola: it offers a framework for the investigation of the motives and harmonies that arise in these configurations, and it provides tools and instruments to model the ways in which these dimensions intersect. I believe that it will prove invaluable to the study of Dallapiccola's music.
Please send comments, questions, and suggestions about Oberlin Online news and feature articles to Linda.Grashoff@oberlin.edu.
