The Musician's Guide to Theory and Analysis
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WebFacts Chapter 34 WebFacts   
WebFacts 2
Although Schoenberg and his students wrote out precompositional sketches of their rows and selected transformations, these sketches do not show complete matrices of all fortyeight row forms written out in a 12 x 12 array. There has been considerable recent interest in the question of who was the first to discover the matrix. According to composer and music theorist Robert Morris, Milton Babbitt may have published the first matrix in his 1957 liner notes to Columbia Records' recording of Schoenberg's Moses and Aaron.

WebFacts 3
For Schoenberg, combinatoriality was an exciting discovery. It gave him yet another way to control the rate at which aggregates were formed, an idea similar to harmonic rhythm. As he wrote in Style and Idea, "Later, especially in the larger works, I changed my original idea . . . to the following conditions: . . . the inversion a fifth below of the first six tones, the antecedent, should not produce a repetition of one of these six tones, but should bring forth the hitherto unused six tones of the chromatic scale. Thus, the consequent of the basic set, the tones 7 to 12, comprises the tones of this inversion, but, of course, in a different order" (p. 225). He goes on to explain the basic principle that if the first hexachord is inversionally related to the second, then P- and I-related rows when paired will create hexachordal aggregates.

WebFacts 4
There are six all-combinatorial hexachords.



These hexachords are sometimes labeled with letters, given in the far right column (introduced by composer Donald Martino).

WebFacts 5
To find an RI-combinatorial pair, again circle the first hexachord of the P form on the left side of the matrix. This time, hunt for the equivalent content of this unordered hexachord as a column in the same quadrant. Why? We are looking at columns because we are looking for I forms, and we look in the same quadrant because we will be running this row in retrograde order for RI-combinatorial pairs. We begin with the same hexachord.





Now read the row forms from the matrix: P5 and RI0 are RI-combinatorial.

We would follow the same procedure for any other pair of P- and RI-forms in the matrix.

WebFacts 6
We can find hexachordally combinatorial pairs without a matrix. For P-combinatoriality, hexachords A and B must map onto each other by transposition. That is, there must be some TnI of hexachord A that produces hexachord B. For I-combinatoriality, hexachords A and B must map onto each other by inversion (which may be followed by transposition).

If we have no row matrix, we can use interval-class vectors to help us find transposition levels that produce P-combinatoriality. The first hexachord of Webern's row, {7 8 9 t e 0} (in normal order), belongs to set-class 6-1 [0 1 2 3 4 5]. Its ic vector is [543210]. The zero in the ic 6 position of the vector tells us that when the set is transposed by six semitones, it will produce no common tones (that is, it will produce its complement-the other six tones of the aggregate). If we try this, we do indeed get the hexachord’s complement:

     {7 8 9 t e 0}

+    6 6 6 6 6 6

     {1 2 3 4 5 6}
This, in turn, means that P0 paired with P6 will create combinatoriality, as will any T6-related rows (P1 and P7, P2 and P8, etc.):

Look again at the row <0 8 7 e t 9 3 1 4 2 6 5> to determine its I-combinatorial potential. Is hexachord A inversionally equivalent to hexachord B? We can find out by placing each hexachord in normal order, as before, and comparing them to see whether we can find consistent sums between them. The normal orders are {7 8 9 t e 0} and {1 2 3 4 5 6}. They are clearly related by T6, as already discussed, but they are also inversionally related, by T1I:

     {7 8 9 t e 0}

+   {6 5 4 3 2 1}

     1 1 1 1 1 1
This, in turn, means that P0 paired with I1 will create combinatoriality, as will any T1I-related rows (P1 and I2, P2 and I3, etc.):

The simplest kind of combinatoriality to write is R-combinatoriality. Any row when paired with itself in retrograde can produce R-combinatoriality.

Since it works for any row, we sometimes call this type "trivial" R-combinatoriality, but there are nontrivial types as well. To create an R-combinatorial row, hexachord A of your row must map onto itself under some transposition. Likewise, to create an RI-combinatorial row, hexachord A must map onto itself under some TnI. That operator can then be used to find a combinatorial row pair.

There are other types of combinatoriality. Robert Morris focuses on a variety of such designs in his book Composition with Pitch Classes (New Haven: Yale University Press, 1987). An example of trichordal combinatoriality adapted from his Class Notes for Atonal Music Theory (Hanover, N.H.: Frog Peak Music, 1991) follows. Here we see four rows, which when combined produce trichordal aggregates:

This type of trichordal combinatoriality, as well as tetrachordal combinatoriality (making aggregates from tetrachordal divisions of three rows), are favorites of Webern, who inclined toward row segments smaller than Schoenberg's preferred hexachords.

WebFacts 7
We looked at some properties of TnI cycles in WebFact 3 of Chapter 33. Read it again for information about how cycles are constructed. Webern's pc recurrences between row pairs can also be explained and predicted by TnI cycles. In the case of the opening of the second movement of the Variations for Piano, look at the row pairs chosen: R3 with RI3 and Rt with RI8. In all cases, a row and its inversion are paired, and the index number is a constant 6 (3 + 3 = 6, and t + 8 = 6, mod12). This means that we can use the T6I cycles to predict the relationships Webern builds in Op. 27. The T6I cycles are (0 6) (1 5) (2 4) (3) (7 e) (8 t ) (9). Cycles (3) and (9) are "singleton" cycles; if we perform T6I on pc 3, the operation produces 3. Likewise, T6I on pc 9 produces 9. This is why Webern's rows work as they do, with pc 3 and 9 "reappearing" in the same order position when the two rows are related by T6I. It also explains why, for example, pcs 8 and t appear repeatedly as a dyad: because they are found within the same parenthetical group in the T6I cycle. You may enjoy comparing the cycle to the score to see how other dyads are realized by Webern.