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 Irelated rows when paired will create hexachordal aggregates.
WebFacts 4
There are six allcombinatorial hexachords.
These hexachords are sometimes labeled with letters, given in the far right column (introduced
by composer Donald Martino).
WebFacts 5
To find an RIcombinatorial 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 RIcombinatorial pairs. We begin with the same hexachord.
Now read the row forms from the matrix: P5 and RI0 are RIcombinatorial.
We would follow the same procedure for any other pair of P and RIforms in the matrix.
WebFacts 6
We can find hexachordally combinatorial pairs without a matrix. For Pcombinatoriality,
hexachords A and B must map onto each other by transposition. That is, there
must be some T_{n}I of hexachord A that produces hexachord B. For Icombinatoriality,
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 intervalclass vectors to help us find transposition
levels that produce Pcombinatoriality. The first hexachord of Webern's row, {7 8 9 t e 0} (in
normal order), belongs to setclass 61 [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 complementthe 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 P_{0} paired with P_{6} will create combinatoriality, as will any T_{6}related
rows (P_{1} and P_{7}, P_{2} and P_{8}, etc.):
Look again at the row <0 8 7 e t 9 3 1 4 2 6 5> to determine its Icombinatorial 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 T_{6}, as already discussed, but they are also inversionally related, by T_{1}I:
{7 8 9 t e 0}
+ {6 5 4 3 2 1}
1 1 1 1 1 1
This, in turn, means that P_{0} paired with I_{1} will create combinatoriality, as will any T_{1}Irelated
rows (P_{1} and I_{2}, P_{2} and I_{3}, etc.):
The simplest kind of combinatoriality to write is Rcombinatoriality. Any row when
paired with itself in retrograde can produce Rcombinatoriality.
Since it works for any row, we sometimes call this type "trivial" Rcombinatoriality, but there
are nontrivial types as well. To create an Rcombinatorial row, hexachord A of your row must
map onto itself under some transposition. Likewise, to create an RIcombinatorial row,
hexachord A must map onto itself under some T_{n}I. 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 T_{n}I 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 T_{n}I cycles. In the case of the opening of the second
movement of the Variations for Piano, look at the row pairs chosen: R_{3} with RI_{3} and R_{t} with RI_{8}.
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 T_{6}I cycles to predict the relationships
Webern builds in Op. 27. The T_{6}I cycles are (0 6) (1 5) (2 4) (3) (7 e) (8 t ) (9). Cycles (3) and
(9) are "singleton" cycles; if we perform T_{6}I on pc 3, the operation produces 3. Likewise, T_{6}I 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 T_{6}I. 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 T_{6}I cycle. You may enjoy comparing the cycle to the score to see how other dyads
are realized by Webern.

