Applications of serial pitch organisation in Urgynes (taken from another forum)
The numbers along the outside detail the
Prime and
Inverse rows, so when I say "P
11" I'm referring to the 0 at the very start of the matrix, when I say "I
25" I'm referring to the 7 which is the second interval in P
5.
From that matrix, or grid, there are endless possibilities for generating material. What I did, in addition to simply following a row in a straightforward manner, was to apply various patterns and shapes and non-standard (e.g.: diagonal) directions to movements about the grid, sometimes I would even jump from one instance of a number to another somewhere else (e.g.: reaching the 8 at P
12 and jumping to the 8 at P
39 and continuing in any direction). Deriving material in this way became a game, and I was often devising different rules for getting from one end of the board to the other and seeing what the resulting melodies, harmonies, chords etc. were, and naturally I would alter or discard the results I was not pleased with.
One technique I used throughout the piece was to apply shapes to the matrix, and the portions of the matrix above highlighted in bold red resemble a particular division I used in the very first movement; two large isosceles triangles from P
61 to P
127 and I
61 to I
127. Within these triangles I used all my other techniques to generate more limited melodic material while I derived chords from the space in between them. In the final movement I used isosceles triangles again, this time to form eight equal divisions of the matrix which operated independently of each other.
Of course, the whole time I was combining all of this serially derived material with free writing and applying transformative techniques independently of the matrix. So ultimately my work was not serial in the strict sense, the matrix was only one tool of many used to reach the end result. I did not apply any complex mathematical processes to the matrix, certainly nothing like Set Theory, which I must confess I do not understand, so for me at least working with serialism was nothing close to an algorithmic kind of composition.
In response to further questions about construction and uses of rows/matrices in the pieceEach row was constructed differently, sometimes I wanted particular intervals to be emphasised, but some were almost created blindly. This row in particular is the most extreme example of doing it blind, I just asked friends in an IRC chatroom to call out numbers between 0 and 11 and wrote them in the order they came up, ignoring repeat numbers. If it had moved a little too predictably I would have scrapped it, but as it came out it seemed pretty interesting on paper. Initially I was not too happy working with it, however, so for most of the second movement I completely ignored it, but then I had something of an epiphany as I was working it back into the ending, then I went back and reset a lot of the material using the matrix as a guide and it sounded a lot better, more unified. So that one in particular was a case of having to get away from the row to write the music and then coming back to it later on. If I had rewritten the row, the movement would have become far too laborious a working process, and I think the end result would have suffered because of that.
The construction of the matrix itself is quite simple, all you need to do is invert the prime row to create the inverse row, then use each interval in the inverse row as the beginning of a transposed prime row. In that matrix I
12 is 4, so P
2 is the prime row transposed up two whole steps. The rest falls into place the same way: P
3 is P
1+9 half steps, P
4 is P
1+ 5 and so on until the entire thing is filled out. Every 12-tone matrix is crawling with patterns, whether intentional or accidental, just take a look at those two red triangles on the example I posted, notice how they are diagonal inversions of each other. P1-6 to P
712 (10, 10, 11, 9, 10, 7, 3) is I
61 to I
127 (2, 2, 1, 3, 2, 5, 9) inverted, the same is true of all those left-to-right diagonal lines. The right-to-left diagonal lines offer up some interesting prospects as well: I
8 to P
8 is 2, 10, 7, 6 followed by its own retrograde inversion 6, 5, 2, 10, and the same is true of all diagonals in that direction. Simply put, using the left-to-right line of 0s as a dividing line, the left side is the inverse of the right. There are lots of other recurring figures, in this one the relationship between 7 and 3 is strongly emphasised, in most instances you can find a 3 right next to a 7, whether straight or diagonally. I think it's an exciting feature of the 12-tone matrix, the way patterns inevitably emerge, recur, invert and transpose each other etc.
Awareness of the results one will get from a matrix, that's something Milton Babbitt talked about, I think in the documentary Portrait of a Serial Composer, and he's lamenting composition students trying to use serial techniques without considering the musical outcome of the rows they create, their lack of understanding means they end up scrapping a lot of unsatisfactory pieces. Of course, Babbitt was very strict in his application of serial organisation, to the extent that the piece was determined by the rows before it was composed (if he was answering [the] question about patterns, I have no doubt he would talk about planning them out meticulously when he constructs a row), so when he talks about that awareness it is within the context of strict application, my applications are much looser and do not underpin the entire work so much as supplement it. Each movement begins with an exploration of the row but is soon enough suffused with free writing, so the construction of the row itself is not so important as the application from then on, but even in those initial explorations the vertical spacing and ordering of the notes makes all the difference, some sections will benefit from a more lyrical treatment while others will require large leaps from one register to another and so on.