A hidden non-hierarchical structure. Block Designs and The Weather Riots

Below, an exchange with musicologist Gilbert Delor, on my composition The Weather Riots and on Block Designs.

The Weather Riots, for variable instrumentation, is one of my most performed pieces. Written in 2002, it constitutes my first compositional response to John Cage's idea of structuring pieces with 'flexible time brackets', which he did in his late series of compositions, the so-called Number Pieces. Links to performances of the piece are given at the end of the post.

Block Designs are structures from the mathematical field of combinatorics. I used one such structure to determine the harmonies in The Weather Riots. Upon learning of this technique, composer Tom Johnson, who is a good friend of mine, explored the subject much more deeply, which led to a whole series of wonderful pieces.




Hi Samuel

I have a question for you: when it comes to combinatorial design, Tom often refers to a piece of yours he heard "around 2003", which was based on a series of 11 five-note chords, each one having two notes in common with all the others. Can you tell me which piece it was, and, with some precision, when this took place? Thanks for your help.

Gilbert




Dear Gilbert,

Block designs appeared in my music briefly in 2002, as basic harmonic structures that were used to develop elaborate contrapuntal possibilities. First this happened in Seasons (2001-2002) which used systems of 9 chords of 4 notes each. The (11, 5, 2) system appeared in The Weather Riots (2002) and Krise (2002); Tom heard The Weather Riots in performance in 2003. The Weather Riots remains among my works the strongest composition to use block designs; the piece has been played more than any other piece of mine so far.

Samuel




Dear Samuel,

Shall I conclude from what you say that you were already aware of block design theory in 2001? Tom says in his writings that you're the one who led him toward this field of mathematics, but that you probably didn't know about it yourself by this time. As if you had settled your collection of chords for The Weather Riots only through empiric approach. Is he wrong?

G.




Dear Gilbert,

No, that was more or less the case. I do have some background in mathematics, but I never studied the field of combinatorics deeply so I didn't know about block designs. I was simply looking for a harmonic system which would have certain symmetrical properties, and I found a few block designs. These I used in my pieces. Then I showed my structures to some music theorists & mathematicians online, and people told me that these were in fact "block designs".

Shortly after that, I told Tom about the structures when he happened to be in Amsterdam, at a concert I organized, on which the ensemble HPS Band performed their version of The Weather Riots (this was in fact on march 11, 2004). In our correspondence, I find the first mention of Tom's investigating "Steiner triple systems" a few weeks later, on April 1, 2004.

I never used block designs seriously in my later works after that; Tom more or less "took over" with the technique, and applied them differently than I did.  For me, block designs were a solution to a compositional problem about harmony. For Tom, the block designs were given, and the compositional problem was how to translate them into music! As a consequence, I think Tom's approach to block designs is much purer than what I did.

The real subject matter of The Weather Riots is not block design harmony as such, but how to integrate the time structure of the Cage number pieces (with flexible time brackets) with what you could call a neo-baroque kind of contrapuntal language.

My problem was this: all the motives in The Weather Riots (each with its own contour & metrical feel) have a similar harmonic structure, being all based on five notes. The time structure in The Weather Riots was supposed not to be about hierarchical relations between the harmonies. That being given, how could I find harmonic families that would relate chords to one another in a minimally hierarchical way - i.e. no two chords were supposed to have a "stronger" relation than any other pair, so that any progression would have the same structural meaning - while also allowing for a maximum of variation of harmonic quality - so that the harmonic feel of the piece would have something "chancy" about it?

Block designs were the solution to the problem. In The Weather Riots, everything is based on eleven sets of five pitch classes (you could say "chords"), with every two such chords having exactly two notes in common. Finding the (11, 5, 2) system (though not yet knowing that it had a mathematical name!), I found a specific set of 11 chords in which I would integrate the two maximally opposed harmonic qualities possible in 12-tone tuning. That is, one of the eleven chords was to be a pentatonic scale (stacks of fifths), another one was to be a chromatic segment (stacks of minor seconds). Taking those two chords as my starting point, much of the choice for the other 9 chords was already fixed, and among the options that were left I chose what I felt would give me the most varied harmonic quality.

For Tom, however, it was a question of taking known block designs and then figuring out the best musical forms for them, which led him to quite different problems, such as how to order them, which scales to choose, what instrumentation, what types of phrasing. He's more interested in choosing all that with an eye to making the block design as such musically clear, whereas for me block designs were more like a hidden structure for governing non-hierarchical relations within a free and highly varied polyphony.

Nowadays, I do feel that, after all of Tom's excellent work on them, I should perhaps go back some time to block designs and explore them again in the context of my own interests (which is often more about "musical games" than about "musical structures"). They remain highly suggestive models for organizing musical actions.

Samuel



Some performances of The Weather Riots:

Number Night Ensemble, Amsterdam 2002 (first performance)

Dante Boon & Samuel Vriezen, Vienna 2005

Ensemble of Moments Musicaux, Aarau 2011