Brouwer’s Combinatorial Art

When we understand Brouwer’s thinking, our performances become more vivid

Leo Brouwer’s output is often divided into three periods: folkloric, avant-garde and neoromantic. When I first started playing the guitar, he was in his avant-garde phase, and works such as La espiral eterna and Canticum helped me to understand how music could be an adventure for the ears, a kind of musical science fiction. Having got to grips with the dissonant sound world of the avant-garde pieces, I well remember the shock of hearing his first neoromantic works, such as El decamerón negro. It was difficult at first to come to terms with their sweetness and apparent rejection of the sonorities of the previous music.

And yet this period-based narrative is not so simple. One complication is that each of Brouwer’s compositional phases incorporates elements of the previous ones: a neoromantic work such as the Fifth Sonata might contain folkloric and avant-garde elements. Another is that Brouwer’s music is characterised not so much by its style as by the way he thinks: whatever he might have rejected is not as important as what he has retained. No wonder, then, that I eventually came to love his later works.

What unites Brouwer’s compositions is a constructive way of thinking that starts with short, incisive ideas and then combines them into larger sections. He can write a long line, but the typical Brouwer idea is a memorable cell: a riff, a dance rhythm, a flourish, a fanfare, a call, a peal of bells; an epigraph, a reminiscence, a quotation; a cry, catchphrase, invocation or curse.

So far, so good, but once a striking idea has been sounded, what comes next? The answer is simple: it can repeat; it can evolve or dissolve; above all, it can be put into conversation with other such ideas. The result might be hypnotic and minimalist, or it might be a glittering collage.

‘Fandangos y boleros’

Let’s look at an example of Brouwer’s collage technique. In this twelve-minute video, I explore the construction of ‘Fandangos y boleros’, the opening movement of the First Sonata. I break down the hypnotic ostinato towards the end of the movement, then explore the play of ideas at the beginning, and close with the role of the Beethoven quotation that ushers in the conclusion of the movement.

In my experience, when students become aware of the kind of construction described in this video, their performance becomes more vivid, more engrossing. Take the beginning of the First Sonata:

What happens if, in the practice room, we play only the gestures highlighted in blue? These are bell sounds in harmonics: first one note, then two notes, then three, always forte, always accented. It’s very satisfying if we can make each strike of the bell resound equally boldly, as though reveling in the power of the guitar’s low harmonics. Each note should sound as though it is calling for attention.

Now let’s play only the gruppetti highlighted in red. The first one is quiet, well under the forte of the first harmonic; the second grows louder as it grows longer and is more expressive; the third gruppetto, the longest, achieves the same forte as the harmonics. It’s as though the gruppetti are coming from far away, and the third one is in our face, so to speak.

As the passage unfolds, the space between the bells and the gruppetti gets filled in. The conclusion of this process comes in bar 9, when the gruppetto interrupts the flow of the music without a break, and here I suggest to students that they observe the rhythm strictly:

The example of Stravinsky

This lego-like approach to composition is characteristic of many twentieth-century composers: just listen to the end of Bartók’s Fourth String Quartet (starting at 19:56 on this video and going on for a minute or so):

Brouwer most likely learned collage techniques principally through an intensive study of Stravinsky. He has said, for example, that a work such as Elogio de la danza refers directly to the Russian composer.1 Indeed, the fact that Stravinsky also went through radical changes of style during his sixty-year career shows the strength and flexibility of the modular mindset.

It was Edward Cone who first explained how Stravinsky’s granular thinking persists through changes in style, so one way to learn more about Brouwer is to look at Cone’s analyses of Stravinsky. It helps that Cone’s 1962 article, ‘Stravinsky: Progress of a Method’, contains some of the most striking musical graphics ever made. In this video, I’ve animated Cone’s diagram of Stravinsky’s Symphonies of Wind Instruments to allow the listener to follow it in real time:

A repertoire of ideas

In 2013, I was fortunate enough to receive an early copy of Brouwer’s Fifth Sonata, commissioned by the Julian Bream Trust. Its title, Ars combinatoria (The Combinatorial Art), describes more than the sonata; it sums up the composer’s entire output. Still, I was curious to know whether Brouwer intended to refer to Leibniz’s 1666 dissertation, De arte combinatoria. The German mathematician’s attempt to define a logic of invention using tiny building blocks of language seemed to me to be a rather apt way of listening to the sonata. I sent Brouwer an email through his office and received the following reply:

The so-called ‘Latin flavor’ is quite far from Julian’s and also my aims. I’m really far from nice rhythms and ‘pretty melodies’ as predominant in musical codes.…
Euclides, Pitágoras, Kepler, Leibniz (of course) and so many others should be studied by composers to enrich the ‘repertoire of ideas’, as well as Paul Klee, Moholy Nagy, Bauhaus (complete) and writers [such] as Carpentier, Octavio Paz (‘The Arc and the Lyre’), Cortázar (‘Rayuela’), Borges, Cummings (the poet) and so on.2

I love the indirectness of this response. Brouwer acknowledges the relevance of Leibniz (‘of course!’), but instead of confining the sonata to a single reference, he opens up a world of art, architecture, philosophy, mathematics and writing. His first reference – to Euclid – was particularly unexpected. For me, it called to mind Oliver Byrne’s 1847 visual version of the Elements, in which the logic of Euclid’s definitions and proofs is expressed in visual building blocks. Here, for instance, is Euclid’s proof of the Pythagorean theorem as rendered by Byrne:

Perhaps it’s no accident that the thought of both Euclid and Brouwer is so easily depicted through colorful building blocks, as in Byrne’s illuminated diagrams and the graphics presented in the videos above.

We’ve seen that the constant play of contrasting ideas can result in strange revelations, as when Beethoven’s Pastoral Symphony interrupts the hypnotic section in the ‘Fandangos y Boleros’. This, though, is one of the glories of Brouwer’s approach: when a tiny block is introduced and developed, unexpected things emerge. Here’s a different kind of example, this time from the Fifth Sonata:

In the heart of the first movement, there is a twelve-tone row (highlighted in red). Notice how distinctive is its presentation, in pizzicato, in the lowest register and in a quirky rhythm: we’ve already noted that this kind of writing requires ideas to be immediately striking. And then notice how Brouwer builds it up starting on the line above: first three notes, E–F–D, then five, eight, and finally twelve notes, getting louder each time. What is this idea doing in the movement? It’s not that Brouwer has suddenly become a twelve-tone composer, but rather that he interrupts the progress of the ideas on either side with a mischievous gesture that quickly exhausts itself through the very simple process of using up all the available pitch classes, after which it has nowhere else to go. In this way, Brouwer creates a microscopic story inside the piece, like the story of an insect that lives for only a day. Here, in a movement full of driving rhythms and virtuosic flourishes, is a moment of marvellously Euclidean construction.

If these excerpts and discussions have led you to think about your own favourite passages of Brouwer, or if you have questions, do let us know in the comments below.

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Guitar discusses Brouwer in several places. Sonata no.1 is used in the section on practising to show how one can use improvisation to learn its language and master its technical challenges (see pp. 138–40 and 146–47). Chapter 9 puts Brouwer in conversation with Villa-Lobos (pp. 231–50), while chapter 12 discusses some of his music for guitar ensemble.

  1. In Brouwer’s own words: ‘The piece is in two movements because the choreographer wanted to pay homage to the Grand Adagio from classical ballet, and to Ballet Russes, including Stravinsky. That is why the second movement makes reference to this composer.’ Clive Kronenberg, ‘Leo Brouwer’s Elogio de la Danza (1964): Imprints of Dance, Stravinsky and the Unison of Contraries’, Researchgate.net, 2011, 46. ↩︎
  2. Email via Oficina Leo Brouwer, September 17, 2013. ↩︎

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