Denis Lorrain

Construction 042

Due cose, mille cose

Algorithmic composition for MIDI piano

(2013, total duration: *ca.* 21 mins)

Programme Note

   This Construction consists of three movements — two of 8 minutes and a third of 4'30". It essentially rests on two materials: the Prouhet-Thue-Morse sequence and a theorem by Erdös and Szekeres.

   The Prouhet-Thue-Morse sequence [1] is a fractal structure qualified of ubiquitous by certain authors. It is composed of two symbols—e.g. A and B—and created by starting from one of the two symbols and recursively applying, ad infinitum, two very simple production rules:

      1) A generates A B

      2) B generates B A

The resulting sequence is "a remarkable example of coincidence of nearly contradictory features in a same mathematical object: utter simplicity and great complexity" [2]. Concretely, I use this sequence by translating A and B into two pitches, for instance, or two durations, two loudnesses, etc.

   The Erdös-Szekeres [3] theorem of monotonous subsequences states: Out of any sequence of n²+1 different numbers, one can extract an increasing subsequence of length n+1 or a decreasing subsequence of length n+1. This theorem demonstrates that a certain degree of order always resides in everything, even in a random phenomenon. I apply this paradox in the domain of pitches: within a sequence of n²+1 different random pitches, the n+1 constituting a monotonous subsequence are marked—for instance by contrasting loudnesses and/or durations. I use limited values of n from 2 to 5. The random source is simply rectangular uniform, with a heuristic control eliminating undue repetitions of identical numbers. The subsequences are extracted by means of a dynamic pattern matching language I have developed.

   A third material episodically appears in the piece. It consists in fast melismatic lines inspired by the vegetal growth of lianas, simulated by a fractal process distorted by a certain degree of randomness [4].

   The Construction itself is freely elaborated with these materials: concrete musical elements are realised, interrelations are created, layouts are organised, forms are deployed on various levels...

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[1] This sequence is most commonly named Thue-Morse. It was rediscovered many times in different contexts. The most famous works are by Eugène Prouhet (1817-1867) in 1851, Axel Thue (1863-1922) in 1906, and Marston Morse (1892-1977) in 1921.

[2] Delahaye, Jean-Paul, 2006: "Des mots magiques infinis", Pour la science, 347: 92 (my translation). The Prouhet-Thue-Morse sequence is traditionally composed with the two symbols 0 and 1, and starting with 0. It can also be obtained by other means. But these specifics are of no concern in the present case.

[3] The hungarian Paul Erdös (1913-1996) was one of the great mathematicians of his time. He published more than a thousand five hundred articles. His friend George Szekeres (1911-2005) was also hungarian. Erdös, P. and Szekeres, G., 1935: "A Combinatorial Problem in Geometry", Compositio Mathematica, 2: 463-470.

[4] Cf. my Construction 041, Toccata et Fugue (2012).

DLO

Karlsruhe

21/08/2013