Denis Lorrain

Construction 061

Pleins et déliés

Algorithmic composition for MIDI piano

(2021, total duration: 16 mins)

Programme Note

   This Construction essentially rests on melodic and harmonic material resulting from nowadays quite usual musical concepts: couples of hexachords form dodecaphonic sequences, which are varied and multiplied through *generalised axial symmetry* operations in a circular representation of the chromatic space. I have used this same basic algorithm in my preceding Construction 053, "... somewhere south of silence" (2016). This process originates in an *universe* consisting of the set of all possible couples of all chromatic hexachords comprising all intervals smaller than a tritone (all-intervals-hexachords, algorithm of Jan Vandenheede). A subset of this universe form dodecaphonic sequences.

   This unique source of melodic and harmonic material is put into play by six different algorithms:

      - Akors (chords)

      - Melos (melodies)

      - Tchakak (alternating jerky chords)

      - Erdoes (complex and contrasted sequences, cf. below)

      - Girland (homorhythmic melodic lines)

      - Hoktaf (homorhythmic melodic lines exploded in octave-jumps)

These names, purely functional, are either explicit, more or less humorous, onomatopoeic or phonetic. Compressed and juxtaposed, they form the titles of the piece's three movements, of which the second and third are linked:

        I — ErdHokGirlAkr   5'29"

       II — MlsAkr          4'44"

      III — MMTchkkAkr      5'43"

            Total duration 15'56"

   In addition, an altogether independent algorithm plays a primordial role in the last movement. Named MM, the listener will easily identify it.

   Nine different couples of hexachords are used as dodecaphonic sources, according to the various algorithms and parts and movements where they are used. All these couples share one common hexachord — the thirteenth delivered by Jan Vandenheede's algorithm.

   The algorithm named Erdoes applies the theorem of monotonous sub-sequences of Paul Erdös and George Szekeres, published in 1935. This theorem demonstrates that out of any sequence of n²+1 different numbers, one can always extract an increasing or decreasing sub-sequence of length n+1. Therefore, a certain part of order inevitably exists in any phenomenon, even though random. I resorted to this theorem in my preceding Construction 042, Due cose, mille cose (2013). It was then applied to homorhythmic random sequences. In the present case, more complex sequences are concerned -- regarding pitch structure as well as rhythm.

   Rhythms are issued from the sturmian sequence of Thomas Noll, on which my Construction 047, Eight Sturmian Studies (2015) was entirely based. This sequence was then the universal source for the production of melodies, harmonies and rhythms. It is here restricted to the creation of rhythms in the above first four algorithms.

   In certain respects, this Construction represents a development of principles and algorithms experimented in the preceding I have mentioned.

DLO

Karlsruhe

February 2021