Denis Lorrain

Construction 047

Eight Sturmian Studies

Algorithmic composition for MIDI piano

(2015, total duration: *ca.* 22 mins)

Programme Note

   Sturmian sequences (or words) are named after the french mathematician Jacques Sturm. They are a specific kind of infinitely long sequences of symbols (e.g. digits or letters). The number of different constituent symbols is commonly limited: typically two -- for instance the two digits 0 and 1. In such case, the qualification binary is added. Sturmian sequences form a subset of an infinity of possible symbol sequences. For instance, I have used the Prouhet-Thue-Morse binary sequence in my Construction 042 (2013).

   The particular sturmian sequence used here is also binary, and has been proposed by Thomas Noll in 2008. It is defined by

      sn = [(n+1)g] - [ng], for n = 0, 1, 2, ...

where [x] denotes the integral part of x, and g = log2(3/2) = 0.5849625..., which is the interval ratio of a physical fifth relatively to one octave. The sequence begins with

      01010110101101010110101101010110 ...

and of course extends infinitely.

   Aside from the raw sequence itself, taken literally one symbol (bit) at a time, I have also experimented with the perusal of the sequence as a succession of binary numbers of various bit-lengths. Grouped for instance two bits at a time, the same initial part of the sequence becomes

      01 01 01 10 10 11 01 01 01 10 10 11 01 01 01 10 ...

Translating from binary to decimal numbers, one gets

      01 01 01 10 10 11 01 01 01 10 10 11 01 01 01 10 ...

       1  1  1  2  2  3  1  1  1  2  2  3  1  1  1  2 ...

The resulting sequence is thus composed of numbers 1, 2 and 3. Taken three bits at a time, it gives

      010 101 101 011 010 101 101 011 010 101 101 ...

        2   5   5   3   2   5   5   3   2   5   5   3   2   5   5   3 ...

By four bits groups:

      0101 0110 1011 0101 0110 1011 0101 0110 ...

         5    6   11    5    6   11    5    6   11    5    6   11    5   10   11 ...

And so forth for other bit-lengths... The preceding examples, except the last, are unfortunately too short to show aperiodicities which appear in longer samples.

   When the original Noll sturmian binary sequence is used in these short studies, its two symbols point to two different musical elements (e.g. pitches or durations). In the case of longer binary number interpretations of the sequence, the resulting decimal numbers are implemented as intervals (e.g. relative to another pitch). Only relatively small binary word lengths are used to peruse the sequence, namely 1, 2, 3, 4, 5, 6 and 8.

   The cryptic subtitles of the studies consist of digits showing the bit-lengths (from 1 to 8 bits out of the original Noll sturmian sequence) creating the number sequences used to generate algorithmically the pitches of the components of each piece. Operator symbols affecting these digits represent various relations between the corresponding sequences, or operations on the resulting components, which would be much too tedious to describe here. These subtitles however show that musical characteristics are not simply proportional to complexity.

   Strangely, surprisingly, something very familiar arises when the sequence is applied four bits at a time on pitch intervals! This is so obvious that it may amaze the listener. It is most clearly heard "solo" in study C, simply entitled 4.

      A — 4/4,4^2/1                                1'09"

      B — (((6>5>3)'3)|((6>5>3)'2));/(1^2)         1'50"

      C — 4                                        2'43"

      D — 1/1/1/1                                  3'20"

      E — 4/3/2/3/3/2                              1'20"

      F — 4/(3'4) nipohc                           2'58"

      G — 4|4/2/3                                  2'08"

      H — 1;/((6>5>3)'3);/((6>5>3)'2);/(3>8);/1;   6'14"

DLO

Karlsruhe

03/12/2015