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Fibonacci Loops

for any set of instruments op.12 (1993)

In the famous Fibonacci sequence, each term is the sum of the two preceding it. Beginning with two terms equal to 1, the first terms of the sequence are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ...

This sequence has a lot of remarkable properties. The most famous is that the ratio of two consecutive terms gets closer and closer to the Golden Section Number equal to 1.6180339887...

An interesting variant consists to calculate the Fibonacci sequence in the finite sets Z_n = {0, 1, 2, ..., n-1}. This yields periodic sequences, the first of them are:

n=2 : 1, 1, 0, 1, 1, 0, 1, 1, 0, ... (period 3)

n=3 : 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... (period 8)

n=4 : 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, ... (period 6)

I calculated these sequences for each value of n between 2 and 25 and made a simple correspondence between the numbers and the notes of the chromatic scale: 0 for C, 1 for C#, 2 for D and so on until 24 in order to cover all the notes on two entire octaves. Each value of n yields in this way a melodic pattern that is played as a loop. For instance, here is the pattern obtained when n = 12 (period 24):

The work is "scored" for any set of instruments able to play this kind of pattern and having a range that encompasses the notes from C3 to C5, the most suitable being keyboard or mallet instruments (but violin, flute or clarinet can also be used).

The scenario of the piece is as follows: the first instrument plays the first pattern (n=2) in a loop, then the second starts some time after with the second pattern (n=3), and so on until the last instrument is entered. At this point, the first instrument switches to the next pattern that remains to play, and so on until all patterns have been used. Then, the instruments stop playing one by one.