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Music and Graphics of the Fibonacci String

The Fibonacci string is a binary -word generated by the morphism . It is aperiodic. One interesting property of the Fibonacci string is that substrings repeat with spaces between repetitions that follow the exact pattern of the Fibonacci string itself. That is, the intervals are equal to in succession where "a" and "b" are integers. The following program demonstrates this property by generating the string, identifying the positioning of a specified substring, determining the spacing between them, and then converting the resulting integers to the symbols "a" and "b". When the last step is performed, the Fibonacci string re-emerges.

Music can also be generated from the Fibonacci string. This next program creates music from the Fibonacci string by means of the following steps:

1. Generation of the string to a particular length;
2. Division of the string into substrings of length (n);
3. Generation of a list of all possible substrings of length (n);
4. Assignment of notes of ascending frequency in the scale of C Major to the previous list with a logical ordering system;
5. Transformation of the divided Fibonacci string according to the rules established in the last step.

This program produces a unique melody for each specified substring length (n). Click on the graphic to hear the generated song.

The previous music can be represented graphically by the next program. It produces a unique graph for each specified length of substring (n). Of particular interest is the apparently regular oscillation between larger and smaller sets of values for any specified substring length. It functions by means of the following steps:

1. Generation of the string;
2. Division of string to specified substring length (n);
3. The substrings of the divided string are interpreted as numerical values and plotted with their specific position in the string to produce a graphic representation of the succession of substrings.

Copyright © 2001 Wolfram Media, Inc. All rights reserved.

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