Born in Pisa, Italy, Leonardo Fibonacci (c1170-c1240) was one of the first mathematicians to introduced the Hindu-Arabic number system to Europeans. While studying in India he became interested in a series of numbers first described in 1150 by Indian mathematicians wherein each value is derived by adding the previous two. In a book published in 1202 entitled Liber Abaci (literally: Book of the Abacus), Fibonacci explores this series and its practical application by posing the following problem: Starting with one mating pair of rabbits, how many pairs exist each month if, after every month, another pair is born to every pair old enough to mate (at least one month old)? The solution is the series of values (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...) that would thereafter become known as Fibonacci numbers.

Several hundred years passed before something strange happened: people began noticing these numbers appearing everywhere in the structure of living things. The number of petals on a flower, the number of spirals on a pine cone, the dimensions of a seashell, the number of leaves sprouting from a stem, even the number of digits on your hand ... Fibonacci numbers appear everywhere. It was as if nature had used Fibonacci numbers as a blueprint for life itself. (click here to view a short PowerPoint slide show).

The main theme of The Fibonacci Waltz is derived by converting Fibonacci numbers into musical notes. The conversion is a simple substitution cipher. Hence, the first six Fibonacci numbers (1, 2, 3, 5, 8, 13) point to the first, second, third, fifth, eighth and thirteenth notes on a scale. Using the familiar 7-tone chromatic, or diatonic scale - the white keys on a keyboard - these notes are C, D, E, G, C and A.

To accomodate larger numbers in the sequence, we repeat the scale indefinitely and ignore octaves. Listed below are the notes derived from the first 48 Fibonacci numbers:

If you listen to these notes played, you'll discover something interesting:

Because of their recursive property, Fibonacci numbers produce an infinitely repetitive melody of just sixteen notes. The numbers being what they are, one could argue that these sixteen notes represent nature's melody - at least in the diatonic major scale. Listen again to just the 16-note Fibonacci Melody:

Here is the same melody, with added rhythm and harmonies. Listen, as the Fibonacci Waltz begins to take form:

There are, of course, other scales. One is the pentatonic, found all over the world and often associated with folk music of the Orient. As the name suggests, the pentatonic scale consists of five notes which appear on a piano as black keys, C#, D#, F#, G# and A#. Using the pentatonic scale and the same substitution method, the Fibonacci sequence once again yields a repetitive melody, although in this case the melody is twenty notes in length. Listen and see if you recognize the secondary theme of the Waltz:

In fact, no matter how many notes the scale contains, its Fibonacci melody will be a finite set of notes that repeats indefinitely.

Oddly, every Fibonacci melody ends with a similar 3-note pattern consisting of the first, last and first notes of the scale. For example, in the diatonic major scale, C D E F G A B, the final three notes of the melody are C B C. Likewise, the Fibonacci melody in the five-note pentatonic scale ends with C# A# C#. Regardless of the underlying scale, every Fibonacci melody ends this way ... which is precisely why the Waltz ends this way.

Finally, The Fibonacci Waltz is exactly five minutes in length - a Fibonacci number of course!