In the Fibonacci sequence, each term (after the first couple) is the sum of the previous two terms. We can denote this as:
Fib(n) = Fib(n-1) + Fib(n-2)
The Padovan sequence is a simple variation on this:
P(n) = P(n-2) + P(n-3)
So the first few members of the sequence, after arbitrarily setting the initial terms to 1, are:
1 1 1 2 2 3 4 5 7 9 12 16 21 28 37 49 65 86
Self-similarity rears its head here... If you look at the differences between terms, starting with P(6) - P(5), you will find that they form the Padovan sequence.
To make Padovan music we can take advantage of another way of generating the sequence using an L-system which I'll define as follows:
- Take the three notes G, A, B.
- Begin the progression with G.
- Follow the rules G -> A; A->B; B->GA
(note: I'm using GAB here rather than the more usual ABC, so that I can be lazy with my JFugue code and stay in the same octave)
Applying these rules the sequence builds like this:
The length of the note sequence at each stage is:
1 1 1 2 2 3 4 5 7
which is the Padovan sequence.
Here's what it sounds like, rendered using JFugue and arranged with two voices, one just behind the other. Self-similarity in the sequence results in the voices falling into step every so often. Keen listeners will also notice that I've exercised a little artistic licence and set the duration of the Bs to be twice that of the Gs and As.