Fibonacci Music Generator

Posted on 25 February 2009

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Eric Archer has come up with a new music sequencer based on the Fibonacci sequence.

Here are the details behind it:

Fans of the Golden Ratio know that it is built-in to the Fibonacci Sequence, being the ratio of S(n)/S(n-1) when n is very large.

But lets make it more complicated.

The Lagged Fibonacci Generator is a generalized version that looks at a wider “history window”; instead of the two previous numbers, it lets you go back farther in the list with parameters j and k.

S(n) = S(n-j) + S(n-k)

These numbers are limited to whatever bit depth you choose to represent them. Whenever the addition generates a “carry”, it rolls over at zero again and outputs a carry flag. For example, if you’re using 8 bits, the m value is 2^8, or 256. What happens then is magic because instead of an ever-increasing string of numbers, you get a set of numbers that hop around back and forth between zero and m. The complexity of the sequence grows rapidly with larger j and k values, but its never random or chaotic.

The patterns are synthesized with a Lagged Fibonacci Generator (LFG) circuit built with discrete logic chips (CD4000 CMOS). The LFG is an algorithm sometimes used in cryptography to generate encryption keys.

I’m exploring it as a pattern generator for algorithmic art and music.

Although cryptographers like the LFG because its output is similar to white noise, I find that through creative electronic filtering, the intricate details of its cyclic patterns become visible, and we can appreciate the details.

Digital patterns from the LFG are converted to two analog outputs with separate hardware of my design. These analog waveforms were processed through an Electrix Filter Queen resonant filter, and visualized on a high resolution Tektronix X-Y monitor. The video was recorded to VHS with a camera pointed at the screen. The audio was recorded directly, and contains the same waveforms being displayed.

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Posted in: New Instruments