Dan Slater's Article in CMJ, and X to the kth power - how do you do it?

[Dan Slater]

Sean Costello wrote:
> 
> Hi all:
> 
> The new Computer Music Journal (Volume 22, Number 2, Summer 1998)
> finally hit the stands.  Dan Slater has a fascinating article in it -
> "Chaotic Sound Systems."  Y'all gotta track it down.
> 
> Anyway, Mr. Slater describes a hypothetical Buchla module (complete with
> an "artist's conception" that points out how truly beautiful the Buchla
> 200 designs were visually) that would implement a Ueda Attractor. As
> described, the Ueda attractor circuit is very similar to a
> state-variable filter, but with the inverting stage replaced with an x^3
> stage (x cubed) and the signal input connected to the first integrator
> stage.  Slater's hypothetical module uses an x to the k function, where
> the circuit can be converted from a state variable topology to an Ueda
> Attractor by changing the value of k (for a state-variable filter, k=1;
> for the Ueda attractor, k=3).
> 
> How the heck would you do this?  Is there some common circuit that
> produces an x to the k function, where k can smoothly vary (i.e. by
> non-integer values) between 1 and 4?  x^1 is simple; x^2 could be a
> multiplier where the input is multiplied by itself; x^3 could be created
> by a second multiplier multiplying the input by the product of an x^2
> stage.  But how do you get, say, x^2.74?  And how do you vary these
> smoothly?  Could you use feedback from a multiplying stage, where a
> signal is multiplied by the output of the multiplier?  Or maybe feedback
> in a 2-multiplier configuration?
> 
> As always, any and all advice welcome.  I think that this would be a
> cool circuit to own.
> 
> Thanks,
> 
> Sean Costello

Hi Sean;

	Thanks for the kind words;

	A relatively simple approach for the x^k is to have a voltage
controlled multiplexer that selects between the various integer exponent
values from 0, 1 or 2 multipliers. Even this can be interesting as the
exponent is swept between different quantized values.

	Non integer exponents can be handled by taking the log of the signal,
multiplying by the exponent and then taking the inverse log
(exponential) of the signal.

	x^n = exp (n * ln(x))

	A problem here is that you can not take the log of a negative number.
One possible solution is to full wave rectify the signal prior to the
log operation and restoring the sign after the exponential operation.

	In my experience to date, I have just used the x^3 configuration as
formed by a pair of multipliers. I have not yet implemented the variable
exponent.

Good luck;

Dan Slater