I've always liked the sound of the Varitone® circuit that Gibson used in their ES355s, and the Blues Hawk.  Basically, this is a notch filter that permits a narrow range of frequencies to be deleted from the overall signal.  Electrically, they comprise an LCRcircuit, consisting of a  capacitor, and an inductor.

The formula for calculating the resonant frequency of an LC circuit is:

f₀ = 1/2π(LC)^1/2

L is the measure of inductance, given in Henries

C is the measure of capacitance, given in Farads.

The reasonable place to start, of course, was with Gibson's schematics..

(Click on image for detail.)

Gibson's Varitone

 

Using their values gives the following frequency centers:

L (H)	1.5		Full Coil
C (microfarads)	0.001		1306.394529
	0.003		754.2472333
	0.01		413.1182236
	0.03		238.5139176
	0.22		88.07710121

 

These seem like pretty reasonable numbers.  The frequencies given for the Blues Hawk are as follows:

1875

1090

650

350

130

 

One problem with this circuit, however, is that 1.5H is a very large inductance, and inductors in that range are difficult to find.

In his book Electronic Projects for Musicians, Craig Anderton presenets a "Passive Tone Control" project that is, in essence, identical to a Varitone.  One nice added feature is that he specifies a center-tapped inductor, effectively doubling the number of sounds for the circuit.  The problem with his circuit is that the numbers are all wrong!  He specifies a 5-6H center tapped inductor (claiming that an audio transformer from Mouser Electronics is the preferred part).  Since neither Anderton nor Mouser provide precise inductance figures for this transformer, the builder is left guessing.

Let's assume that the identified transformer provides 5.5 H inductance.  Using the capacitance values given by Anderton leads to the following frequency table:

L (H)	5.5		Full Coil	Half Coil
C (microfarads)	0.01		215.743956	305.1080286
	0.02		152.5540143	215.743956
	0.05		96.48363026	136.4484585
	0.1		68.22422923	96.48363026
	0.22		45.99676597	65.04925025

However, the figures he claims are as follows:

			Full Coil	Half Coil
C (microfarads)	0.01		540	1015
	0.02		380	755
	0.05		240	560
	0.1		170	430
	0.22		107	260

Quite a discrepancy!  Mr. Anderton must have left his slide rule home that day.  

Since we don't know the actual value of the inductor he's using, we might assume that he actually measured the frequency response, and since we do know the values for the capacitors, we can try to work backwards.  Unfortunately, the numbers he provides are not even consistent with each other, but a value of 0.5 H (still pretty large) gets us into the ballpark:

L (H)	0.5		Full Coil	Half Coil
C (microfarads)	0.01		715.5417528	1011.928851
	0.02		505.9644256	715.5417528
	0.05		320	452.54834
	0.1		226.27417	320
	0.22		152.5540143	215.743956

Clearly, Anderton's numbers are not to be trusted.  Nevertheless, I built the circuit, using his specified inductor, and the results sound pretty good.

Since building the circuit, I  discovered a source of larger inductors:

http://www.acksupply.com/catalog/inductor.pdf

Since they offer 1.5H chokes (inductors), I've repopulated my table assuming a 3-H center-tapped inductor, adjusting the capacitance values to retain a similar frequency range:

L (H)	3		Full Coil	Half Coil
C (microfarads)	0.00047		1347.443074	1905.572269
	0.001		923.7604307	1306.394529
	0.0022		622.7991553	880.7710121
	0.022		196.9463856	278.5242495
	0.1		92.37604307	130.6394529

This gives me a range pretty close to Gibson's, with a bit finer granularity.