How this tone stack works

This is the tone control from the Boss DS-1, and it is basically the Big Muff tone stack with one extra resistor. A potentiometer pans between a low pass filter and a high pass filter. The treble path runs from the input through C1 and R2 to the top of the pot, with R1 holding that node to ground: an RC highpass, corner at 1/(2π·(R1+R2)·C1), about 804 Hz stock. The bass path runs through R3 to the bottom of the pot, with C2 shunting that node to ground: an RC lowpass, corner at 1/(2π·R3·C2), about 234 Hz stock. The pot track spans the two paths and the wiper is the output. R2 is the only real difference from a Big Muff. Type 0 into it (a wire link) and you have the stock Big Muff tone stack, exactly.

The scoop comes from the gap between the corners. With the stock parts at noon the response bottoms out at −14.0 dB around 684 Hz, and the band edges hold about −6.1 dB on the bass side and −9.2 dB on the treble side. Notice the treble side sits lower than the bass side. That is R2 doing its job: even when the highpass is wide open, the highs still have to divide through R2 against R1 and the pot, so at full treble they stop about 3 dB short of unity. That is also why the dip never fully disappears at the treble end of the knob the way it does on a Big Muff. Compared to the stock Big Muff (−12.7 dB at 934 Hz), this scoop is deeper and sits lower in the band. And like any passive tone stack it throws away signal at every setting; the pedal has gain in front of this network and makes up the level after it.

At high frequencies C1 is effectively a short and C2 has already grounded the bass path, so the treble level is the knob fraction times R1∥pot over (R2 + R1∥pot): −9.1 dB at noon, −3.1 dB at full treble. At the bass end the caps drop out and the whole thing is a resistive divider, (R1 + (1−t)·pot) over (R1 + pot + R3), which works out to exactly −6.0 dB at noon because R1 and R3 are both 6.8k. In between, turning the knob toward treble drags the dip down the band: about 1.8 kHz at 25%, 684 Hz at noon, 421 Hz at 75%, 238 Hz at full treble. The pot value mostly moves the center, not the depth: 5k puts the noon dip at 1.1 kHz, the stock 20k at 684 Hz, 100k at 554 Hz, all within a fraction of a dB of each other. A 1k pot loads both paths so hard the noon scoop disappears completely.

The numbers above assume a buffered, low-impedance source in front and a high-impedance stage behind the wiper. This stack runs at much lower impedances than the Big Muff’s, so the load barely matters here but the source really does. A 100k load on the wiper only deepens the noon dip to −14.5 dB and moves the band edges less than a dB. But 10k of source resistance sinks the noon dip to −23.2 dB and drags the treble edge down about 12 dB, because 10k is bigger than every resistor in the network. The dashed curve is the nearest-E12 build; the stock values are already E12, so it hides under the solid curve until you type in something odd.